Spatial Patterns of an SIS Epidemic Model with Diffusion

Abstract

We consider an SIS spatial epidemic model with nonlinear incidence rate and diffusion, that consists of susceptible S and infective I individual populations which interact randomly in their physical environment. We use pattern formation to explain the spread and control of the epidemic over a course of time. Turing instability conditions are established and analysed for the model to exhibit spatial patterns. We find the exact Turing space in the parameter regime for these conditions to hold. An implicit pseudo-spectral method is used to numerically approximate the system and the patterns form reveal that the susceptible and infected populations behave of the same way. In some examples these populations are in isolation from each other. This happens because the susceptible individuals diffuse or move away from the infected individuals to avoid contact and the possibility of getting infected with the disease. Rigorous numerical experimentation reveals that the model has rich dynamics. We find that whenever the transmission rate β is less than the treatment r, there is no outbreak but for β ≥ r there is a possibility of having an outbreak. The results obtained extend well the findings of pattern formation in epidemic models and may have direct implications for the study of disease spread and control and perhaps the mechanistic impact of public health interventions on epidemics.