^{*}

Conceived and designed the experiments: CTB. Performed the experiments: AP. Analyzed the data: AP. Wrote the paper: AP CTB.

The authors have declared that no competing interests exist.

Certain theories suggest that it should be difficult or impossible to eradicate a vaccine-preventable disease under voluntary vaccination: Herd immunity implies that the individual incentive to vaccinate disappears at high coverage levels. Historically, there have been examples of declining coverage for vaccines, such as MMR vaccine and whole-cell pertussis vaccine, that are consistent with this theory. On the other hand, smallpox was globally eradicated by 1980 despite voluntary vaccination policies in many jurisdictions. Previous modeling studies of the interplay between disease dynamics and individual vaccinating behavior have assumed that infection is transmitted in a homogeneously mixing population. By comparison, here we simulate transmission of a vaccine-preventable SEIR infection through a random, static contact network. Individuals choose whether to vaccinate based on infection risks from neighbors, and based on vaccine risks. When neighborhood size is small, rational vaccinating behavior results in rapid containment of the infection through voluntary ring vaccination. As neighborhood size increases (while the average force of infection is held constant), a threshold is reached beyond which the infection can break through partially vaccinated rings, percolating through the whole population and resulting in considerable epidemic final sizes and a large number vaccinated. The former outcome represents convergence between individually and socially optimal outcomes, whereas the latter represents their divergence, as observed in most models of individual vaccinating behavior that assume homogeneous mixing. Similar effects are observed in an extended model using smallpox-specific natural history and transmissibility assumptions. This work illustrates the significant qualitative differences between behavior–infection dynamics in discrete contact-structured populations versus continuous unstructured populations. This work also shows how disease eradicability in populations where voluntary vaccination is the primary control mechanism may depend partly on whether the disease is transmissible only to a few close social contacts or to a larger subset of the population.

Interest in infectious disease models that incorporate the effects of human behavior has been growing in recent years. However, most of these models predict that it should never be possible to eradicate a disease under voluntary vaccination, due to nonvaccinating “free riders” that emerge when vaccine coverage is high. This prediction contradicts the fact that smallpox was eradicated under a voluntary vaccination policy in many jurisdictions, and that other diseases such as polio are likewise near eradication. These previous models assumed that populations mix homogeneously. However, for some diseases, such as HIV and smallpox, individuals are more likely to get the disease from certain social contacts. Here we show that using a network model that captures this social structure can reconcile the previous theories to the empirical fact that diseases can be eradicated under voluntary vaccination. When infection is transmitted only through close contacts in the network, then an outbreak can be quickly contained using only voluntary vaccination. However, when infection can potentially be transmitted to almost everyone in the network (such as for measles), a disease outbreak can never be contained using voluntary vaccination. This latter observation may have some relevance to the Measles–Mumps–Rubella autism “vaccine scare.”

Model-based analyses of vaccination programmes have often concluded that it should be difficult or impossible to eradicate a vaccine-preventable disease under a voluntary vaccination policy without other incentives

To date, smallpox is the only vaccine-preventable disease ever to have been globally eradicated

Most, if not all, previous mathematical models that analyze discrepancies between individually and socially optimal vaccination strategies under voluntary vaccination have considered populations without spatial or social contact structure (e.g. social networks), and where populations are large enough for the continuum approximation to apply. This also appears to be true of behavior-infection models more generally

Here, we show that disease dynamics under a voluntary vaccination policy are substantially and qualitatively altered by the introduction of individual-level social contact structure. We analyze a social contact network model, where each node represents an individual, and each link represents a close contact through which infection may spread. Individuals decide whether or not to vaccinate based upon their expected payoffs for vaccinating versus not vaccinating. We assume the vaccine is free to individuals, which describes the situation for many major pediatric vaccines in advanced countries, as well as the situation under the WHO smallpox eradication program in the 1970s. We first study epidemics on this contact network for a general vaccine-preventable infection with simplified SEIR-type (Susceptible-Exposed-Infectious-Recovered) disease history. At baseline parameter values, for small neighborhood sizes, outbreaks are quickly contained using only voluntary ring vaccination. As the neighborhood size increases while infection risk (force of infection) is held constant, a threshold neighborhood size is reached. Above this threshold, voluntary vaccination fails and the population experiences both a considerable final epidemic size and a large number vaccinated. Hence, the limit of large neighborhood size in this model recovers dynamics similar to those of homogeneous mixing models. Because the force of infection is held constant as neighborhood size increases, the failure of voluntary vaccination is attributable solely to a decrease in how localized disease transmission is on the network. We associate smaller neighborhood sizes with close contact infections such as smallpox, and larger neighborhood sizes with diseases that do not require close contact for transmission, such as measles. We carry out a similar investigation for smallpox-specific disease history and vaccine properties, with similar results. This analysis illustrates the importance of considering discrete, contact-structured populations for modeling vaccinating behavior for close contact infections, and provides a framework for reconciling previous theoretical predictions concerning the ineradicability of infectious diseases under voluntary vaccination to the empirical fact of the global eradication of smallpox and local elimination of many other infectious diseases through voluntary vaccination.

