Frequency distributions of the parameters including Rd when sampling 7 from uni- form distributions ..... According to the Diagnostic and Statistical Manual-IV (DSM-IV), diagnosis criteria for Alco- ...... are bounded functions for t e 5R. Also, we ...
THE ROLE OF ENVIRONMENTAL CONTEXT IN THE DYNAMICS AND CONTROL OF ALCOHOL USE by Anuj Mubayi
A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
ARIZONA STATE UNIVERSITY December 2008
UMI Number: 3340900
INFORMATION TO USERS
The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion.
®
UMI UMI Microform 3340900 Copyright 2009 by ProQuest LLC. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 E. Eisenhower Parkway PO Box 1346 Ann Arbor, Ml 48106-1346
THE ROLE OF ENVIRONMENTAL CONTEXT IN THE DYNAMICS AND CONTROL OF ALCOHOL USE by Anuj Mubayi
has been approved September 2008
Graduate Supervisory Committee: Carlos Castillo-Chavez, Co-Chair Priscilla Greenwood, Co-Chair Laurie Chassin Yang Kuang Gerardo Chowell
ACCEPTED BY THE GRADUATE COLLEGE
ABSTRACT Alcohol consumption is a function of social dynamics, environmental contexts, individuals' preferences and family history. Empirical surveys have not been able to rank, identify, or quantify the mechanisms responsible for heavier drinking. There have been a few attempts to apply nonlinear models to the study of the dynamics of alcohol use at the population level, but there are no systematic approaches for evaluating the impact of proposed control programs over time. In this dissertation, the role of environmental context in the dynamics of alcohol drinking is studied. First, a simple deterministic model that focuses on the impact on drinking of individuals' traffic between low- and high-risk environments, is introduced. The model studies drinking as a socially contagious process, that is, it is assumed that "contacts" between drinkers and non-drinkers in the appropriate context, may lead to drinking. The strength of the contagion process is mediated by the drinking environment. Results show that under certain assumptions, increases in the movement of drinkers between distinct drinking environments can increase the prevalence of heavy drinkers. However, the local culture of drinking environments as measured by the strength of social contacts within a given environment may in fact change the outcomes significantly. Model parameters from college data that include the results of a drinking survey at 14 state college campuses in California carried out by researchers at the Prevention Research Center at Berkeley, are estimated. The basic drinking reproduction number that measures the "reproductive" impact of moderate drinkers on light drinkers is derived from the model. Uncertainty and sensitivity analysis is performed to assess the variability in basic drinking reproduction number that results from error in measurement in social and environmental residence times parameters. An extended model is then considered by assuming that random effects arise in the process of social interactions of drinkers, and increasing/decreasing of drinking levels may occur by chance
iii
mechanisms. This stochastic model is used to capture variability in drinking patterns as a result of these intrinsic naturally occurring random effects. The effects of prevention and intervention programs on the distribution of drinking levels are investigated using the deterministic and stochastic models. Results from this systematic study of control programs suggest that prevention is better than intervention in controlling heavier drinking. However, exclusive implementation of prevention policies with perfect efficacy may not be feasible. Simultaneous implementation of prevention programs, and interventions in high-rsik environments is the next best option. Higher rates of intervention in low-risk environments than intervention rates in high-risk environments are also better when the aim is to control high level of drinking in the community.
IV
ACKNOWLEDGMENTS The printed pages of this dissertation hold far more than the culmination of years of study. These pages also reflect the relationships with many generous and inspiring people I have met not only since beginning my graduate work but during my whole life. The list is long, but I cherish each contribution to my development as a scholar and teacher: It is difficult to overstate my gratitude to my two Ph.D. supervisors, Drs. Carlos CastilloChavez and Priscilla Greenwood. With their enthusiasm, inspiration, and great efforts to explain things clearly and simply, mathematics has definitely become fun for me. Professor Castillo-Chavez has been a huge influence in shaping my thinking about mathematical modeling and preparing me to be a strong applied mathematician. He advised and supported me not only in academics, but also in extracurricular activities. His mentorship was paramount in providing a well rounded experience consistent with my long-term career goals. Professor Greenwood has taught me a great deal about scientific writing. I have learned from her many concepts in mathematics, especially stochastic modeling and parameter estimation. I think she has been correctly described by one author as the "queen of probability." Throughout my dissertation research, they both provided encouragement and support. Both believed in me throughout. I would have been lost without them. I thank both of them for taking the time to share their wisdom and knowledge so generously with me. I also would like to thank Professors Christopher Kribs-Zaleta and Gangaram Ladde, who taught me some crucial applied mathematics courses that were instrumental in the choice of my current field of research. I am very thankful to my friend-cum-mentor Dr. Kribs-Zaleta for offering me a postdoctoral position in his group even before completion of my studies. I am also grateful to Professors Laurie Chassin and Yang Kuang for their advice, suggestions, and role on my Ph.D. committee. Professor Gerardo Chowell, also on my Ph.D. committee, taught me many statistical
v
techniques and helped me a great deal with parameter estimation, for which I am grateful. I also wish to thank Dr. Xiaohong Wang for helping with many tools and techniques during my Ph.D. program. Dr. Paul Gruenewald from the Prevention Research Center, Berkeley, and Dr. Dennis Gorman deserve special mention for helping me understand concepts in this interesting field of alcohol research. For their kind assistance with writing letters, giving wise advice, and helping with various applications, I wish to thank Professors Peeyush Chandra, Mohan K. Kadalbajoo, D. Bahuguna, Pravir Dutt, Manjul Gupta, Debashish Kundu, P.C. Das, P.C. Joshi, Goujun Liao, Chaoquin Liu, and Andrzej Korzeniowski. Thanks to the Mathematical and Theoretical Biology Institute and my Department staff for assisting me with the administrative tasks necessary for completing my doctoral program: Lee Cruz, Dawn Conklin, Stephen Tennenbaum, Preston Swan, Shanae Blunt, Debbie Olson, Alissa Ruth, and Bate Agbor-Baiyee. I wish to thank my best friends, from my hockey group, Steve Hilton, Alissa Donaldson, Mohammad Riyaz, Kristen Mohammad, Abhishek Swain, Shekhar Bhagat, and Roma Bhagat for all the emotional support, camaraderie, entertainment, and caring they provided. I am especially indebted to my comrade Marcela Naciff, who assisted me in many different ways. She deserves special mention for her untiring and unselfish help in getting me through the difficult times. For everything you have done for me, Marcela, I thank you. I finish with India, where the most basic source of my life energy resides: my family. I have an amazing family, unique in many ways, and the stereotype of a perfect family in many others. I wish to thank my entire family for providing me a loving environment and unconditional support. My brother, Ashwani Mubayi, definitely has been an inspiration for me in higher education and helping me in achieving this level of success. My sister, Anamika Mubayi, has been the most caring and motivating person in my life. The love and support of my brother and sister always
vi
kept me focused. They are my true idols and I would not have been to this level without them. My brother's wife, Shivangi Mubayi, who took care of me during my trips to India, and my little nephews, Anirudh Mubayi and Amogh Mubayi, were a great encouragement during my research. They really told me the truth about love. Lastly and most importantly, I wish to thank my parents, Surabhi Mubayi and Arun Kumar Mubayi. They bore me, raised me, supported me, taught me, and loved me. It was under their watchful eye that I gained so much drive and an ability to tackle challenges head on. My whole-hearted gratitude goes to my parents and their sacrifices for my better life. To them and the memory of my loving grandmother, Shanti Mubayi, who passed away in 2007, I dedicate this dissertation.
Funding for this work has been provided in part by NIAAA grant on "Ecosystem Models of Alcohol-Related Behavior", Contract No.
HHSN2S1200410012C, ADM Contract No.
NolAA410012 through Prevention Research Center, PIRE, Berkeley, National Science Foundation (Grant DMS-0441114), National Security Agency (Grant H98230-05-1-0097), The Alfred P. Sloan Foundation (through the ASU-Sloan National Pipeline Program in Mathematical Science), The Office of the Provost of Arizona State University and Los Alamos National Laboratory.
vu
TABLE OF CONTENTS
Page LIST OF TABLES
xii
LIST OF FIGURES
xiii
CHAPTER 1 INTRODUCTION
1
1.
Alcohol Use, Abuse and Addiction
1
2.
Social Interactions, Environmental Influences, Prevention and Intervention Programs at US Colleges and Universities
4
3.
Review of Mathematical Models of Alcohol Drinking Dynamics
7
4.
Definitions and Concepts in Alcohol Research
8
CHAPTER 2
THE IMPACT OF RELATIVE RESIDENCE TIMES IN HIGHLY DISTINCT
ENVIRONMENTS ON THE DISTRIBUTION OF HEAVY DRINKERS
14
1.
Introduction
14
1.1.
16
A Mathematical Approach
2.
Model Description
18
3.
Results
21
4.
3.1.
Model Equilibria and Stability
21
3.2.
Computing the Effect of Residence Times
24
3.3.
Numerical Results
27
Discussion
32
CHAPTER 3 AN APPLICATION OF A MATHEMATICAL MODEL TO COLLEGE DRINKING USING STATISTICAL METHODS
36
1.
36
Introduction viii
Page
1.1.
The Ecological Problem .
37
1.2.
A Mathematical Approach
39
2.
Data Sources
44
3.
Uncertainty and Sensitivity Analyses
49
4.
Results
52
5.
Discussion
56
CHAPTER 4
IMPACT OF ENVIRONMENTAL SPECIFIC CONTROL PROGRAMS:
STOCHASTIC MODELING APPROACH
59
1.
Introduction
59
2.
Stochastic Model
61
3.
Analysis and Approximations
67
3.1.
Corresponding Deterministic Model
67
3.2.
Quasi-Stationary Distribution
69
3.3.
Time of Elimination of Moderate and Heavy Drinking in the Presence of
4.
Prevention and Intervention Programs
73
3.4.
Critical Community Size
76
3.5.
Establishment-Time for Moderate and Heavy Drinking in Naive Communities 76
Numerical Experiments
78
4.1.
Quasi-Stationary Distribution
79
4.2.
Elimination-Time of Moderate and Heavy Drinkers and Critical Community Size
4.3.
82
Establishment-Time of Moderate and Heavy Drinkers in a Naive Community 86 ix
Page
5.
Discussion
CHAPTER 5
88
CONCLUSIONS AND FUTURE DIRECTIONS
92
1.
Conclusions
92
2.
Future Directions
94
2.1.
Sustained Oscillations through Power Spectral Density
94
2.2.
Parameter Estimation in Stochastic Differential Equations Model
97
2.3.
Model Extension: Structured Populations
97
REFERENCES APPENDIX A
99 BASICS AND MATHEMATICAL TOOLS AND TECHNIQUES USED IN
THIS RESEARCH
117
A Historical Note
118
Simple Mathematical Epidemic Models
118
Simple Stochastic Epidemic Models
120
Understanding a Simple Stochastic Model
122
Difference between Deterministic and Stochastic Models
124
Poission Process
126
Reproduction Numbers
129
Parameter Estimation
129
Uncertainty and Sensitivity Analysis
130
APPENDIX B
SIMPLE DETERMINISTIC MODEL (CHAPTER 2)
Model: System of Equations
134 135
x.
Page
Analytical Results APPENDIX C
135
MODEL ANALYSIS (CHAPTER 2)
137
Derivation of Drinking Reproduction Number and Local Stability of the Abusive Drinking Free Equilibrium
,
Global Stability Analysis of the Abusive Drinking Free Equilibrium
138 140
Existence of Endemic Drinking Equilibrium (i.e., State of Mixed Drinking Community)
142
Proof of Theorem B.l in Appendix B
145
Local Stability of the Endemic Equilibrium
146
APPENDIX D
PARAMETER ESTIMATION; UNCERTAINTY AND SENSITIVITY
ANALYSIS ON THE REPRODUCTION NUMBER (CHAPTER 3)
149
Model Results from Chapter 2
150
Steps of Uncertainty and Sensitivity Analysis
151
APPENDIX E
ANALYSIS OF DETERMINISTIC MODEL CORRESPONDING TO
STOCHASTIC MODEL (CHAPTER 4)
154
Deterministic Model
155
Deterministic Model Analysis
156
APPENDKF
ANALYSIS OF STOCHASTIC MODEL (CHAPTER 4)
160
Computing Quasi-Stationary Distribution
161
Time to Extinction: Proof of Theorems 3.2 and 3.3 in Chapter 4
161
xi
LIST OF TABLES
Page The definition of the model variables and parameters
22
Definition of the parameters and their estimates for two values of 7
45
Percentage of drinkers in three drinking-level classes from 1982 to 1991. Data source: Engs et al., 1994
46
Percentage of drinkers in four class-years along with their increase from previous class-year during academic year 1993-94. Data source: Engs etal., 1997
47
Use of Drinking Venues by Drinker Class
47
Notations used in the text. I use notations Xt and X ( t ) interchangeably.
61
Definition of the parameters and their estimates for two values of 7
62
Transitions by which stochastic processes can change and their corresponding rates.
65
University of California UC Digest 2003: University of California system schools and intake. I assume that the annual intake of schools is 1.5 times the fall semester intake. Engs et al. (1994; 1997) suggested that 65% of freshman are drinkers. . . .
78
Comparison of Quasi-Stationary Distribution. Sample size 1000 and total population of 10000 individuals
81
Covariances of Quasi-Stationary Distribution. Sample size 1000 and total population of 10000 individuals
82
xii
LIST OF FIGURES
ure
Page
1.
"Drink" in US (source: NIAAA's NIH publication no. 07-3769, May 2007) . . . .
2.
Heterogeneity of types of drinkers (Mark L. Willenbring, MD, Director, Division of Treatment & Recovery Research NIAAA/NIH, Bethesda, MD, USA, presentation)
3.
11
12
Drinking Population Model with Three Types of Drinkers (light (S), moderate {M\ and M2) and heavy (H)) and Two Distinct Drinking Environments (low-risk (E\) and high-risk (£^2))- T n e S - M2 interactions in the flow chart diagram are drawn within the environment type E\, but their nature is differentiated by the use of a dotted line in order to show that it captures the interactions happened outside E\ and Ei drinking environments
4.
20
Using five different initial conditions, these graphs show that the endemic drinking state, D\, appears asymptomatically stable. Here, Rd = 2.67 which is greater than one (condition for the existence of D{). The parameters used for these graphs are fi = 0.28, A) = 0.71, fo = 2.29, 71 = 0.49, 72 = 0.49 and a = 0.54. For this set of parameters 7 = 0.27
5.
25
Plot of the surface 72 = / ( 7 i , a ) (function ' f obtained from Rd = 1); Fixed parameters are (3Q = 2, fo = 2 and /i = 0.25. The parameter definitions are in Table 1
27
xm
Page
6.
Equilibrium class sizes for various values of a (per capita progression rate of moderate drinkers in high-risk environments to the heavy drinking class), as a function of the proportion, 7, of residence time in high-risk environments; with fa — 2, fa = 2, fi = 0.25 and N = 1000; On the vertical axis I have the number of individuals in the S*, M* and H* classes; S* represented by Solid, Green curve, M* by Dotted, Red curve, and H* by Dashed, Blue curve; Parameters are defined in Table 1. 29
7.
Equilibrium class sizes for various values of fa (peer pressure coefficient related to moderate drinker in low-risk environments), as a function of proportion, 7, of residence time in high-risk environments; with fa = 2, a = 1.5, \i = 0.25 and N = 1000. On the vertical axis I have number of individuals in S*, M* and H* classes; S* represented by Solid, Green curve, M* by Dotted, Red curve, and H* by Dashed, Blue curve. Parameters are defined in Table 1
8.
30
Equilibrium class sizes for various values of /i (per capita arrival or departure rate of drinkers population), as a function of proportion, 7, of residence time in high-risk environments; with J3Q = 2, fa = 2, a — 1.5 and N = 1000, On the vertical axis I have number of individuals in S*, M* and H* classes; S* represented by Solid, Green curve, M* by Dotted, Red curve, and H* by Dashed, Blue curve, Parameters are defined in Table 1
32
xiv
Page
9.
Frequency distributions of the parameters including Rd when sampling 7 from uniform distributions with mean 0.28. The overall shape of the distribution remains same for the cases when mean of 7—distribution was 0.50 and 0.83. Distributions were calculated from one of the 10 Monte-Carlo Samples, each of size 105 sampled parameter values using approach described in the text. Horizontal axis has parameter values, and vertical axis represents frequencies in the graphs
50
10.
Variation in estimates of parameters with change in distribution of 7
53
11.
Sensitivity (measured by PRCC values) of the Rd with respect to its parameter when Mean(7) is varied
55
12.
Model Flow Chart
67
13.
Reproduction Number as Intervention rates vary
79
14.
A sample path of the stochastic differential system and solution of the corresponding deterministic system
80
15.
Marginal Q-S Distribution approximation
81
16.
Variation in mean of QSD with change in £1 and 62 • The graphs in first two rows also show the plane with Z-component equal to zero. With increase in v the shape of the graphs remain the same, but their magnitude decreases
17.
Probability density function of elimination-time of M- and H-drinkers for specific control values
18.
19.
82
83
Expected elimination time of M- and H- drinkers as efficacy of prevention program varies
84
Expected elimination-time of M-and H-drinkers as control parameters vary
85
xv
Figure
20.
21.
Page
Expected elimination-time of M- and H-drinkers as initial population size N varies in the presence of controls
87
Establishment-time of M-and H-drinkers
88
xvi
CHAPTER 1 Introduction 1. Alcohol Use, Abuse and Addiction Although there is no concrete evidence of when alcoholic beverages were introduced, the use of intentionally fermented beverages has been documented as far back as the Neolithic period (ca. 10,000 B.C.) (Hanson, 1995). Historically alcohol, has served as a source of nutrients and has been used for its medicinal, antiseptic, and analgesic properties. It is known to facilitate relaxation and increases the pleasure of eating. Sensible drinking has been documented to reduce the risk of heart attacks, a major cause of death in the United States (US), by about 40%. Alcohol also reduces the risk of cardiovascular diseases since it decreases coronary artery spasm. Other benefits of drinking alcohol moderately includes reduction in the risk of hypertension or high blood pressure, Alzheimer's disease, diabetes, Rheumatoid arthritis, bone fractures and csteoporosis, kidney stones, digestive ailments and liver disease, parkinson's disease, Hepatitis A, Pancreatic cancer, etc. Alcohol abuse, a well established problem, not only affects individuals' health, but also endangers the health and safety of others in society. In many parts around the world, alcohol consumption is socially acceptable regardless of health risks and related social consequences. The continuous abuse of alcohol leads to chronic diseases, that is, diseases that develop over time due to excessive drinking with early death a likely outcome. Alcohol consumption resulted in 85,000 deaths (3.5% of total deaths) in 2000 in the US (Mokdad et al., 2000). The American Psychiatric Association (APA) defines criteria for Alcohol abuse and Alcohol dependence to facilitate the clinical diagnosis of Alcohol Use Disorders (AUD). In 2003, the NSDUH 2004 survey1 reported that more than 14 million adults aged 21 or older 'The National Survey on Drug Use and Health (NSDUH) is an annual survey sponsored by the Substance Abuse & Mental Health Services Administration (SAMHSA). The 2003 data are based on information obtained from 67,784 persons aged 12 or older, including 36,309 persons aged 21 or older.
2
(roughly 7.4% of the adult population) can be classified as alcohol dependent or abusers of alcohol. The American Medical Association (AMA) considers alcoholism a disease, an idea put forward by E. Morton Jellinek in his famous 1960 book "The Disease Concept of Alcoholism" (Jellinek, 1960; Mann et al., 2000). In this dissertation, I model alcohol drinking as a socially contagious disease, evaluate the role of heterogeneous environmental contexts on its dynamics, and explore the effectiveness of specific control programs. In the US, alcohol consumption has been declining for decades. The National Institute on Alcohol Abuse and Alcoholism (NIAAA) reports that the per capita consumption of alcohol by Americans age 14 and older dropped over time from a per capita yearly consumption of 2.76 gallons in 1980, to 2.43 in 1990 to 2.18 in 2000 (Lakins et al., 2007). Moreover, the recovery analysis conducted on a national survey2 that considered a subgroup of 4,422 adults3 (individuals who met the clinical criteria for alcohol dependence a year or more before the 2001-2002 survey) shows encouraging results (NIAAA press release, 2005). According to the survey data analysis,
• only 25.0% of these individuals are still (in 2002) alcohol dependent;
• 27.3% are in partial remission (that is, exhibit some symptoms of alcohol dependence or alcohol abuse); • 11.8% are asymptomatic risk drinkers with no symptoms, but whose consumption increases the chances of relapse (for men, more than 14 drinks per week or more than 4 drinks on any day; for women, more than 7 drinks per week or more than 3 drinks on any day); • 35.9% fully recovered, a subgroup that includes 17.7% low-risk drinkers and 18.2% abstain2 The largest survey ever conducted in the alcohol use, 2001-2002 National Epidemiologic Survey on Alcohol and Related Conditions (NESARC) by the NIAAA. 3 This subgroup is taken from a sample of 43,000 total U.S. adults aged 18 years and older in the survey.
