Elsevier

Mathematical Biosciences

Volume 180, Issues 1–2, November–December 2002, Pages 29-48
Mathematical Biosciences

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Dedicated to the memory of John Jacquez
https://doi.org/10.1016/S0025-5564(02)00108-6Get rights and content

Abstract

A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R0>1, then it is unstable. Thus, R0 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 R0 near one. This criterion, together with the definition of R0, 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.

Keywords

Basic reproduction number
Sub-threshold equilibrium
Disease transmission model
Disease control

Cited by (0)

1

Research supported in part by an NSERC Research Grant, the University of Victoria Committee on faculty research and travel and MITACS.

2

Research supported by an NSERC Postdoctoral Fellowship tenured at the University of Victoria.

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