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Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. (English) Zbl 1015.92036
Summary: A precise definition of the basic reproduction number, $\cal R_0$, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if $\cal R_0<1$ , then the disease free equilibrium is locally asymptotically stable; whereas if $\cal R_0>1$, then it is unstable. Thus, ${\cal R}_0$ 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 $\cal R_0$ near one. This criterion, together with the definition of $\cal R_0$, 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.

##### MSC:
 92D30 Epidemiology 34C60 Qualitative investigation and simulation of models (ODE)
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##### References:
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