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.