Mathematical analysis of a model for HIV-malaria co-infection
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1.
Department of Applied Mathematics, National University of Science and Technology, Box AC 939 Ascot, Bulawayo
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2.
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2
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3.
Department of Mathematics and Applied Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950
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4.
Mathematics Department, University of Dar es Salaam, P.O.Box 35062, Dar es Salaam
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Received:
01 December 2007
Accepted:
29 June 2018
Published:
01 March 2009
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MSC :
Primary: 92D30; Secondary: 92B05; 34D23.
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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[J]. Mathematical Biosciences and Engineering, 2009, 6(2): 333-362. doi: 10.3934/mbe.2009.6.333
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Abstract
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.
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