The basic reproduction number for infection dynamics models and the global stability of stationary points

We consider a system of ordinary differential equations representing a large class of mathematical models concerning the dynamics of an infection in an organism or in a population and we show that the study of the linearized stability leads to some conditions on the basic reproduction number which are sufficient, not only for the local asymptotic stability of a stationary point, but also for its global asymptotic stability. For a more restricted class of problems also the necessity is proved and the limiting behaviour of the model is described.