# American Institute of Mathematical Sciences

• Previous Article
Examination of a simple model of condom usage and individual withdrawal for the HIV epidemic
• MBE Home
• This Issue
• Next Article
Epidemic models with differential susceptibility and staged progression and their dynamics
2009, 6(2): 333-362. doi: 10.3934/mbe.2009.6.333

## Mathematical analysis of a model for HIV-malaria co-infection

 1 Department of Applied Mathematics, National University of Science and Technology, Box AC 939 Ascot, Bulawayo, Zimbabwe 2 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada 3 Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa 4 Mathematics Department, University of Dar es Salaam, P.O.Box 35062, Dar es Salaam, Tanzania

Received  December 2007 Revised  August 2008 Published  March 2009

A deterministic model for the co-interaction of HIV and malaria in a community is presented and rigorously analyzed. Two sub-models, namely the HIV-only and malaria-only sub-models, are considered first of all. Unlike the HIV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the malaria-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for malaria, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, using centre manifold theory, that the full HIV-malaria co-infection model undergoes backward bifurcation. Simulations of the full HIV-malaria model show that the two diseases co-exist whenever their reproduction numbers exceed unity (with no competitive exclusion occurring). Further, the reduction in sexual activity of individuals with malaria symptoms decreases the number of new cases of HIV and the mixed HIV-malaria infection while increasing the number of malaria cases. Finally, these simulations show that the HIV-induced increase in susceptibility to malaria infection has marginal effect on the new cases of HIV and malaria but increases the number of new cases of the dual HIV-malaria infection.
Citation: Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333-362. doi: 10.3934/mbe.2009.6.333
 [1] Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934/mbe.2015.12.23 [2] Kazeem Oare Okosun, Robert Smith?. Optimal control analysis of malaria-schistosomiasis co-infection dynamics. Mathematical Biosciences & Engineering, 2017, 14 (2) : 377-405. doi: 10.3934/mbe.2017024 [3] Salihu Sabiu Musa, Nafiu Hussaini, Shi Zhao, He Daihai. Dynamical analysis of chikungunya and dengue co-infection model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1907-1933. doi: 10.3934/dcdsb.2020009 [4] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [5] Yijun Lou, Li Liu, Daozhou Gao. Modeling co-infection of Ixodes tick-borne pathogens. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1301-1316. doi: 10.3934/mbe.2017067 [6] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [7] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [8] Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525 [9] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [10] Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297 [11] Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934/mbe.2018026 [12] Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995 [13] Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934/dcdsb.2009.12.827 [14] Stephen Pankavich, Deborah Shutt. An in-host model of HIV incorporating latent infection and viral mutation. Conference Publications, 2015, 2015 (special) : 913-922. doi: 10.3934/proc.2015.0913 [15] Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103 [16] Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229 [17] M. Hadjiandreou, Raul Conejeros, Vassilis S. Vassiliadis. Towards a long-term model construction for the dynamic simulation of HIV infection. Mathematical Biosciences & Engineering, 2007, 4 (3) : 489-504. doi: 10.3934/mbe.2007.4.489 [18] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020044 [19] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [20] Lih-Ing W. Roeger, Z. Feng, Carlos Castillo-Chávez. Modeling TB and HIV co-infections. Mathematical Biosciences & Engineering, 2009, 6 (4) : 815-837. doi: 10.3934/mbe.2009.6.815

2018 Impact Factor: 1.313