The Vaccine Dilemma: Epidemiological Game Theory of Vaccination in Set Structured Populations

Abstract

Vaccination is an effective measure for curtailing the spread of an epidemic, and while this process is very beneficial for a population, it inherently provokes a dilemma. The vaccine dilemma stems from two primary setbacks. Vaccines impose a non-negligible cost on vaccinators, and additionally, the eradication of a disease is possible with imperfect vaccine coverage (i.e. herd immunity). Both of these reasons, the cost of vaccination and knowledge of herd immunity, could prompt an individual to defect from vaccination. These individuals, also known as free riders, do not contribute to herd immunity but benefit from its protection. Thus, vaccinators pay a cost, contributing to their interests (their protection) and the group interest, which is the complete eradication of the disease. Free riders, on the other hand, defect from this behavior, ultimately aiding in the persistence of the disease. This dilemma forms the basis of a classic payoff game between vaccinators and free riders in the midst of an epidemic. This thesis explores the behavioral interplay of these individuals and their impact on the process and outcome of an epidemic. These interactions take place in set structured populations, in which individuals interact exclusively with other individuals of the same set. The dynamics are assessed in two different contexts: a single, isolated population (a one-set system) and two sub-populations, which are linked by migration (a two-set system). Ultimately, we show that the vaccinator equilibrium stabilizes at a value below the minimum threshold for elimination. Additionally, if either the cost of vaccination is reduced or if the tendency to vaccinate is enhanced, the resulting vaccinator equilibrium will near the threshold, decreasing the infected population. Finally, when migration is permitted, vaccinators make a set more appealing to outsiders, which increases the flux of individuals into the set.