3 ers. But what is alarming is that among the above group of individuals, more than 50% experienced the onset of alcohol dependence between the ages of 18 and 24, that is, during their college ages. Moreover, only 25.5% of them received some form of treatment for their alcohol problems. Although alcohol consumption in the US seems to be on the decline, a culture of abusive drinking which includes a large percentage of students, particularly on college campuses (mostly from the age group 18-24), continues to thrive. The existence of these culture challenges our ability to develop effective and lasting alcohol abuse prevention programs. According to the research summarized in a NIAAA's College 2002 Task Force report, excessive drinking by college students has significant, destructive, and costly consequences, and the impact is on all students (whether they drink or not). Statistics from this and other reports show that drinking by college students aged 18 to 24 contributes to an estimated 1,700 student deaths, 599,000 unintentional injuries, and 97,000 cases of sexual assaults per year (NIAAA College report, 2002; NIAAA College report, 2007). A study based on 12 consecutive months of drinking among college students concludes that 31% of college students may meet the criteria of alcohol abuse and 6% may be diagnosed as alcohol dependent (Knight et al., 2002). Since Straus and Bacon (1953) first reported that alcohol on college campuses presented serious problems to college administrators, several alcohol prevention and intervention policies have been put in motion at various levels. The Drug Free Schools and Communities Act Amendments of 1989 are attempts to legislate change. Despite several years of increasing efforts, a recent study
by Weschler et al., 2002 reported approximately 44% of college students engage in high-risk or "binge" drinking, defined as the consumption of 5 or more drinks in a row for males and 4 or more for females in one sitting. Alarmingly, the percentage of frequent binge drinkers among college
4 students has also gone up from 20% in 1993 to 21% in 1997 to 23% in 2001 (Wechsler et al.(2002).
2. Social Interactions, Environmental Influences, Prevention and Intervention Programs at US Colleges and Universities As a result of continued high-risk drinking on college campuses, schools have increased the number and intensity of educational (e.g., lectures, meetings) and/or intervention programs addressing alcohol use (Wechsler et al., 2002, Tevyaw et al., 2007). These efforts have not reduced the consequences of alcohol use by students (Hingson, Berson, & Dowley, 1997), and not much research has been done that evaluates the effectiveness of alcohol control programs or that identifies useful principles that can help in the development of policies or programs that reduce drinking on campuses over time (NIAAA College Report 2002). In this dissertation, I study the impact of control programs through the use of simple dynamic mathematical models. Models analyzed here can be used to evaluate the efficacy of particular control programs. Programs for reducing drinking problems can be classified into ones that are directed to changing alcohol-related socio-cultural beliefs or programs oriented towards the control (reductions) of alcohol consumption. Socio-culturally directed approaches (focus on educating drinkers to drink responsibly) assume that alcohol abuse is the problem and not alcohol itself. On the other hand, control of alcohol consumption oriented policies tend to assume that alcohol is the cause of all drinking problems (Hanson, 1995). The NIAAA's 2002 Task Force report suggests that prevention and intervention programs in colleges should be directed towards changing the culture of drinking and not towards drinking prohibition. The belief is that within the current environmental contexts socio-cultural approaches are superior and that short-term reductions in alcohol problems can be better achieved with control programs that target high-risk groups. Effective long-term con-
5
trol programs must reinforce joint interventions between colleges and surrounding communities. These programs must have some understanding of the community as a dynamical system (Holder, 1999, Gorman et al., 2004). Among a large percentage of students there is a belief that alcohol is a necessary ingredient for social success. Alcohol is present at most college students' social functions (Thombs, 1999). It is known that peer pressure is closely linked to heavy drinking and that peer pressure can vary from direct (mere overt offers of drinking) to indirect (imitating one's drinking behavior or perceived social norms) influences (Borsari and Carey, 2001). Many students lack self-confidence and maturity to make appropriate decisions while in group events. Borsari & Carey (2001) insists that drinking rates of groups should be monitored over time to document the emergence of certain members of a peer group as more influential. Peers are important social force in college environments since parents' influence diminishes at this time (Perkins, 1997). Student behavior, including alcohol use and abuse is directly related to mutually shaping interactions between students and the college environments (Huebner, 1979). Social learning theory, which focuses on learning within a social context supports the view that environmental contexts, reinforces the social learning of alcohol use (Maisto etal., 1999). Because of the changing culture on campuses, a great deal must be learned about the characteristics of college drinking environments, particularly when it comes to environments that can influence the use and abuse of alcohol. In fact, Shore et al. (1983) remark that the ability to cope with peer pressure linked to drinking is more strongly related to college environmental factors than to personal background variables. It is important to determine the extent to which college drinking can be attributed to college environmental factors (Presley et al., 2002). People in settings where alcohol is present, such as drinking establishments or fraternity parties, feel an obligation to drink.
Resistance to too much drinking in such settings is tied to characteristics of each environment, that is, the conditions in it including the support of the people "living" in such environments. It has been observed that certain environments protect college students while others make students vulnerable to heavy drinking. Moreover, it seems obvious that the time and frequency that individuals spend in environments that promote alcohol use reinforce its use and increases the likelihood that individuals will engage in heavy drinking. How do residence times in environments effect drinking patterns at the population level? This is the question that has not been systematically studied. Environmental and peer influences combine to create a culture of drinking. In particularly, drinking in college is influenced by social factors, social contexts and environmental residence times. My goal here is to address the following questions: 1. How can I conceptualize the ecological dynamics of alcohol drinking, relating choices of drinking places and social influences within it to drinking levels? 2. How can a conceptual model be used to understand the effect of residence times in distinct risk-level drinking environments on drinking patterns? 3. How can I capture variability in drinking levels because of random intrinsic factors'! 4. How can these models and their results be used to evaluate the impact of environmentspecific prevention and intervention programs? To provide insights into these questions, I model drinking as a socially contagious process and incorporate two distinct risk-level drinking environments. The modeling framework includes three
discrete "epidemiological" states representing number of light, moderate, and heavy drinkers in a population where some drinkers are at risk of becoming heavy drinkers. The framework is used to explore the impact of the "movement" of drinkers between heterogeneous drinking environments
7 on drinking outcomes at the population level. The results of this work are highlighted in the context of college drinking. The dissertation is structured as follows. I end this chapter with a review of mathematical dynamic models on substance abuse. Appendix A lays out the differences between deterministic and stochastic approaches. A deterministic contagion model of social influence that incorporates three different drinking levels and two contrasting drinking environments is examined in Chapter 2 (Mubayi et al., submitted). Chapter 3 focuses on drinking patterns in a setting that uses US college population data as a reference point. The efficacy of prevention and intervention programs is evaluated in Chapter 4 using a stochastic model. The role of stochasticity is explored following, in part, approaches of Nasell (2002) and Anderson and Britton (2000). Mathematical details from the chapters are collected in Appendices B-F. Some mathematical background related to tools and techniques used in this dissertation are gathered in Appendix A.
3. Review of Mathematical Models of Alcohol Drinking Dynamics Mathematical models have a long history of successes in ecology, immunology, population biology, and epidemiology (Anderson & May, 1991; Brauer & Castillo-Chavez, 2001). Most recently, applications of a similar framework to the study of substance abuse problems have been carried out : studies of drug related problems by Behrens et al. (1999) and Caulkins and Reuter (2006); alcohol related aspects by Braun et al. (2006) and Sanchez et al. (2007); smoking related issues by Bynre, Mazanovv and Gregson (2001); dynamics of substance abuse in general by Hawkins and Hawkins (1998). In particular, studies by Braun et al. (2006) and Sanchez et al. (2007) showed the importance of social interactions on the dynamics of alcohol consumption. Sanchez et al. (2007) considered a dynamical model consisting of a population with ab-
8
stainers, drinkers, and recovered (treated) individuals. Their study focused on the role of relapse on drinking levels over time. They found that if the relapse rate is high, it is difficult to develop successful treatment programs even if the recovery rates are high. Braun et al. (2006) studied the evolution of alcohol dependence in populations with various social network characteristics. Their study focuses on the influence of social network neighbors within structured populations. Their work suggests that the design of treatment and intervention policies must account for detailed social structures. These studies on alcohol drinking and other research on substance abuse have not incorporated environmental heterogeneity. I have chosen to look at the role of environmental heterogeneity on drinking patterns because there are numerous statistical studies associated with alcohol use among college students, but just a few mathematical studies on the dynamics of drinking. There has been no systematic method for evaluating the impact of specific interventions over time in heterogeneous drinking environments. The models presented here incorporate heterogeneity in a simple way, following the approaches or perspectives of Patten and Arboleda-Florez (2004), Gonzalez et al. (2003), Song et al. (2006), and Winkler et al. (1996).
4. Definitions and Concepts in Alcohol Research In this section, I introduce some alcohol-related terminology that may help non-experts understand the context of this study. There is no consensus on key definitions among researchers in the field. Consequently, this section is by no means definite.
1. Alcoholism: The ingestion of alcohol is necessary in the development of alcoholism, but alcohol use does not per se lead to alcoholism. The quantity, frequency, and regularity of alcohol consumption required to develop alcoholism varies greatly from person to person. Although
9 the biological mechanisms underpinning alcoholism are uncertain, some risk factors, including social environment, emotional health, and genetic predisposition, have been identified as contributors. The Journal of the American Medical Association defines alcoholism as "a primary, chronic disease characterized by impaired control over drinking, preoccupation with the drug alcohol, use of alcohol despite adverse consequences, and distortions in thinking." 2. Alcohol abuse and alcohol dependence: The American Psychiatric Association has developed strict criteria for the clinical diagnosis of Alcohol Use Disorders (AUD). Clinical diagnostic criteria for AUD are Alcohol abuse and Alcohol dependence. According to the Diagnostic and Statistical Manual-IV (DSM-IV), diagnosis criteria for Alcohol dependence requires the satisfaction of 3 of the following 7 characteristics: (a) increased tolerance to alcohol; (b) symptoms of withdrawal after discontinuing alcohol use; (c) increased amount of alcohol intake; (d) quitting of previously chosen controls; (e) spending significant amounts of time to obtain, use or recover from alcohol use; (f) continuing even though aware of destructive consequences; (g) the neglect of regular activities. On the other hand, diagnosis criteria for Alcohol abuse requires the satisfaction of 1 of the following 4 characteristics: (a) initial disregard of school, work, or social responsibilities because of alcohol use; (b) engagement in dangerous activities while intoxicated; (c) drinkingrelated legal problems; (d) drinking-related social or interpersonal problems.
3. Alcohol seeking behaviour: An individual is considered to show this behavior if he/she has a craving for alcohol and continue drinking despite adverse effects.
4. Binge use: Binge use of alcohol is defined as drinking five or more drinks on the same occasion (i.e., at the same time or within a couple of hours) at least once in the past 30 days.
10
This definition is based on drinking behaviors as measured in an individual with blood alcohol concentration (BAC) up to or above the level of 0.08% gm. This level is typically reached by men with 5 or more drinks in about 2 hours and by women with 4 or more drinks in the same time.
5. College Environments: Environmental contexts can include living places (e.g., parents' house and residential halls), social context of drinking (e.g. house parties), places that sell/promote alcohol (e.g., bars and restaurants), places for extra curricular activities (e.g. sports events), or organizational events (e.g. fraternity and sororities events). Some collegiate environments are powerful enough to influence almost everyone (Moos, 1979). Environments that promote and encourage excessive drinking can be termed as high-risk.
6. Drink: A drink in the US is 12 grams of alcohol or 0.5 oz of absolute ethanol (e.g., 12-oz can or bottle of beer, a 5-oz glass of wine, or 1.5 oz of liquor; see Figure 1). Each contains the same amount of alcohol and they are all the same to a Breathalyzer. The liver breaks down approximately 95% of all alcohol consumed, requiring about one hour to metabolize the alcohol in one standard drink. The remaining 5% passes out via the urine, the breath, and perspiration.
7. Drinkers: Individuals are considered "drinkers" provided they consume alcohol in the previous 12 months. 8. Heavy drinkers: The heavy drinking class is composed of individuals who drink 3 to 4 drinks at least once a week or consume 5 or more drinks at any one sitting more than once a week. Hence, binge drinkers are included in the heavy drinking category.
11
12 oz. of beer or cooler
8-9 oz. of malt liquor
5 oz. of table wine
8.5 oa. ahown la a 12-os. gU» that, if full, would hold about I.5 itantud ctriola of oalx liquor
I
3-4 oz. of fortified wine (sud) as s b m y or port) 3>5 oa. mown
2-3 oz- of cordial. liqueur, or aperitif
1.5 oz. of brandy
1.5 oz. of spirits
(a single jigger)
(a single jigger of 80-proof gin, vodka, whiskey, etc.) Sham Knight and in * highball gb* wi.hicrlo.faow the Jfln berate
f
"
- 5 % okohol
-7%«IK.IIO1
-U aiiolioi
- 1 " % ajcliol
- 2 4 % akoI.ol
^40*o i l u i h o l
- H > % akaliol
12 oz.
8.5 oz.
S oz.
3.5 oz.
2.5 oz.
1.5 oz.
1.5 oz.
Figure 1. "Drink" in US (source: NIAAA's NIHpublication no. 07-3769, May 2007) 9. Heavy use: Heavy use of alcohol is defined as drinking five or more drinks on the same occasion (i.e., at the same time or within a couple of hours of each other) on each of 5 or more days in the past 30 days. Heavy alcohol users also are defined as binge users of alcohol. 10. Heterogeneity of types: In general, a typical drinker fits one of the four types shown in Figure 2. Most college students are early onset reaching a peak during their college years, followed by diminished drinking that is, first (blue) or second (green) curve in this figure. 11. Light drinkers: Light drinkers are defined as those who drink at least once a month, but consume no more than 3 drinks at any one sitting. 12. Moderate drinkers: Moderate drinkers are individuals who drink at least once a month and consume 3 to 5 drinks per sitting or who drink at least once a week but consume no more than 3 drinks per sitting. 13. Remission and Recovery: Remission is often used to refer to a state in which an alcoholic is no longer showing symptoms of alcoholism. The American Psychiatric Association considers
12
12
18
25
32 Age
40
50
60
Figure 2. Heterogeneity of types of drinkers (MarkL. Willenbring, MD, Director, Division of Treatment & Recovery Research NIAAA/NIH, Bethesda, MD, USA, presentation) remission to be a condition in which the physical and mental symptoms of alcoholism are no longer evident, regardless of whether or not the person is still drinking. They further subdivide those in remission into early or sustained, and partial or full. Some groups, most notably Alcoholics Anonymous, do not recognize remission. Instead, these groups use the term recovery to describe those who have completely stopped consumption of alcohol and are addressing underlying emotional and social factors. 14. Tolerance: It is the ability to drink progressively larger amounts without apparent intoxication. 15. Treatment Protocol: Treatment respondents are classified as those needing treatment for an alcohol use problem, provided they meet at least one of the following three criteria over the past year: (1) alcohol dependence, (2) alcohol abuse, or (3) received treatment for an alcohol
13 use problem at a specialty facility (i.e., drug and alcohol rehabilitation facilities [inpatient or outpatient], hospitals [inpatient only], and mental health centers). Treatments for alcoholism are quite varied because there are multiple perspectives on the condition itself. Those who approach alcoholism as a medical condition or disease recommend different treatments than, for instance, those who approach the condition as one of social choice.
16. Withdrawal: It defines a state in which anxiety, agitation, confusion, shakes, fits, etc., occur when alcohol consumption is suddenly stopped.
CHAPTER 2 The Impact of Relative Residence Times in Highly Distinct Environments on the Distribution of Heavy Drinkers 1. Introduction Approximately 40% of US college students engage in "binge" drinking, defined as five or more drinks in a row for males and four or more for females (Wechsler et al., 2002). Close to one third meet the DSM-IV diagnostic criteria for abuse and 6% for dependence (Knight et al., 2002). The availability of alcohol around college campuses seems strongly linked to drinking risks (Weitzman et al., 2003). Certain living arrangements seem to protect college students from heavy drinking (e.g., living in parents' homes) (Gfroerer et al., 1997), while high-risk freshmen tend to group themselves into high-risk drinking environments upon their enrollment in college (Baer et al., 1995). Increasing the time spent in high-risk environments reinforces the use of alcohol and increases the likelihood of engaging in risky drinking. It is important to identify the interpersonal and social risk factors associated with established heavy drinking, if nothing else, to evaluate the impact of targeting treatment interventions. However, we can not ignore that the preponderance of problems reported by students occur among "moderate drinkers" (i.e., those consuming 2-3 drinks per occasion) (Gruenewald et al., 2003). Attention must be paid to the context in which less extreme forms of drinking occur, with emphasis on the contextual conditions under which the drinking of moderate drinkers becomes "risky" (increasing the probability of the occurrence of alcohol-related problems and/or increased levels of drinking). Within the micro-environments in which college drinkers use alcohol, the factors most often identified as primary influences upon drinking are cognitive variables related to social norms (Sher et al., 2001) and alcohol expectancies (Darkes et al., 2004; Goldman et al., 2004). The influence of family diminishes dramatically during this time. Drinking levels of college students are most affected by
15
the drinking patterns of their peers and are strongly impacted by positive and negative expectancies about drinking outcomes (Borsari et al., 2001; Orford et al., 2004; Sher et al., 2001). Social influences and expectancy effects are context dependent, that is, they differ between settings (Goldman et al., 2004). For college students, these settings are extremely heterogeneous. For example, student-living arrangements range from those in which alcohol is normally banned (e.g., university dormitories) through partially regulated environments (e.g., fraternities and apartment complexes) to entirely unregulated settings (e.g., cooperatives and private apartments). Since drinking patterns segregate across the residences of drinkers, new residents will be exposed to different influences and pressures depending on where they live (e.g., a fraternity versus a university dormitory) (Gruenewald et al., 2003). Drinking patterns also segregate across different venues for use, with, for example, off-campus parties being the preferred venue for becoming intoxicated (Harford et al., 2002). Thus, different places for drinking will also expose users to different influences and pressures for high-risk drinking, resulting in both levels of intoxication and alcohol-related problems being context specific (Harford et al., 2003; Usdam et al., 2005). The influence of such settings may also vary according to age. Underage students are more likely to drink in private than public settings (as well as to drink less often, but more per occasion) than students who have reached the legal drinking age (Wechsler et al., 2000) Previous research has clearly identified shifting patterns of alcohol use that include high peak drinking levels among freshman and sophomore students, especially white males, became more moderate as students progress through their junior and senior years as they reach the legal drinking age (Gruenewald et al., 2002, 2003). These observations lead us to consider two questions. First, how does one conceptualize the ecological dynamics relating choices of drinking places, and social influences in operation at those places, to drinking levels and their evolution over time? Second, what are the implications
16
of these dynamics for high-risk drinking among various groups including college students? While the current work does not purport to provide a comprehensive answer to either of these questions it does provide a framework, a way of thinking, in which they can begin to be addressed. As befits the study of any ecological problem that exhibits a large degree of complexity, a great deal of initial conceptual work is required before appropriate models can be constructed that effectively inform subsequent empirical inquiries (Gorman et al., 2004). 1.1. A Mathematical Approach There are numerous statistical examinations of the development of and change in alcohol use among college students. However, there are few mathematical studies of the dynamics of drinking. Consequently, there is little insight into the impact of specific dynamic interventions on problem drinking. The model presented herein builds upon previous work in mathematical epidemiology (Anderson et al., 2001; Brauer et al., 2001). The framework provides a starting point for the development of specific population level models whose mathematical analysis may provide partial answers or additional insights to questions of the type posed earlier within a restrictive, but clearly identified, set of assumptions. Derived from classical disease models, these contemporary models of contagion and transmission have been used to study patterns of social influence and their effects upon problem behaviors such as violence (Patten et al., 2004), eating disorders (Gonzalez et al., 2003), alcohol use (Braun et al., 2006; Sanchez et al., 2007) and use of illicit drugs (Behrens et al., 1999; Song et al., 2006; Winkler et al., 2004). The dynamics of drinking are the result of a set of interacting factors that take place in various levels of social organization and over distinct temporal scales. In the present analysis, a simple compartmental model with two distinct drinking environments and three classes of drinkers is employed to examine the effects of residence time upon the differential development and persistence
17 of heavy drinking. College-student drinking provides the context (reference frame) in which the model and its analysis is discussed. I focus on the effect of the proportion of time spent by moderate drinkers in low-risk (e.g., alcohol-free dormitories) versus high-risk (e.g. bars, nightclubs, sports events) drinking environments. The underlying assumption is that in the former, alcohol is not being actively promoted, whereas in the latter it is (through happy hours, targeted marketing, special pricing, etc.). That is, I presume that drinking in active drinking environments will have long term effects on the drinking behaviors of participants in these venues, since contacts between drinkers are known to promote heavier drinking (Collins et al., 1985; Van de Goor et al., 1990). In other words, the social influences operating in drinking places can catalyze problem drinking to different extents. My primary goal is to illustrate the importance of heterogeneous drinking environments in the prevalence of high-risk drinking using a simple model of drinking. The focus is not on the mathematics or on generality detailed and highly technical analyses of similar or related models can be found in earlier work (Castillo-Chavez et al., 2003; Song et al., 2006) but rather on highlighting the added dimension that simple models bring to the discussion of the role of environmental context on drinking dynamics. Of course, the potential usefulness of my results (just like those of any field study) is a function of the assumptions made. The experiences collected through over a century of model-generated theories in ecology and epidemiology support the view that model results tend to do one or more of the following things. They confirm the results of equally limited field studies, generate novel hypotheses, raise new questions, and/or increase our attention on missed factors. The remainder of the chapter is organized as follows: section 2 (Model Description) introduces a simple compartmental model with two drinking environments and two highly stylized forms of social mixing; section 3 (Results) focuses on the partial analysis of the model, computes
18 the conditions necessary for the establishment and persistence of a drinking culture, and provides summaries of the results of extensive simulations; section 4 discusses the results and their implications, and outlines possible extensions; the Appendix provides enough details of the mathematical analysis of the model to follow the discussion.
