Oct 28, 2008 - [14] Daniel A Salmon, Stephen P Teret, C Raina MacIntyre, David Salis- bury, Margaret A Burgess, and Neal A Halsey. Compulsory vaccination.
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
Dynamic Modeling of Vaccinating Behavior as a Function of Individual Beliefs Flávio Codeço Coelho and Cláudia Torres Codeço
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
Abstract Individual perception of vaccine safety is an important factor in determining a person's adherence to a vaccination program and its consequences for disease control. This perception, or belief, about the safety of a given vaccine, is not a static parameter but a variable subject to environmental in uence. To complicate matters, perception of risk (or safety) does not correspond to actual risk. In this paper we propose a way to model the dynamics of such beliefs in the context of a realistic epidemiological scenario. The methodology proposed is based on Bayesian inference, and can be extended to model more complex belief systems associated with decision models.
Introduction Since early vaccination campaigns against smallpox, vaccination policies have been a matter of debate
10
: mass vaccination versus blocking strategies; compul-
sory versus voluntary, are some highly debated issues. Despite these early controversies - and consequent alternative policies implemented in dierent countries - high disease scare in the past has led to very high vaccine coverage and consequent successful eradication of smallpox, as well as very low incidence of measles, polio, tetanus, diphtheria, etc, resulting in over 98% mortality reduc7
tion by vaccine preventable diseases in developed countries . In recent years, after complete or almost complete elimination of these diseases, the debate is shifting towards issues of vaccine safety. Increased perception of vaccine risks and lowered perception of disease risks has challenged previous willingness to vaccinate (fundamental for the success of any immunization
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
program, either voluntary or compulsory)
14
.
In this scenario, understanding
and predicting individual's willingness to vaccinate is paramount for estimating vaccine coverage and compare strategies to achieve coverage goals. Willingness to vaccinate is highly dependent on the perceived risk of acquiring a serious disease
15
. When (perceived) disease risk is low, however small risk
of adverse events from the vaccine become relatively important and may lead to vaccine coverage lower than required to control transmission
15
. When (per-
ceived) serious disease risk is too high, on the other hand, vaccine coverage may 3
increase above that required to guarantee population protection . We illustrate these behaviors with two examples:
The MMR vaccine scare
In the UK, MMR vaccine uptake started to de9
cline after a controversial study linking MMR vaccine to autism . In a decade, vaccine coverage went well below the target herd immunity level of 95%. Despite the con dence of researchers and most health professionals on the vaccine safety, the con dence of the public was deeply aected. In an attempt to nd ways to restore this con dence, several studies were carried out to identify factors associated with parent's unwillingness to vaccinate their children.
They
found that 'Not receiving unbiased and adequate information from health professionals about vaccine safety' and 'media's adverse publicity' were the most common reasons in uencing uptake
12
. Other important factors were: 'lack of
belief in information from the government sources'; 'fear of general practitioners promoting the vaccine for personal reasons'; and 'media scare'. Note that during this period the risk of acquiring measles was very low due to previously high vaccination coverage.
The Brazilian Yellow Fever disease scare
Sylvatic yellow fever (SYF) is
a zoonotic disease, endemic in the north and central regions of Brazil. Approximately 10% of infections with this avivirus are severe and result in hemorrhagic 6
fever, with case fatality of 50% . Since the re-introduction of A. aegypti in Brazil (the urban vector of dengue and yellow fever), the potential reemergence of ur5
ban yellow fever is of concern . In Brazil, it is estimated that approximately 95% of the population living in the yellow fever endemic regions have been vaccinated. In this area, small outbreaks occur periodically, especially during the rainy season, and larger ones are observed every 7 to 10 years
1
11
, in response to increased
1
New confirmed cases Number of notified serious adverse events Vaccine coverage in the epicenter
25
0,8
20 0,6 15 0,4 10
0,2
5
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
0
0 0
Figure 1: events.
vaccine coverage in DF
number of new confirmed cases
30
10
20
30 40 50 days after 15 dec 2007
60
70
80
Yellow fever cases, vaccination coverage, and adverse vaccination
Coverage data is from Brasilia (DF), which was the epicenter of the
outbreak.
