Positive steady states of a SI epidemic model with cross diffusion

Abstract

We propose a spatiotemporal SI epidemic model with cross diffusion. The cross diffusion term is proposed for the first time in literature and through our analysis we have confirmed that coexistence of the participating species is possible under its influence. We study the model using coupled upper and lower solutions for an elliptic partial differential equations with Dirichlet boundary conditions. We derive sufficient conditions for the coexistence of the susceptible and infected populations. Mathematical analysis is performed to prove that at least one coexistence state will exist for the proposed model system. We further proved that the system exhibits Turing instability in the presence of the proposed cross diffusion and carried out numerical simulation to observe the patterning behavior. This work provides a significant insight on how epidemic models can be influenced by cross diffusion effects.