2. Model Description I consider a stratified population of drinkers that interact in "low-risk" (Ei) and "highrisk"(i?2) drinking environments. Drinkers are classified according to their levels of drinking as susceptible (light or occasional drinkers, S), moderate drinkers (Mi and M2) or M-drinkers, and heavy drinkers (H). The classes S, M and H are not defined precisely here, albeit such definitions do exist (see Chen et al., 2005). It is assumed that individuals in the S- and if-drinking states socialize primarily (here exclusively) with individuals in a single environmental class, that is, only within low- and high-risk drinking environments, respectively. S and H individuals, within my caricature model, are prisoners of their preferred drinking environments, a situation that changes only when their drinking status changes. A minimal degree of "drinking environmental" heterogeneity is achieved through the inclusion of two classes of moderate drinkers: M\ and M2. M»-individuals socialize, in the context of drinking, exclusively in environment Ei. M-drinkers are allowed to make transition between both M classes at the rates 71 and 72 as shown in Figure 3. Epidemiological frameworks (Brauer et al., 2001; Busenberg et al., 1991; Castillo-Chavez et al., 2003) can handle high and detailed levels of heterogeneity. However, the goal here is not to introduce a general model within a contagion framework (as has been done for other problems, for example, Busenberg et al., 1991; Castillo-Chavez et al., 1994; 2003), but rather to show that this approach can generate (under clearly specified assumptions) insights or hypotheses of the kind that are not evident from the direct
19 statistical analyses of the data. This framework allows for the possibility of systematically exploring the impact of hypothetically effective intervention measures in a "virtual" setting under rather general assumptions if one so desires. My model incorporates four mechanisms or drivers associated with drinking dynamics and progression. The first driver comes from the local social interactions between drinking types in low-risk drinking environments. The strength of this effect is incorporated through the parameter /3Q. The second driver is a function of the residence time spent by moderate drinkers while in highrisk environments. This effect is incorporated via the parameter a. The third driver is a function of environment-specific M-residence times (functions of 71 and 72). The fourth driver (model via the parameter fii) is the result of the social interactions between S— and M%— individuals outside both drinking environments E\ and E2. Alternate ways of modeling relevant interactions may include a non-drinking environment where drinking classes and non-drinkers coexist. The 5* — M2 interactions in the flow chart diagram are drawn within the environment type E\, but their nature is differentiated by the use of a dotted line (see Figure 3). Transitions which might have lesser effect in the college time-frame, such as regression of heavy drinkers back to the M or S classes, are not included in this model. The denominators in the non-linear rate expressions in Figure 3 express the total population relevant to the interaction being modeled. The larger the denominator, the less likely that a social encounter between the selected types may occur. In the rate involving /30 the total population involved is assumed to be S + M\, whereas in the rate involving #2 the total population involved is larger, e.g., S+M\ + M2. There are alternative and even rigorous ways of modeling such influences (Blythe et al., 1995; Busenberg et al., 1991; Castillo-Chavez et al., 1994), but the goal here is only to address, in as simple a setting as possible, the impact of environment-dependent influences on
20
drinking dynamics. Hence, I model in a broad manner distinct, environment-specific contact rates between individuals. Social interactions have been modeled before in great generality (Busenberg et al., 1991; Castillo-Chavez et al, 1994; Castillo-Chavez et al., 2003). The goal here is to highlight the dramatic potential impact of heterogeneous contact rates on drinking dynamics.
Low-Risk Drinking Environments (Et)
1 \MX
A
L_
nMi
S+Mh
;S
*"*
i
'
_ _ _ _ _
*
SM,
s
High-Risk Drinking Environment (12)
M» »1
H:
Mi ^Mj
i
1 j
1
H, with the total population at risk being N = S + Mi + M2 + H. Here, ~ indicates circulation between Mi and M2. The
21
system (E.l) of equations for the model are stated in Appendix B. Definitions of model variables and parameters are collected in Table 1. The term abusive drinkers is used to refer collectively to moderate and heavy drinkers. The community comprised of light, moderate, and heavy drinkers is called the mixed drinking community. The parameters /%, /%, 71, and 72 are non-negative, whereas the parameters \i and a are positive. The variables S, M\, Mi, and H take on only non-negative values (i.e. zero or positive). Although I could build more elaborate models of drinking as a process of contagion in established ways (Busenberg et al., 1991; Castillo-Chavez et al., 1994; Castillo-Chavez et al., 2003), I have chosen not to follow such a path here. I have constructed a model (see Equations in System (E.l)) that incorporates "touches" of heterogeneity in a simple way via four drivers. I have learned from this theoretical study that characterizing the environments where people drink and measuring the average-relative residence times of drinkers in such environments (not done in current studies) may be a worthwhile path in alcohol research. I hope that social scientists interested in the study of alcohol dynamics at the community, regional, or national level will find the results of this study insightful. This study will be a sucess if the characterization of the places where people drink and the individuals' distributions of drinking times are eventually identified and if their role in drinking is fully explored.
3. Results 3.1. Model Equilibria and Stability System (E.l) has an abusive drinking "free" (or trivial) equilibrium (i.e., S = N, M\ = 0, Mi = 0, H = 0), which represents a population with only S-individuals. The possibility of the propagation of high levels of drinking within the S-community is analyzed through the examina-
22
Table 1. The definition of the model variables and parameters. Variables & Parameters t S(t) M\(t) M2(t) H(t) 00
02
fi a 71 72 7
Definitions
Units
Time variable. Number of light or occasional drinkers (or susceptibles) at time t. Moderate drinkers in low-risk drinking environments (£1) at time t. Moderate drinkers in high-risk drinking environments (E2) at time t. Number of heavy drinkers (or abusive drinkers) at time t.
Years. Number of individuals. Number of individuals. Number of individuals. Number of individuals. Average number of social influential contacts (environment Number of condependent) of one susceptible per unit time with M\ indi- tacts x per unit viduals (also called as peer pressure coefficient). time x per individual. Average number of social influential contacts (environment Number of condependent) of one susceptible per unit time with M2 in- tacts x per unit dividuals mediated by Mi individuals (also called as peer time x per indipressure coefficient). vidual. Per capita arrival and departure rate for the system (can be Per unit time x referred as turnover rate in college context). per individual. Per capita progression rate of M% — m moderate drinkers to Per unit time x H class (also referred as linear transition rate). per individual. Per capita leaving rate of Mi-moderate drinkers from low- Per unit time x risk environments. per individual. Per capita leaving rate of M2 -moderate drinkers from high- Per unit time x risk environments. per individual. Proportion of time individual resides in the high-risk Dimensionless. environments (Residence time parameter).
tion of the impact generated by the introduction of a small number of "typical" moderate drinkers ("invasion" process) in a large population of susceptibles. That is, the proportion of S individuals is approximately 1. The theoretical characterization of "typical" moderate drinkers must include their "residence" habits, that is, the proportion of time such an individual spends in E\ and Ei. The construct of a "typical" individual in a population is a useful construct (Diekmann et al., 2000). A "typical" individual may not exist in my population (the mean of a sample is not necessarily represented by any member of the population). Following the approach outlined in Van den Driessche et
23
al., (2002), I compute the average number of conversions (5 —> Mi), Ra, generated by a "typical" moderate drinker in a population where the proportion of S-individuals is approximately one. In fact, Rd{l)
where 7 =
+
^
+a.
=
/ i ( l - 7 ) + (/x + a ) 7 ( 1 "
7) +
/ i ( l - 7 ) + (M + a ) 7 7 '
^
The role of the parameter 7 is clarified later in Section 3.2. R^, the drinking
reproductive number, gives the average number of new moderate drinkers generated by a typical moderate drinker during his/her time as moderate drinker, where the proportion of S-individuals is approximately one. If 7 « 0 then R ~~P° -Krf —> that is, individuals primarily drink in E\ while most moderate drinkers do not move to E2. If 7 ~ 1 then Rd ~
—;—•,
[i + a
that is, most moderate drinkers spend their drinking time in £2. The basic drinking reproductive number can be rewritten as the weighted sum Rail) = A)(l - 7)^(7) + ^27^(7), where the 7-average period of M-influence is
Dh) =
. /*(1 - 7) + Ox + a ) 7
If R 1 a drinking
"outbreak" takes place, that is, the population of moderate (and consequently heavy) drinkers begins to grow (for analogous situations in epidemiology see Brauer et al., 2001). Numerical simulations support the existence of a unique ("stable") mixed drinking community, that is, a community with 5,
24 Mi, M2, and if positive when Rd > 1. This will be referred to as the endemic drinking equilibrium,
Numerical solutions (Figure 4) and experience with similar models strongly support the hypothesis that if Rd > 1 then the unique endemic drinking state, D\, is locally asymptomatically stable1. Whether or not it is globally stable2 is not obvious, albeit extensive simulations using parameters derived from college data suggest that for such populations it is likely to be.This would mean that a mixed culture of drinking is likely to become established regardless of the size (> 0) of the initial population of drinkers. Furthermore, such a community would persist (re-organize itself) even after major perturbations, that is, large increases or reductions in the number of drinkers. I have proved local asymptomatical stability of D\ only when Rd is slightly larger than one. The Theorem in Appendix B states some mathematical results3. 3.2. Computing the Effect of Residence Times I derive the endemic drinking equilibrium state D\ by setting the derivatives in the System E. 1 equal to zero. At the equilibrium, the number of individuals in the classes (or compartments) of the model are not changing with time, but individuals continue to circulate among the compartments, that is, they enter and leave the classes at the prescribed rates. Because the system is at equilibrium the class sizes are constant. Let M* = M{ + M%. A constant 7, 0 < 7 < 1, can be found such that Mf = (1 - 7)M* and M | = 7M*. That is,
'A drinking equilibrium point or steady state is locally asymptomatically stable if solutions starting from nearby initial conditions (values of state variables at time 0) remain nearby. 2 A drinking equilibrium point or steady state is globally stable if starting from any initial condition that includes moderate drinkers the solution approaches this equilibrium as time becomes large. 3 The focus is not on the complete analysis of this model, but rather on its potential role in generating at least some new hypothesis. Some mathematical details are provided for those with interest on addressing relevant technical challenges.
25
M,(0) •0.21, M,(0) (0.03, H«}) >0.C _^sic)-e.B. M,(O)-O.«, MJ[0)-O.M,
Htn-n.1
^-S(01 -US, M,(0] -0.3, M-(O) .0,1, H(0] - a i .,...sio).i).«, M , | o ) i a . i , M;(0).o.i, H|a]-a.z -^_S[O)-0.i, M,«l)'(l.4, Mg(0)-0.1, H( 0 there is movement of moderate drinking individuals from E\ to £"2- Since at any time, in equilibrium, the fraction of members of M who are in E% is 7, we see that members of M spend a proportion (7) of their time in E^. Hence, 7 denotes the proportion of time, the relative residence time, spent by a generic (M*) moderate drinking individual in E2 (high-risk drinking environments) while 1 — 7 denotes the proportion of time spent by the generic M* individual in Ei (low-risk drinking environments). We see that 7 captures, on a 0 to 1 scale, the effect of the rates of movement between environments, 73 (i = i, 2). I say that the drinking environments are connected if 0 < 7 < 1 and highly connected when 7 is near 0.5. The relative residence times in distinct drinking environments as parameterized by 7 £ (0,1) have therefore been computed. I observe that R4 captures the essence of the "typical" moderate (M*) drinker who must be part of both worlds (i.e. high- and low-risk environments) or who must have an average (i.e. weighted by two environments) preference. Now that I have expressed the endemic drinking equilibrium state values in terms of 7 (see Equation (E.4) in Appendix C), I proceed to evaluate the impact of the proportion of residence time (7) of M* individuals on the prevalence of different levels of drinkers.
27
3.3. Numerical Results This subsection has two parts. The first part (Figure 5) explores the impact of variations of model parameters 71, 72, and a on the basic drinking reproduction number, Rj. The second part (Figures 6,7, and 8) focuses on the impact that relative residence times, 7, have on endemic drinking levels. Parameter estimates using available national college data give fa ~ 0.71, fa ~ 2.20, a ~ 0.51, p. ~ 0.27 and 7 ~ 0.28 (see Chapter 3). The parameter values used to illustrate the results of numerical studies were selected purposefully. The community size is fixed at 1000 individuals. The social influence of moderate drinkers from low-risk and high-risk environments (fa and fa) are assumed to be equal (fa = fa = 2). The average residence time in the system is assumed to be 4 years (p = 0.25). The per capita progression rate from moderate to heavy drinking, a, is taken to be 1.5. I fix the values of three of the above four parameters in order to focus on the study of the effect of changing 7 and either a, fi, or fa (Figures 6-8).
Figure 5. Plot of the surface 72 = /(71, oc) (function 'f obtained from Rd = I); Fixed parameters are (3Q = 2, fa = 2 and JX = 0.25. The parameter definitions are in Table 1.
28 3.3.1. Effect of Movement and Progression Parameters on the Reproduction Number. 7, is the per capita transfer rate of moderate drinkers from environments Ei to Ej (i ^ j , i = 1,2). a is the per capita progression rate of moderate drinkers (in E2) to the heavy drinking class. The graph of the surface 72 = /(71, a), obtained by equating Ra to 1 and fixing the values of fJo, P2, and /x is shown in Figure 5. This figure captures the effect of 71, 72 and a on Rj. The parameter regions where Rd > 1 and Rj < 1 are highlighted also in Figure 5. The graph of / is the boundary on which Rd = 1. Figure 5 shows in what region (either side of the surface 72 = / ( 7 1 , a)) the combination of the parameters (71, 72, and a) lie. For drinking communities with (71, a, 72) = (5,4.5,0.5), I have that 7 = 0.4878 and 1 - 7 = 0.5122. For this last example, even though the values of 7 and 1 — 7 are close to each other, abusive drinking will eventually die out, since Rj < 1. For the community given by (71, a, 72) = (4,4.5,8), I have 7 = 0.2388 and 1 — 7 = 0.7612. In this last community mixed drinking will grow (and most likely persist) since Rd > 1. 3.3.2. Drinking Patterns at Equilibrium, Endemic drinking equilibrium values for all class of drinkers as a function of 7, the proportion of time spent by moderate drinkers in high risk drinking environments, is plotted in Figures 6 through 8. Figure 6 shows the impact on drinking patterns from changes in a. Figure 7 highlights the impact of variations in /3o on the composition of the drinking community. Figure 8 focuses on the impact of variations in /x. Definitions of all the parameters are collected in Table 1. In Figure 6 I fix the values of f3o, fh, V* and vary 7 and a. If the rate of progression from moderate to heavy drinking is low (i.e. a — 0.01), the endemic equilibrium is nearly constant in 7 (the proportion of time a moderate drinker spends in high-risk environments, E2) and the class M* is large (Figure 6). Larger values of a (0.2 and 0.6) increase the sensitivity of equilibrium values to 7, with H* increasing as 7 increases at the expense of M*. For the selected values of a, S* remains
29
a =0.01 1000
a =0.2
a =0.6
800 600 400 200
0
1000 800 600 400 200
0
Figure 6. Equilibrium class sizes for various values of a (per capita progression rate of moderate drinkers in high-risk environments to the heavy drinking class), as a function of the proportion, 7, of residence time in high-risk environments; with fio = 2, fo = 2, p, = 0.25 and N = 1000; On the vertical axis I have the number of individuals in the S*, M* and H* classes; S* represented by Solid, Green curve, M* by Dotted, Red curve, and H* by Dashed, Blue curve; Parameters are defined in Table 1. small. For all a < 1, increases in 7 result in an increase in the number of heavy drinkers. That is, the more time moderate drinkers spend in high-risk environments, the larger is the number of heavy drinkers in the community even when the progression rates from moderate to heavy drinking are low. Further increases in a (from 1.2 to 1.7), with 7 values near 1, result in sharp increases in the class S* and decreases in the class H*. In this last situation, the M* class changes slowly becoming almost extinct. That is, with low movement between environments, decreases in the average moderatedrinking-period of a drinker from 10 months (a = 1.2) to 7 months (a = 1.7) result in decreases in the number of heavy drinkers from 60% to 20%. When 1 < a < 1.8 heavy drinkers "dominate" for most 7 values, becoming about 70% of the total population for some intermediate value of 7. High
30 values of a correspond to fast progression from M to H, which among college students for example may be defined as moving from moderate to recurrent high-risk (binge) drinking. Fast progression of drinkers from a moderate to a heavy drinking level increases the proportion of heavy drinkers in communities with high "connectivity" (i.e., 7 near 0.5) between environments. P0=0.6
Po= 0 - 2 1000
P0=1.8
1000
1000
\ s ^ ^ 500
500
^c
t
/ ,
0
0.5
ft
:
': 0.5
|
..„„,. nun
1
0.5 1
P0=7
3 1000
1000
P0=50 1000
,
: \
; „----" "X
',t 500
500
/ ' [
J .—L—^-^"-
y
v, A
500
i 0.5
"~--s
..'%...j/r*
Y
Y
P0=
500
ffj^lMP'-**wri!C
TsTW) dM2 dt
s p2
,,
~
i3A)
+ M1 + M2y n
SM2
7 l M l + P2
+
,,
M
(S + M1 + M 2 ) -» *
,„„,
(3 2)
-
= 71-Mi - 72M2 - \xM2 - aM 2 ,
(3.3)
=
(3.4)
aM2-fiH,
where iV = S + Mx + M2 + H.
In summary, the model is a simple caricature that incorporates, in as simple mathematical terms as possible, four mechanisms: the local social interactions between drinking types in low-risk drinking environments (the strength of this effect is incorporated in /%), the residence time spent by
41
moderate drinkers while in high risk environments (this effect is incorporated in a), environmentspecific M—residence times (functions of 71 and 72), and the social interactions between S— and M2— individuals outside of both drinking environments E\ and E2 (its strength is incorporated in fti)- The total recruitment rate is fiN, with a per capita average departure rate \i within each drinking class. Thus, the average time a drinker spends in the system is 1/fx units of time and the population N remains constant. Progression from S to Mi, defined by terms containing /3b and (32, results from environmentally dependent encounters between S and Mj—individuals (i = 1,2). Specifically, PQ describes "effective" encounters between S and M\ while in E\, and fe describes "effective" encounters of M2 with S-individuals outside the E-environments. In other words, the fa are used to quantify effective social contacts/interactions of a moderate drinker in E^-environment with light drinkers. The parameter a reflects the progression of moderate drinkers in high-risk drinking environments to heavy drinking. The true values of the parameters of this model are unknown, but estimable from a variety of sources. This estimation is a subject of the analysis presented in this study. The values shown in the Table 2 represent the best estimates from the analysis. Clearly, I only have a caricature of a highly complex and heterogeneous social process. There has been extensive research on ways of modeling from first principles highly heterogeneous, complex processes (Blythe et al., 1995; Busenberg & Castillo-Chavez, 1991; Castillo-Chavez & Hsu Schmitz, 1997; Castillo-Chavez, 2003). However, the inclusion of such a level of detail at this stage would have serious drawbacks. First, it would increase the number of parameters that must be estimated and would make the model less tractable mathematically. I choose a simplified approach, common in past studies of the dynamics of highly complex systems: HIV immunology (Perelson et al., 1996) and HIV epidemiology (Anderson & May, 1988). It may be argued that through computer
42
simulations one could handle more sophisticated models. This may be possible if my goal was to make specific short-term predictions from a full set of data. My goal, however, has been to identify the consequences of the movement of drinkers between highly distinct environments. Parameter estimation for my model using available college data (collected for alternate purposes) has been possible because of my choice of a parameter-scarce model. The per capita rates of movement of moderate drinkers between drinking environments (71 and 72) are incorporated into my analysis at non-trivial steady state by the use of a single parameter 7 defined by (D.2) in Appendix D (derived in Chapter 2). Parameter 7 is interpreted as the proportion of the relative time that moderate drinkers spend in high-risk drinking environments. Consequently, 1 — 7 represents the proportion of the relative time that moderate drinkers spend in low-risk drinking environments. A value of 7 near 0 or 1 corresponds to no movement, while a value of 0.5 indicates highly connected risk environments. A key aspect of drinking dynamics within the current model is captured by the basic reproductive number, RQ, a widely used threshold in mathematical epidemiology (Diekman, Heesterbeek & Metz, 2000; Driessche & Watmough, 2002). It can be interpreted as the average number of secondary case conversions from light to moderate drinking generated by a generic or typical moderate drinker in a population of "predominantly" light drinkers at a demographic steady state. A typical moderate drinker captures the average behavior and characteristics found in a population of moderate drinkers in pre-specified environments and may not exist as a member of a modeled population. The role of R^ is to quantify the likelihood that a community of light drinkers will be destabilized in the event that a "typical" moderate drinker (or a small number of drinkers whose mean characteristics represent that of a typical moderate drinker) is (are) introduced into a population of mostly light drinkers. If its value is less than one, then moderate to heavy drinking will eventually die out in die
43 population. However, if Rd is greater than one, all levels of drinking will spread until a particular mixed community of drinkers becomes established. Hence, I refer to Rd as a measure of the impact of effective social influence on light drinkers by a typical mobile drinker. Effective intervention programs are based on Rd, a predictor not only of the ability of an "invasion" to succeed and alter the drinking community structure, but also a measure of the severity of the outcome (size and distribution of the "endemic" drinking state). A policy that modifies the ^-parameter will effectively replace Rd by R*d (basic reproductive number under intervention) with R*d < Rd- The magnitude of R*d relative to Rd will determine the success of the intervention program. If R*d < 1, eventually the M and H classes will diminish to zero, as long as such "policies" remain in effect long enough. From the perspective of the current model, the amount of drinking that will be observed within a population will depend, at least in part, on model parameters that lead to increases or reductions in Rd- However, the parameters in such models are never measured with precision. Hence, uncertainty and sensitivity analyses are needed to quantify the variability in Rd that result from errors in their measurement. To the extent that estimates of underlying parameters can be provided by statistical studies, these can be entered into the model and plausible variations in distributions of model parameters explored. The effect of uncertainties in parameter estimates are investigated by examining their impact on the distribution of Rd and hence on model outcomes. I introduce a Monte-Carlo sampling-based procedure to perform uncertainty and sensitivity analysis. Sensitivity analysis assesses the amount and type of change produced in the model predictions (in terms of Rd) that result from small changes in Rd—parameter values. Sensitivity indexes that use partial rank correlation coefficients are involved in my sensitivity analysis. If, for example, I find that Rd is particularly sensitive to a particular model input, changing greatly in response to changes in one parameter rather than another, then the model suggests that conditions that affect that parameter may
44
be particularly effective in changing drinking in college. Thus, the model can lead to (a) specific statements of hypotheses about the effects of parameter change over time and (b) estimates of the sensitivity of model outcomes to certain changes in model inputs. Although uncommon in the study of alcohol-related behaviors, the analyses are by no means unique and are commonly employed in the study of questions associated with the dynamics of populations that lie at the interface of the social and natural sciences (Nuno et al., 2006; Blower & Dowlatabadi, 1994). In the present analysis, I address whether the movement of moderate drinkers between low- and high-risk drinking environments, and their contacts with light drinkers, will establish and maintain a mixed drinking community. I do so using data pertaining to the drinking patterns and drinking settings of college students in the United States.
