Global stability for an SEI model of infectious disease with immigration

•We study an SEI model of an infectious disease, including immigration of infecteds.•There is no disease-free equilibrium.•There is a unique endemic equilibrium and it is globally asymptotically stable.•The disease cannot be eliminated unless immigration of infecteds is blocked.

We study an SEI model of disease transmission with immigration into all three classes. For incidence, we allow for a nonlinear response to the number of infectives, including mass action and saturating incidence as special cases. There is no disease-free equilibrium and therefore no basic reproduction number. For all parameter values, the only equilibrium is an endemic equilibrium. Using a Lyapunov function, we show that this equilibrium is globally asymptotically stable.