A mathematical analysis of prey-predator population dynamics in the presence of an SIS infectious disease

In this paper, we propose and analyze a detailed mathematical model describing the dynamics of a prey-predator model under the influence of an SIS infectious disease by using nonlinear differential equations. We use the functional response of ratio-dependent Michaelis-Menten type to describe the predation strategy. In the presence of the disease, prey and predator population are divided into two disjointed classes, namely infected and susceptible. The first one is governed through due predation interaction, and the second one is governed through the propagation of disease in the prey and predator population via predation. Our aim is to analyze the effect of predation on the dynamic of the disease transmission. Important mathematical results resulting from the transmission of the disease under influence of predation are offered. First, results concerning boundedness, uniform persistence, existence and uniqueness of solutions have been developed. In addition, many thresholds have been computed and used to investigate local and global stability analysis by using Routh-Hurwitz criterion and Lyapunov principle. We also establish the Hopf bifurcation to highlight periodic fluctuation with persistence of the disease or without disease in the prey and predator population. Finally, numerical simulations are carried out to illustrate the feasibility of the theoretical results.