Dynamics of a Two-Strain Model With Vaccination, General Incidence Rate, and Nonlocal Diffusion

Abstract

In this paper, we studied a two-strain model with vaccination, general incidence rate, and nonlocal diffusion. In this model, it is considered that those vaccinated can become infected with either strain and that those removed individuals with respect to strain 1 can become infected with strain 2 (partial cross-immunity). We prove that solution exists and is unique, bounded, and positive, and there exists a disease-free steady state. The basic reproduction number ${ℛ}_0$ is defined, and the existence of principal eigenvalue is studied. For ${ℛ}_0<1$, we prove the disease-free steady state is globally asymptotically stable, and when ${ℛ}_0>1$, the disease-free steady state is unstable. We also prove the uniform persistence of the system. Four steady states were obtained, and it was shown that the existence of each of the steady state depends on 4 threshold quantities and we used a Lyapunov approach to prove the global stability of the steady state under certain assumptions. The results are confirmed using some graphical representations.