*(English)*Zbl 1015.92036

Summary: A precise definition of the basic reproduction number, ${\mathcal{R}}_{0}$, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if ${\mathcal{R}}_{0}<1$ , then the disease free equilibrium is locally asymptotically stable; whereas if ${\mathcal{R}}_{0}>1$, then it is unstable. Thus, ${\mathcal{R}}_{0}$ is a threshold parameter for the model.

An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for ${\mathcal{R}}_{0}$ near one. This criterion, together with the definition of ${\mathcal{R}}_{0}$, is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.