Global Structure and Asymptotic Profiles of the Endemic Equilibria of a Diffusive Epidemic Model with Mass-Action

Infectious diseases are a great challenge to the health and successful function of society. Therefore, it becomes crucial to develop methods and tools that would allow us to be able to control an infectious disease once it starts spreading within an environment. In this regard, mathematical research on epidemic models has provided important tools in the qualitative and quantitative analysis of the spread and control of infectious diseases. Each mathematical epidemic model incorporates important factors that could affect the spread of a disease, such as population movement and temporal or environmental heterogeneity. This dissertation focuses on a susceptible-infected-susceptible (SIS) model in the form of a system of diffusive partial differential equations that takes into account a moving population within a spatially heterogeneous environment. Our goal is to assess the effectiveness of disease control strategies aimed at restricting population movement. To this end, we first consider basic fundamental questions such as existence, uniqueness, and global stability of solutions to the model. Next, we discuss how population movement may affect the disease dynamics by looking at the asymptotic profiles of endemic equilibrium (EE) solutions of the model. Consequently, we determine conditions leading to a multiplicity of EE solutions, which demonstrate that the disease can become difficult to control when movement is included in the model. In doing so, we discover various bifurcation curves describing multiple EE solutions for the diffusive SIS epidemic model.