An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations

The thresholds for mathematical epidemiology models specify the critical co nditions for an epidemic to grow or die out. The reproductive number can pr ovide significant insight into the transmission dynamics of a disease and c an guide strategies to control its spread. We define the mean number of con tacts, the mean duration of infection, and the mean transmission probabilit y appropriately for certain epidemiological models, and construct a simplif ied formulation of the reproductive number as the product of these quantiti es, When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heter ogeneity, and the expressions for the reproductive number become correspond ingly more complex. We formulate several models with different heterogeneou s structures and demonstrate how to define the mean quantities for an expli cit expression for the reproductive number. In complex heterogeneous models , it seems necessary to define the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers w eighted by their heterogeneity to estimate the reproductive number for the total population. (C) 2000 Elsevier Science Inc. All rights reserved.