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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, 0 , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if 0 <1 , then the disease free equilibrium is locally asymptotically stable; whereas if 0 >1, then it is unstable. Thus, 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 0 near one. This criterion, together with the definition of 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:
92D30Epidemiology
34C60Qualitative investigation and simulation of models (ODE)