viral activity within the environmental reservoir. In 2007, increased detection of dead monkeys in the endemic zone, led the government to implement vaccine campaigns targeting travellers to these areas and the small fraction of the resident population who were still not protected by the vaccine. The goal was to vaccinate 10 - 15% of the local population. Intense noti cation in the press regarding the death of monkeys near urban areas, and intense coverage of all subsequent suspected and con rmed human cases and death events led to an almost 3
country-wide disease scare, incompatible with the real risks , which caused serious economic and health management problems, including waste of doses with already immunized people (60% of the population was vaccinated when only 10-15% would be su cient), adverse events from over vaccination (individuals taking multiple doses to 'guarantee' protection), national vaccine shortage and international vaccine shortage, since Brazil stopped exporting YF vaccine to supply domestic vaccination rush (www.who.int/csr/don/2008_02_07/en/).
The importance of public perceptions and collective behavior for the outcome of immunization campaigns are starting to be acknowledged by theoreticians
2,13,5
. These factors have been examined in a game theoretical framework,
where the in uence of certain types of vaccinating behaviour on the stability and equilibria of epidemic models is analyzed. In the present work, we propose a model for individual immunization behavior as an inference problem: Instead of working with xed behaviors, we develop a dynamic model of belief update, which in turn determines individual behavior. An individual's willingness to vaccinate is derived from his perception of disease risk and vaccine safety, which is updated in a Bayesian framework, according the epidemiological facts each individual is exposed to, in their daily life.
We also explore the global eects of individual decisions on vaccination
2
adherence at the population level. In summary, we propose a framework to integrate dynamic modeling of learning (belief updating) with decision and population dynamics.
Methods We set the vaccination decision problem in the context of a population experiencing a vaccine preventable disease outbreak which leads to a mass vaccination campaign. Individuals receive information regarding vaccine and disease events from local and global sources. We assume that 'good' events (prompt recovery from infection or safe vaccine events) are visible locally only while severe cases of disease or potentially adverse events from the vaccine enjoy global visibility due to the natural preference of media channels for scary stories. In order to
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
integrate behavioral and epidemiological dynamics, an individual based model was developed. Individual's behavior regarding vaccination is represented in a belief-decision model which describes the dynamics of belief updates in response to epidemiological events and the decision making based on the person's current beliefs. The epidemiological model determines the disease dynamics in a population with hierarchical contact structure, representing a large urban setting.
Belief Model The belief model describes the temporal evolution of each individual's willingness to vaccinate,
vt ,
in response to his evaluation of vaccine safety and disease
risk. To account for the uncertainties regarding vaccinating behavior,
vt
is mod-
eled as a random variable, whose distribution is updated weekly as the individual observes new events. The update process is based on logarithmically pooling
vt
with other random variables as described below. The belief update model takes the form:
v α1 sα2 dα3 rα4 vt+1 = R t α1 t α2 t α3 t α4 vt st dt rt where
α1 = α2 = 1/4
with remaining
αi
(1)
taking values according to the
following conditions:
if nsd > 0 : α3 = 1/2, α4 = 0 otherwise : α3 = 0, α4 = 1/2
nsd is the number of serious disease cases witnessed by the individst and dt are random variables describing individual's belief regarding
where ual, and
vaccine safety and disease risk, respectively. Every individual starts o with a very low expected value for
vt : v0 ∼ Be(1, 20), E[v0 ] ≈ 0.047.
Since we assume that under no perceived disease risk, willingness to vaccinate should decrease, the last term in (1), in
E[vt ]
up to the minimum value of
rt is a drift E[v0 ].
force which causes a reduction 8
Equation (1) combines the dierent beliefs through a logarithmic pooling , which is a standard way of combining probability distributions with the purpose of obtaining a consensus of multiple beliefs.
In this operation, the
αi
act as
weights, modifying the importance of each distribution, and must add up to 8
one .
