Research Article Disease Control through Voluntary Vaccination Decisions Based on the Smoothed Best Response Fei Xu and Ross Cressman Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5 Correspondence should be addressed to Fei Xu; fxu.feixu@gmail.com Received 18 July 2013; Accepted 16 December 2013; Published 16 February 2014 Academic Editor: Travis Porco Copyright © 2014 F. Xu and R. Cressman. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate game-theory based decisions on vaccination uptake and its efects on the spread of an epidemic with nonlinear incidence rate. It is assumed that each individual’s decision approximates his/her best response (called smoothed best response) in that this person chooses to take the vaccine based on its cost-beneit analysis. he basic reproduction number of the resultant epidemic model is calculated and used to characterize the existence and stability of the disease-free and endemic equilibria of the model. he efects on the spread and control of the epidemic are revealed in terms of the sensitivity of the response to changes in costs and beneits, in the “cost” of the vaccination, and in the proportion of susceptible individuals who are faced with the decision of whether or not to be vaccinated per unit time. he efects of the best response decision rule are also analyzed and compared to those of the smoothed best response. Our study shows that, when there is a perceived cost to take the vaccine, the smoothed best response is more efective in controlling the epidemic. However, when this cost is 0, the best response is the more eicient control. 1. Introduction In modern society, infectious diseases threaten millions of people’s lives each year and, as such, controlling the spread of these diseases is essential. As one of the efective control strategies, vaccination against infectious diseases has been widely used to slow down or eliminate their spread [1–4]. Recent investigations of theoretical models based on diferent vaccination policies [2, 3] indicate that there are many ways an efective vaccine can be used to control an epidemic. hese theoretical models oten consider the “cost” to get vaccinated. Besides the actual monetary cost of the vaccine, there are potential risks to being vaccinated. hus people making rational decisions may avoid vaccinations when the perceived cost of taking the vaccine is higher than its beneits. hat is, individual decisions about the vaccination uptake might follow a cost-beneit analysis. hus, the analysis of the efect of voluntary vaccination decisions is becoming increasingly important as people are now able to obtain up- to-date information about the spread of an epidemic as well as about the cost of vaccination. he aim of this paper is to model how individuals implement their rational decisions on vaccine uptake and investigate the efects of these decisions on the spread and control of the epidemic. On one hand, susceptibles have the risk of being infected. On the other hand, due to the perceived risk of vaccine side efects, susceptible individuals might choose not to receive the vaccination. During an epidemic, a susceptible individual has to make a choice based on the risk of being vaccinated and the risk of getting infected. We use game theory to model this situation since this theory studies how individuals optimize their behavior given their net beneits and the behavior of others (i.e., how individuals make rational decisions). Since the probability that a susceptible individual gets infected decreases as the vaccination level of the population increases, rational decisions may lead to a reduced number of vaccination intakes whereby rational individuals rely on others to maintain the vaccination level of the population. his situation is also known as “free riding” [5]. However, this free riding strategy is not optimal to control the disease spread in the long run. hat is, these rational decisions will lead to an increase in the number of Hindawi Publishing Corporation Computational and Mathematical Methods in Medicine Volume 2014, Article ID 825734, 14 pages http://dx.doi.org/10.1155/2014/825734