Here we describe the social contact network, infection transmission, and assumptions regarding individual vaccinating behavior. Definitions of parameters and corresponding parameter values appear in

Parameter | Meaning | Value | Reference |

Population size | 5,000 | ||

Initial number of individuals inoculated with smallpox | 10 | ||

Mean neighborhood size | 10 | ||

Probability of node-to-node transmission | 0.02 day^{−1} |
||

Perceived probability of node-to-node transmission | 0.02 day^{−1} |
||

Mean duration of latent period | 12 days | ||

Variance of latent period | 4 days^{2} |
||

Mean duration of infectious period | 19 days | ||

Variance of infectious period | 4 days^{2} |
||

Probability of death due to infection | 0.3 | ||

Probability of death due to vaccine-related complications | 0.001 | ||

Vaccine efficacy | 0.95 | ||

Payoff for individuals with continued susceptibility | 40 life-years | ||

Payoff for individuals with lifelong immunity | 40 life-years |

On any given day, each individual who has not already vaccinated decides whether or not to vaccinate depending upon their perceived risk of infection from neighbors versus the perceived risks of vaccination. We assume that infectiousness coincides with the onset of symptoms, and that individuals base vaccination decisions upon the presence of symptoms in a neighbor. If an individual has

For the vaccine-preventable SEIR infectious disease, we assume that a newly-infected person remains in the latent stage for a duration of time drawn from a gamma probability density function (PDF) with a mean of ^{2}. After this time, the individual becomes infectious and remains so for a duration of time drawn from a gamma PDF with a mean of ^{2}. Subsequently, the individual either dies with perceived probability

We first consider the payoff

The parameter

Over time horizons longer than the current epidemic, other factors may contribute to what assumptions must be made about

Due to uncertainties in exactly how much

We next compute the payoff

By comparison, if the vaccine is not efficacious, then either the individual dies from vaccine complications, accruing zero additional life years (probability

On any given day, if

To tailor the model to the case of smallpox, we incorporate a more realistic description of the natural history of smallpox and the effects of vaccination. We incorporate the fact that conventional smallpox vaccine can work in individuals who are already infected: those who receive the vaccine within 3 days of infection usually do not become infectious and subsequently recover with long-term immunity, those who receive the vaccine 4–7 days post-exposure experience less severe symptoms but will be as infectious as an unvaccinated person, and those who receive the vaccine more than 7 days post-exposure gain no benefit

To compute payoffs for the case of smallpox, we accounted for both fatal and non-fatal outcomes, such as pockmarks or blindness due to vaccine or disease. In order to compare fatal and non-fatal health outcomes in the same payoff function, the payoffs were expressed in terms of quality-adjusted life years (QALYs), i.e., the number of life years accrued, multiplied by a utility score between 0 (death) and 1 (perfect health) to reflect quality of life during those years

It is possible to show from Equations 3 and 6 that an individual with at least one infectious neighbor will vaccinate (i.e.,

The derivation of this expression appears in

Low values of

In the present model, as the average neighborhood size increases and the node-to-node transmission rate declines correspondingly, we should therefore expect to find conditions where Equation 10 is violated and voluntary vaccination fails to contain the outbreak, as predicted by models that assume homogeneous mixing. To test for this, we simulated the spread of the SEIR-type and smallpox infections on the network under a wide range of possible values for the average neighborhood size ^{−1} for each value of ^{−1} = 0.2 day^{−1} for the baseline parameters). Hence, a doubling of the number of social contacts would reduce the node-to-node transmission rate by half, which should maintain approximately the same infection risk per susceptible node as in the baseline scenario. However, higher-order effects relating to network structure are not accounted for in this approach and could change the infection risk in subtle ways. Therefore, we also controlled for infection risk using Method II, which involved calibrating the value of

For the SEIR-type infection, the simulation results under Method I indicate that voluntary vaccination contains the outbreak until the average neighborhood size

For smallpox infection, voluntary vaccination contains the outbreak for all

We also carried out a univariate sensitivity analysis with respect to the payoff

It is now a well-established principle in theoretical population biology that the dynamics of spatially structured populations can differ significantly from the dynamics of unstructured populations

We expect that our results for SEIR infections should generally apply to vaccine-preventable infectious diseases where the SEIR disease history is a good approximation to the actual disease history, and where appearance of symptoms approximately coincides with the start of infectiousness. Examples of such diseases include smallpox (for the small neighbourhood limit), and measles, rubella, chickenpox, perhaps pertussis (for the large neighbourhood limit).

The basic reproductive number,

Some previous game theoretical models have suggested that eradication is less likely to occur under voluntary vaccination when

It has previously been observed that spatial structure can also encourage the persistence of cooperative behavior in classical games such as the Prisoner's Dilemma

There were several limitations to the present analysis. First, we assumed that

Subject to the above limitations, there are

Additional details on mathematical derivations and smallpox parameter values.

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Dependence of final epidemic size and final number vaccinated on vaccine efficacy epsilon (A), payoff alpha for continued susceptibility (B), and node-to-node transmission probability beta per day (C) for smallpox infection.

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Representative realizations from the simulation: time series of number infected and number vaccinated for the SEIR-type infection.

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Representative realizations from the simulation: time series of number infected and number vaccinated for smallpox infection.

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CTB and AP are grateful for comments from three anonymous reviewers.

_{0}.

_{0}: an implicit treatment of transmission in networks.