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Zbl 1015.92036
van den Driessche, P.; Watmough, James
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.
(English)
[J] Math. Biosci. 180, 29-48 (2002). ISSN 0025-5564

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. \par 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 2000:
*92D30 Epidemiology
34C60 Applications of qualitative theory of ODE

Keywords: basic reproduction number; sub-threshold equilibrium; disease transmission model; disease control

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