2. Data Sources No single study in the published literature could be found that provides all the data necessary to derive point estimates and reasonable ranges for all the parameters of the model. Hence, I use data from two studies conducted by Eng et al. (1994; 1997) as well as data from a survey of college drinking from Saltz and Gruenewald (2007). Parameter estimates and their ranges are derived from these sources. Having established approximate point estimates and variances for some of the parameters, I proceed to assign them "reasonable" probability distributions centered on the computed point estimates. Samples from these distributions are substituted in the non-trival steady state expression of the model via the Monte-Carlo sampling-based approach to estimate other parameters and to generate distributions of the values of Rd (see Section 3 for details). Point estimates of the parameters and variances of these estimates appear in Table 2. The data sources used to estimate drinking classes include student university surveys from
Rd
a
H
/?2
/3o
Parameter
Per capita progression rate of moderate drinkers in high-risk environments to heavy drinking class (per capita linear transition rate) Average number of secondary conversions of light to moderate drinkers generated by the introduction of a "typical" moderate drinker into a population of primarily light drinkers
0.27 (0.06) 0.17 (0.04) 2.14 (0.47)
0.27 (0.06) 0.51 (0.12) 2.69 (0.16) per person per year per person per year Dimensionless
0.75 (0.47) 2.20 (1.40)
Interactions per person per year
1.69 (1.08)
Mean (Std.) 0.83 (0.02)
0.71 (0.45)
Dimensionless
Proportion of residence time of a moderate drinker in high-risk environments Complement of this proportion will be their time in low-risk environments (1 — 7) Average number of "effective" interactions (environment dependent) of one light drinker per unit time with moderate drinkers in low-risk environments Average number of "effective" interactions (environment dependent) of one light drinker per unit time with moderate drinkers of highrisk environments mediated by moderate drinkers of low-risk environments Per capita arrival and departure rate for the system
Mean (Std.) 0.28 (0.01)
Interactions per person per year
Unit
Definition
Table 2. Definition of the parameters and their estimates for two values of7.
LA
46 all 48 contiguous US states. They were conducted during the 1980's and early 1990's by Engs et al. (1994; 1997). Students are considered "drinkers" provided they consumed any amount of alcohol during the previous 12 months. Quantity and frequency of drinking are assessed and used to divide drinkers into the light, moderate, and heavy categories. Light drinkers are denned as those who drink at least once a month, but consume no more than 3 drinks at any one sitting. Moderate drinkers are individuals who drink at least once a month and consume 3 to 5 drinks per sitting or who drink at least once a week, but consume no more than 3 drinks per sitting. The Heavy drinking class is composed of individuals who drink 3 to 4 drinks at least once a week or consume 5 or more drinks at any one sitting more than once a week. Hence, binge drinkers are included in the heavy drinking category. Data from these surveys are used to calculate the average percentage of light, moderate, and heavy drinkers in the college drinking population, that is, the proportion of S, M, and H classes in my model. These are estimated at 25%, 49%, and 26%, respectively (Table 3). Table 3. Percentage of drinkers in three drinking-level classes from 1982 to 1991. Data source: Engs et al, 1994.
Classes S M H
1982 (%) 240 51.2 24.8
Years 1985 1988 (%) (%) 219 253 51.0 48.5 25.1 26.2
1991 (%) 26\8 45.9 27.3
Average (%) 25.00 49.15 25.85
In a later study, Engs et al. (1997) collected data from a population of university students from every state during the 1993-1994 academic year. Their study involves more than 12,000 students and includes data on the frequency distribution of student drinkers by year in college. Among freshmen, 64.2% fell into one of the three drinking classes (light, moderate, and heavy). This percentage rose to 71.4% among sophomores, 76.1% among juniors, and 80.6% among seniors (Table 4). I choose to interprete these data as follows: 64.2% of drinkers drink for 4 years, an additional
47
7.2% remain drinkers for 3 years, 4.7% more for only 2 years, and 4.5% more for only 1 year. Specifically, the mean and variance of the average time a student remains a drinker while in college (i.e. 1/n) are estimated as 3.63 and 0.69, respectively. Hence, on average a typical student remains a drinker for approximately 3 years and 7 months while in college. My estimate of the mean of /x (the per capita arrival and departure rate for the system) is 0.27, and its variance, calculated using the delta method, is 0.0039. Table 4. Percentage of drinkers in four class-years along with their increase from previous classyear during academic year 1993-94. Data source: Engs etal, 1997. All Students Abstainers Light+Moderate+Heavy % Increase In Drinkers Class Year (%) (%) From Previous Class Year Freshman Sophomore Junior Senior
35.8 28.6 23.9 19.4
64.2 71.4 76.1 80.6
7.2 4.7 4.5
Table 5. Use of Drinking Venues by Drinker Class.
Drinking Venue Pubs, Bars, or Restaurants
Visits by all Drinkers ini Class Heavy Light Moderate 3664 1815 2925 (0.330) (1.198) (2.321)
Total Visits 8404 (0.856)
Residence hall
643 (0.117)
1268 (0.415)
1302 (1.033)
3213 (0.327)
Campus events
142 (0.026)
321 (0.105)
307 (0.244)
770 (0.078)
Off-campus Parties
3338 (0.607)
5891 (1.926)
5050 (4.008)
14279 (1.455)
"Greek" Parties
969 (0.176)
1950 (0.638)
1966 (1.560)
4885 (0.498)
Outdoors
300 (0.055)
698 (0.228)
761 (0.604)
1759 (0.179)
5497 (56.0%) 7207 (1.311)
3058(31.2%) 13792 (4.510)
1260 (12.8%) 12321 (9.779)
Total 9815 33320 (3.395)
Number of Drinkers (%) Number of Visits (Visits per Drinker type)
48
I was not able to find any study that characterized college drinking environments, their relative rates of use by college students, or the residence time that drinkers spend in these places. For this purpose I drew upon data available from the Safer Colleges Survey conducted in 14 University of California and California State University campuses in 2003 (Saltz & Gruenewald, 2008). A representative sample of respondents from these universities were interviewed using direct mail or web-based survey methods. They reported whether or not they had consumed alcohol since they arrived at college, their drinking frequencies and quantities over the past 4 weeks or since they started the semester, and the frequencies with which they had consumed alcohol at different venues around the campus (this excludes drinking in their own homes). The drinking data were used to classify drinkers into light, moderate, and heavy drinking groups (defined above), then the frequencies that these different drinkers used different venues for drinking standardized to a four week period and tabulated. Table 4 presents the results of this tabulation for the 9,815 drinkers surveyed at these campuses. For each drinking venue and drinker class the number of times alcohol was consumed at the different venues over 4 weeks is reported with utilization rates per drinker (venue visits per drinker type) and indicated in parentheses. As the table shows, there was considerable variation in the use of different places for drinking, with heavy drinkers consuming very frequently in many different places, but most often at off-campus parties. Two procedures were used to identify or define the top three high risk settings in which moderate drinkers would encounter heavy drinkers. In the first definition I consider just the rate at which heavy drinkers drank in different places, high-risk settings are Residence Hall Parties, "Greek" Parties (fraternities and sororities), and Outdoor Drinking. Heavy drinkers were to be found in these environments on 32.7% of their drinking occasions. Moderate drinkers were to be found in these environments on 28.4% of these occasions (95% CI: 26.7% to 30.1 %). In the second definition I estimate the rate at which moderate drinkers may en-
49 counter heavy drinkers in different environments. I get a different distribution of high risk places, Pubs/Bars/Restaurants, Off-Campus Parties, and Greek Parties (again). Moderate drinkers spend much more time in these places and are most likely to encounter heavy drinkers at these places. 83.4% of moderate drinker visits are to venues which are also frequented by heavy drinkers (95% CI: 80.5% to 86.3%). The first simply states.where heavy drinkers are most likely to be found. The second states where moderate drinkers are most likely to encounter heavy drinkers (taking activities of both moderate and heavy drinkers into account). The parameter 7 reflects the rate at which moderate drinkers shift between low- (without heavy drinkers) and high-risk environments (with heavy drinkers). This parameter is estimated using the above two percentages obtained from the different definitions. In addition, it is assumed that drinking occasions at different venues are proportional to the number of moderate drinkers in these places. Since the data do not capture average drinking quantity per environment type, I choose to interpret the data as follows: 28.38% and 83.43% of moderate-drinkers' drinking occasions are spent in high-risk environments and the remainder in low-risk. Hence, two point estimates (one from each definition of high-risk places) are 7 = 0.28 and 7 = 0.83. Since the estimated values of 7 are extreme and since different sets of data are used for estimating different model parameters, I carry out analyses in this research by varying 7 from zero to one.
3. Uncertainty and Sensitivity Analyses Uncertainty and sensitivity analyses of the effect of parameters /3b, (h, /J, a, and 7 on the drinking reproduction number Rj, are the focus of this section. Monte-Carlo sampling is used to perform both analyses. I extend the procedures employed in earlier studies of disease transmission (Blower & Dowlatabadi, 1994; Chowell et al., 2003; Shanchez & Blower, 1997; Mishra, 2004). I
50 use point estimates that can be obtained from available data. The procedure used here involves the use of non-trivial steady state equations, that is, the equations obtained by setting the right hand side of the System (E.l) to zero. The steady state equations have a unique solution with non-zero components (see Appendix D for expression of components). Distribution of j i
0.2
O.J
0.4
• 1 Distribution of y
(.5
0.0
0.7
0.265 0.265
Distribution of j3-
Distribution of a
0.27
0.275
0.20
0.265
6.29
6.2*5
0.3
0.105
0.27
0.275
13.28
0.265
0.29
0.2*5
0.3
0.195
Distribution of p .
0
0.2
M
0.0
0,0
1
1.2
1.'
Distribution of R .
L I B Figure 9. Frequency distributions of the parameters including Rd when sampling 7 from uniform distributions with mean 0.28. The overall shape of the distribution remains same for the cases when mean of7—distribution was 0.50 and 0.83. Distributions were calculatedfrom one of the 10 MonteCarlo Samples, each of size 105 sampled parameter values using approach described in the text. Horizontal axis has parameter values, and vertical axis represents frequencies in the graphs.
A systematic sampling procedure is carried out, before performing uncertainty and sensitivity analyses, as follows: I assign probability density functions (PDFs) to the parameters 7 and p. Estimates of the mean and variances of these parameters can be obtained from data sources described in the previous section. The uniform distribution is chosen to model the distribution of 7, and a truncated (to ensure positivity of the parameter) normal distribution is selected to model the distribution of \i. A sample is chosen from the hypothesized distributions of 7 and p. This sample and estimates of the steady state values of the sizes of the S, M, and H classes are substituted in one
51 of the steady state (non-trivial) equations in order to get estimates of a. A linear relation between fa and fa is obtained from the remaining steady state equations. The sampled values of 7, /i, and a, the linear relation (between fa and fa), and the non-negativity requirement of the parameters fa and fa are used to obtain bounds on fa. Next a sample of fa is chosen from the uniform distribution defined on the evaluated interval (obtained from bounds on fa). A value of fa is evaluated by substituting sample values of 7, //, a, and fa in the linear relationship linking fa and fa. The details of these steps are collected in Appendix D. Specific steps associated with the uncertainty and sensitivity analyses are carried out separately. I consider available regional data for estimating 7, which might differ from campus to campus (across US). Hence, I extend the uncertainty and sensitivity analyses for all possible estimates of 7 and not only for 0.28 and 0.83. That is, I vary the estimated value of 7 from 0 to 1. However, results when 7 equal to 0.28, 0.5, and 0.83, are specifically noted. From the pre-specified point estimate of 7 and its resulting distribution, uncertainty and sensitivity analyses are performed using samples collected, as described above, of the parameters (7, fi, a, fa, and fa). The uncertainty analysis is carried out as follows: I take an independent sample (without replacement) of 105 values for each parameter following the above steps. From the sample I derive a frequency distribution for Rd using Equation (D.l) given in Appendix D. This sampling procedure is repeated 10 times (i.e., 10 realizations are performed) to provide a measure of stability of results in the analysis. Hence, an empirical distribution of Rd is constructed from generated simulated data for each realization as a function of the variability of the parameters, including 7. For example, the distribution of Rd obtained in one realization when 7 is sampled from a uniform distribution with mean 0.28 is shown in the last panel of Figure 9. Figure 3.10(a) shows the change in Rd estimates and its variance as the estimated mean of 7 varies from 0 to 1. It also shows that the estimated Rd
52
values mostly lie between one and three. The Partial Rank Correlation Coefficient (PRCC) method for performing sensitivity analysis is used to quantify the sensitivity of R4 to variations in its parameters (Helton, 1993; Saltelli et al., 2000). The sensitivity analysis is carried out as follows: I take a sample of 105 values from empirical distribution of the ^-parameters (as described above). This sample and the corresponding Rd values are used to calculate the PRCC of Rd with respect to each of its parameters. This is done using a sequence of regression models as an intermediate step in the analysis (see Appendix D for details). The PRCCs of Rd, with respect to each of the parameters, when the mean value of 7 varies are shown in Figure 11. For example, when mean 7 is equal to 0.28, then the PRCCs of Rd with respect to /3o, (h, 7> /". and a are -0.69, 0.61, -0.13, 0.03, and -0.01, respectively. The higher the absolute PRCC value of the parameter, the higher the effect of the parameter on Rd- The parameters with |PRCC| > 0.5 are considered strongly influential whenever the correlation between them is significant at a level < 0.05 (i.e. p-value < 0.051 ). I can rank the model parameters according to the size of their effect (based on the absolute value of the PRCC values) on Rd- The signs indicate direct or inverse dependence of the parameters.
4. Results The first step of my calculations here is the estimate of steady state value of class sizes S, M, and H, which I obtain from college drinking data (Table 3). I assume that the data in Table 3 gives stationary state values. Hence, all of my results are dependent on these values. Recall that Rd is the measure of the impact of effective social influence of a typical mobile drinker on a 'This value determines the appropriateness of rejecting the null hypothesis in hypothesis test (Null hypothesis is HQ : two variables are uncorrected; alternate hypothesis is Hi : two variables are corretaled). The smaller the p-value (that lies between 0 and 1), the smaller the probability that rejecting the null hypothesis is a mistake. If a p-value is less than given alpha level (in my case it is 0.01), we reject the null hypothesis.
53
0.4
0.5 Meanty)
o.e
o.9
(a) Uncertainty in Rd when Mean(7) is varied.
(b) Variation in estimated values of /3o and ft with change in 7 estimates.
Figure 10. Variation in estimates ofparameters with change in distribution of'7.
54 population of light drinkers, /% quantifies the social contacts/interactions of a moderate drinker in Ei— environments with light drinkers, and 7 is interpreted as proportion of the relative time that moderate drinkers spend in high-risk drinking environments. For a community with parameters obtained from the data sets used here, my results suggest that the more time a typical moderate drinker spends in high-risk environments, the less drinkinginfluence he/she will have on light drinkers (Rd decreases with 7; see Figure 3.10(b)). But the variability in the estimates of Rd increases with 7. For example, in the system where environments are highly connected (i.e., 7 = 0.5) the estimated Rd value is 2.4 with confidence interval of (1.91, 3.04). If a moderate drinker spends less than 47% of his drinking time in high-risk environments then the probability that estimates of Rd will lie between 2 and 3 is effectively one. This probability is reduced to half if a typical moderate drinker spends nearly all his drinking time in high-risk environments. Rd is roughly uniformly distributed with a slightly skewed left end (Figure 9). The estimated value of/?o increases and /% decreases with increases in 7 (Figure 3.10(b)). This implies that interactions of moderate drinkers with light drinkers in low-risk environments increases, whereas interactions in non-drinking environments (i.e. outside E-environments) between these groups of drinkers decreases with increase in high-risk drinking time of an average moderate drinkers. Distribution of fa and fo are similar in shape and resemble two overlapping distributions: uniformly distributed on left end and normally distributed (with high standard deviation) on their right end (Figure 9). Rd is most sensitive to changes in parameters associated with social interactions. If a mobile drinker spends less than 81% of his drinking time in high-risk environments, then the parameter related to interactions in low-risk drinking environments is the most influential on Rd- But if he/she spends more than 81% of his/her drinking time in high-risk environments, then the parameter asso-
55
Figure 11. Sensitivity (measured by PRCC values) of the Rd with respect to its parameter when Mean('y) is varied. ciated with social interactions outside E-environments is most influential (see Figure 11). Hence, Rd is most sensitive to changes in parameter representing low-risk social contacts when 7 is about 0.28 and to the parameter related to social contacts outside E-environments if 7 is about 0.83. If environments E\ and E
63 The process of entry into the system (or inflow) is a Poisson process with fixed rate parameter A. The inflow may be reduced, as a result of a prevention program, by the factor of (1 — u). The parameter v quantifies the effectiveness of a prevention program, with v = 1 implying perfect prevention. Intervention programs are composed of two Poisson-type processes2 which are environmentally dependent with rate parameter Si times the class sizes in E\ and 1 then DQ becomes unstable and a unique locally stable endemic drinking equilibrium D\ = (S*, M j , M£, H*) (all drinking levels above zero) comes into existence. That is, moderate to heavy drinkers co-exist with light drinkers in this community.
The proof of the Theorem 3.1 is in Appendix E. Note that among the control parameters, 7 depends on the intervention parameter associated with the high-risk environment (£2), but it is independent of Lipschitz continuous on an open subset of 3tn then lim Xfi(0) = XQ ==> for every 5 > 0 and t > 0, Jim P I sup \X^(T) TV—>oo
N—>oo
\r I> ) = 0 .
69 the parameters associated with the prevention program (u) and the intervention program in the lowrisk environment ( 1 and Rd < 1 in which the system supports distinctive qualitative behavior. A similar dichotomy holds for the Stochastic Model (4.2). However, if Rd > 1 the non-trival state (non-zero M and H) in the stochastic model is temporary. This is because eventually the stochastic solution converges to the "trivial state" even though the corresponding deterministic solution converges to an endemic state. I define the quasi-stationary distribution as the distribution after a long time, conditioned on the processes M and H not having gone to zero. This distribution can be approximated in following way. For sufficiently large N and t the distribution of the process ^ / S t - S * M l t - M{ M a t - M2* H t - f f * \ \ JV ' TV ' TV ' JV J can be approximated at fixed t by the stationary distribution of the Ornstein-Uhlenbeck (OU) process that has local drift, J (Jacobian matrix of deterministic model at endemic state) and covariance matrix, C (obtained using diffusion coefficients of System (4.2)). Furthermore, the stationary distribution of this OU process is multivariate, normal with mean 0 and covariance S given by the relation, J X + E J T = -C,
(4.5)
70 (Andersson & Britton, 2000; Nasell, 2002) where u,j (i,j — 1,2,3,4) represents the covariance between ith and j t h variables, and an denotes the variance of ith variable which is written as a\ or a2. 3.2.1. Derivation of an Approximation of the Quasi-Stationary Distribution. Since the deterministic System (E.l) obtained by deleting the dWiS in System (4.2) is smooth, I can first scale it by N and then linearize it around the stable endemic state (s*, m\, m\, h*). The expressions for s*, m\,m2, and h* are given in Equation (E. 12). In the process of linearization, I center the process by the change of variables u = s — s*, v\ = m\ - m\, V2 = m.2 — m^, and w = h — h*. I obtain a diffusion process whose stationary law approximates the quasi-stationary distribution of (4.2) (after shifting) using the linearised deterministic system (at (u*,v^,V2, w*) = (0,0,0,0) = 0). Its diffusion coefficients approximate the corresponding coefficients of System (4.2) near the deterministic endemic state. The diffusion process is
(
\ dWi dW2
u
u
dW3 Vl
+G
= J(0)
dW4
V2
w
K J
w
dW5 dW6 dW7
y dW8 j
(4.6)
71
where / -(P + H + 61)
\
-Q -T
0
T + 72
0
-(/i + £2 + 72 + a)
0
J(0) = 0
71
0
0
a
, (4.7)
-(M+&) y
and Gi
-Gi
-G$
0
0
0
0
0
0
G2
0
-G4
-G5
0
0
0
0
0
0
G4
0
-G6
-G7
0
0
0
0
0
0
G6
0
-Gi
G
\
J
The quantities in the matrix J (Jocabian matrix of the deterministic model at endemic state 0 : (S t , M i t , M 2 t , H t ) = {Su 0,0, Ht) \ initial distribution L }. This distribution of TL is
P{TL
1, insuring the existence of the quasi-stationary state. For R^ > 1, increases in intervention rates result in decreases in the mean sizes of Mi-, M2-, and H- drinkers. The mean size of S-drinkers increases faster with the increase in high-risk intervention rates {82). Reducing the time until treatment in high-risk environments reduces moderate and heavy drinkers at a faster rate (than in low-risk environments), but at the expense of having a higher proportion of S-drinkers. On the other hand, reducing the time until treatment in low-risk environments reduces all types of drinkers, but at a slower rate than in high-risk environments. The total population size at
82
Table 11. Covariances of Quasi-Stationary Distribution. Sample size 1000 and total population of 10000 individuals. Cov(S,Mi) Cov(S,M2) Cov(S,H) Co\(MuM2) Co\(MuH) Cov(M2,H) -292.7 -382.5 -193.1 186.2 25.5 38.6 quasi-stationarity is lower than the initial population size (100 individuals). The values are shown in the second panel in the last row of Figure 16. Since Rd does not depend on the prevention parameter O), increases in its value do not affect the critical curve in 5\ - 5)
(a) When average time until treatment is 10 years and prevention efficacy 90%. Probabtty Density Fuictlon of EtimiMtlonTlma of M - & H-Mnkers: Mean-11.8 year If 1 0 % Efficacy of Prevention Program and A v e n g e Treatment Seeking T i m * of 2 year
Tlii»(MyaMt)
(b) When average time until treatment is 2 years and prevention efficacy 10%.
