# American Institute of Mathematical Sciences

2008, 5(1): 145-174. doi: 10.3934/mbe.2008.5.145

## Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment

 1 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada, Canada, Canada 2 Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043

Received  May 2007 Revised  November 2007 Published  January 2008

This paper addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic model, which incorporates many of the essential biological and epidemiological features of the two dis- eases. In the absence of TB infection, the model (HIV-only model) is shown to have a globally asymptotically stable, disease-free equilibrium whenever the associated reproduction number is less than unity and has a unique endemic equilibrium whenever this number exceeds unity. On the other hand, the model with TB alone (TB-only model) undergoes the phenomenon of back- ward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. The analysis of the respective reproduction thresholds shows that the use of a targeted HIV treatment (using anti-retroviral drugs) strategy can lead to effective control of HIV provided it reduces the relative infectiousness of individuals treated (in comparison to untreated HIV-infected individuals) below a certain threshold. The full model, with both HIV and TB, is simu- lated to evaluate the impact of the various treatment strategies. It is shown that the HIV-only treatment strategy saves more cases of the mixed infection than the TB-only strategy. Further, for low treatment rates, the mixed-only strategy saves the least number of cases (of HIV, TB, and the mixed infection) in comparison to the other strategies. Thus, this study shows that if resources are limited, then targeting such resources to treating one of the diseases is more beneficial in reducing new cases of the mixed infection than targeting the mixed infection only diseases. Finally, the universal strategy saves more cases of the mixed infection than any of the other strategies.
Citation: Oluwaseun Sharomi, Chandra N. Podder, Abba B. Gumel, Baojun Song. Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. Mathematical Biosciences & Engineering, 2008, 5 (1) : 145-174. doi: 10.3934/mbe.2008.5.145
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