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

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. © 2002 Elsevier Science Inc. All rights reserved.