Mathematical analysis for a multi-group SEIR epidemic model with age-dependent relapse

We consider a multi-group SEIR epidemic model in which recovered population relapse back to infectives depending on the time elapsed since the recovery. This leads to a hybrid system for which we can determine the basic reproduction number by the spectral radius of the next generation matrix and prove the threshold behaviors. The key idea to prove the global asymptotic stability of each equilibrium is the usage of the graph-theoretic approach to construct suitable Lyapunov functionals. The necessary arguments, including the existence of an endemic equilibrium, the asymptotic smoothness of the semiflow, the uniform persistence of the system, and the existence of a global attractor are also addressed.