Global dynamics of a multi-group epidemic model with general exposed distribution and relapse

Abstract

In this paper, we investigate a class of multi-group epidemic models with general exposed distribution and relapse. Nonlinear incidence rate is used between compartments. It is showed that global dynamics are completely determined by the threshold parameter $R_{0}$ under suitable conditions. More specifically, the disease will die out if $R_{0} \leq 1$ and that if $R_{0} > 1$, the disease persists in all groups. The approaches used here, are the theory of non-negative matrices, persistence theory in dynamical systems and graph-theoretical approach to the method of Lyapunov functionals. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behavior of the SIR model with general exposed distribution and relapse. On the other hand, our global dynamical results exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions.