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November  2009, 12(4): 827-864. doi: 10.3934/dcdsb.2009.12.827

A mathematical analysis of malaria and tuberculosis co-dynamics

Expeditho Mtisi 1, , Herieth Rwezaura 2, and  Jean Michel Tchuenche 2,

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.
Keywords: sensitivity., co-dynamics, basic reproduction number, Malaria-TB model, stability, equilibrium.
Mathematics Subject Classification: Primary: 92C60, 92B05; Secondary: 92D30, 93D05, 93D2.
Citation: Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934/dcdsb.2009.12.827
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