A mathematical study of an eco-epidemiological system on disease persistence and extinction perspective

• A prey–predator system with nonlinear incidence rate of the disease in prey is described.
• The dynamical behavior of the system is investigated at each equilibrium point.
• Sufficient conditions are derived for the uniformly strong persistence of the system.
• Results are obtained to extract the criteria for disease persistence and extinction.

A prey–predator system with disease in prey is proposed. The proposed system is an extension of the model analyzed by Bhattacharyya and Mukhopadhyay (2011) which did not consider the density of fish population as a dynamic variable which significantly influence the dynamics of the system. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Incidence rate of the disease is considered as a bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the co-existing equilibria. Sufficient conditions are derived for the global stability of the system around coexistence equilibria. Uniform strong persistence of the system is discussed in order to ensure long-term survival of the species. The obtained results are useful to extract the criteria for disease extinction and persistence. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.