Journal of Mathematical Biology

, Volume 67, Issue 4, pp 963–987

The effect of population heterogeneities upon spread of infection


DOI: 10.1007/s00285-012-0578-x

Cite this article as:
Clancy, D. & Pearce, C.J. J. Math. Biol. (2013) 67: 963. doi:10.1007/s00285-012-0578-x


It has often been observed that population heterogeneities can lead to outbreaks of infection being less frequent and less severe than homogeneous population models would suggest. We address this issue by comparing a model incorporating various forms of heterogeneity with a homogenised model matched according to the value of the basic reproduction number \(R_0\). We mainly focus upon heterogeneity in individuals’ infectivity and susceptibility, though with some allowance also for heterogeneous patterns of mixing. The measures of infectious spread we consider are (i) the probability of a major outbreak; (ii) the mean outbreak size; (iii) the mean endemic prevalence level; and (iv) the persistence time. For each measure, we establish conditions under which heterogeneity leads to a reduction in infectious spread. We also demonstrate that if such conditions are not satisfied, the reverse may occur. As well as comparison with a homogeneous population, we investigate comparisons between two heterogeneous populations of differing degrees of heterogeneity. All of our results are derived under the assumption that the susceptible population is sufficiently large.


Basic reproduction numberSIR epidemicSIS epidemic Outbreak sizeEndemic prevalenceFade-out of infection

Mathematics Subject Classification (2000)


Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK