Global Dynamics of an SEIR Model with Two Age Structures and a Nonlinear Incidence

In this paper, we study an SEIR model with both infection and latency ages and also a very general class of nonlinear incidence. We first present some preliminary results on the existence of solutions and on bounds of solutions. Then we study the global dynamics in detail. After proving the existence of a global attractor $\mathcal{A}$ , we characterize it in two cases distinguished by the basic reproduction number $R_{0}$ . When $R_{0}<1$ , we apply the Fluctuation Lemma to show that the disease-free equilibrium $E_{0}$ is globally asymptotically stable, which means $\mathcal{A}=\{E_{0}\}$ . When $R_{0}>1$ , we show the uniform persistence and get $\mathcal{A}=\{E_{0}\}\cup C \cup \mathcal{A}_{1}$ , where $C$ consists of points with connecting orbits from $E_{0}$ to $\mathcal{A}_{1}$ and $\mathcal{A}_{1}$ attracts all points with initial infection force. Under an additional condition, we employ the approach of Lyapunov functional to find that $\mathcal{A}_{1}$ just consists of an endemic equilibrium.