Theoretical Population Biology 56, 325335 (1999) Host Heterogeneity and Disease Endemicity: A Moment-Based Approach Jonathan Dushoff Institute of Physics, Academia Sinica, Nankang 11529, Taipei, Taiwan Received November 10, 1998 This paper investigates the possibility of understanding the effects of host heterogeneity on disease levels through the use of moment approximations. The approach is to avoid assump- tions about the distribution of mixing rates (or other parameters) in the population, by treating the low-order moments of the distribution as estimable parameters. This approach, while approximate, can greatly reduce the number of parameters needed to explore the effects of population heterogeneity on disease dynamics. This makes the approach useful for both inference and prediction, and also for gaining insight into the qualitative effects of heterogeneity on the spread of disease. This paper focuses on populations with variations in mixing rate and random mixing. It is shown that moment-based approximations can provide good quantitative estimates of disease dynamics, as well as aiding in qualitative under- standing, over a respectable range of parameters. It is hoped that this approach will provide a useful complement to more traditional box models of heterogeneity. ] 1999 Academic Press 1. INTRODUCTION Ross (1908; 1910) described the link between mos- quito densities and disease prevalence in a malaria model and concluded that it should be possible to eliminate malaria by reducing mosquito density below a threshold level, without necessarily eliminating mosquitoes. His work implied a direct connection between risk levels and endemic levels of disease. Under homogeneous mixing, if an infected individual in a totally susceptible population infects an average of R 0 individuals, then in general an infected individual would infect R 0 SN individuals, where SN represents the proportion of the population susceptible. At an endemic equilibrium, each infected individual should generate an average of one new infection, giving R 0 S N =1. (1) Thus, the proportion of the population not susceptible to disease at endemic equilibrium, V, is given by: V=1&1R 0 . (2) Many diseases follow patterns which are consistent with (2). Other diseases, however, notably many sexually transmitted diseases (STDs) and helminth infections show patterns that are not consistent with (2) and hence not consistent with simple homogeneous-population dis- ease models. For example, gonorrhea is stably endemic in a large number of cities throughout the world, usually with prevalences of around 5 0 to 20 0 of the sexually active population. Since gonorrhea does not cause lasting immunity, this would imply that a wide variety of places have R 0 ranging from about 1.05 to 1.25, and remaining within this range over time. This level of similarity between places and time periods is not realistic. The solution to this paradox is reasonably well under- stood. Yorke, Hethcote, and co-workers (1978; 1984) have shown that variation in sexual mixing rates among Article ID tpbi.1999.1428, available online at http:www.idealibrary.com on 325 0040-580999 K30.00 Copyright ] 1999 by Academic Press All rights of reproduction in any form reserved.