A compartmental model for epidemiology with human behavior and stochastic effects

We propose a compartmental model for epidemiology wherein the population is split into groups with either comply or refuse to comply with protocols designed to slow the spread of a disease. Parallel to the disease spread, we assume that noncompliance with protocols spreads as a social contagion. We begin by deriving the reproductive ratio for a deterministic version of the model, and use this to fully characterize the local stability of disease free equilibrium points. We then append the deterministic model with stochastic effects, specifically assuming that the transmission rate of the disease and the transmission rate of the social contagion are uncertain. We prove global existence and nonnegativity for our stochastic model. Then using suitably constructed stochastic Lyapunov functions, we analyze the behavior of the stochastic system with respect to certain disease free states. We demonstrate all of our results with numerical simulations.