Pii: S0025-5564(00)00025-0

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide signi®cant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We de®ne the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for certain epidemiological models, and construct a simpli®ed formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate several models with di€erent heterogeneous structures and demonstrate how to de®ne the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it seems necessary to de®ne the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the total population. Ó 2000 Elsevier Science Inc. All rights reserved.