Social Distancing and Markov Process for Infectious Diseases and Epidemics

In the epidemiology, the infected group has been well understood by applying the game theory and Nash equilibrium analysis. But most mathematical models of epidemiology games have focused on the infected groups and made a strong assumption that the susceptible group is homogenous. That is, the susceptible individuals are assumed to share the same risk preference and adopt the identical strategy to protect themselves. This work proposes a new epidemic model, in which we define a hierarchical structure of the cost for adopting strategies, and study how different levels of cost stratify the susceptible group and hence affect the equilibria of the endemic dynamics. In this work, we find different sets of the equilibrium solutions to each cost level. Interestingly, the result shows that the equilibria produced by the classical epidemiological models are special cases of the our model as cost goes to zero. This new model also discover more equilibria as cost vary. Nash equilibrium analysis of the new model produces the same equilibrium solutions to the population of the susceptible and infected groups as the usual DE equilibrium analysis.