The final size of an epidemic and its relation to the basic reproduction number

Bull Math Biol. 2011 Oct;73(10):2305-21. doi: 10.1007/s11538-010-9623-3. Epub 2011 Jan 6.


We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number R0. There is a unique positive solution giving the size of the epidemic if and only if R0 exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number R0. For other mixing patterns, the relation may be reversed.

Publication types

  • Research Support, N.I.H., Extramural
  • Research Support, Non-U.S. Gov't

MeSH terms

  • Epidemics / statistics & numerical data*
  • Humans
  • Linear Models
  • Mathematical Concepts
  • Models, Biological
  • Nonlinear Dynamics
  • Population Density