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.


92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
Full Text: DOI


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