Optimal control analysis of malaria-schistosomiasis co-infection dynamics

Kazeem Oare Okosun; Robert Smith?

doi:10.3934/mbe.2017024

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2017, Volume 14, Issue 2: 377-405. Doi: 10.3934/mbe.2017024

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Optimal control analysis of malaria-schistosomiasis co-infection dynamics

Kazeem Oare Okosun^1, and
Robert Smith?^2,

1.
Department of Mathematics, Vaal University of Technology, Andries Potgieter Boulevard, Vanderbijlpark, 1911, South Africa
2.
Department of Mathematics, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N6N5, Canada

Received: August 05, 2015

Revised: June 02, 2016

Published: July 2016

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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Abstract

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.

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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

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

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

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

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

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

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

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

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

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

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

Parameter	Description	Sensitivity index	Sensitivity index
		if $R_{0m} <1$	if $R_{0m} >1$
$\mu_{sv}$	snail mortality	$-1$	$-1$
$\mu_v$	mosquito mortality	0.56	0.07
$\lambda_s$	prob. of snail getting infected with schisto	0.5	0.5
$\Lambda_s$	snail birth rate	0.5	0.5
$\beta_h$	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 death	0.12	$-0.31$
$\omega$	recovery from schisto	$-0.10$	0.26
$m$	schisto-induced death	$-0.02$	0.05
$\psi$	recovery from malaria	0.003	$-0.0084$

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Table 2. Sensitivity indices of $R_{0m}$ expressed in terms of $R_{sc}$

Parameter	Description	Sensitivity index	Sensitivity index
		if $R_{sc} <1$	if $R_{sc} >1$
$\beta_v$	prob. of mosquito getting infected	0.5	0.5
$\Lambda_v$	mosquito birth rate	0.5	0.5
$\lambda$	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 schisto	0.41	0.41
$m$	schisto-induced death	0.09	0.09
$\psi$	recovery from malaria	$-0.01$	$-0.01$
$\mu_{sv}$	snail mortality	0.0000002	0.000007
$\Lambda_h$	human birth rate	0.0000001	0.000004

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Table 3. Parameters in the co-infection model

Parameter	Description	value	Reference
$\phi$	malaria-induced death	0.05-0.1 day$^{-1}$	[43]
$\beta_h$	malaria transmissibility to humans	0.034 day$^{-1}$	assumed
$\beta_v$	malaria transmissibility to mosquitoes	0.09 day$^{-1}$	[5]
$\lambda$	schistosomiasis transmissibility to humans	0.406 day$^{-1}$	[45]
$\lambda_s$	schistosomiasis transmissibility to snails	0.615 day$^{-1}$	[9]
$\mu_h$	Natural death rate in humans	0.00004 day$^{-1}$	[5]
$\mu_v$	Natural death rate in mosquitoes	1/15-0.143 day$^{-1}$	[5]
$\mu_{sv}$	Natural death rate in snails	0.000569 day$^{-1}$	[9,45]
$\alpha$	malaria immunity waning rate	1/(60$\times$365) day$^{-1}$	[5]
$\epsilon$	schistosomiasis immunity waning rate	0.013 day$^{-1}$	assumed
$\Lambda_h$	human birth rate	800 people/day	[9]
$\Lambda_v$	mosquitoes birth rate	1000 mosquitoes/day	[5]
$\Lambda_s$	snail birth rate	100 snails/day	[13]
$\delta$	recovery rate of co-infected individual	0.35 day$^{-1}$	assumed
$\omega$	recovery rate of schistosomiasis-infected individual	0.0181 day$^{-1}$	assumed
$\psi$	recovery rate of malaria-infected individual	1/(2$\times$365) day$^{-1}$	[5]
$\tau$	co-infected proportion who recover from malaria only	0.1	assumed
$\eta$	schistosomiasis-induced death	0.0039 day$^{-1}$	[9]

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