Control of an Epidemic Spreading in a Heterogeneously Mixing Population DAVID GREENHALGH* Co,zrrol zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA and Munugemenr Systems Division, Engineering Department, Mill Lane. Cambridge, CB2 IRX, England Received 29 March 1985; revised 6 January 1986 ABSTRACT A theoretical epidemic model for a heterogeneously mixing community is examined in the context of control by immunization of susceptibles and removal of infected people. Under certain assumptions an optimal control scheme is developed using the ideas of dynamic programming, and is found to have a particularly simple form for either method of control. Counterexamples are discussed to illustrate why the simplifying assumptions made cannot be relaxed. Applications of these results are considered with respect to the accuracy of the assumptions of the model. 1. INTRODUCTION Most classical epidemic models assume that the population amongst which a disease is spreading mix homogeneously as outlined in Bailey [l]. However an alternative, more realistic, assumption would be to assume that the population is divided into several distinct groups with homogeneous mixing within the groups but with heterogeneous mixing between groups. This heterogeneity may be due to a variety of factors such as an age structure or some other geographical or social structure in the population. Another possible cause of heterogeneity may be differences in the genetic composition of the population. Examples of this type of approach are given by Watson [2], Rushton and Mautner [3], and Cane and McNamee [4]. Watson considers a stochastic heterogeneously mixing epidemic model by using simulations and analytic approximations, whereas Rushton and Mautner consider the case of a deterministic simple epidemic in m equivalent classes. The model discussed in this paper is based on Cane and McNamee’s model, which will therefore be discussed in detail later. *Address for correspondence: Department of .Mathematics, University of Strathclyde, Glasgow, Gl lXH, Scotland. MA THEMA TICA L BIOSCIENCES 80:23-45 (1986) OElsevier Science Publishing Co., Inc., 1986 52 Vanderbilt Ave., New York, NY 10017 23 0025-5564/86/$03.50