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Optimal control analysis of malaria-schistosomiasis co-infection dynamics

The authors are grateful to two anonymous reviewers whose comments greatly improved the manuscript. KOO acknowledges the Vaal University of Technology Research Office and the National Research Foundation (NRF), South Africa, through the KIC Grant ID 97192 for the financial support to attend and present this paper at the AMMCS-CAIMS 2015 meeting in Waterloo, Canada. RS? is supported by an NSERC Discovery Grant. For citation purposes, please note that the question mark in "Smith?" is part of the author's name.
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  • This paper presents a mathematical model for malaria-schistosom-iasis co-infection in order to investigate their synergistic relationship in the presence of treatment. We first analyse the single infection steady states, then investigate the existence and stability of equilibria and then calculate the basic reproduction numbers. Both the single-infection models and the co-infection model exhibit backward bifurcations. We carrying out a sensitivity analysis of the co-infection model and show that schistosomiasis infection may not be associated with an increased risk of malaria. Conversely, malaria infection may be associated with an increased risk of schistosomiasis. Furthermore, we found that effective treatment and prevention of schistosomiasis infection would also assist in the effective control and eradication of malaria. Finally, we apply Pontryagin's Maximum Principle to the model in order to determine optimal strategies for control of both diseases.

    Mathematics Subject Classification: Primary: 92B05, 93A30; Secondary: 93C15.


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  • Figure 1.  Flow diagram for the co-infection model. Dashed curves represent cross-species infection

    Figure 2.  Simulations of the submodels to illustrate the occurrence of a backward bifurcation

    Figure 3.  Simulations of the malaria-schistosomiasis model with varying initial values

    Figure 4.  Simulations of the malaria-schistosomiasis model showing the effect of malaria prevention and treatment on transmission

    Figure 5.  Simulations of the malaria-schistosomiasis model showing the effect of schistosomiasis prevention and treatment on transmission

    Figure 6.  Simulations of the malaria-schistosomiasis model showing the effect of prevention of both infections on transmission

    Figure 7.  Simulations of the malaria-schistosomiasis model showing the effect of treatment of malaria and schistosomiasis transmission

    Figure 8.  Simulations of the malaria-schistosomiasis model showing the effect of both prevention and treatment

    Figure 9.  Simulations of the malaria-schistosomiasis model showing the effect of varying transmission rates

    Figure 10.  Simulations of the malaria-schistosomiasis model showing the effect of varying the mosquito death rate

    Table 1.  Sensitivity indices of $R_{sc}$ expressed in terms of $R_{0m}$

    ParameterDescriptionSensitivity indexSensitivity index
    if $R_{0m} <1$if $R_{0m} >1$
    $\mu_{sv}$snail mortality$-1$$-1$
    $\mu_v$mosquito mortality0.560.07
    $\lambda_s$prob. of snail getting infected with schisto0.50.5
    $\Lambda_s$snail birth rate0.50.5
    $\beta_h$prob. of human getting infected with malaria$-0.28$$-0.03$
    $\beta_v$prob. of mosquito getting infected$-0.28$$-0.03$
    $\Lambda_v$mosquito birth rate$-0.28$$-0.03$
    $\Lambda_h$human birth rate$-0.22$$-0.47$
    $\phi$malaria-induced death0.12$-0.31$
    $\omega$recovery from schisto$-0.10$0.26
    $m$schisto-induced death$-0.02$0.05
    $\psi$recovery from malaria0.003$-0.0084$
     | Show Table
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    Table 2.  Sensitivity indices of $R_{0m}$ expressed in terms of $R_{sc}$

    ParameterDescriptionSensitivity indexSensitivity index
    if $R_{sc} <1$if $R_{sc} >1$
    $\beta_v$prob. of mosquito getting infected0.50.5
    $\Lambda_v$mosquito birth rate0.50.5
    $\lambda$prob. of human getting infected with schisto$-0.5$$-0.5$
    $\lambda_s$prob. of snail getting infected with schisto$-0.5$$-0.5$
    $\Lambda_s$snail birth rate$-0.5$$-0.5$
    $\phi$malaria-induced death$-0.49$$-0.49$
    $\omega$recovery from schisto0.410.41
    $m$schisto-induced death0.090.09
    $\psi$recovery from malaria$-0.01$$-0.01$
    $\mu_{sv}$snail mortality0.00000020.000007
    $\Lambda_h$human birth rate0.00000010.000004
     | Show Table
    DownLoad: CSV

    Table 3.  Parameters in the co-infection model

    $\phi$malaria-induced death0.05-0.1 day$^{-1}$[43]
    $\beta_h$malaria transmissibility to humans0.034 day$^{-1}$assumed
    $\beta_v$malaria transmissibility to mosquitoes0.09 day$^{-1}$[5]
    $\lambda$schistosomiasis transmissibility to humans0.406 day$^{-1}$[45]
    $\lambda_s$schistosomiasis transmissibility to snails0.615 day$^{-1}$[9]
    $\mu_h$Natural death rate in humans0.00004 day$^{-1}$[5]
    $\mu_v$Natural death rate in mosquitoes1/15-0.143 day$^{-1}$[5]
    $\mu_{sv}$Natural death rate in snails0.000569 day$^{-1}$[9,45]
    $\alpha$malaria immunity waning rate1/(60$\times$365) day$^{-1}$[5]
    $\epsilon$schistosomiasis immunity waning rate0.013 day$^{-1}$assumed
    $\Lambda_h$human birth rate800 people/day[9]
    $\Lambda_v$mosquitoes birth rate1000 mosquitoes/day[5]
    $\Lambda_s$snail birth rate100 snails/day[13]
    $\delta$recovery rate of co-infected individual0.35 day$^{-1}$assumed
    $\omega$recovery rate of schistosomiasis-infected individual0.0181 day$^{-1}$assumed
    $\psi$recovery rate of malaria-infected individual1/(2$\times$365) day$^{-1}$[5]
    $\tau$co-infected proportion who recover from malaria only0.1assumed
    $\eta$schistosomiasis-induced death0.0039 day$^{-1}$[9]
     | Show Table
    DownLoad: CSV
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