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

Math Biosci. 2002 Nov-Dec;180:29-48. doi: 10.1016/s0025-5564(02)00108-6.


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.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Animals
  • Disease Transmission, Infectious*
  • Endemic Diseases*
  • Epidemiologic Methods
  • Humans
  • Models, Biological*
  • Models, Statistical