A general approach for population games with application to vaccination

Abstract

Reconciling the interests of individuals with the interests of communities is a major challenge in designing and implementing health policies. In this paper, we present a technique based on a combination of mechanistic population-scale models, Markov decision process theory and game theory that facilitates the evaluation of game theoretic decisions at both individual and community scales. To illustrate our technique, we provide solutions to several variants of the simple vaccination game including imperfect vaccine efficacy and differential waning of natural and vaccine immunity. In addition, we show how path-integral approaches can be applied to the study of models in which strategies are fixed waiting times rather than exponential random variables. These methods can be applied to a wide variety of decision problems with population-dynamic feedbacks.

Figure 1

Asymptotic expected utility gain per unit time U′ (π̄, π̄) for an individual using the population’s average strategy π̄. There is a corner in the utility gain at vaccination rates of about π̄ = .25 that is barely sufficient to ensure eradications. This is the social optimum, as faster or slower rates yield slower gains to the average individual. Parameter values: γ = 1, β = 6, a = .05, c_I = 4, c_V = 2.

Figure 2

An individual’s asymptotic expected utility loss U′ (π, π̄) as a function of the individual’s strategy and the population’s average vaccination rate for the same parameters as presented in Figure 1. The absolute utility loss (left) is minimized for different individual choices, depending on the population’s average behavior. The relative utility (right), calculated as the ratio of the absolute utility loss divided by the average utility loss, shows that vaccination rates around .195 are self-consistent in the sense that no one can unilaterally improve on their situation by deviating from the average behavior. All contour values are negative; smaller values represent higher utilities.

Figure 3

The equilibrium vaccination rate π^* as a function of the waning rates a_R and a_V. The shaded region indicates parameter values where π^* = 0. Increases in the waning rate of natural immunity a_R increase Nash equilibrium vaccination. However, the response to changes in the waning rate of vaccine immunity a_V may either increase or decrease equilibrium vaccination. If natural immunity lasts longer than vaccine immunity, the equilibrium is to refuse vaccination. Parameter values β = 4, γ = 1, c = .1.

Figure 4

Dependence of the Nash equilibria vaccination rates π^* upon the relative cost of vaccination (c = c_V/c_I). The Nash equilibrium is unique when c < .393 or c > .426. If .393 < c < .426, there are three Nash equilibria. Parameter values γ = 1, β = 6, a_R = a_V = 0.05, σ = 0.15.

Figure 5

Parameter-space diagrams of the bifurcation structure in the Nash equilibria of Eq. (3.32) as functions of the relative probability of infection σ and the relative cost of vaccine c = c_V/c_I. The right plot is a magnification of the intersecting region of the left plot. The curves represent sets of parameters where bifurcations occur. The plot is divided into 5 regions. If the cost of vaccine is very high, no vaccination (π^* = 0) is the only equilibrium. If the cost of vaccine is very small and the relative probability of infection under vaccination is small, there is a unique equilibrium 0 < π^* < ∞. If the relative probability of infection under vaccination is larger, individuals get vaccinated instantly on entering the susceptible compartment (π^* = ∞). For vaccine costs near the threshold for no vaccination, there may be two locally evolutionarily stable equilibria: No or instant vaccination (π^* = 0 and π^* = ∞) if the relative probability of infection is sufficiently high, and no or a finite vaccination rate if the relative probability of infection is small enough. Parameter values γ = 1, β = 6, a_R = a_V = 0.05.

Figure 6

An individual’s asymptotic expected absolute utility losses (left) and relative utility losses (right) calculated from Eq. (3.47) as a function of the individual’s strategy and the population’s resident strategy. Parameter values are the same as those used in Figures 1 and 2.