3
Perceived vaccine safety (st ):
Regularly, during a mass vaccination cam-
paign, individuals will try to infer the value of vaccinating based on available information regarding vaccine events. During a campaign, the number of safe vaccine events,
V can naturally be modeled as a binomial distributed variable, η and π standing for the number of doses given and the proba-
with parameters
bility of safe event, respectively
16
. However, since data available to individuals
is biased and incomplete, the observed variable inferential process is a Binomial governed by
n
v
that feeds each individual's
and
s,
the perceived number of
doses and perceived probability of a safe vaccine event, respectively:
V ∼ Bin(η, πv ) v ∼ Bin(n, s)
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
Note that
v
is not the true number of safe vaccine doses applied in the
population, but represents a subset of these events which the individual is aware of. This means that the perceived safety of the vaccine will always be a biased estimate, and will vary from individual to individual generating variation in the population belief distribution. Each individual will make inference of
s
based on
n
and
v,
as a iterative Bayesian inference. Let the prior distribution for
This is modelled
st
be a
Be(α, β),
which is the natural conjugate for a binomial process. The posterior distribution
st+1
is then given by:
st+1 (n, v) ∼ Be(α + v, β + n − v) The posterior
st+1
(2)
is used in the following iteration cycle as the prior.
To better emulate the biased availability of good versus bad news in real populations, we assume that vaccine adverse events are visible globally, while safe events are visible only within their neighborhoods. To include the eects of an exaggerated media coverage of vaccine adverse events, we considered scenarios where the the observed number of adverse events constant
a
(n − v)
is ampli ed by a
in equation (2). We call this factor the media-ampli cation factor,
varying from 1 (no ampli cation) to 16.
Perceived disease risk (dt ):
In this model, we try to emulate a scary disease,
that is, a disease severe enough that a few cases will lead to a high willingness to get a vaccine shot. Disease scare is de ned as an increase in the individual's
vt , upon witnessing
disease cases with serious consequences. It must be noted, however, that this probability refers to the decision of getting a vaccine, since eectively getting vaccinated will also depend on the availability of the vaccine. This eect enters
vt
update equation (1) as the variable
dt ∼ Be(α, β),
where:
E[dt ](1 − E[dt ]) −1 α = E[dt ] σ2 E[dt ](1 − E[dt ]) −1 β = (1 − E[dt ]) σ2
E[dt ] = E[vt ] + (1 − E[vt ])tanh(0.3nsd) 4
(3)
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
More adverse effects
More safe vaccinations
Figure 2: Updating the perceived probability of a safe vaccination. The expected value of
st
will change towards one with more safe vaccines witnessed and in
the opposite direction with the accumulation of VAE events.
where how
nsd
E[dt ]
is the number of serious disease cases witnessed. Figure 3 shows
varies with the number of serious disease cases witnessed.
disease cases are visible globally. Here, pooling between
vt
and
dt
dt
has a xed variance
is done as in (1). In equation (1)
α3
Serious
σ 2 = 0.005.
The
can be modi ed
to make the disease more or less scary to individuals.
Drift (rt ):
The drift is a slow but continuous reduction of the mean willingness
to vaccinate,
vt
towards its initial distribution
vt=0 .
This will happen only in
the absence of perceived serious disease cases. The drift enter as the variable
rt
E[rt ] = and parameters derived from
Decision:
vt
update cycle
in (1) and has a Beta distribution with mean given by:
5E[vt ] + E[v0 ] 6
E[rt ]
in the same way as in (3).
Once a week, during the simulation, susceptible and exposed in-
dividuals decide whether to go vaccinate with a probability sampled from
vt ,
updated according to equations (1) and (2). This update is based on evidence collected during the past seven days ( gure 4). Only non-infectious individuals make the decision to whether or not they should go vaccinate. We consider that exposed individuals do not know they have been infected, so they also may seek vaccination. This is important because there is a limited amount of vaccine doses available per week and exposed individuals will compete with susceptibles for them. Only susceptibles are successfully immunized by the vaccine.
5
1
0,9 scare-induced new E[v]
E[v]
0,8
0,7
0,6 Previous E[v]
0,5
0,4
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
0,3 0
2
4
6
Serious disease cases
8
10
Figure 3: Mean scare as a function of Serious disease cases witnessed.
Figure 4: Daily and weekly events that determine individual activity. Movement decisions take place on a daily basis.
6
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
Figure 5: Design of the hierarchical population model. Arrows show the possible patterns of daily movement of individuals.
Population model We model disease spread in a hypothetical city represented by a multilevel metapopulation individual-based model where individuals belong to groups that in turn belong to groups of groups, and so on, forming a hierarchy of scales
17
.
In this hypothetical city, individuals live in households with exactly 4 members each; neighborhoods are composed by 100 households and sets of 10 neighborhoods form the city's zones. During the simulation, individuals commute between home and a randomly chosen neighborhood anywhere in the population graph. Each individual has a probability
0.25
of leaving home daily.
