The first birth cohort for which the vaccine becomes available (
) typically faces a situation where the disease is highly endemic. Hence, there is a greater probability of being infected at a young age, where vulnerability to complications of infection is highest (Text S1
). Therefore, unless vaccine risk is unacceptably high, we expect the Nash equilibrium strategy to be full vaccination. The second cohort (
) may also opt for full vaccination, although herd immunity provided by the first birth cohort has lessened their risk of infection somewhat. Similar reasoning applies for
. Eventually there is a birth cohort
for which a pure vaccinator strategy
is no longer optimal, due to herd immunity generated by previous vaccinating cohorts. Cohort
may therefore free-ride on the herd immunity provided by previous vaccinating cohorts. However, this strategy may not be optimal if subsequent cohorts
also exempt themselves from vaccination, and/or if any outbreak that results from the non-vaccinating behavior of cohort
occurs when individuals in cohort
are at an age where infection penalties are still high. Subsequent cohorts
have the disadvantage that, if they do attempt to free-ride by not vaccinating, they will be less protected since the next epidemic is more likely to occur when they are at an age at which disease risk is higher, and there is already less herd immunity for them to rely on due to the non-vaccinating behavior of cohort
. Therefore, subsequent cohorts are more likely to return to a strategy of vaccinating at higher coverage levels, in order to prevent an outbreak that may affect them disproportionately.
This strategic interaction through time between individuals born in different cohorts can result in a rich variety of behavior (). However, in every scenario, free-riding behavior develops surprisingly quickly, within the first 2–5 years of the program. In all eight scenarios, Nash equilibrium vaccine coverage is
in the first birth cohort. In the scenarios of high disease risk, vaccine coverage also remains close to
in the first 4–5 birth cohorts; in subsequent cohorts it drops but remains relatively high, with minor adjustments in coverage between cohorts in response to intermittent periods of epidemic activity; disease prevalence is low but non-zero due to free-riding. By comparison, when disease risk is small, vaccine coverage is highly variable from one cohort to the next, resulting in irregular patterns and more frequent outbreaks (). In some cases, birth cohorts free-ride by adopting pure non-vaccinator strategies, followed by birth cohorts that adopt pure or intermediate vaccinator strategies. The average vaccine coverage across birth cohorts is also lower for the scenarios of lower disease risk.
Figure 1 Nash equilibrium vaccine coverage by cohort, and incidence per 100,000 by year, for under the 8 scenarios (see main text).
The model dynamics across the eight scenarios can be visualized by surface plots of the Nash equilibrium coverage
as a function of birth cohort
and vaccine risk
(). The destabilizing effect of lower disease risk observed in is also apparent in the surface plots of . For very low levels of vaccine risk, vaccine coverage is high and relatively stable across cohorts, whereas increasing levels of vaccine risk quickly destabilize coverage across cohorts and reduce average coverage levels. The depth of non-vaccinating troughs increases as vaccine risk
increases, although not necessary their width (number of adjacent non-vaccinating cohorts). The first cohort to reduce its vaccine coverage, which dictates the frequency of non-vaccinating behavior in subsequent cohorts, varies with vaccine risk. The resulting alternation of low and high coverage levels resemble natural patterns such as sand bars or zebra stripes, which are also attributable to nonlinear feedbacks () 
. The range of vaccine risks analyzed here was chosen to capture the spectrum of possible dynamical behaviors, rather than being based on actual or perceived vaccine risks from real populations.
Nash equilibrium coverage levels versus cohort and vaccine risk under the 8 scenarios (see main text).
When disease risk is high, the effect of reduced vaccine efficacy is to increase vaccine coverage (). This occurs because a sufficiently imperfect vaccine cannot interrupt disease transmission, even at
coverage, and thus the persistent risk of infection continues to encourage individuals to vaccinate. However, low vaccine efficacy also makes the vaccine less attractive since it is less likely to protect against infection. This competing effect appears to dominate in the case of low disease risk, where low vaccine efficacy reduces overall coverage without significantly changing the pattern of wide variability in vaccine coverage across birth cohorts.
