Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate

In this paper, we investigate a class of multi-group epidemic models allowing heterogeneity of the host population and that has taken into consideration with general relapse distribution and nonlinear incidence rate. We establish that the global dynamics are completely determined by the basic reproduction number R ₀. The proofs of the main results utilize the persistence theory in dynamical systems, Lyapunov functionals and a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach. Biologically, the disease (with any initial inoculation) will persist in all groups of the population and will eventually settle at a constant level in each group. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behaviors of the basic SIR model. On the other hand, the global dynamics exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions.