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Discontinuity, Nonlinearity, and Complexity 3(1) (2014) 49--58 | DOI:10.5890/DNC.2014.03.004

Carla MA Pinto$^{1}$; Ana Carvalho$^{2}$

$^{1}$ Department of Mathematics, School of Engineering, Polytechnic of Porto, and Center of Mathematics, University of Porto, and GECAD - Knowledge Engineering and Decision Support Research Center Rua Dr António Bernardino de Almeida, 431, 4200-072 Porto, PORTUGAL

$^{2}$ Department of Mathematics, Faculty of Sciences, University of Porto, Rua do Campo Alegre s/n, 4440–452 Porto, Portugal

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Abstract

In this paper, it is proposed a mathematical model for co-infection of HIV/AIDS and tuberculosis. The model includes treatment and vertical transmission for HIV/AIDS. The treatment for tuberculosis is not included. The disease-free equilibrium of the model is computed and local stability is proved. The reproduction numbers of the full model and of its two sub- models, concerning single infection by HIV/AIDS and single infection by tuberculosis, are also calculated. Numerical simulations show the effect of the variation of the recruitment rate, of the movement rate, and of the tuberculosis infection rate on the variables of the model. Results are as expected. Namely an increase in the recruitment rate increases the suscep- tible population. As the movement rate is decreased, the individuals single infected with HIV decrease. Moreover, an increase in the tuberculosis in- fection rate translates in an increase of the single infected with tuberculosis and dually-infected with tuberculosis and HIV/AIDS individuals. Future work will consider the inclusion of treatment of tuberculosis.

Acknowledgments

Authors which to thank Fundação Gulbenkian, through Prémio Gulbenkian de Apoio à Investigação 2003, and the Polytechnic of Porto, through the PAPRE Programa de Apoio à Publicação em Revistas Científicas de Elevada Qualidade for financial support.

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