CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 14, Number 3, Fall 2006 GLOBAL STABILITY OF THE ENDEMIC EQUILIBRIUM OF MULTIGROUP SIR EPIDEMIC MODELS HONGBIN GUO, MICHAEL Y. LI AND ZHISHENG SHUAI ABSTRACT. For a class of multigroup SIR epidemic mod- els with varying subpopulation sizes, we establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasi- ble region. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. 1 Introduction Multigroup models have been proposed in the lit- erature to describe the transmission dynamics of infectious diseases in heterogeneous host populations. Heterogeneity in host population can be the result of many factors. Individual hosts can be divided into groups according to different contact patterns such as those among children and adults for Measles and Mumps, or to distinct number of sexual partners for sexually transmitted diseases and HIV/AIDS. Groups can be geo- graphical such as communities, cities, and countries, or epidemiological, to incorporate differential infectivity or co-infection of multiple strains of the disease agent. Multigroup models can also be used to investigate infectious diseases with multiple hosts such as West-Nile virus and vec- tor borne diseases such as Malaria. For a recent survey of multigroup models, we refer the reader to [34]. A multigroup model is, in general, formulated by dividing the pop- ulation of size N (t) into n distinct groups. For 1 ≤ k ≤ n, the k-th group is further partitioned into three compartments: the susceptibles, infectious, and recovered, whose numbers of individuals at time t are AMS subject classification: 34D23, 92D30. Keywords: Multigroup SIR model, basic reproduction number, endemic equilib- rium, global stability. Copyright c Applied Mathematics Institute, University of Alberta. 259