MATHEMATICAL BIOSCIENCES http://math.asu.edu/˜mbe/ AND ENGINEERING Volume 1, Number 1, June 2004 pp. 81–93 INFLUENCE OF BACKWARD BIFURCATION ON INTERPRETATION OF R 0 IN A MODEL OF EPIDEMIC TUBERCULOSIS WITH REINFECTION Benjamin H. Singer and Denise E. Kirschner Department of Microbiology and Immunology The University of Michigan Medical School Ann Arbor, MI 48109-0620 (Communicated by Carlos Castillo-Chavez) Abstract. There is significant disagreement in the epidemiological literature regarding the extent to which reinfection of latently infected individuals con- tributes to the dynamics of tuberculosis (TB) epidemics. In this study we present an epidemiological model of Mycobacterium tuberculosis infection that includes the process of reinfection. Using analysis and numerical simulations, we observe the effect that varying levels of reinfection has on the qualitative dynamics of the TB epidemic. We examine cases of the model both with and without treatment of actively infected individuals. Next, we consider a variation of the model describing a heterogeneous population, stratified by sus- ceptibility to TB infection. Results show that a threshold level of reinfection exists in all cases of the model. Beyond this threshold, the dynamics of the model are described by a backward bifurcation. Uncertainty analysis of the parameters shows that this threshold is too high to be attained in a realistic epidemic. However, we show that even for sub-threshold levels of reinfection, including reinfection in the model changes dynamic behavior of the model. In particular, when reinfection is present the basic reproductive number, R 0 , does not accurately describe the severity of an epidemic. 1. Introduction. Tuberculosis (TB), an infectious disease caused by the bac- terium Mycobacterium tuberculosis, is estimated to infect one-third of the world’s population and results in nearly 3 million deaths per year [1, 2, 3]. The high burden of TB infection in regions of Southeast Asia, Africa, and Russia has highlighted the need for global TB control [4, 5]. The emergence of drug-resistant strains of M. tuberculosis [5] and TB/HIV coinfection [6, 7, 8] will likely impact TB treatment and control strategies [9, 4]. The long period of latency in M. tuberculosis infection prior to the onset of ac- tive disease introduces additional ambiguity into understanding disease progression. Since initial infection is separated so dramatically in time from the development of disease, it is unclear whether the transition from latency to active disease is due to endogenous reactivation or exogenous reinfection [10, 11]. The relative impor- tance of these two pathways to the development of active disease has significant 2000 Mathematics Subject Classification. 93D30, 34G20, 92D20. Key words and phrases. TB, Epidemic, backward bifurcation, reinfection, R 0 , heterogenous populations. 81