This same hierarchical structure is used to de ne local and global events. Locally visible events can only be witnessed by people living in the same neighborhood while globally visible events are visible to the entire population regardless of place of residence.
Epidemiological Model The epidemiological model describes a population being invaded by a new pathogen.
This pathogen causes an acute infection, lasting 11 days (incuba-
tion period of 6 days and an infectious period of 5 days). Once in the infectious period, individuals have a xed probability,
psd
of becoming seriously ill. After
recovery, individuals become fully immune. The proportion of the population in each immunological state at time
t
is labeled as
S(t), E(t), I(t)
and
R(t),
which
stands for susceptibles, exposed, infectious and recovered states. At the same time the disease is introduced in the population, a vaccination campaign is started, making available
nd doses per week to the entire population,
meaning that individuals may have to compete for a dose if many decide to vaccinate at the same time. Once an individual is vaccinated, if he/she has not been exposed yet, he/she
7
Table 1: Parameter values used for the simulations. Symbol
Meaning
Value
psd (1 − π) βnh βhh c a
prob. of serious illness
[10−6 , 10−2 ] [10−6 , 10−2 ] 0.2 0.3 4 [1, 16]
prob. of vaccine adverse eects prob. of transmission/contact prob. of transmission/contact at home number of contacts per day media ampli cation factor
moves directly to the recovered class, with full immunity (thus, a perfect vaccine is assumed). If the individual is in the incubation period of the disease, disease
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
progression is unaected by vaccination. chance
(pae = 1 − π)
Vaccination carries with it a xed
of causing adverse eects.
Transmission dynamics is modelled as follows: at each discrete time step,
1, 2, 3, ..., 70,
t=
each individual contacts others in two groups: in his residence and
in the public space. The probability of getting infected at home is given by p0 = 1−(1−βhh )ihh where βhh is the probability of transmission per household contact and ihh is the number of infected members in the house. In the public space, that is, in the neighborhood chosen as destination for the daily commutations, each infected person contacts
c
persons at random, and if the contact is with a
susceptible, infection is transmitted with probability
βnh .
Results We ran the model as described above for 100 days, under various scenarios to reveal the interplay of belief and action under the proposed model. Figures 6 and 7 show a summary output of the model dynamics under contrasting conditions. In gure 6, we have VAE (Vaccine adverse events) preceding the occurrence of severe disease events. As expected, VAE become the strongest in uence on
vt ,
keeping
E[vt ]
low with consequences to the attained vaccination coverage at
the end of the simulation.
We characterize this behavior as a 'vaccine scare'
behavior. In a dierent scenario, gure 7, we observe the eect of severe disease events occurring in high frequency at the beginning of the epidemics.
In this case,
disease scare pushes willingness to vaccinate (vt ) to high levels. The impact of individual beliefs on vaccine coverage is highly dependent on the visibility of the rare VAE. Figure 8 shows the impact of the media ampli cation factor on infectious disease with
E[vt ] and psd = 0.1
vaccination coverage after and
pae = 0.075.
≈ 14
weeks, for a
If no media ampli cation
occurs, willingness to vaccinate and vaccine coverage are high, as severe disease events are common and severe adverse events are relatively rare.
As vaccine
adverse events are ampli ed by the media, individual's willingness to vaccinate at the end of the 14 weeks tend to decrease. has a low impact on the vaccine coverage.
Such belief change, however,
The explanation for this is that
vaccine coverage is a cumulative measure and, when VAE appear, a relatively large fraction of the population had already been vaccinated.
8
These results
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
Figure 6:
Impact of occurrence of adverse vaccination events at the begin-
ning of a vaccination campaign.
Top-left:
vaccination coverage and vaccine
uptake (×8000 doses);top-right: adverse vaccination and serious disease events; Bottom-left:
willingness to vaccinate (vt ) and perceived vaccine safety (st );
Bottom-right: Epidemiological time-series.
suggest that VAE should not strongly impact the outcome of an ongoing mass vaccination campaign, although it could aect the success of future campaigns. Fixing ampli cation at
a = 8
psd = 0.1,
and
we investigated how
vt
(at
the end of the simulation) and vaccine coverage would be aected by increasing the rate of vaccine adverse events,
psd ,
pae
( gure 9).
