Dynamics analysis of a diffusive SIRI epidemic system under logistic source and general incidence rate

This study presents and examines a diffusive SIRI epidemic model incorporating logistic source and a general incidence rate. Differing from existing works, the system incorporates two factors: the general incidence rate and the logistic source. We first consider the well-posedness of the system. Then, utilizing the construction of four Lyapunov functions, we thoroughly examine the global asymptotic stability of equilibria in both specific and general scenarios, assuming all coefficients remain constant. In addition, we establish the basic reproduction number, denoted as Ro, and subsequently derive the correlation between Ro and the local basic reproduction number. Furthermore, we provide a detailed discussion of the persistence and extinction of the infective population. In particular, in the case where Ro equals one and certain assumptions are met, we demonstrate the global asymptotic stability of the disease-free equilibrium. Lastly, we substantiate the validity of our theoretical findings through the presentation of five illustrative examples.