# American Institute of Mathematical Sciences

2004, 1(1): 81-93. doi: 10.3934/mbe.2004.1.81

## Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection

 1 Department of Microbiology and Immunology, The University of Michigan Medical School, Ann Arbor, MI 48109-0620, United States, United States

Received  February 2004 Revised  March 2004 Published  March 2004

There is signiﬁcant disagreement in the epidemiological literature regarding the extent to which reinfection of latently infected individuals contributes 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 susceptibility 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.
Citation: Benjamin H. Singer, Denise E. Kirschner. Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences & Engineering, 2004, 1 (1) : 81-93. doi: 10.3934/mbe.2004.1.81
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