Figure 17. Probability density function of elimination-time ofM- and H-drinkers for specific control values.
Increases in the efficacy of the prevention program (u) decreases the elimination-time of Mand H-drinkers (time to extinction), initially rapidly and then gradually, from the community with a mixed drinking pattern (i.e. all drinking types of individuals are present). This in part because of the decrease in size of the population. However, if efficacy is increased beyond 50%, the decrease
84 in expected time to extinction are effectively negligible (see Figure 18). ,X104 | 7
6
i
—1
1
1
i
1
i
1
r
\ \\ \
5" \
I \ s«- \ ! \ 1 \
V-
\
Efficacy of Prevention Program (%)
Figure 18. Expected elimination time of M- and H- drinkers as efficacy of prevention program varies.
I carry out numerical experiments in my study of the effect of environment-specific intervention programs on drinking outcomes. Figure 19 shows the effect on elimination time for 0%, 50%, and 90% prevention efficacy when average time until treatment in low-risk (Figure 4.19(a)) and high-risk (Figure 4.19(b)) environments varies. Thesefiguresshow that when comparing results of treatment efforts in environments, elimination times of M- and H- drinking are lower in the case when intervention efforts are applied in high-risk. However, increasing the intervention effort in low-risk environments results in better rates of reduction in elimination-time than increasing such effort in high-risk environments. The expected elimination-time is an order in magnitude shorter when improved intervention rates are applied in low-risk than in high-risk environments (Figure 20). The expected eliminationtime increases with increases in initial population size N. However, the rate of this increase is
85
1 I6,: Average Treatment Seeking Time in Low-Risk Environments (In years)
(a) When there is no treatment in high-risk environments.
1/6 : Average Treatment Seeking Time In Hlgh-Rlsk Environments (In years)
(b) When there is no treatment in low-risk environments.
Figure 19. Expected elimination-time ofM- and H-drinkers as control parameters vary.
86 higher in the latter case (i.e., when the better intervention rate is applied in high-risk than low-risk environments). If the probability of extinction (p*) and the time horizon (t*) are known, then the critical community size can be computed using Figures 4.20(a) or 4.20(b), and the equation
jEfElimination-Timel 1 J = ——
t*
ln(l-p*)
-.
In summary, if my aim is to eliminate heavier drinking from the community, prevention are better than intervention programs, but the exclusive implementation of prevention policies may not even be feasible (inadequate infrastructure or implementation of policies incorrectly). Moreover, it may be too costly to implement prevention programs with 100% efficacy. The optimal benefit one can achieve is by implementing it at an efficacy greater than 50% and putting the remaining effort into intervention programs. On the other hand, if the choice is of implementing intervention programs in either low- or high-risk environments exclusively, then it is better to apply controls in high-risk environments. However, when they are implemented in both environments, I find that interventions in low-risk give at least two times better results than in high-risk. 4.3. Establishment-Time of Moderate and Heavy Drinkers in a Naive Community I consider a case where prevention programs are implemented in a naive community (a community where drinking culture has not been established). The time of establishment of heavier drinking in such a community can be delayed if efficacy of the prevention program is at least 50%. Lower efficacy prevention programs may be disadvantageous since they reduce the time of establishment (Figure 4.21(a)). Variability in establishment-time follows similar trends when seen as a function of the varying efficacy of prevention program (Figure 4.21(a)). Not surprisingly, the establishment-time of drinking in naive community increases linearly with increases in initial population size, but the variance in establishment-time decreases with initial
87
Efficacy of Prevention Program when: 8, > d 2 -> Average Treatment Seeking Time in Low-risk and High-risk Environments are 1.1 and 1.7 years, respective!
,l 500
I 1000
I 1S00 Population size (N;
I 2000
I 2500
(a) When time until treatment is better in low-risk environments. Efficacy of Prevention Program when: 6 2 > by Average Treatment Seeking Time in Low-risk and High-risk Environments are 1.7 and 1.1 years, respective
>00
1000
1500 Population Size (N
2000
2500
(b) When time until treatment is better in high-risk environments.
Figure 20. Expected elimination-time ofM- and H-drinkers as initial population size N varies in the presence of controls.
88
Eff«leyPwwntiiBeofiheP»vMi!tonPn>8r»in
\
(a) Mean and variance of establishment time as efficacy of prevention programs, u, varies.
i,„
BOO
900
soo
eoo
800
900
1000
(b) Mean and variance of establishment time as initial population size, N, varies.
Figure 21. Establishment-time ofM- and H-drinkers.
population size (Figure 4.21(b)).
5. Discussion Alcoholic beverages and the problems they engender have long been known in human societies, but the problems associated with alcohol use seem to have escalated in recent years. According to recent studies, a mere 2.5% of drinkers accounts for 27% of the alcohol consumed by adults in the US (Greenfield & Rogers, 1999). Especially alarming are alcohol-related problems among the age
89
group 18-25 years old, which consists mostly of college students. Approximately 44% of college students engages in binge or heavy episodic drinking patterns (Wechsler et al., 2002), and some 1700 die annually due to unintentional alcohol-related injuries (Hingson et al., 2002). Although excessive drinking by college students is accepted as a rite of passage by many, alcohol-related tragedies never fail to shock us and prompt us to immediate action. When schools respond to wellintentioned programs, but the problem persists, it is natural to ask, how should a particular school select an appropriate program as well as how can I understand their role in reducing drinking consequences on campus? I provide here a framework based on mathematical models with which these or similar questions can be posed and studied systematically. I also evaluate the variability in drinking patterns in the presence of various prevention and intervention regimes, extending my previous work on alcohol consumption in heterogeneous environments. That is, my study analyzes the effect of residence times in distinct risk environments on drinking patterns (see Chapter 2). Various mathematical experiments, in the presence of prevention and intervention programs, are considered in this research. The threshold quantity, controlled drinking reproduction number, denoted by Rj is derived. The value of Rj depends on the parameters of the population at risk that ensure the elimination/persistence of heavier drinking in a community. If Rj < 1 the populations of moderate and heavy drinkers do not survive in the community, whereas if Rd > 1 there is a high probability that the mixed drinking culture will prevail. The quantity Rd is affected by the magnitude of the intervention programs. I used estimates of the model parameters from my research in Chapter 3. I found that in the absence of intervention programs in the high-risk environment, the average time until treatment in the low-risk environment should be better than 9 months in order to rapidly eliminate moderate and heavy drinking from the community. On the other hand, when
90
interventions were only applied to the high-risk environment, the average time until treatment in the high-risk environment is shorter than 3 months if the goal is the elimination of moderate and heavy drinking. Current data trends support endemic drinking patterns. I compute a quasi-stationary distribution, that is, the distribution of drinkers after a long time, conditioned on non-extinction of moderate and heavy drinkers from the community. I study the effect of prevention and intervention programs on the variability of drinkers at quasi-stationarity. Improving time until treatment in high-risk environments reduces moderate and heavy drinkers at a faster rate (than in low-risk environments), but at the expense of increasing the proportion of S-drinkers. On the other hand, improving the time until treatment in low-risk environments turns out to reduce all types of drinkers, but at a slower rate than in high-risk environments. Increases in the efficacy of the prevention program reduce the sizes of all types of drinkers at quasi-stationarity. I compute a distribution of the establishment-time of a mixed drinking culture in a naive community and a distribution of the elimination-time of moderate and heavy drinking in a community with a mixed drinking culture. Mixed drinking culture means that it supports a sizable proportion of all types (light, moderate, and heavy) of drinkers. The effect of changes in the size of the community on these distributions was studied. I proved that the elimination-time is exponentially distributed, but I can only compute numerically the distribution of establishment-time as a function of changes in prevention and intervention efforts. I found that when the aim is to eliminate heavier drinking from the community, prevention programs are better than intervention programs, but the exclusive implementation of prevention policies with perfect efficacy may not be feasible. The optimal benefit one can achieve is by implementing a prevention program at around 50% efficacy, putting the remaining efforts in intervention programs. Moreover, it is better to implement
91
intervention programs in high-risk environments. The implementation of interventions in both environments with rates greater in low-risk than in high-risk leads to an elimination-time that is at least half of elimination-time when rates are greater in high-risk environments. The eliminationand establishment-times are longer in larger communities. That is, moderate and heavy drinking can be eliminated faster at smaller colleges, but it will take longer for mixed drinking cultures to become established in larger colleges. The actual data needed for the validation of model discussed here is not available. Yet, modeling has been crucial in investigating how changes in various assumptions and parameter values affect drinking patterns. The modeling framework used here can help in our understanding of the mechanism behind the spread of drinking in college populations. This framework helps evaluate the potential effectiveness of various intervention and prevention programs.
CHAPTER 5 Conclusions and Future Directions 1. Conclusions There have been few attempts to use models in the study of the dynamics of drinking, a situation quite in contrast to the status quo in ecology, epidemiology, economics, and other fields. My primary goal has been to show that models can be useful in the study of the dynamics and controls of drinking. I hope that this work shows that if models are going to be useful, they must be built not as descriptors of detailed social processes (data rich models), but rather in response to specific questions (as is done in the life sciences, for example). Here, I have built models in response to specific questions, including: What is the role of individual movement between heterogeneous drinking environments on the distribution of abusive drinkers? How do prevention and intervention efforts affect drinking patterns? I found that there are no data-driven characterizations of the places where people drink. Therefore, I was "forced" to use proxy data "generated" by the Prevention Research Center in Berkeley that captured some of the needed information. That is, the available college data was re-organized to test the potential usefulness of my approach. I cannot validate the models in a traditional sense because there are no specific studies that have collected the required data. Aggregated data, from a synthesis of recent drinking studies that took place in specified environments, were used instead. The completeness of this approach and our understanding of the mathematical possibilities should be questioned in the context of my goal of highlighting the value of models in the study of drinking dynamics. For example, I believe that this work strongly supports the need for specific field studies. A glance at the work of Robert May (FRS and former President of the Royal Society) and
93 Roy Anderson (FRS) on HIV (starting in 1987) shows the value that simple models have had in increasing our understanding of the HIV epidemic. I model mixing in a simple ad-hoc way, as Perelson et al. (1996), Anderson and May (1987; 1988), and many others have done in immunology and epidemiology. I model the social interactions in a way that it becomes clear that the mixing is different in each environment. My goal is to build a simple system with clear differences in social mixing so that I can explore in a rather simple setting the impact of such differences on the dynamics of drinking. Similar results are likely to be obtained using the rigorous work of Blythe et al. (1995); Busenberg et al. (1991); and Castillo-Chavez et al. (1994; 1997; 2003). Here, I model mixing as frequency-dependent phenomena and in a phenomenological (ad hoc) way. I build mixing differences by letting the frequency-dependence effect be stronger in one environment than in the other. I assume that only one type of individuals float between both environments. I also assume that interactions outside both environments are possible between floaters and some non-floaters. In this crude way, I achieve the goal of building two highly heterogeneous (and artificial) mixing/drinking environments. In fact, I proceed to study the consequences of variations in floaters' environmental-specific residence times on drinking outcomes. I further study the effects of various environment-dependent preventive and treatment schemes for floaters on population drinking patterns. I found that environmental mixing strength (nobody has measured this), the distribution of residence times of floaters (nobody has measured this) and the rate of drinking progression (there does not seem to be direct measures of this), can lead to either similar or highly distinct drinking outcomes. I conclude that it is important to characterize the places where people drink and to estimate the distribution of residence times of drinkers in such environments. Moreover, among the
94 three broad categories of control policies (prevention, interventions in low-risk environments and interventions in high-risk environments), prevention programs are more effective, but all have some limitations. The use of a simple model meant that I had to deal with few parameters. Parameter-scarce models are useful because they allow us to explore the relevance of the approach with minimal data. Here, I was able to use college data and the model to highlight the potential value of this approach in a specific population. Consequently, I put tremendous effort into trying to identify drinking residence times in highly distinct drinking environments from studies that did not measure such times directly. I found some reasonable estimates, but not exactly what I needed. However, the full "exercise" has value because it identifies the importance of collecting data that is not on the radar.
2. Future Directions 2.1. Sustained Oscillations through Power Spectral Density Analysis of the corresponding deterministic system shows that its trajectories start close to the endemic drinking state and spiral into the endemic state, if Rj > 1. On the other hand, the dynamics of the analogous stochastic model may show small but sustained, nearly regular oscillations' around the deterministic stable endemic state when Rj > 1. These oscillations have a narrow frequency distribution and stochastically varying amplitude. I have not yet been able to evaluate the existence of sustained oscillations in drinking patterns. Knowing the frequency of sustained oscillation in drinking patterns might help in adequately 'The phenomenon of having stochastic fluctuations sustain nearly periodic oscillations in a system which has a stable endemic equilibrium in the deterministic limit is called coherence resonance or autonomous stochastic resonance. These oscillations were shown by Kuske, Gordillo, and Greenwood (2007) in a simple SIR model.
95 planning the use of control mechanisms. Numerically, oscillation can be evaluated with the help of the Power Spectral Density of the simulated stochastic process. The Power Spectral Density describes how the power of the sample path of a simulated quasi-stationary stochastic process is distributed in frequency. The Power Spectral Density (or simply, power spectrum) of a random signal can be estimated using the Discrete Fourier Transform (DFT). There are two basic approaches: one is called periodogram analysis, and the other (an indirect approach) is based on the use of an autocorrelation sequence. If x(t) is a sample path of a stationary process X(i), then the Power Spectral Density of the process (by periodogram analysis) is given by
Sx(f)
= lim ±-E
[|X T (/)| 2 ] = lim
i—>oo 11
±-E
/
x{t)e,-2«iftdt
t—>oo Zl
On the other hand, the Power Spectral Density with given frequency, based on autocorrelation sequence, is given by oo
/
R(T)e-^frdT -oo
where the independent variable r represents time and the transform variable / represents ordinary frequency (in hertz) (Gardiner, 2004). Numerically, the Welch method can be used to estimate Power Spectral Density. The Welch method is based on the concept of using periodograms, which convert a signal from the time to the frequency domain. Due to the noise caused by imperfect and finite data, noise reduction from the Welch method is often desired. The Welch method is carried out by dividing the time signal x(t) into successive blocks, forming the periodogram for each block and averaging. The steps in the method are as follows:
96 1. The original data segment is split up into L data segments of length M, overlapping by D points.
• If D = M / 2, the overlap is said to be 50% • If D = 0, the overlap is said to be 0%. This is the same situation as in the Barlett's method.
2. After the data is split into overlapping segments, the individual L data segments have a window applied to them (in the time domain). That is, the mth windowed, zero-padded2 frame3 of the signal x is
xm(n)
= w(n) * x{n + mR),
n = 0,1,
, M — 1; m = 0,1,
, L — 1;
where w(n) is the window function and R is window hop size. • Most window functions4 afford more influence to the data at the center of the set than to data at the edges, which represents a loss of information. To mitigate that loss, the individual data sets are commonly overlapped in time (as in the above step). • The windowing of the segments is what makes the Welch method a "modified" periodogram. 2
Zero-padding means that you append an array of zeros to the end of your input signal before you FFT it. Two reasons that you might want to zero-pad is to increase the number of data points to a power of 2. Traditionally, the FFT algorithm is more efficient when it is dealing with signals that contain 2 ^ data points. Another reason for zero-padding is for a "better" resolution in the frequency spectrum. 3 Used for performing FFT. 4 A window function (also known as an apodization function or tapering function) is a function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation. When another function or a signal (data) is multiplied by a window function, the product is also zero-valued outside the interval: all that is left is the "view" through the window.
97
3. The periodogram is calculated by computing the discrete Fourier Transform and then computing the squared magnitude of the result. The periodogram of the mth segment is given by N-l PxnMuk)
= jf
I FFTN,k{*m)
|2 =
2
jj n=0
4. The individual periodograms are then time-averaged, which reduces the variance of the individual power measurements. The end result is an array of power measurements vs. frequency "bin." The Welch estimate of Power Spectral Density is given by 1
L_1
m=0
2.2. Parameter Estimation in Stochastic Differential Equations Model The stochastic model used in Chapter 4 introduced some new parameters which are not yet estimated. My future efforts will use ideas similar to those used by Ditlevsen (2004) and Picchini (2007) for estimating parameters for stochastic differential equations. The estimation method that they use compares the observed and simulated time series and minimizes a distance function, constructed using Power Spectral Density (PSD) function, over the parameters. To use spectral density for inference purposes, first estimates of both the empirical and the model spectral densities are required (see Section 2.1 for estimation of PSD). 2.3. Model Extension: Structured Populations I have focused on issues related to invasions and persistence, times to extinction, thresholds and critical community sizes. A particular challenge for the future is to extend these results in the context of structured populations (for example, age-structured models), that is, to situations where homogeneous mixing does not apply. We need to devise better methods and tools for estimating
98 parameters using available data. In addition, for some questions it may be essential to incorporate host heterogeneity, including variation in susceptibility due to genetics or immune status and the role of heterogeneity of mixing, when the rate of contact between individuals depends on their spatial separation or relative positions in a social network.
REFERENCES
[1] Allen, L. (2003). An Introduction to Stochastic Processes with Applications to Biology. Upper Saddle River, N. J.: Pearson Education-Prentice Hall.
[2] Andersson, H. and Britton, T. (2000). Stochastic epidemics in dynamic populations: quasi-stationarity and extinction. J. Math. Biol., 41, 559-580.
[3] Andersson, H. and Britton, T. (20Q0). Stochastic Epidemic Models and their Statistical Analysis. New York: Springer. Lecture Notes in Statistics, no. 151.
[4] Andersson, H. and Djehiche, B. (1998). A threshold limit theorem for the stochastic logistic epidemic. Journal of Applied Probability, 35, 662-670.
[5] Anderson, R. and May, R. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press.
[6] Anderson, R. M. and May, R. M. (1988). Epidemiological parameters of HIV transmission. Nature, 333(6173), 514-519.
[7] Arriola, L. M. and Hyman, J. M. (2007). Being Sensitive to Uncertainty. Computing in Science & Engineering, 9(2), 10-20.
[8] Baer, J. S., Kivlahan, D. R., and Marlatt, G. A. (1995). High-risk drinking across the transition from high school to college. Alcoholism: Clinical and Experimental Research, 19(1), 54-61.
[9] Bailey, N.TJ. (1975). The Mathematical Theory of Infectious Diseases and its Applications. London: Griffin.
[10] Barbour, A. D. (1972). The principle of diffusion of arbitrary constants. Journal of Applied Probability, 9, 519-541.
[11] Barbour, A. D. (1974). On a functional central limit theorem for Markov population processes. Advances in Applied Probability, 6, 21-39.
100 [12] Barbour, A. D. (1975). The duration of the closed stochastic epidemic. Biometrika, 62, 477-82.
[13] Bartlett, M. S. (1949). Some evolutionary stochastic processes. Journal of the Royal Statistical Society, B l l , 211-229.
[14] Bartlett, M. S. (1956). Deterministic and stochastic models for recurrent epidemics. In Proc. Third Berkeley Symp., Mathematics, Statistics and Probability, Vol. 4, pp. 81-109. Berkeley: University of California Press.
[15] Bartlett, M. S. (1957). Measles periodicity and community size. Journal of the Royal Statistical Society, A120, 48-70.
[16] Bartlett, M. S. (1960). Stochastic Population Models in Ecology and Epidemiology. London: Methuen.
[17] Barton, S. (1994). Chaos, self-organization, and psychology. American Psychologist, 49, 5-14.
