Global dynamics of a SEIR model with varying total population size

A SEIR model for the transmission of an infectious disease that spreads in a population through direct contact of the hosts is studied. The force of infection is of proportionate mixing type. A threshold σ is identified which determines the outcome of the disease; if σ⩽1, the infected fraction of the population disappears so the disease dies out, while if σ>1, the infected fraction persists and a unique endemic equilibrium state is shown, under a mild restriction on the parameters, to be globally asymptotically stable in the interior of the feasible region. Two other threshold parameters σ′ and σ are also identified; they determine the dynamics of the population sizes in the cases when the disease dies out and when it is endemic, respectively.