## 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, $$\mathcal R_0$$, 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.

### MSC:

 92D30 Epidemiology 34C60 Qualitative investigation and simulation of ordinary differential equation models
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### References:

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