[18] Basen-Engquist, K., Edmundson, E. W. and Parcel, G. S. (1996). Structure of health risk behavior among high school students. Journal of Consulting and Clinical Psychology, 64, 764-775.
[19] Bates, M. E., and Labouvie, E. W. (1997). Adolescent risk factors and the prediction of persistent alcohol and drug use into adulthood. Alcsm Clin Exp Res., 21, 944-950.
[20] Behrens, D. A., Caulkins, J. P., Tragler, G., Haunschmied, J. L., and Feichtinger, G. (1999). A dynamic model of drug initiation: implications for treatment and drug control. Mathematical Biosciences, 159, 1-20.
[21] Bernoulli, D. (1760). Essai dune nouvelle analyse de la mortalite causee par la petite verole et des advantages de linculation pour la prevenir. Mem. Math. Phys. Acad. Roy. Sci., Paris, 1-45.
[22] Blower, S. M. and Dowlatabadi, H. (1994). Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model as a example. International Statistical
101 Review, 62, 229-243.
[23] Blythe, S. P., Busenberg, S., and Castillo-Chavez, C. (1995). Affinity and Paired-Event Probability. Mathematical Biosciences, 128, 265-84.
[24] Bonabeau, E. (2002). Agent-based modeling: Methods and techniques for stimulating human systems. Proceddings of the National Academy of Science, 99, 7280-7287.
[25] Booth, B. M., Fortney, S. M., Fortney, J. C , Curran, G. M , and Kirchner, J. E. (2001). Short-term course of drinking in an untreated sample of at-risk drinkers. Journal of Studies on Alcohol, 580-588.
[26] Borsari, B. and Carey, K. B. (2001). Peer influences on college drinking: A review of the research. Journal of Substance Abuse, 13/14, 391-424.
[27] Brack, C. J., Brack, G. and Orr, D. P. (1994). Dimensions underlying problem behaviors, emotions, and related psychological factors in early and middle adolescents. Journal of Early Adolescence, 14, 345-370.
[28] Brauer, F. and Castillo-Chavez, C. (2001). Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 40, Springer-Verlag, New York.
[29] Brauer, E, van den Driessche, P., and Wu, J. (Eds.) (2008) Mathematical Epidemiology: Lecture Notes in Mathematics, Mathematical Biosciences Subseries, Vol. 1945.
[30] Braun, R. J., Wilson, R. A., Pelesko, J. A., Buchanan, R. B., and Gleeson, J. P. (2006). Application of small-world network theory in alcohol epidemiology. Journal of Studies on Alcohol, 67, 591-599.
[31] Busenberg, S. and Castillo-Chavez, C. (1991). A General Solution of the Problem of Mixing Sub-Populations, and its Application to Risk-and Age-Structured Epidemic Models for the Spread of AIDS. IMA Journal of Mathematics Applied in Medicine and Biology, 8, 1-29.
[32] Byrne, D. G., Mazanov, J., and Gregson, R. A. M. (2001). A cusp catastrophe analysis of changes to adolescent smoking behaviour in response to smoking prevention programs.
102 Non-linear Dynamics, Psychology and the Life Sciences, 5, pp 115-137.
[33] Castillo-Chavez, C , Feng, Z., and Huang, W. (2002). On the computation of 7tLo and its role on global stability. In, Mathematical Approaches for emerging and re-emerging infectious diseases: an introduction (C. Castillo-Chavez, P. van den Driessche, D. Kirschner, A.-A. Yakubu, (Eds.)). Part I, IMA, 125, 224-250. New York: Springer-Verlag.
[34] Castillo-Chavez, C. and Hsu Schmitz, S., (1997). The evolution of age-structured marriage functions: It takes two to tango. In, Structured-Population Models Marine, Terrestrial, and Freshwater Systems (S. Tuljapurkar and H. Caswell, (Eds.)), 533-550. New York: Chapman & Hall.
[35] Castillo-Chavez, C , and Song, B. (2004). Dynamical Models of Tuberculosis and applications. Journal of Mathematical Biosciences and Engineering, 1(2), 361-404.
[36] Castillo-Chavez, C. and Song, B. (2003). Models for the transmission dynamics of fanatic behaviors. In, Bioterrorism-Mathematical modeling applications in homeland security (H.T. Banks, C. Castillo-Chavez (Eds.)), 155-172. Philadelphia: SIAM Frontiers in Applied Mathematics.
[37] Castillo-Chavez C , Song B., and Zhang, J. (2003). An Epidemic Model with Virtual Mass Transportation: The Case of Smallpox in a Large City. In, Bioterrorism-Mathematical modeling applications in homeland security (H.T. Banks, C. Castillo-Chavez (Eds.)), 173-197. Philadelphia: SIAM Frontiers in Applied Mathematics.
[38] Castillo-Chavez, C , Velasco-Hernandez, J. X., and Fridman, S.,(1994). Modeling Contact Structures in Biology. In, Frontiers of Theoretical Biology, Lecture Notes in Biomathematics 100 (S.A. Levin (ed.)), 454-91. Berlin-Heidelberg-New York: Springer-Veralg.
[39] Catalano, R.F. and Hawkins, J.D. (1996). The Social Development Model: A theory of antisocial behavior. In: Delinquency and Crime: Current Theories (J.D. Hawkins (ed.)). New York: Cambridge University Press.
[40] Caulkins, J.P, and Tragler, G. (2004). Dynamic drug policy: An introduction and overview. Socio-Economic Planning Sciences, 38, 1-6.
103 [41] Caulkins, J.P. and Reuter, P. (2006). Illicit drug markets and economic irregularities. Socio-Economic Planning Sciences, 40, 1-14.
[42] Chassin, L., Flora, D., and King, K. (2004). Trajectories of Alcohol and Drug Use and Dependence from adolescence to adulthood: The effects of familial alcoholism and personality. Journal of Abnormal Psychology, 113, 577-595.
[43] Chen, C. M., Dufour, M. C , and Hsiao-ye, Y. (2005). Alcohol consumption among young adults ages 18-24 in the United States: Results from the 2001-2002 NESARC Survey. Alcohol Research and Health, 28, 269-280.
[44] Chowell, G., Castillo-Chavez, C , Fenimore, P.W., Kribs-Zaleta, CM., Arriola, L., and Hyman, J.M. (2004). Model parameters and outbreak control for SARS. Emerging Infectious Diseases, 10, 1258-1263.
[45] Chowell, G., Fenimore, P.W., Castillo-Garsow, M.A., and Castillo-Chavez, C. (2003). SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism, Journal of Theoretical Biology, 224, 1-8.
[46] Clancy, D. (1996). Carrier-borne epidemic models incorporating population mobility. Mathematical Biosciences, 132, 185-204.
[47] Clancy, D. (2005). A stochastic SIS infection model incorporating indirect transmission. J. Appl. Probab., 42, 726-737.
[48] Clancy, D. and Pollett, P. K. (2003). A note on the quasi-stationary distributions of birthdeath processes and the SIS logistic epidemic. Journal of Applied Probability, 40, 821-825.
[49] Clapp, J.D., Lange, J.E., Min, J.W., Shillington, A., Johnson, M., and Voas, R.B. (2003). Two studies examining environmental predictors of heavy drinking by college students. Prevention Science, 4, 99-108.
[50] Clapp, J.D., Reed, M.B., Holmes, B.A., Lange, J.E., and Voas, R.B. (2006). Drunk in public, drunk in private: the relationship between college students, drinking environments and alcohol consumption. American Journal of Drug and Alcohol Abuse, 32, 275-285.
104 [51] Collins, R. L., Parks, G. A., and Marlatt, G. A. (1985). Social determinants of alcohol consumption: The effects of social interaction and model status on the self-administration of alcohol. Journal of Consulting and Clinical Psychology, 53(2), 189-200.
[52] Curran, G.M., White, H.R., and Hansell, S. (1997). Predicting Problem Drinking: A Test of an Interactive Social Learning Analysis. Alcoholism: Clinical and Experimental Research, 21(8), 1379-1390.
[53] Daley, D.J. and Gani, J. (1999). Epidemic Modelling. Cambridge: Cambridge University Press.
[54] Daley, D.J. and Kendall, D. G. (1965). Stochastic rumours. JIMA, 1,42-55.
[55] Daniels, H.E. (1991). A look at perturbation approximations for epidemics. In, Selected Proc. Sheffield Symp. Appl. Probab. (I. V. Basawa and R. L. Taylor (Eds.)), 48-65. Hay ward, Ca.: IMS Monograph Series, Vol. 18.
[56] Darkes, J., Greenbaum, P.E., and Goldman, M.S. (2004). Alcohol expectancy mediation of biopsychosocial risk: Complex patterns of mediation. Experimental and Clinical Psychopharmacology, 12, 27-38.
[57] Darroch, J. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuoustime finite Markov chains. Journal of Applied Probability, 4, 192-196.
[58] Dawson, D.A., Grant, B.F., Stinson, F.S., Chou, P.S., Huang, B., and Ruan, W.J. (2005). Recovery from DSM-IV alcohol dependence: United States, 2001-2002. Addiction, 100, 281-292.
[59] DeJong, W. and Langford, L.M. (2002). A typology for campus-based alcohol prevention: moving toward environmental management strategies. Journal of Studies on Alcohol, 63 (Supplement 14), 140-147.
[60] Demers, A., Kairouz, S., Adlaf, E., Gliksman, L., Newton-Taylor, B., and Marchand, A. (2002). Multilevel analysis of situational drinking among Canadian undergraduates. Social Science and Medicine, 55, 415-424.
105
[61] Diekmann, O., and Heesterbeek, J.A.P. (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Chichester: John Wiley.
[62] Diekmann, O., Heesterbeek, A.A.P., and Metz, J.A.J. (2002). On the definition and computation of the basic reproductive ratio R0 in models for infectious diseases in heterogeneous populations. Journal of mathematical Biology, 28, 365-382.
[63] Dietz, K. (1988). The first epidemic model: a historical note on P. D. Enko. Austr. J. Statist. A3, 56-65.
[64] Dietz, K. (1989). A translation of the paper On the course of epidemics of some infectious diseases by P. D. Enko. International Journal of Epidemiology, 18, 749-755.
[65] Donovan, J.E. (2004). Adolescent alcohol initiation: A review of psychosocial risk factors. Journal of Adolescent Health, 35, 529.e7.
[66] Duerr, H.P., Dietz, K., Schulz-Key, H., Buttner, D.W., and Eichner, M. (2002). Infection associated immunosuppression as an important mechanism for parasite density regulation in onchcerciasis. Transactions of the Royal Society of Tropical Hygiene and Medicine, 97, 242-250.
[67] Duncan, G.J., Boisjoly, J., Kremer, M , Levy, D.M., and Eccles, J. (2005). Peer effects in drug use and sex among college students. Journal of Abnormal Child Psychology, 33, 375-385.
[68] Engs, R. C. and Hanson, D.J. (1994). Boozing and Brawling on campus: a national study of violent problems associated with drinking over the past decade. Journal of Criminal Justice, 22(2), 171-180.
[69] Engs, R. C , Hanson, D.J., and Diebold, B. (1996). The Drinking Patterns and Problems of a National Sample of College Students, 1994: Implications for Education. Journal of Alcohol and Drug Education. 41(3), 13-33, spring.
[70] Engs, R. C , Hanson, D.J., and Diebold, B. (1997). The Drinking Patterns and Problems of a National Sample of College Students, 1994: Implications for Education. Journal of Alcohol and Drug Education. Spring.
106
[71] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes:Characterization and Convergence. New York; J. Wiley & Sons.
[72] Feng, Z. and Castillo-Chavez, C. (1998). Global stability of an age-structure model for TB and its applications to optimal vaccination strategies. Mathematical Biosciences, 151,135154.
[73] Gardiner, C.W. (2004). Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Berlin-Heidelberg-New York: Springer-Verlag.
[74] Gfroerer, J., Wright, D.,and Kopstein, A. (1997). Prevalence of youth substance use: The impact of methodological differences between two national surveys. Drug and Alcohol Dependence, 47, 19-30.
[75] Goldman, M.S. and Darkes, J. (2004). Alcohol expectancy multi-axial assessment (A.E.Max): A memory network-based approach. Psychological Assessment, 16, 4-15.
[76] Gonzalez, B., Huerta-Sanchez, E., Ortiz-Nieves, A., Vazquez-Alverez, T., and Kribs-Zaleta, C. (2003). Am I too fat? Bulimia as an epidemic. Journal of Mathematical Psychology, 47, 515-526.
[77] Gorman D.M. (in press). "On the Other Hand": Rejoinder to Frankford. American Journal of Public Health.
[78] Gorman, D.M., Gruenewald, P.J., Hanlon, P.J., Mezic, I., Waller, L.A., Castillo-Chavez, C , Bradley, E., and Mezic, J. (2004). Implications of systems dynamic models and control theory for environmental approaches to the prevention of alcohol- and other drug use-related problems. Subst Use Misuse. 39(10-12), 1713-50.
[79] Grant, B., Dawson, D., Stinson, E, et al. (2004). The 12-Month Prevalence and Trends in DSM-IV Alcohol Abuse and Dependence: United States, 1991-1992 and 2001-2002. Drug
and Alcohol Dependence, 74(3), 223:234.
[80] Grasman, J. and Van Herwaarden, O.A. (1999). Asymptotic Methods for the Fokker-Planck equation and the Exit Problem in Applications. Heidelberg: Springer-Verlag.
107 [81] Gruenewald, P.J., Johnson, F.W., Light, J.M., Lipton, R., and Saltz, R.F. (2003). Understanding College Drinking: Assessing Dose Response from Survey Self-Reports. Journal of Studies on Alcohol, 64(4), 500-514, July.
[82] Gruenewald, P.J., Russell, M., Light, J., Lipton, R., Searles, J., Johnson, R, Trevisan, M., Freudenheim, J., Muti, P., Carosella, A.M., and Nochajski, T.H. (2002). One Drink to a Lifetime of Drinking: Temporal Structures of Drinking Patterns. Alcoholism: Clinical and Experimental Research, 26(6), 916-925.
[83] Hamer, W. H. (1906) Epidemic disease in England.The Lancet, 1, 733-739.
[84] Hanson, D.J. (1995). Preventing Alcohol Abuse: Alcohol, Culture, and Control. Greenwood Publishing Group, Westport, Connecticut.
[85] Harford, T.C., Weschler, H., and Muthen, B.O. (2003). Alcohol-related aggression and drinking at off-campus parties and bars: a national survey of current drinkers in college. Journal of Studies on Alcohol, 64, 704-711.
[86] Harford, T.C., Wechsler, H., and Seiberg, M. (2002). Attendance and alcohol use at parties and bars in college: a national survey of current drinkers. Journal of Studies on Alcohol, 63, 726-733.
[87] Hawkins, R.C., and Hawkins, C.A. (1998). Dynamics of substance abuse: implications of chaos theory for clinical research. In, Clinical Chaos: a Therapists Guide to Non-Linear Dynamics and Therapeutic Change (L. Chamberlain & M. Butz (eds.)), 89-101. Philadelphia: Brunner/Mazel.
[88] Hethcote, H. (2000). The mathematics of infectious diseases. SIAMRev., 42, 599-653.
[89] Helton, J.C. (1993). Uncertainty and Sensitivity Analysis Techniques for Use in Performance Assessment for Radioactive Waste Disposal. Reliability Engineering and System Safety, 42, 327-367.
[90] Hingson, R., Berson, J., and Dowley, K. (1997). Interventions to reduce college student drinking and related health and social problems. In, Alcohol: Minimising the harm (M. Plant, E. Single, & T. Stockwell (Eds.)),143-170. New York: Free Association Books.
108 [91] Hingson, R., Heeren, T., Zakocs, R., et al. (2002). Magnitude of alcohol-related morbidity, mortality, and alcohol dependence among U.S. college students age 18-24. Journal of Studies on Alcohol, 63, 136-144.
[92] Holder, H. (1999). Alcohol and the Community: A Systems Approach to Prevention. Cambridge: Cambridge University Press.
[93] Huang, W. and Castillo-Chavez, C. (2002). Age-structured Core Groups and their impact on HIV dynamics. In, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, (C. Castillo-Chavez, P. van den Driessche, D. Kirschner, A.-A. Yakubu, (Eds.)), IMA Vol. 126, 261-273. Berlin-Heidelberg-New York: Springer-Veralg.
[94] Huebner, L.A. (Ed.). (1979). Redesigning campus environments: New directions for student services (No. 6). San Francisco: Jossey-Bass.
[95] Isham, V. (2005). Stochastic models for epidemics: current issues and developments. In Celebrating Statistics (A.C. Davison, Y. Dodge and N. Wermuth. (Eds.)), 27-54. Oxford: Oxford University Press.
[96] Jellinek, E.M. (1960). The Disease Concept of Alcoholism. New Haven: Hillhouse.
[97] Jessor, R. and Jessor, S.L. (1977). Problem Behavior and Psychosocial Development: A Longitudinal Study of Youth. New York: Academic Press.
[98] Johnston, L.D., O'Malley, P. M , and Bachman, J. G. (1997). National survey results on drug use from the Monitoring the Future Study, 1975-1995. Volume II: College students and young adults. Publication No. 98-4140. Rockville, Md.: National Institute on Drug Use.
[99] Johnson, R.A., and Wichern, D. W. (1998). Applied Multivariate Statistical Analysis. Upper Saddle River (4th edition). Chapter 4, pg 157. NJ: Prentice-Hall.
[100] Kiel, L.D. (1992). The nonlinear paradigm: advancing paradigmatic progress in the policy sciences. Systems Research, 9, 27-42.
109 [101] Keeling, M. J. and Grenfell, B. T. (1997). Disease extinction and community size. Science, 275, 65-67.
[102] Kermack, W.O. and McKendrick, A. G. (1927). Contributions to the mathematical theory of epidemics, part I. Proceedings of the Royal Society of London, A115, 700-721.
[103] Kleinbaum, D., and Kupper, L., Muller, K., Nizam, A. (1998). Applied Regression Analysis and Other Multivariate Methods (3rd edition). North Scituate, Massachusetts: Duxburg Press.
[104] Knight, J.R., Wechsler, H., Kua, M., Seibring, M., Weitzman, E.R., and Schuckit, M.A. (2002). Alcohol abuse and dependence among US college students. Journal of Studies on Alcohol, 63, 263-270.
[105] Kribs-Zaleta, C M . (2002). Center manifolds and normal forms in epidemic models. In, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, (IMA Vol. 125), 269-286. Berlin: Springer-Verlag.
[106] Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability, 7,49-58.
[107] Kurtz, T. G. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability, 8, 344-356.
[108] Kurtz, T.G. (1973). The relationship between stochastic and deterministic models in chemical reactions. / Chem. Phys., 57, 2976-2978.
[109] Kurtz, T. G. (1981). Approximation of Population Processes. Philadelphia: SIAM.
[110] Kuske, R., Gordillo, L.F., and Greenwood, P. (2007). Sustained oscillations via coherence, resonance in SIR. J TheorBiol, 245(3), p. 459-469.
[ I l l ] Lakins, N.E., LaVallee, R.A., Williams, G.D., and Yi, H. (2007). Alcohol Epidemiologic Data System. Surveillance Report #82: Apparent Per Capita Alcohol Consumption: National, State, and Regional Trends, 1970-2005. Bethesda, MD: National Institute on Alcohol
110 Abuse and Alcoholism, Division of Epidemiology and Prevention Research, August.
[112] Mann, K., Hermann, D., and Heinz, A. (2000). One hundred years of alcoholism: the Twentieth Century. Alcohol Alcohol. 35, 10-5. PMID: 10684770.
[113] Maisto, S.A., Carey, K. B., and Bradizza, C. M. (1999). Social learning theory. In, Psychological theories of drinking and alcoholism (K. E. Leonard, & H. T. Blane (Eds.), 2nd ed.). New York: Guilford Press.
[114] May, R.M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459-467.
[115] McKendrick, A.G. (1926). Applications of mathematics to medical problems. Proceedings of the Edinburgh Mathematical Society, 44, 98-130.
[116] Mishra, S. (2004). Sensitivity Analysis with Correlated Inputs - An Environmental Risk Assessment Example. Proceedings of the 2004 Crystal Ball User Conference.
[117] Mokdad, A.H., James, S.M., Donna, F.S., and Julie, L. G. (2004). Actual Causes of Death in the United States, 2000 Journal of the American Medical Association, 291(10), p. 1242.
[118] Mubayi, A., Greenwood, P., Castillo-Chavez, C , Gruenewald, P., and Gorman, D.M. (in review). Effects of residence time in drinking environments for populations with high turnover rates.
[119] Mubayi, A., Wang, X., Castillo-Chavez, C , Greenwood, P., Gorman, D.M., and Gruenewald, P. (in progress). Types of Drinkers and Drinking Settings: Application of a Mathematical Model.
[120] Nasell, I. (1985). Hybrid Models of Tropical Infections. Lecture Notes in Biomathematics, No. 59. Berlin: Springer.
[121] Nasell, I. (1995). The threshold concept in stochastic epidemic and endemic models. In Epi- demic Models: Their Structure and Relation to Data, (D. Mollison (Ed.)), 71-83. Cambridge: Cambridge University Press.
Ill
[122] Nasell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Advances in Applied Probability, 28, 895-932.
[123] Nasell, I. (1999). On the time to extinction in recurrent epidemics. Journal of the Royal Statistical Society, 61, 309-330.
[124] Nasell, I. (2002). Stochastic models of some endemic infections. Mathematical Biosciences, 179, 1-19.
