Epidemic Population Games for Policy Design: Two Populations with Viral Reservoir Case Study

We extend to two populations a recently proposed system theoretic framework for studying an epidemic influenced by the strategic behavior of a single population's agents. Our framework couples the well-known susceptible-infected-susceptible (SIS) epidemic model with a population game that captures the strategic interactions among the agents of two large populations. This framework can also be employed to study a situation where a population of nonstrategic agents (such as animals) serves as a disease reservoir. Equipped with the framework, we investigate the problem of designing a suitable control policy that assigns dynamic payoffs to incentivize the agents to adopt costlier and more effective mitigating strategies subject to a long-term budget constraint. We formulate a non-convex constrained optimization program for minimizing the disease transmission rate at an endemic equilibrium, and explain how to obtain an approximate solution efficiently. A solution to the optimization problem is an aggregate strategy distribution for the population game which minimizes the basic reproduction number, hence the disease transmission rate, at the corresponding endemic equilibrium. We then propose a dynamic payoff mechanism and use a Lyapunov function to prove the convergence of i) the aggregate strategy distribution, ii) infection levels, and iii) the dynamic payoffs; the aggregate strategy distribution of the population converges to an (approximate) solution to the optimization problem, and the infection levels in the two populations converge to the endemic equilibrium associated with the solution of the optimization.