Threshold dynamics of an SEIR epidemic model with a nonlinear incidence rate and a discontinuous treatment function

Abstract

We consider a susceptible-exposed-infected-removed (SEIR) epidemic model with a nonlinear incidence rate and a discontinuous treatment function. By applying the LaSalle-type invariance principle for differential inclusions and the Lyapunov theory for discontinuous differential equations, we show that the threshold dynamics are completely described by the basic reproduction number $$R_0$$, under some mild assumptions on the incidence rates and the treatment functions. The disease-free equilibrium state is globally asymptotically stable when $$R_0\le 1$$ and there is endemic equilibrium state which is globally asymptotically stable when $$R_0>1$$. Furthermore, we prove that the disease described by the SEIR epidemic model will die out in finite time. Finally, four illustrative numerical examples are given to support our theoretical results.