# American Institute of Mathematical Sciences

• Previous Article
Dependence of propagation speed on invader species: The effect of the predatory commensalism in two-prey, one-predator system with diffusion
• DCDS-B Home
• This Issue
• Next Article
Distributed susceptibility: A challenge to persistence theory in infectious disease models
2009, 12(4): 827-864. doi: 10.3934/dcdsb.2009.12.827

## A mathematical analysis of malaria and tuberculosis co-dynamics

 1 Dar es Salaam Institute of Technology, P.O.Box 2958, Dar es Salaam, Tanzania 2 Department of Mathematics, University of Dar es Salaam, P.O. Box 35062, Dar es Salaam, Tanzania, Tanzania

Received  October 2008 Revised  February 2009 Published  August 2009

We formulate and analyze a deterministic mathematical model which incorporates some basic epidemiological features of the co-dynamics of malaria and tuberculosis. Two sub-models, namely: malaria-only and TB-only sub-models are considered first of all. Sufficient conditions for the local stability of the steady states are presented. Global stability of the disease-free steady state does not hold because the two sub-models exhibit backward bifurcation. The dynamics of the dual malaria-TB only sub-model is also analyzed. It has different dynamics to that of malaria-only and TB-only sub-models: the dual malaria-TB only model has no positive endemic equilibrium whenever $R_{MT}^d<1$, - its disease free equilibrium is globally asymptotically stable whenever the reproduction number for dual malaria-TB co-infection only $R_{MT}^d<1$ - it does not exhibit the phenomenon of backward bifurcation. Graphical representations of this phenomenon is shown, while numerical simulations of the full model are carried out in order to determine whether the two diseases will co-exist whenever their partial reproductive numbers exceed unity. Finally, we perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission.
Citation: 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
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] Scott W. Hansen. Controllability of a basic cochlea model. Evolution Equations & Control Theory, 2016, 5 (4) : 475-487. doi: 10.3934/eect.2016015 [9] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [10] Youshan Tao. Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2705-2722. doi: 10.3934/dcdsb.2013.18.2705 [11] Hui Wan, Jing-An Cui. A model for the transmission of malaria. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 479-496. doi: 10.3934/dcdsb.2009.11.479 [12] Surabhi Pandey, Ezio Venturino. A TB model: Is disease eradication possible in India?. Mathematical Biosciences & Engineering, 2018, 15 (1) : 233-254. doi: 10.3934/mbe.2018010 [13] Marcello Delitala, Tommaso Lorenzi. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Mathematical Biosciences & Engineering, 2017, 14 (1) : 79-93. doi: 10.3934/mbe.2017006 [14] 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 [15] Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5 (4) : 789-801. doi: 10.3934/mbe.2008.5.789 [16] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [17] Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457 [18] Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 [19] G.A. Ngwa. Modelling the dynamics of endemic malaria in growing populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1173-1202. doi: 10.3934/dcdsb.2004.4.1173 [20] Yanyu Xiao, Xingfu Zou. On latencies in malaria infections and their impact on the disease dynamics. Mathematical Biosciences & Engineering, 2013, 10 (2) : 463-481. doi: 10.3934/mbe.2013.10.463

2016 Impact Factor: 0.994