[125] National Institute on Alcohol Abuse and Alcoholism, The Task Forces report (2002). A Call to Action: Changing the Culture of Drinking at U.S. Colleges.
[126] National Institute on Alcohol Abuse and Alcoholism (2005). 2001-2002 Survey Finds That Many Recover From Alcoholism, Press release 18 January.
[127] National Institute on Alcohol Abuse and Alcoholism (2007). Fall Semester - A Time For Parents To Revisit Discussions About College Drinking, NIH Publication, No. 07-5640, August.
[128] Nuno, M., Feng, Z., Martcheva, M., Castillo-Chavez, C. (2005). Dynamics of Two-Strain Influenza with Isolation and Partial Cross-Immunity. SIAM Journal on Applied Mathematics, 65(3) 964-982.
[129] NSDUH (2004). Office of Applied Studies. (2003) Results from the 2003 National Survey on Drug Use and Health: National findings (DHHS Publication No. SMA 04-3964, NSDUH Series H-25). Rockville, MD: Substance Abuse and Mental Health Services Administration.
[130] Orford, J., Krishnan, M., Balaam, M., Everitt, M., and Van der Graaf, K. (2004). University Student Drinking: The Role of Motivational and Social Factors. Drugs Education Prevention Policy, 11(5), 407-421.
[131] Patten, S.B. and Arboleda-Florez, J.A. (2004). Epidemic theory and group violence. Social Psychiatry & Psychiatric Epidemiology, 39, 853-856.
[132] Patton, G.C., Coffey, C , Lynskey, M.T., Hemphill, S., Carlin, J.B., and Hall, W. (2007). Trajectories of adolescent alcohol and cannabis use into young adulthood. Addiction, 102,
112 607-615.
[133] Perelson, A. S., Neumann, A. U., Markowitz, M , Leonard, J. ML, and Ho, D.D. (1996). HIV-1 dynamics in vivo: Virion clearance rate, infected cell lifespan, and viral generation time. Science, 271, 1582-1586.
[134] Perkins, H. W. (1997). College Student Misperceptions of Alcohol and Other Drug Norms among Peers: Exploring Causes, Consequences, and Implications for Prevention Programs. Pp. 177-206 in Designing Alcohol and Other Drug Prevention Programs in Higher Education: Bringing Theory into Practice. The Higher Education Center for Alcohol and Other Drug Prevention, U.S. Department of Education.
[135] Pollett, P.K. (2001) Diffusion approximations for ecological models. In (Ed. Fred Ghassemi) Proceedings of the International Congress on Modelling and Simulation, Vol. 2, Modelling and Simulation Society of Australia and New Zealand, Canberra, Australia, pp. 843-848.
[136] Presley, C.A., Meilman, P.W., and Leichliter, J.S. (2002). College Factors That Influence Drinking. Journal of Studies on Alcohol/Supplement No. 14.
[137] Ross, R. (1911) The Prevention of Malaria, 2nd edition. London: Murray.
[138] Ross, R. (1916) An application of the theory of probabilities to the study of a priori pathometry, I. Proceedings of the Royal Society of London, A92, 204-30.
[139] Ross, R. and Hudson, H. P. (1917a) An application of the theory of probabilities to the study of a priori pathometry, II. Proceedings of the Royal Society of London, A93, 212-25.
[140] Ross, R. and Hudson, H. P. (1917b) An application of the theory of probabilities to the study of a priori pathometry, III. Proceedings of the Royal Society of London, A93, 225-40.
[141] Saltelli, A., Chan, K., and Scott, M. (Eds.). (2000). Sensitivity Analysis. Chichester: John Wiley & Sons.
[142] Sanchez, M. A., and Blower, S. M. (1997). Uncertainty and Sensitivity Analysis of the Basic Reproduction Number: Tuberculosis as an example. American Journal of Epidemiology,
113 145(12), 1127-1137.
[143] Sanchez, R, Wang, X, Castillo-Chavez, C , Gorman, D.M., and Gruenewald, P.G. (2007). Drinking as an epidemic - a simple mathematical model with recovery and relapse. In, Guide to Evidence-Based Relapse Prevention, (Witkiewitz & G.A. Marlatt (Eds.)). New York: Elsevier.
[144] Sani, A., Kroese, D.P., and Pollett, P.K. (2006). Stochastic models for the spread of HIV in a mobile heterosexual population. Mathematical Biosciences, 208(1), 98-124.
[145] Sher, K. J., Bartholow, B. D., and Nanda, S. (2001). Short- and long-term effects of fraternity and sorority membership on alcohol use: A social norms perspective. Psychology of Addictive Behaviors, 15, 42-51.
[146] Skinner, H.A. (1989). Butterfly wings flapping: Do we need more chaos in understanding addictions. British Journal of Addiction, 84, 353-356.
[147] Sobell L.C., Ellingstad, T.P., and Sobell, M.B. (2000). Natural recovery from alcohol and drug problems: methodological review of the research with suggestions for future directions. Addiction, 95, 749-764.
[148] Song, B., Castillo-Garsow, M., Rios-Soto, K.R., Mejran, M., Henso, L., and Castillo-Chavez, C. (2006). Raves, clubs and ecstacy: the impact of peer pressure. Mathematical Biosciences & Engineering, 3, 249-266.
[149] Soper, H.E. (1929). Interpretation of periodicity in disease prevalence. Journal of the Royal Statistical Society, 92, 34-73.
[150] Swaim, R.C. (1991). Childhood risk factors and adolescent drug and alcohol abuse. Educational Psychology Review, 3, 363-398.
[151] Tevyaw, T.O., Borsari, B., Colby, S. M., and Monti, P. M. (2007). Peer enhancement of a brief motivational intervention with mandated college students. Psychology of Addictive Behaviors, 21(1), 114-119.
114 [152] Thieme, H. R. (2003). Mathematics in Population Biology. Princeton Series in The- oretical and Computational Biology. Princeton and Oxford: Princeton University Press.
[153] Thombs, D.L. (1999). An introduction to addictive behaviors (2nd ed.). New York: Guilford Press.
[154] Toomey, T.L. and Wagenaar, A.C. (2002). Environmental policies to reduce college drinking: options and research findings. Journal of Studies on Alcohol, 63 (Supplement 14), 193-205.
[155] UC Information Digest (2003). A reference guide on student access & performance at the University of California.
[156] Usdan, S.L., Moore, C.G., Schumacher, J.E., and Talbott, L.L. (2005). Drinking location prior to impaired driving among college students: implications for prevention. Journal of American College Health, 54, 69-75.
[157] Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180, 29-48.
[158] Van de Goor, L. A., Knibbe, R. A., and Drop, M. J. (1990). Adolescent drinking behavior: an observational study of the influence of situational factors on adolescent drinking rates. Journal of Studies on Alcohol, 51(6), 548-555.
[159] Van Herwaarden, O.A. and Grasman, J. (1995). Stochastic epidemics: major outbreaks and the duration of the endemic period. J Math Biol.;33(6), 581-601.
[160] Vik, P.W., Cellucci, T., and Ivers, H. (2003). Natural reduction of binge drinking among college students. Addictive Behaviors, 28, 643-655.
[161] Wechsler, H„ Kelley, K., Weitzman, E.R., San Giovanni, J.P. and Seibring, M. (2000). What colleges are doing about student binge drinking: a survey of college administrators. Journal of American College Health, 48, 219-226.
115 [162] Wechsler, H. and Kuo, M.(2003). Watering down the drinks: the moderating effect of college demographics on alcohol use of high-risk groups. American Journal of Public Health, 93, 1929-1933.
[163] Wechsler, H., Kup, M., Lee, H. and Dowdall, G.W. (2000). Environmental correlates of underage alcohol use and related problems of college students. American Journal of Preventive Medicine, 19, 24-29.
[164] Wechsler, H, Lee, J.E., Kuo, M, and Lee, H. (2000). College binge drinking in the 1990s: A continuing problem. Results of the Harvard School of Public Health 1999 College Alcohol Survey. Journal of American College Health, 48(5), 199-210.
[165] Wechsler, H., Lee, J.E., Kuo, M, Seibring, M, Nelson, TF, Lee, H. (2002). Trends in College Binge Drinking During a Period of Increased Prevention Efforts: Findings From 4 Harvard School of Public Health College Alcohol Study Surveys: 1993-2001. Journal of American College Health, 50( 5), p203.
[166] Wechsler, H., Nelson, T.F., Lee, J.E., et al. (2003). Perception and reality: a national evaluation of social norms marketing interventions to reduce college students heavy alcohol use. Journal of Studies on Alcohol, 64, 484-494.
[167] Weiss, G.H. and Dishon, M. (1971). On the asymptotic behaviour of the stochastic and deterministic models of an epidemic. Math. Biosci., 11, 261-265.
[168] Weitzman, H., Nelson, T.F. and Wechsler, H. (2003). Taking up binge drinking in college: the influences of person, social group, and environment. Journal of Adolescent Health, 32,26-35.
[169] Weitzman, E.R., Folkman, A., Folkman, K.L. and Weschler, H. (2003). The relationship of alcohol outlet density to heavy and frequent drinking and drinking-related problems among college students at eight universities. Health & Place,9, 1-6.
[170] Werch, C.E., Pappas, D.M., Carlson, J.M., et al. (2000). Results of a social norm intervention to prevent binge drinking among first-year residential college students. Journal of American College Health, 49, 85-92.
116
[171] Whittle, P. (1957). On the use of the normal approximation in the treatment of stochastic processes. Journal of the Royal Statistical Society, B19, 268-281.
[172] Wilson, E.B. and Burke, M. H. (1942). The epidemic curve.Proceedings of the National Academy of Science of the USA, 28, 361-367.
[173] Wilson, E.B. and Burke, M. H. (1943). The epidemic curve, II. Proceedings of the National Academy of Science of the USA, 29, 43-48.
[174] Witkiewitz, K. and Marlatt, G.A. (2007). High-risk situations: relapse as a dymanic process. In, Therapists Guide to Evidence-Based Relapse Prevention (K. Witkiewitz & G.A. Marlatt (Eds.)), 19-33. New York: Academic Press.
[175] Winkler, D., Caulkins, J.P., Behrens, D.A. and Tragler, G. (2004). Estimating the relative efficiency of various forms of prevention at different stages of a drug epidemic. SocioEconomic Planning Sciences, 38, 43-56.
[176] Zucker, R.A. (1994). Pathways to alcohol problems and alcoholism: A developmental account of the evidence for multiple alcoholisms and for contextual contributions to risk. In, The Development of Alcohol Problems: Exploring the Biopsychosocial Matrix of Risk (R.A. Zucker, G. Boyd & J. Howard (Eds.)), NIAA Research Monograph No. 26, 255-289. US Government Printing Office; Washington, DC.
APPENDIX A BASICS AND MATHEMATICAL TOOLS AND TECHNIQUES USED IN THIS RESEARCH
118
A Historical Note Mathematical modelling of infectious diseases has a long history, and the starting point is generally taken to be a paper by Daniel Bernoulli (in 1760) on the prevention of smallpox by inoculation. An account of Bernoulli's model-based analysis of data can be found in Daley and Gani (1999). However, as Bailey (1975) points out, the birth of the first model took place a hundred years or so before the physical basis for the cause of infectious diseases became well-established. Perhaps, because of our ignorance of the causes of disease, the pace of modeling progress picked up until early in the 20th century, with the work or ideas of Hamer (1906), Ross (Ross, 1911, 1916; Ross and Hudson, 1917), and Kermack and McKendrick (1927). These researchers postulated the principle of mass action (homogeneous mixing) as the key to generation of new infections per unit of time. The disease's incidence rate was assumed to be proportional to the current numbers of susceptibles and infectious in the population. The now-familiar deterministic equations for the general epidemic model (mean field models; Isham, 2001) was thus developed. Stochastic epidemic models were also being developed early in the 20th century, alongside the deterministic ones. McKendrick (1926) in fact discussed a stochastic version of the general epidemic model. The best-known chain binomial model is the one proposed by Reed and Frost during a series of lectures that they gave in 1928 (Wilson and Burke, 1942; 1943), but their work had been preceded some 40 years earlier by Enko in 1989 (Dietz, 1988; 1989).
Simple Mathematical Epidemic Models The suitability of a model and the appropriateness of the assumptions on which it is based depend entirely on its purpose. The benefits of simple models are immense. They can provide robust approximations and help in qualitative understanding of specific issues of biological or social
119 systems. Complex models can incorporate more detailed assumptions of the nature of the underlying process, but at the expense of correspondingly more unknown parameters. The SIR (Susceptible - Infectious - Recovered) model with constant population and no demographic process is the prototype of the useful simple model. Here is the detailed description: S(t), I(t), and R(t) denotes the susceptible, recovered, and infectious populations at time t. The number of new cases of infection is proportional to SI, and the total population N — S + I + R is assumed constant. There are no births or deaths. The system is given by
f - *4-* -
=
-,1
1, protection from an epidemic outbreak is only possible as long as the proportion of susceptibles is kept below the threshold by vaccination; this effect is known as herd immunity.
• A population of fixed size, but subject to demographic changes in the absence of control strategies, will be subject to recurrent epidemics if RQ > 1.
120 Simple Stochastic Epidemic Models The sizes of sub-populations of a natural system are usually random because of the intrinsic stochasticity which has many essential consequences. These include effects on such issues as the possible fade-out and re-emergence of a problem, the prediction of the course of an individual realization of an epidemic, and the determination of the appropriate period of application of a control treatment. The incorporation of uncertainty requires the use of a stochastic SIR model where, for example, the expected value that there is a change in number of infectious individuals is given by
E[I{t + dt) - I(t) | S{t) = s(t), I(t) = i(t), R(t) = r(t)} = (/3s(t)/n - -y)i(t)dt + o{dt)
where S(t), I(t), and R(t) are now seen as random processes. Even for the simple stochastic model, explicit results are often difficult to obtain, and much effort and ingenuity have been spent in deriving methods of solution or approximations. For large populations, if the initial number of infectives is small, essentially all contacts of infectives are with susceptibles, and a branching process approximation is appropriate. If RQ < 1, there will be only a small outbreak of infection, and the distribution of the final size of the epidemic will be J-shaped (i.e., with a mode at or close to zero and decreasing monotonically thereafter). The time to extinction of the infection will be O(l) 1 as n —> oo. If Ro > 1 and I0 denotes the initial number of infectives, then with probability 1 —
{1/RQ)IQ
the branching process explodes, corresponding to
'Suppose f(x) and g(x) are two functions defined on some subset of real numbers. We say, f(x) = 0(g(x))
as x —> oo
iff there exists a M > 0 and a XQ e 3? s.t. \f{x)\ < M\g(x)\
forx>x0.
121 a major outbreak of infection. As the number of infectives builds up, the approximation ceases to be appropriate and a central limit effect takes over. In this case, the final size distribution will be U-shaped (i.e. bimodal), and the time to extinction will be 0(log n) as n —> 00 (Barbour, 1975). Nasell (1995) suggests defining the threshold as the value of RQ for which the distribution of the total size of the epidemic switches from J-shape to U-shape. Based on numerical work, he conjectures that this threshold has the form RQ = 1 + 0{n~ll^)
(Isham, 2005). There is a large literature on
stochastic SIR. Stochastic effects play an essential role in questions of recurrence and extinction (or fadeout) of infections. Bartlett (1949; 1956; 1957; 1960) introduced the idea of a critical community size below which an infection will rapidly go extinct unless it is reintroduced from outside. In a data-analytic investigation he describes the critical size as that for which there is a 50:50 chance of fade-out following a major outbreak. Damped oscillations were first noted by Soper (1929) in a deterministic (ill posed) model variant with no latent class and an exponentially growing population since no deaths were included in the model. The corresponding stochastic formulation used by Bartlett (1956) showed by simulation that the stochastic formulation of the model does lead to undamped oscillations, at a time scale that depends on the population size, and to the possibility of fade-out. Hence, the concept of quasi-stationary distribution becomes relevant. The quasi-stationary equilibrium (of the stochastic model) is an equilibrium distribution conditional on non-extinction (Darroch and Seneta, 1967). It is computed around the endemic equilibrium of the deterministic model. The process remain at quasistationary state for some random time, before eventually a sufficiently large random fluctuation will lead to extinction (Kuske et al., 2008). Andersson and Djehiche (1998) consider initial conditions so that IQ goes to infinity with n
122 (population size) to show that the time to extinction is asymptotically exponentially distributed with a mean that grows exponentially with n if i?o > 1. Nasell (1996, 1999) discusses approximations for the mean time to extinction when the initial distribution is the quasi-stationary distribution and also shows that the quasi-stationary distribution itself exhibits a threshold behaviour, being approximately geometric for RQ < 1 and approximately normal when Ro > 1 (see also Clancy and Pollett, 2003).
Understanding a Simple Stochastic Model In order to explain our approach we include a section that should facilitate the understanding of our use of stochastic models. If we consider the differential equation: dx , - = a(z,*), add a random component, // dx —
=a(x,t)+b(x,t)ri(t),
then we see that the solution to this stochastic differential equation is problematic because the inclusion of randomness prevents the equation from having bounded measure. In fact, the derivative does not exist. One way to deal with such random equations is to look at them in differential form, dx = a(x, t)dt + b(x, t)r](t)dt (A.4) = a(x,t)dt
+
b{x,t)dW(t).
Typically, such systems incorporate white noise2 which can be thought of as the derivative of Brownian motion (or the Wiener process), but other types of random fluctuations are possible. 2
White noise is a random signal (or process) with a flat Power Spectral Density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency. White noise is considered analogous to white light which contains all frequencies. A continuous time random process (white noise), w(t) where t £ 5ft, is a white noise process if and only if its mean function and autocorrelation function satisfy the following:
123 If b is constant, the system is said to be subject to additive noise otherwise it is said to be subject to multiplicative noise. Additive noise is the simpler and correct solution of the equation can often be found using ordinary calculus (particularly, the ordinary chain rule of calculus). However, in the case of multiplicative noise, such a system is not a well-defined entity on its own, and it must be specified whether the system should be interpreted as an Ito Stochastic Differential Equations (Ito SDE) or a Stratonovich SDE. This is because two versions of stochastic calculus can be used, the Ito stochastic calculus and the Stratonovich stochastic calculus, to interpret the solution of SDE. The SDE equation can be interpreted as an informal way of expressing the corresponding integral equation: t+s
rt+s
/ a(x,u)du+
/
b(x,u)dW(u)
(A.5)
The above equation characterizes the behavior of the continuous time stochastic process x(t) as the sum of an ordinary Lebesgue integral and an Ito integral. A heuristic (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length At the stochastic process x(t) changes its value by an amount that is normally distributed with expectation a(x(t), t)At and variance b(x, u)2At that are independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The function a is referred to as the drift coefficient, while b is called the diffusion coefficient. The stochastic process x(t) is called a diffusion process,3, and it is usually a Markov process. If the 1. pw{t) = E[w{t)\ = 0 2. Ry,(t, t + At) = E[w{t)w(t + At)} =
{N0/2)6{At).
That is, it is a zero mean process for all time and has infinite power at zero time shift since its autocorrelation function is the Dirac delta function. So its Power Spectral Density is NQ/2. 3 A diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with continuous sample paths. A stochastic process x adapted to a filtration F is a diffusion when it is a strong Markov process with respect to F, homogeneous in time, and has continuous sample paths. Having said that, some authors do not insist that diffusions be homogeneous, and some even do not insist that they be strong Markov processes. But this is the general sense in
124 coefficients depend not only on the present value of the process x(t), but also on previous values of the process and possibly on present or previous values of other processes too, then the solution process, x, is not a Markov process, and it is called an Ito process and not a diffusion process. When the coefficients depend only on present and past values of x, the defining equation is called a stochastic delay differential equation. A typical existence and uniqueness theorem for Ito SDEs taking values in n-dimensional Euclidean space 3?" and driven by an m-dimensional Brownian motion, B, can be found in Stochastic Differential Equations: An Introduction with Applications (0ksendal, 2003).
Difference between Deterministic and Stochastic Models Let Xx{t)
be a vector of stochastic processes that is governed by a system of stochastic
differential equations which defines population of interest of size N. Let x(t) is the solution of corresponding system of differential equations. If
^ ' —> XQ then —ff
—> x(t) in proba-
bility i.e., convergence is almost surely and uniformly on compact time intervals as N —> oo. In other words, the normed jump Markov vector process for a large population N is approximately equal to the deterministic vector function x(t). Since the scaled Poissonprocess converges (weakly) to the standard Brownian motion, the scaled stochastic process, \/~N I ^>' — x(t) 1, may be approximated by a Gaussian process whose covariance matrix at some fixed large time is similar to the covariance matrix of the Ornstein-Uhlenbeck process.4
That is, the fluctuations around the
which the term is used. A The Ornstein-Uhlenbeck process (an example of a Gaussian process that has a bounded variance), also known as the mean-reverting process, is a stochastic process given by the following stochastic differential equation drt = -6(rt
- p)dt + adWt
where Wt is the Wiener process. Note that the drift term is linear and the diffusion coefficient is constant. For O-Uprocess, E(rt) = r0e-et+p(l -e~et) andcov(rs,rt) = f ^ e " e ( s + t ) ( e 20 ( sAt ) - l ) ,
125 deterministic solution are asymptomatically Gaussian (Andersson and Britton, 2000). In summary,
1. The quasi-stationary distribution of the original stochastic process (given by SDE) can be approximated by the stationary distribution of the Ornstein-Uhlenbeck process. 2. If the total population size parameter N appears in the deterministic model, then it can be eliminated from the system by re-scaling the state variables, whereas stochastic models account for demographic stochasticity. A stochastic model is a general model, and the deterministic model can be derived from it by dividing the state variables by constant N and taking a limit as N approaches infinity. 3. In general, in a deterministic model there is a threshold value i?o which identifies two parameter regions (RQ > 1 and i?o < 1), with different qualitative results. Nasell (2002) showed that for some simple stochastic models there are three parameter regions whose boundary depend on the value of the parameter N. The additional region in the stochastic version of the model considered by Nasell is near i?o = 1 whose size decreases to zero as N goes to infinity. The three different qualitative behaviors can be seen by deriving asymptotic approximations of the quasi-stationary distribution and of the expected time to extinction from the quasi-stationary distribution as N —> oo .
