Stochastic Partial Differential Equation SIS Epidemic Models: Modeling and Analysis

Abstract

The study on epidemic models plays an important role in mathematical biology and mathematical epidemiology. There has been much effort devoted to epidemic models using ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). Much study has been carried out and substantial progress has been made. In contrast to the development, this work presents an effort from a different angle, namely, modeling and analysis using stochastic partial differential equations (SPDEs). Specifically, we consider dynamic systems featuring SIS (Susceptible-Infected-Susceptible) epidemic models. Our emphasis is on spatial dependent variations and environmental noise. First, a new epidemic model is proposed. Then existence and uniqueness of solutions of the underlying SPDEs are examined. In addition, stochastic partial differential equation models with Markov switching are examined. Our analysis is based on the use of mild solution. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.