An aspect of behavior-prevalence dynamics that has previously been neglected is the role of case importation. This is the only source of disease burden once local chains of transmission have been interrupted through herd immunity, therefore, inclusion of case importation should potentially modify predicted vaccinating behavior once herd immunity has been built through immunization programs. In the scenarios of high disease risk in the present model, including case importation stabilizes patterns of vaccine coverage, creating greater consistency in predicted behavior for various values of
. Case importation also increases the frequency of outbreaks moderately.
An important question in the design of phase III vaccine trials and phase IV safety studies is: how small a vaccine risk must be ruled out before the vaccine is considered acceptable for licensing? From the standpoint of an individual's decisions, this question becomes: how small does the vaccine risk have to be for a rational person to decide it is in their interest to vaccinate? The answer to this will depend on the birth cohort, since different birth cohorts face different epidemiological landscapes that have been shaped by the vaccination decisions of older birth cohorts.
To address this question, we define
as the smallest vaccine risk for which the Nash equilibrium vaccine coverage in cohort
will be at least
is conceived of as a desirable target coverage by a public health decision maker. The predicted dependence of
on model parameters () mirrors the dependence of Nash equilibrium coverage
on model parameters ( and ). The first few birth cohorts are generally willing to tolerate a relatively high vaccine risk
. However, subsequent birth cohorts differ considerably in how much vaccine risk they will tolerate, with cohorts that are able to free-ride on herd immunity tolerating lower vaccine risk. In the scenario of high disease risk and low vaccine efficacy with
target coverage, the first seven birth cohorts tolerate vaccine risk
. At the other extreme lies the scenario of low disease risk and low vaccine efficacy, where only the first birth cohort is willing to tolerate vaccine risk
Figure 3 Largest tolerated risk versus cohort k under the 8 scenarios (see main text).
Lowering disease risk lowers the average tolerance for vaccine risk across birth cohorts, which is not surprising (). However, the way that tolerance is distributed across birth cohorts can change in surprising ways as disease risk is lowered. For instance, when vaccine efficacy is high, lower average tolerance takes the form of more cohorts with intermediate tolerance, rather than a few cohorts with zero tolerance. In the scenario of low efficacy vaccine, the decrease in tolerance that occurs as disease risk becomes lower is manifested by decreased vaccine tolerance in all birth cohorts (whereas when disease risk is high, reduced tolerance for vaccine risk is concentrated in early birth cohorts).
Increasing vaccine efficacy also has some surprising and paradoxical effects. In the scenarios of high disease risk, increasing vaccine efficacy decreases tolerance for vaccine risk in most birth cohorts. This occurs because use of a higher efficacy vaccine makes it possible to achieve very low incidence through herd immunity, and the resulting low force of infection reduces the tolerance for vaccine risks. By comparison, in the scenarios of lower disease risk, increasing vaccine efficacy results in widely variable vaccine risk tolerance across cohorts.
There are few qualitative differences in
when disease risk is low. However, when disease risk is high, the largest tolerable vaccine risk is much smaller (sometimes close to zero) for
. An exception occurs in the case of low vaccine efficacy and high disease risk, where the tolerated vaccine risk remains high only in the first two cohorts when
, but the tolerated vaccine risk remains high in the first seven cohorts when
The social optimum was defined as the vaccine coverage
such that the total burden in all cohorts from either vaccine or infection is minimized, and was was assumed to be the same in all cohorts. When vaccine risk is zero,
). As vaccine risk increases, the
, since additional vaccination beyond the herd immunity threshold does not prevent significant additional infection but does impose some burden due to vaccine risks.
declines with increasing vaccine risk, but not always monotonically. In some cases, it is constant and then declines abruptly with increasing vaccine risk. This occurs because of assumptions about decreasing disease risk with increasing age.