4. Generally, the disease in stochastic epidemic models with finite state spaces and at least one absorbing state, ultimately tends to extinction regardless of the value of RQ. However, the time of extinction may be very long. If RQ > 1, the probability distribution conditioned on
non-extinction approaches is a quasi-stationary probability distribution whose mean agrees where s A t = min(s,t). If we have a multivariate O-U process (i.e., dXt = —AXtdt + BdWt where A and B are constant matrices), then under the condition that all the eigen values of A have positive real parts, a stationary solution of the process exists. The mean of the stationary solution is 0 and its covariance matrix is given by S, where AT. + T,AT = BBT.
126
with the deterministic endemic equilibrium (Nasell, 2002).
5. The time of extinction is a random variable whose distribution depends on the distribution of the initial states. If the process is going on for a long time and has not gone extinct, then the quasi-stationary distribution can be used to approximate the distribution of the states. The time of extinction starting from the quasi-stationary distribution measures the persistence of the infection (Nasell, 2002). Ro(N) gives the persistence threshold when the initial distribution equals the quasi-stationary distribution and the invasion threshold when initially we have one infected individual (Nasell, 1999). If N is large and i?o is strictly greater than one, then the quasi-stationary (corresponding to the stationary state in the deterministic model) distribution of the stochastic model can be approximated by a multivariate normal distribution (Nasell, 2002).
In order to make this research accessible to a wider audience we include some definitions and concepts that have been used in the derivation and analysis of the stochastic model (Chapter 4).
Poission Process A Poisson process, Nt 6 Z+, (with parameter A) is a counting process5 such that
(i). P(Nt+At
-Nt
= l) = XAt + o(At).
(ii). P(Nt+At
-Nt
= 0) = l-XAt
(iii). P{Nt+At
-Nt?0orl)
5
+ o{At).
= o(At).
A stochastic process is a counting process when every sample function is zero if t < 0, integervalued and nondecreasing with time.
127
(iv). Nt is a Markov events, that is, (Nt+At — iV* = 1) and (Ns+As - Ns = 1) are independent (i.e. P(Nt+8 = k\Nu;u• E[j:iNn (ii).
= l* XAt + 0 * (1 - XAt) + o(At) = XAt + o(At). - Nt)2] - E2[Nt+At
- Nt] + o(At2) = XAt + o(At).
, n are independent Poisson process, then =
j2iE[Ni].
Var[YJiNt]=YJiVar[Nl]. The parameter, A, of the Poisson process may or may not change over time. The process
is called non-homogeneous Poisson process (or Poisson-type process) if it does depend on time, otherwise it is called homogeneous. The expected number of events between time 'a' and time 'b' for the non-homogeneous process is
K,b = [ X(t)dt. Ja 'Essentially, the probability of an arrival during any instant is independent of the past history of the process, i.e., it is memoryless.
128
Thus, the number of arrivals in the time interval (a, b], given as N(b) - N(a), follows a Poisson distribution with associated parameter Aa>(>. For a stochastic process X(t), X ( t i ) , the value of a sample function at time instant t\ is a random variable. It has PDF /x(ti)( a ; )
an
d expected value £'[X(ti)]. The expected value of a
stochastic process X(t) is the deterministic function fix{t) = E[X.(t)]. The autocorrelation function of the stochastic process X(t) is # X ( M + At) = £ [ X ( t ) X ( t + A t ) ] . If X ( t ) is stationary, then Rx(t, t + At) = Rx(At)
and px{t) = fix, for all t. The average power1 of a stationary pro-
cess X ( t ) is Rx(t, t) = i?[X 2 (i)]. The Fourier transform offers another view of the variability of functions of time. A rapidly varying function of time has a Fourier transform with high magnitudes at high frequencies, and a slowly varying function has a Fourier transform with low magnitudes at high frequencies. The Power Spectral Density function, Sx{f),
is the expected value of the squared
magnitude of the Fourier transform of a sample path of X( M%°, as n —> oo and limsup n Mi(e„) < Mf°. By using M2 equation of System (E.l), we get lim M 2 (e„) = 71 lim Mi(e n ) - (/z + 72 + a) lim M 2 (e„), n—*oo
n—-»oo
n—>oo
0 < 7 i M 1 0 0 - ( / i + 72 + a)M 2 00 , A*2° < -, — (/* + 72 + a) Since S+M
(C.l)
TM?°.
< 1 and S+Af1+M2 < 1, therefore the Mi equation of System (E.l) yields
Mi < /30Mj - (/* + 7i)Mi + (ft + 72)M 2 . Similarly, there is a sequence {£„} such that M\{tn)
(C.2)
—> 0, Mi(i n ) —> Mf°, as n —> 00 and
limsup n M2{tn) < M2 0 , and so by using inequality (C.l) and (C.2) we get 0 < ftMf0 - (/x + 7 i ) A ^ ° + (ft + 72)M20C (C.3)
o< U-(/i+ 71) + 7 ^ 2 ^ ) ^ fOO
V
(/x + 72 + a)y
Simplification gives us
0
< / > + 7i)fr + 7a + «)-7i72\ ~ V (M + 72 + a) )
(jR
_
1}
TO
(C4)
X
If i? d < 1 then inequality (C.4) can hold only if Mf° = 0. But we know 0 < Mi < Mf 0 , so Mi = 0. Hence, we also get M2°° = 0 (or M 2 = 0) by (C.l). We can use the same approach in the S equation of System (E.l). That is, there is a sequence {sn}
142 such that S(sn) —> 0, S(sn) —> S ^ , as n —> oo, limsupMi(s„) < Mf° and lim sup M 2 (s n ) < M%°. Since — 1 < — S+M
and — 1 < - g + M s + M , therefore the 5 equation becomes
(c-5)
S>-p0M1 + fiN-nS-j32M2. Hence, by the fluctuation method
0 > -ftAff 0 + /iTV - ^tSoo -
fcM?.
(C6)
But we have proved above that if Rd < 1 then Mf° = 0 and M | ° = 0. Therefore, we get S^ > N. Since the total population N in our system is bounded, hence S ^ = N. So 5 = N. This proves the global stability of the drinking free steady state, if Rd < 1.
|
Existence of Endemic Drinking Equilibrium (i.e., State of Mixed Drinking Community) Lemmas E.7 and E.8 prove the existence of a unique endemic drinking equilibrium. In order to find endemic drinking equilibrium solutions we set ^ = 0, ^ ^ = 0, ^f2 = 0, and ^f = 0 in the System (E.l) and assume H ^ 0. We then solve for the endemic values S*, Mf, M£, and H* using these equations. The endemic values are given as follows
S* = N - VH*, Ml = £0H*, M2* = OH*,
(C.7)
where TJ, £, 0 are given in lemma E.7, H* = HN, and H satisfies the quadratic equation aH1 -bH + c = 0.
The coefficients a, b, and c of the quadratic equation (E.5) are given in Lemma E.7.
(C.8)
143 Lemma C.4 Let H ^ 0, then the coeffients 'a', 'b', and 'c' of the quadratic equation (E.5) are given by
b = eU+^)+(l c=(Rdwhere Q = £ , V = u±&, £ = 1 ^ ,
7
+ r1)(Rd-l),
(C9)
1), = ^ i _ _ ,
and
# = f . If i ^ > 1, then the coefficients
a, b and c are positive.
Proof Setting derivatives equal to zero in the System (E.l) leads to four different equations in variables S, M\, Mi, and H. Let any solution of these equations be denoted by 5*, M{, M^ and H*. Then the equation ^
= 0 gives
MS = -H* (= 9H*), a
(CIO)
and the equation ^ j j 2 = 0 gives
M\ = I i - ^ 1 M2* (= £M2*).
(C.l 1)
By substituting (E.7) in (E.8) we get M{ = Z9H*.
(C.l 2)
Adding the equations 4f = 0 and ^ . = 0 and using values of Ml and Mjf from (E.9) and (E.7), respectively, gives S* = N-r]H*.
(C.13)
144 Using values from (E.10), (E.9) and (E.7) in the equation ^ = 0, we get the quadratic equation in H*,
i + e-v-
2 — + (i- Rd)v H* a
NH* + (1 - Rd)N2 = 0. (C.14)
After further simplification, and using H = |£, we get the quadratic equation (E.5) with coefficients given in (E.6). Since all the parameters in the System (E. 1) are positive and 7 lies between 0 and 1, therefore 8, 77 and £ are positive. This implies the sign of the coefficients a, b, and c will depend on the factor (Rd - 1). Hence, if R& > 1 then all coefficients will be positive.
|
Clearly by (E.4), M{ > 0 and M2* > 0 provided H* > 0. So in order to have a positive endemic equilibrium we must have S* > 0 and H* > 0. Let H* > 0 (i.e., / P 0 ) . If if* < ^ (or H< i ) , then from (E.4), S* > 0.
Lemma C.5 IfR^>l
then there exists exactly one positive real root H* of the quadratic equation
(E.5) such that H* e ( o , ^ ) .
Proof By lemma E.7, if R^ > 1 then a > 0, and so the parabola defined by (E.5) will be concave up. Let f(x)
— ax2 — bx + c be a function with a, b, and c given by expressions in (E.6). Then
/(0) = c and / ( ^ ) — %(l - rj). Using the expression for 77, we find / ( ^ ) < 0. Since Rj > 1, therefore by (E.6), /(0) > 0. Hence, by the intermediate value theorem there exists at least one root between 0 and K But since f(x) is a parabola facing up, there exists exactly one positive real value 'z' between 0 and ^ such that f(z) = 0. This is equivalent to saying that there exists exactly one positive real root H* of the quadratic equation (E.5) such that H* G (0, - J.
|
145 Proof of Theorem B. 1 in Appendix B Proof Lemma C.2 shows that the abusive drinking free steady state is locally asymptotically stable if Rd < 1 and unstable if Rj > 1. If Rd > 1, then by Lemmas E.7 and E.8 we have a unique H* such that H* > 0 and S* > 0. Therefore, there exists a unique endemic drinking equilibrium for the model. If Rd = 1, then the quadratic equation (E.5) with coefficients given in (E.6) becomes a linear equation if H ^ 0. Solving linear equation gives H* > - (i.e., S* < 0). Hence, in this case no endemic drinking equilibrium in the positive quadrant exists. If Rd < 1, then global stability of the drinking free steady state rules out the existence of an endemic drinking equilibrium for the model. We now prove the claimed stability at and near Rd = 1 (including the case when Rd is slightly larger than one). For simplicity we choose /3b as the bifurcation parameter instead of Rd- If Rd = R*d (i-e. Rd = 1) or (30 = $J (i.e., j30 = a - ^ ) , where 6 = fi + 72 + a, a = \i + 71 and 6 = (52 + 12, then the drinking "free" equilibrium (DFE) is non-hyperbolic, and Center Manifold Theorem (CM Theorem) and Reduction Principle can be used to obtain the stability of the DFE. According to the CM Theorem and Reduction Principle, under these conditions the original system will have the same dynamics as the system reduced on the center manifold. Calculations lead us to the following normal form of the reduced system on the center manifold: dX
where X=* m
~
= m(pX + -X2) + h.o.t. V m l
U 1 _= M +n 72 + a, p = % - ft, n = - ^ ^P?+7i(fti-H»)] l ^ l V ^ . ' ft t7 1 (e2 ^ + ^S1 < 5 ) At r ^ . '"' ^ "•• " • ^ ~ ^ 0 - M). » - -
and
(02+7i 0 and 5 > 0 such that for 0 < p < e (i.e., e < Rd < 1) the DFE is locally asymptotically stable and for —6 < p < 0 (i.e., 1 < Rd < S) the DFE is unstable, and a locally asymptotically stable endemic equilibrium exists (Kribs-Zaleta, 2002; Van den Driessche et al., 2002).
Local Stability of the Endemic Equilibrium If 7 denotes the proportion of time that the mobile M-individuals spend in the high-risk environment E2, then we derive /3i(l-7) , fol A) V»\ fa (p + a)7i d = —T. \—>~ —/ /Ti( l - 7 )T+T ~(M7 +i « T) 7- = —-T"n—T~7 \—T + (A* + « ) ' [A*^I + (M + Q )7i] fj,(l - 7)T+7 ~{n7 +1 0)7 A* ' M I + (/*;+ a)7i] (C.15)
K
where 71 A + 7i + 72 + 4
a
(C.16)
and 61 = A* + 72 + a. Theorem C.6 Let Rd be given by Expression (D.l). If Rd > 1, A/ie« £>i = (5*, M{,
M^H*)
(all drinking levels above zero) is locally asymptotically stable, that is, the numbers in the classes approach the values in D\ as long as they are already close, as time increases.
Proof By setting the right hand side of the System (E.l) to zero and solving, we obtain the steady state solution. The normalized (with respect to the total population) steady state solution for the model is given by
147
. where, s* +m\+m*2 Z=^,a=%(t
+ h* = 1, s* = §, m* = f (= *&$&),
+
h* = f, rj =
^ ,
%)+r,(Rd-l),b=%(t+&)+(l+v)(Rd-l)andc=(Rd-l).
Since the System (E.l) is smooth, we can linearize it around the stable endemic state, (s*, m\, m*2, h*), of the deterministic system. In the process of linearization, we first center them by the change of variables u = s - s*, v\ = m\ — m*, V2 = rri2- m\, and w = h-h*.
Then, linearis-
ing at (u, v\,V2, w) = (0,0,0,0) and substituting the values at the endemic state in the diffusion coefficients we get
I .^
u
Vl
d_
Vl
= J(s*, ml, m2l h*)
Jt
(C.18)
V2
V2
\W J
\W J
where J is the Jacobian matrix evaluated at endemic steady state of the deterministic system, /
-(P + /x) P
-Q Q-{n
+ 6 + -n)
-T
0
T + 72
0
\
(C.19)
J ( s , m i , 7712) =
V
0
7i
-(/x +(5 + 72 + a)
0
0
a
0 —fi I
p=tk (^)2+/^ffigsg:.. Q=-h {^y+/v+ffV'and ^2(s*+m*+m*)2- s *' rni Let a = -r_ a)(l -a-b),Q
an
0, C > 0 and AB > C. Note that 7
=
g ^ ,
£ =
%, r, =
1 + £ (^ +1)
=
1 + £ (£ + 1 ) and
Special Cases: i) If fa = 0 then P = (3l (j^f, a
Q = P! (^f,
T = 0,A,B,C>0,
and Rd =
ftg^g,
"d i f >>* > (i + g(Wn))' t h e r e f o r e P + Q = & (fel) ^ °"
ii) ffft = OthenP = / % ( l - a ) ( l - a - & ) , Q = / % a ( l - a - & ) , P + Q = / 3 2 ( l - a - b ) > 0, , ^ , a n and d i f iff r/i** >>( 1n +, a L* , »)} therefore P - T = /32(1 - 2a - 6) > 0. Rd = P2^zz,
APPENDIX D PARAMETER ESTIMATION; UNCERTAINTY AND SENSITIVITY ANALYSIS ON THE REPRODUCTION NUMBER (CHAPTER 3)
150 Model Results from Chapter 2 If 7 denotes the proportion of time that the mobile M-individuals spend in the high-risk environment E%, then we derive
Rdiy) =
MU-THVW1 - 7) + Ml-7)+V + a)77
(D 1}
-
where
7 =
• A* + 7i + 72 + a
(D.2)
tf Rd < 1 ^ e n o«/y light drinkers exist in the community, and if Rd > 1 then moderate to heavy drinkers co-exist with light drinkers in the community. By setting the right hand side of the system (E.l) to zero and solving we obtain the steady state solution. The normalized (with respect to the total population) steady state solution for the model is given by
s*
=
,x h —
h
_ S
1 - rjh*,
(D.3) (D.4)
a b- Vb2 - 4ac
(D.5)
2a * _ M (_
Mt+M2\ h* = S . Tl. _ M+7«
f
ja
a
= £ (^ + £ ) + v(Rd - 1), & = £ J ) + g ) + (1 + v)(Rd - 1) and c=(Rd~
:-l=2
1).
151
Steps of Uncertainty and Sensitivity Analysis Analyses were performed using Monte Carlo simulations in Matlab. 10 3 samples were collected 10 times (10 realization) from each assigned or evaluated pdf of .R^-parameters as described below.
1. We use estimates of 7 from data to assign a uniform distribution to it. Normal distribution is assigned to \x using its estimates from data. We truncated the normal distribution at zero.
2. Using steady state equation from H' we obtain the expression for a as follows: fih*
where h* = ^ and m\ = ^
(D.6)
= 7 ^ = 7m*. Hence, the expected value and variance of a
are
*"•>=(£) xB (?)
mi Var(o)
=(£)'x"-(7) •
Var ( ^ ) is calculated using delta method and estimates of 7 and /x. A sample is chosen independently from assigned distribution of 7 and /x. Equation (D.6) is then used along with this sample to obtain an estimate of a.
3. Using steady state expression for h* (see Equations D.3) we get a linear relationship between Po and P2 as follows f)n
Or,
* = £*>+£'
(D.7)
where 01 = £ \-h* (h* - l)g
CO]
rj(h*-l) +
- \-q{h* - 1) + ^P-
^P
, 02 =
X
at]
Tl(h*-l) +
1-ft*)
^
, and 03 =
. Since 0X < 0, 02 > 0 and 0 3 < 0, therefore Equation
152 (D.7) is a line with negative slope and positive y-intercept. Hence, fa and fa will be nonnegative if their values will lie in the intervals 0, — ^
and 0, ^ , respectively.
For each sample from 7, /i, and estimate of a, we evaluate value of 0$, i = 1,2,3 (Equation (D.7), where #3 < 0, #2 > 0). Uniform distribution is assigned to fa defined on interval 0 U
'
-2302
and a sample is chosen from this distribution. The sample of fa thus obtained is
substituted into Equation (D.7) to obtain a value of fa.
Uncertainty analyses were performed using samples of parameters 7, fi, a, fa, and fa obtained as described above. 1. The value of all parameters are used in the Equation (D.l) to get estimate of Rd- Since the steps are repeated 105 times, we obtain the distribution of each of the parameters including Rd using one realization.
2. Statistics were carried out on the obtained synthetic data of the parameters including Rd. In the sensitivity analysis, 10 3 samples of each parameters and corresponding Rd values were taken from one of the realizations from the uncertainty analysis and used for calculating Partial Rank Correlation Coefficients (PRCCs). A parameter (say, 'p', one of fa, fa, 7, \i, and a) is first chosen with respect to which sensitivity on the Rd is to be computed. Then the following steps are used to calculate its PRCC value with respect to Rd.
1. The samples of Rd and 'p' are first ranked in order to avoid any discrepancies that may arise as a result of a non-linear relationship between these two variables. 2. Regression curves, between Rd and the rest of the parameters except 'p' ('p' is fixed), and
153 similarly between 'p' and the rest of the model parameters except Rd, were constructed using the sample values. Thus, we obtain 105 residuals from the ^-regression curve and 105 residuals from the 'p'-regression curve.
3. Correlation coefficients between these two sets of residuals were calculated to get partial correlation coefficient between Rd and 'p'. Scatter plots between these two sets of residuals were also analyzed for verifying the assumption of monotonicity.
4. The value of the correlation coefficient so obtained is called the partial rank correlation coefficient of Rd with respect to 'p'.
5. The P-value for correlation coefficients based on the test statistic, t (t = rJjE^i
where r
is sample correlation coefficient (obtained above) and n is total sample points i.e. 105) is calculated. When HQ : p = 0 is true (i.e., when the population correlation coefficient is equal to zero), the ^-statistic above follows a t distribution with n - 2 degrees of freedom.
The effect of parameters 71 and 72 are noted in the results through a single parameter 7, where expression is given in Equation (D.2). This expression can be rewritten when 72 becomes a linear function of 71. This linear function will have positive slope, passing through point (^4^, 0 j , and y-intercept will be at — {JJL + a). Since all the parameters are assumed positive in the model, we can assume uniform distribution for 71 with its minimum value ^ 4 ^ . We did not mention estimates of 71 and 72 in the main text, although rough idea can be obtained from this linear function.
APPENDIX E ANALYSIS OF DETERMINISTIC MODEL CORRESPONDING TO STOCHASTIC MODEL (CHAPTER 4)
155 Deterministic Model We used a framework similar to the one considered in deterministic model in Chapter 2. We extended the simple model by allowing different recruitment and exit rates of the system and by incorporating parameters associated with control measures. The total recruitment rate into the system is assumed to be (1 — v)A, where v represents the proportion of recruits who do not enter the at-risk population as a result of prevention programs. All new recruits are assumed to be susceptible and to enter the system through E\. Individuals are also removed from classes due to intervention programs at a per-capita rates of 5\ and 52 from E\ and E2, respectively. The assumptions and definitions mentioned above give rise to the following model:
