mathematical, modeling and epidemiological ...

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Proceedings Trim Size: 9in x 6in

CastilloChavez

ROLES OF HOST AND PATHOGEN MOBILITY IN EPIDEMIC OUTBREAKS: MATHEMATICAL, MODELING AND EPIDEMIOLOGICAL CHALLENGES

CARLOS CASTILLO-CHAVEZ Mathematical, Computational and Modeling Sciences Center Arizona State University, Tempe, 85287-1904 School of Human Evolution and Social Changes and School of Sustainability Santa Fe Institute, Santa Fe, NM, 87501 Cornell University, Biological Statistics and Computational Biology, Ithaca, NY 14853 - 2601 E-mail: [email protected] JUAN PABLO APARICIO Instituto de Investigaci´ on en Energ´ıas no Convencionales, CONICET-UNSa Universidad Nacional de Salta, Av. Bolivia 5150, 4400 Salta, Argentina. E-mail: [email protected] ´ FERNANDO GIL JOSE Instituto de Investigaci´ on en Enfermedades Tropicales Universidad Nacional de Salta, Av. Bolivia 5150, 4400 Salta, Argentina.

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2 Finding and developing macroscopic descriptions for the dynamics of large ensembles of highly mobile host and/or pathogens subject to modern forms of dispersal pose tremendous challenges to researchers involved in the development and implementation of local, regional, national and global public health policies in the presence of often unexpected, threats of undetermined or uncertain origin. In addition, the study of the dynamics of highly mobile host and/or pathogens provide opportunities for the computational, modeling and mathematical communities in many directions that include finding new ways of applying existing mathematical methods or through the development of new modeling frameworks. Moreover, the challenges involved in dissecting complex systems have generated opportunities to mathematicians in applied and pure mathematics. Further, recent advances in computation (hardware, software and theory) mean that today “we” are no longer committed just to the study of simple systems or at best the study of complex systems over limited “windows” in time and space. Today, we can, in fact, apply continuous advances in the computational sciences to the solution of problems at the interphase of the life and social sciences. That is, the study of systems whose dynamics are driven by processes that take place over multiple temporal and spatial scales; systems that in addition, must routinely account for the inherent levels of heterogeneity of its biological components. This manuscript, and its companion, “Change in Host Behavior and its Impact on the Transmission Dynamics of Dengue” (F. Sanchez, D. Murillo and Castillo-Chavez, this volume) highlights some of the challenges in the context of the dynamics of vector-transmitted diseases, particularly dengue. It is the hope that these contributions will motivate members of this community, particularly those who are new to the field of mathematical biology, to tackle the challenges posed by the study of complex systems.

1. Introduction Before the era of vaccination measles patterns supported a variety of epidemic regimes that captivated epidemiologists and theoretical biologists alike. For example, in small communities well defined outbreaks at irregular intervals were observed while on the other hand in big cities, childhood diseases most often reached endemic levels. Bartlett, in his classic 1957 paper, Measles Periodicity and Community Size 6 , was able to explain much of the observed dynamics using a metaphorical simple computational metapopulation model. Geographical space was captured via a grid (connected populations, the elements of the metapopulation model) with a fraction of infectious individuals in each patch allowed to migrate to adjacent cells. The observed global persistence was explained as a consequence of restricted pathogen mobility between neighboring cells (patches), a mechanism that provided some time to local populations to recover from previous outbreaks. Bartlett’s pioneering research highlighted the critical importance of spatial heterogeneity, pathogen mobility, and stochasticity on disease dy-

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namics. His research paved the road towards the theoretical advances carried out by a myriad of researchers over the subsequent five decades 2,3,7,8,18,17,28,41,42,44,48,50,56,9,35 . Last decades also has seen advances involving the use reaction-diffusion, integro-differential and integro-difference nonlinear systems12,11,13,14,15,16,37,38,40,43,46,47,55,64,69 . A metapopulation model consist of a network of subpopulations or patches connected through the interactions of, for example, mobile individuals. At the local level each of these patches is described at the population level, that is, only total numbers of individuals in determined states are considered as variables, a feature that significantly speeds-up computational times. Metapopulation models, which offer a compromise between simplicity and realism have been widely used and applied in a great variety of contexts 40 . Additional developments came, in part, as a result of the study of an extinct disease, smallpox (albeit it is kept in some laboratories). For smallpox intimate contacts are fundamental in understanding patterns of transmission and network models account for these local effects 56,20,21,22 . In a network model each individual is in close contact only with a small group which form his/her neighborhood. Social distance is not related with spatial distance. The structure, or topology, of a network depend not only of sociological traits but also on the disease under study. Unveiling such network topology is critical when developing realistic network models. Fortunately in many cases data exist about patterns of human mobility at many different scales, including within cities, or from city to city, or due to international travel. As the fear of the deliberate release of smallpox (or other pathogens) increased, detailed network models (among others) have also been developed29 . A greater detail is incorporated in these models in which the state of each individual in the population is followed through time, for example, such modeling efforts included those that focused in different strategies of containment39,27,66 . In recent years highly virulent avian influenza H5N1 and human influenza H1N1 raised concern about the possibility of a new high mortality influenza pandemic. Influenza has a short generation time (of about three days) and therefore actual high levels of human mobility imply that a pandemic is virtually impossible to avoid as demonstrated by the 2009 influenza pandemic. Modeling pandemics is a formidable task as it involves the largest spatial scale. A series of interesting contributions, primarily

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from Canadian, Mexican and U.S.A based researchers, inspired by the 2009 influenza pandemic has just been published 17 . A glance at the literature show that impressive advances have been achieved in the modeling of directly transmitted diseases albeit comparatively few have been carried out in the context of vector-borne diseases like malaria or dengue even though the mathematical machinery is available11 . Vector transmitted diseases pose additional challenges since not only human but also vector mobility play an important role in the patterns of disease spread. The companion to this manuscript, “Change in Host Behavior and its Impact on the Transmission Dynamics of Dengue” (F. Sanchez, D. Murillo and Castillo-Chavez, this volume) highlights some recent advances and questions. The importance and challenges posed by the study of dengue or other vector-transmitted diseases, for example, can also be partially gauged through, for example, the following recent contributions62,24,25,26 2. Directly transmitted infectious diseases For directly transmitted diseases pathogen’s spread is tied in to host’s mobility which may be modeled in a variety of forms. Most models incorporate host’s mobility implicitly but others involving high resolution simulators have explicitly represent host’s movement. 2.1. Patch Models In patch models the population is divided in subpopulations or patches. The dynamics within a patch are described from a population’s perspective and, consequently, variables are the total number of individuals in the classes used to describe or characterize the members of the population (susceptible, infected, recovered, etc.). Host mobility is included implicitly: infectious individuals of a given patch may infect individuals of other patches. The assumed patterns of host mobility determine the probability of such longdistance contacts. As observed, in Bartlett’s 6 seminal work, infection spreads only to adjacent patches. This form of spatial diffusion is not realistic for human populations in general (particularly in today’s world) since they tend to be highly mobile. In fact, social distance usually is unrelated to geographical proximity. People who live next door to each other may be epidemiologically isolated. A glance at the 2009 influenza pandemic sees the H1N1 virus jumped almost instantly from Mexico to the U.S.A and Canada. In a matter of weeks it had already reached Japan as well, see 17 and references

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there in. For foot-and-mouth, a livestock disease, the probability of long distance infection has been modeled using a decreasing function of the distance49,23 . Human activity seems to be responsible for much of the disease dispersion as road density correlates with risk of infection59 . From a theoretical perspective is worth to mention the work of Watts et al.68 . The population was partitioned in small subpopulations for which homogenous mixing was assumed. These patches were organized in a nested hierarchy. For example, we can think that subpopulations form groups, which in turn form groups or groups, and so on. Infectious individuals may infect individuals in all subpopulations with a probability decreasing with the ‘distance’ in the hierarchy. Consider for example only one infectious individual in a given subpopulation. Members of this subpopulation are all at the same risk of infection. Individuals within other subpopulations in the same level in the hierarchy (or group of patches) are all at the same risk of infection. Risk of infection continues decreasing for individuals belonging to groups higher in the hierarchy. In this simple model the disease dynamics are driven by rare events of colonization of patches far away in the hierarchy, increasing global persistence and possibly the production of multiple outbreaks as observed in real epidemics45 . An interesting application of patch models to influenza pandemics is found in the work of Vespignani et al. 27,4,1 . These researchers use global data on air traffic, representing the network of almost all airports in the world: 3100 nodes with more than 17000 links. Each link was weighted according to the reported number of seats and frequency of the flights. It was assumed that within each city (associated to each airport) that individuals mixed homogeneously. The model was used to asses different containment strategies. Air travel restrictions, which are quite difficult to enforce, produce only a modest delay in the course of the pandemic as observed in the 2009 influenza pandemic. However, a cooperative intervention with antiviral drugs could mitigate the pandemic if timely distributed among the populations hit by the pandemic. While this model provide the most detailed description of the global human mobility, no details of local mobility or land transportation between urban centers is considered, a task left for the high-resolution network models.

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2.2. Static Contact Networks In a static contact network model each individual is always in contact with some individuals in the population albeit the strength of the contact usually depend on its type. For example, households contacts usually are considered more intimate than workplace contacts. Casual contacts which occur, for example, in shopping malls can also be incorporated in static networks. Network models have been mostly used in the study of fast dynamics diseases, that is, for outbreaks that run out of energy in a few months. The disease selection allows the researcher to neglect the population dynamics (but see 67 ). Ferguson et al31,32 developed a large-scale detailed static network model which was parameterized for Thailand, Great Britain and the United States. Population density (for the different countries modeled) was estimated using the the 2003 Landscan1 dataset (Oakridge National Laboratory). Household sizes and age distribution were taken from census data. Additional official sources were used to estimate school size distribution and schools were placed according to population density. Data on workplace size distribution were used to generate workplaces. Allocation of persons in workplaces used data on origin-destination flows for travel-to-work obtained from census data or other sources. Air-travel was modeled in a similar fashion. Disease transmission was modeled as follows: the probability of infection of a susceptible individual was computed as 1 − exp(λi ∆T ) with λi denoting the risk of infection and ∆T the simulation time-step. The risk of infection λi was generated from a weighted contribution of all possible sources, household, workplace, and casual contacts, for example. The contacts are known from the beginning of the simulation and do not change over time, that is, the network is static. This particular way to model the risk of infection speeds-up computational times. Naturally, it is unknown which particular individuals are responsible for the infection (transmission event) in each case. A similar model involving 281 million individuals “living” in the United States was developed and used in a research effort built in collaboration between the University of Washington and Los Alamos National Laboratory52,34. The population was distributed using the data for the 65,334 census tracts used in the 2000 U.S. census. Population in each simulated tract was grouped in individuals communities. Each individual belongs to ‘mixing groups’ defined by households, household clusters, work places, schools, etc. Casual contacts which are assumed to occur in super-

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markets or shopping malls are also included. Census data were also used to generate households size and age distributions as well as in the modeling of persons commuting to work. Some individual may take also long-distance trips, decisions matching data from the Bureau of Transportation Statistics. The time step is fixed at ‘12 hours’ to account for day and night. Thus during the night period, transmission is assumed to take place in households or in household clusters. Within a given time-step the probability of infection is assume to depend on the infectious individuals involved in the mixing groups shared by the susceptible persons and on the probability of infection in each of them. The twelvehour contact probabilities are assumed to be age dependent and their values were fitted to match reported influenza attack rates.

2.3. Dynamic Contact Networks The large-scale high-resolution simulation model, developed at Virginia Tech29,5 , used the Transportation Analysis and Simulation System (TRANSIMS), which was developed at Los Alamos National Laboratory to capture realistic patterns of population mobility. Individuals move between locations houses, schools, work places, supermarkets and more, with their movements determined from data generated from specific surveys. Susceptible individuals in a particular location have a risk of infection which depends on the susceptibility, the number of infectious individuals, their infectivity, and duration of the contact, among other factors. This contact network evolves over time and transmission takes place in well defined spaces. In this model it is possible to known where and by whom an individual got infected. The implementation uses two graphs: one composed by all the locations considered, and other composed of all individuals in the population. Because of its high resolution, simulations for large populations are extremely time consuming. Typical simulations were run for less than six simulated months. The network is dynamic in the sense that contacts of a given individual change over time but because of the short time scale considered population growth is disregarded. Nevertheless, the structure and statistical properties of this network can be studied over time with focus on overall patterns or just on the patterns generated (over time) for specific type of mobility drivers (social or work or school activities). A detailed analysis of the temporal statistical dynamics of this network can be found in 21

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3. Vector-borne diseases The 1911 defining contributions by Ross61 generated from his interests on the study of the dynamics of malaria, provide the most important modeling milestone in field of theoretical epidemiology and yet, the theoretical work on vector-born diseases lags behind the theory developed to address the challenges posed by directly-transmitted diseases. Vector-borne diseases involve additional challenges and levels of complexities. Vector dispersal, for example, is not as well understood as human mobility. Will understanding the mechanism of vector dispersal patterns help us understand and altered the observed patterns of human malaria cases? Vector dispersion introduces a higher degree of spatial diffusion connecting, for example, neighboring houses. However, in most cases its contribution to long distance disease spread is still unquantified. Merging social network dynamics (based on human mobility) with vector mobility has yet to be carried out in powerful theoretical way. It is believed that for highly mobile human populations long-distance vector dispersion has not had, arguably, a significant impact. However, this is not always the case. Myxomatosis, a highly lethal viral disease for European rabbits, transmitted by fleas and mosquitoes provides an interesting system to highlight the lack of universal patterns of spread. Outbreaks of myxomatosis in Great Britain, where the main vector is the flea, travelled with a speed of the order of 500 meters/month54 but in Australia, the speed of propagation was as high as 5 Km/day51 , a value that could only be explained with the assumption of wind aided long-distance mosquito flights58 . The rabbit-myxomatosis system exemplifies the importance of disentangling vector and host movements if we wish to gain a better understanding of disease transmission; a key element in the design of control strategies. Dengue and malaria are two of the most important vector-borne human diseases and both are transmitted by mosquitoes and so far, aggregate ODE models have been mostly used to study their dynamics53 . Although human mobility is recognized as a fundamental factor in vector-borne diseases spread 63 , most researchers have focused their investigations primarily on the role of vector dispersal. Tran and Raffy65 considered a deterministic but spatially explicit model in their study of dengue dynamics. In their model, only adult vectors were considered with mosquito dispersion modeled via a diffusion equation parameterized with environmental data obtained with remote-sensing images.

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Otero et al 57 developed a model for dengue with a detailed description of the mosquito population including temperature-dependent rates for all the mosquito stages. They considered a small population of 40000 persons distributed in a spatial grid of 20 × 20 blocks. Human population was modeled via an agent-based framework but human mobility was no included. The mosquito population was described at the population level. Otero et al 57 in their dengue model assume that spread is due only to between blocks mosquito dispersion with within block-homogenous-mixing between individuals and mosquitoes. Some individual based models have also been developed in the case of malaria as well but for small populations 36,33 . Gatton and Cheng used an agent-based model involving a population of 1000 persons. Their model was used to explore the potential impact of chemotherapy and the use of bed nets on malaria eradication albeit the mobility of hosts and/or vectors was not factored in.

3.1. An example: A Dengue outbreak in a non-endemic area The 2009 Dengue epidemic in San Ramón de la Nueva Orán, northwest Argentina, is an example of how human and vector patterns of mobility shape disease spread in different temporal and spatial scales. Dengue is a vector-borne disease transmitted mainly by Aedes aegypti female mosquitoes. Dengue re-emerged in Argentina in 1997 with the largest epidemic outbreak of this new era recorded in 2009. Dengue is non-endemic in Argentina but it is regularly re-introduced from Brazil, Paraguay, and Bolivia where it is endemic. San Ramón de la Nueva Orán is a city with population of about 100000 inhabitants in the tropical region of Salta, a province of Argentina. Salta is located in one of the main routes connecting Bolivia and northwest Argentina; shopping and leisure trips are common between individuals of both nations. This is the main source of dengue outbreaks in northwestern Argentina. The 2009 outbreak in Orán started in January, in the middle of the rain season, with at least one resident of the city importing the disease from Bolivia. In Fig. 1 the evolution of cases during the first weeks of the outbreak are displayed. The first cases appear to be clustered in the south of the city (plate a). Few days later a new case was reported about 2km away. It is extremely likely that this new focus was started as a result of human

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mobility patterns since Aedes aegypti is known for its short range dispersal. In general, the pattern of disease spread within the city seems to be determined primarily by human long distance movements and the local diffusion of infected vectors. The epidemic ended by June nearly simultaneously with the precipitous drop of the mosquito populations. The total reported cases were found to be distributed almost randomly within the city (see Fig.2) with one spatial cluster located around the first cases (José Gil et al, work in progress). From Figure 2, we see the role of the patterns of mobility in shaping the incidence patterns. At the scale of the city, local diffusion of vectors spread the disease among neighboring individuals. Human mobility, in turn, produces a new foci of infected vectors within the city in what seems to be almost a random way. No correlation was observed between case incidence and socioeconomic factors or larvae indexes. In this region, seasonality plays a major role as it drives vector population dynamics. At a regional scale, the reintroduction of dengue was caused by the regular traffic that exists between Salta and Bolivian cities. But land transportation also seems to have been responsible for mosquito reintroduction and dispersal in Argentina 60 .

4. Discussion and Conclusions Large scale agent-based simulation models have been developed and refined through their use in the study of the transmission dynamics and control of directly transmitted diseases. Most recently the emphasis has turned in to their use in the study of epidemic processes, including pandemics. Research, through the use of agents that literally represent each member of the unit in question, is being carried out. In fact, a model was developed involving around 300 million inhabitants. These models include detailed patterns of human mobility. The possibility of developing large dynamic network models that help identify the key drivers behind the observed patterns of disease spread, must be considered. Additional insights could be gained from the incorporation of higher levels of detail, including intra-host dynamics or history of co-infections, in the new generation of models that is about to emerge. From the world of international travel to the world of intra-host pathogen dynamics enormous differences can be identified in patterns of disease spread or recurrence. The differences trace back to their role/impact over key spatial or organizational or temporal scales. At a worldwide scale an agent-based approach seems impractical, most likely not that useful.

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There are however patch models, including those involving air connections and traffic flow, that will generate important insights and results from the use of agent-based models. Large-scale and high-resolution simulation models are resource demanding, with running times, for example, in the order of ten hours just for a single realization, spanning less than one simulated year. Large-scale, highresolution models are indeed valuable tools in the exploration of scenarios and mitigation strategies. Reliable, lower-resolution models are also critical since, for example, they assist us with real-time assessments that mitigate of a growing number of undergraduates. Disease dynamics’ determinants evolve over significantly different time scale. Climate change and the expansion of vector habitats or coevolutionary processes are, but some of the forces that shape and re-shape, the patterns of disease spread. The time evolution (dynamics) of slow diseases may, for example, have natural scales that change over long intervals in time. Outbreaks generated by fast diseases, on the other hand, tend to be over in months. Understanding of these processes enhances the likelihood of finding ways of linking mathematically processes living over distinct time-scales. The study of slow diseases provide continuous challenges and opportunities for those interested in identifying the drivers behind recurrent outbreaks, endemic situations, and disease evolution. Vector-borne diseases, a promising area for theoretical developments, finds itself lacking large-scale, individual-based models. The kind of models that would help our understanding of, for example, the role of vectors’ life history on the observed patterns of disease spread. The role of vector mobility patterns is in general quite difficult to elucidate. However, we are also aware that a large host population comes in with (in general) a large vector population. Under these circumstances, an agent-based approach would be resource demanding. The systematic use of individual-based models can provide tremendous insights on the mechanisms behind disease spread for directly transmitted and vector-borne diseases across temporal, spatial, and biological scales. Computer intensive vector-borne diseases models are being developed and explored in the context of smaller human populations, than for existing directly transmitted diseases because of the complications that come from the importance of evaluating the role of vectors. Finding ways to scale up theoretical results to meet the demands of larger populations evolving over long periods of simulated times will be extremely beneficial to us. In this manuscript we have traced some of the history behind modeling

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and theoretical advances. It should be obvious from the discussions that we now live in a “hybrid” world where the need to understand processes that have components operating over diverse temporal or spatial scales demands the use of multiple levels that include: theoretical model-constructs that can be analyzed and interpreted, larger models validated with data, and large-scale and high-resolution simulation models.There is not a silver bullet and therefore a multitude of approaches is now more than ever, the best strategy.

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Figure 1. Cumulated distribution of cases during the first weeks of the dengue epidemic in San Ram´ on de la Nueva Or´ an, Salta province, Argentina. a) January 3 to 7, b) Jan 8 c) week 2, d) Jan 18, e) week 4, f) week 8.

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Figure 2. Spatial distribution of cases at the end of the 2009 epidemic in San Ram´ on de la Nueva Or´ an.