As
pae
increases above
willingness to vaccinate drops fastly, while vaccine coverage diminishes but
slightly.
Discussion In the present world of mass media channels and rapid and inexpensive communications, the spread of information, independent of its quality, is very eective, leading to considerable uncertainty in public opinions.
Vaccinating behavior
dynamics has been modelled in dierent ways in the recent literature, from behaviors that aim to maximize self-interest
13
1
to imitation behaviors . Perceived
side-eects and eectiveness of vaccination are important factors in people's de4
cision to vaccinate . In this paper we modeled these perceptions dynamically, and showed its relevance to decision-making dynamics and the consequences to the underlying epidemiological system and e cacy of vaccination campaigns. The process through which people update beliefs which will direct their de-
9
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
Figure 7:
Impact of occurrence of adverse vaccination events at the begin-
ning of a vaccination campaign.
Top-left:
vaccination coverage and vaccine
uptake (×8000 doses);top-right: adverse vaccination and serious disease events; Bottom-left:
willingness to vaccinate (vt ) and perceived vaccine safety (st );
Bottom-right: Epidemiological time-series.
10
0,8 No amplification
0,7
vt Coverage
0,6
0,5
0,4
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
0,3
0,2
0,1 0
2
4
6 8 10 12 Media Amplification Factor (a)
14
16
Figure 8: Eect of media ampli cation factor on vaccination coverage and willingness to vaccinate. Coverage and
vt
values are averages and standard devia-
tions over the population values in the last week of simulation.
0,8
vt Coverage
0,7
0,6
0,5
0,4
0,3
0,2 0
0,02
0,04 0,06 Probability of vaccine adverse effects
0,08
0,1
Figure 9: Vaccine coverage and willingness to vaccinate for vaccines with dierent levels of safety. Coverage and
vt
values are averages and standard deviations
over the population values in the last week of simulation.
11
cisions, was modeled using a Bayesian framework. We trust this approach to be the most natural one as the Bayesian de nition of probability is based on the concept of belief and Bayesian inference methodology was developed as a 8
representation human learning behavior . To further improve the adherence of our model to realistic human learning and decision making behavior, we have modeled the probability of getting vaccinated (vt ) as a separate random variable whose dynamics was determined by Logarithmic pooling with
st
and other
probability distributions representing disease scare (dt ) and reduced willingness to vaccinated in the absence of risk(rt ). The usage of logarithmic pooling to combine probability distributions representing the key variables in the model, allowed for independent modeling of decision making, based on
vt
without decisions interfering with Bayesian in-
ferential process governing the dynamics of individual perception of vaccine safety(st ).
Nature Precedings : hdl:10101/npre.2008.2447.1 : Posted 28 Oct 2008
The framework presented here comprises a set of useful tools for an adequate quantitative representation of a common yet complex public-health issue. These tools include representation of beliefs as Bayesian probabilities, usage of logarithmic pooling to combine probability distributions representing opinions, and usage natural conjugate priors to e ciently compute the Bayesian posterior. This approach allowed a comprehensive treatment of the uncertainty regarding vaccination behavior in a realistic epidemiological model. The tools proposed in this work can be easily used as building-blocks to compose more complex models. Extended models might include multiple beliefs as a joint probability distribution, more layers of decision or multiple, independently evolving belief systems. This study has also provided valuable lessons about the relevance of the availability of unbiased information to the general public through channels which are both o cial and trustworthy. The yellow fever scare in Brazil demonstrated clearly that no o cial press release is taken at face value, it is always colored by 3
political issues . In multiple occasions, people reported to the press that they would do the exact opposite of what was being recommended by public health authorities due to their mistrust of such authorities. Another conclusion of this model, is that if the perception of the disease risks are well established in the population, leading to a higher initial willingness to vaccinate, adverse events of vaccination, even when widely publicized by the media, tend to have less impact on vaccination coverage. VAE are more eective when happening at the beginning of vaccination campaigns, when they can sway the opinions of a larger audience. Although disease scare can counteract, to a certain extent the undesired eects of VAE, public health o cials must also be aware of the risks involved in overusing disease risk information, in vaccination campaign advertisements since this can lead to a rush towards immunization as seen in the 2008 Yellow Fever scare in Brazil.
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