Optimal control of a two-group malaria transmission model with vaccination

doi:10.1007/s13721-022-00403-0

. 2023;12(1):7.

doi: 10.1007/s13721-022-00403-0. Epub 2022 Dec 23.

Optimal control of a two-group malaria transmission model with vaccination

S Y Tchoumi^{1

2}, C W Chukwu³, M L Diagne⁴, H Rwezaura⁵, M L Juga⁶, J M Tchuenche^{7

8}

Affiliations

¹ Department of Mathematics and Computer Sciences ENSAI, University of NGaoundere, P. O. Box 455, Ngaoundere, Cameroon.
² Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa.
³ Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109 USA.
⁴ Departement de Mathematiques, UFR des Sciences et Technologies, Universite de Thies, Thies, Senegal.
⁵ Mathematics Department, University of Dar es Salaam, P.O. Box 35062, Dar es Salaam, Tanzania.
⁶ Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006 South Africa.
⁷ School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa.
⁸ School of Computational and Communication Sciences and Engineering, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania.

PMID: 36575768
PMCID: PMC9780107
DOI: 10.1007/s13721-022-00403-0

Optimal control of a two-group malaria transmission model with vaccination

S Y Tchoumi et al. Netw Model Anal Health Inform Bioinform. 2023.

. 2023;12(1):7.

doi: 10.1007/s13721-022-00403-0. Epub 2022 Dec 23.

Authors

S Y Tchoumi^{1

2}, C W Chukwu³, M L Diagne⁴, H Rwezaura⁵, M L Juga⁶, J M Tchuenche^{7

8}

Affiliations

¹ Department of Mathematics and Computer Sciences ENSAI, University of NGaoundere, P. O. Box 455, Ngaoundere, Cameroon.
² Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa.
³ Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109 USA.
⁴ Departement de Mathematiques, UFR des Sciences et Technologies, Universite de Thies, Thies, Senegal.
⁵ Mathematics Department, University of Dar es Salaam, P.O. Box 35062, Dar es Salaam, Tanzania.
⁶ Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006 South Africa.
⁷ School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa.
⁸ School of Computational and Communication Sciences and Engineering, Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania.

PMID: 36575768
PMCID: PMC9780107
DOI: 10.1007/s13721-022-00403-0

Abstract

Malaria is a vector-borne disease that poses major health challenges globally, with the highest burden in children less than 5 years old. Prevention and treatment have been the main interventions measures until the recent groundbreaking highly recommended malaria vaccine by WHO for children below five. A two-group malaria model structured by age with vaccination of individuals aged below 5 years old is formulated and theoretically analyzed. The disease-free equilibrium is globally asymptotically stable when the disease-induced death rate in both human groups is zero. Descarte's rule of signs is used to discuss the possible existence of multiple endemic equilibria. By construction, mathematical models inherit the loss of information that could make prediction of model outcomes imprecise. Thus, a global sensitivity analysis of the basic reproduction number and the vaccination class as response functions using Latin-Hypercube Sampling in combination with partial rank correlation coefficient are graphically depicted. As expected, the most sensitive parameters are related to children under 5 years old. Through the application of optimal control theory, the best combination of interventions measures to mitigate the spread of malaria is investigated. Simulations results show that concurrently applying the three intervention measures, namely: personal protection, treatment, and vaccination of childreen under-five is the best strategy for fighting against malaria epidemic in a community, relative to using either single or any dual combination of intervention(s) at a time.

Keywords: Malaria; Optimal control; Sensitivity analysis; Vaccination.

© The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2022, Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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Conflict of interest statement

Conflict of InterestNone.

Figures

**Fig. 1**
Model flow diagram of the human component of the model

**Fig. 2**
Model flow diagram of the mosquito component of the model

**Fig. 3**
Tonardo plot showing all the model parameters against $R_{C}^{m} .$ The longer the bar, the more sensitive if the corresponding parameter

**Fig. 4**
Scatter plots simulation showing the PRCC values of parameters $λ_{a}$ $ϑ_{e}, ξ$ and $μ_{h}$ against the vaccination class $V_{e}$

**Fig. 5**
Effect of implementing controls of the model state variables for (a) Susceptible humans under 5-years. (b) Susceptible humans over 5-years. (c) Infectious humans under 5-years. (d) Infectious humans over 5-years. (e) Vaccinated humans under 5-years

**Fig. 6**
Control profiles for the use of optimal control variables (a) $c_{1}$ -personal protective measures. (b) $c_{2}$ -Treatment effort (c) $c_{3}$ -vaccination effort

**Fig. 7**
Effect of implementing only control $c_{1}$ of the model state variables for (a) Susceptible humans under 5-years. (b) Susceptible humans over 5-years. (c) Infectious humans under 5-years. (d) Infectious humans over 5-years. (e) Vaccinated humans under 5-years

**Fig. 8**
Effect of implementing only control $c_{2}$ of the model state variables for (a) Susceptible humans under 5-years. (b) Susceptible humans over 5-years. (c) Infectious humans under 5-years. (d) Infectious humans over 5-years. (e) Vaccinated humans under 5-years

**Fig. 9**
Effect of implementing only control $c_{3}$ of the model state variables for (a) Susceptible humans under 5-years. (b) Susceptible humans over 5-years. (c) Infectious humans under 5-years. (d) Infectious humans over 5-years. (e) Vaccinated humans under 5-years

See this image and copyright information in PMC

References

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[1] Agusto FB, Gumel AB, Parham PE. Qualitative assessment of the role of temperature variations on malaria transmission dynamics. J Biol Syst. 2015;23(04):1550030. doi: 10.1142/S0218339015500308. - DOI

[2] Agusto FB, Gumel AB, Parham PE. Qualitative assessment of the role of temperature variations on malaria transmission dynamics. J Biol Syst. 2015;23(04):1550030. doi: 10.1142/S0218339015500308. - DOI

[3] Atcheson E, Bauza K, Reyes-Sandoval A. A probabilistic model of pre-erythrocytic malaria vaccine combination in mice. PLoS ONE. 2019;14(1):e0209028. doi: 10.1371/journal.pone.0209028. - DOI - PMC - PubMed

[4] Atcheson E, Bauza K, Reyes-Sandoval A. A probabilistic model of pre-erythrocytic malaria vaccine combination in mice. PLoS ONE. 2019;14(1):e0209028. doi: 10.1371/journal.pone.0209028. - DOI - PMC - PubMed

[5] Bala S, Gimba B. Global sensitivity analysis to study the impacts of bed-nets, drug treatment, and their efficacies on a two-strain malaria model. Math Comput Appl. 2019;24(1):32.

[6] Bala S, Gimba B. Global sensitivity analysis to study the impacts of bed-nets, drug treatment, and their efficacies on a two-strain malaria model. Math Comput Appl. 2019;24(1):32.

[7] Bauer AL, Hogue IB, Marino S, Kirschner DE. The effects of hiv-1 infection on latent tuberculosis. Math Model Nat Phenomena. 2008;3(7):229–266. doi: 10.1051/mmnp:2008051. - DOI

[8] Bauer AL, Hogue IB, Marino S, Kirschner DE. The effects of hiv-1 infection on latent tuberculosis. Math Model Nat Phenomena. 2008;3(7):229–266. doi: 10.1051/mmnp:2008051. - DOI

[9] Bernoulli D (1760) Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l’inoculation pour la prévenir. Histoire de l’Acad Roy Sci (Paris) avec Mem, pages 1–45

[10] Bernoulli D (1760) Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l’inoculation pour la prévenir. Histoire de l’Acad Roy Sci (Paris) avec Mem, pages 1–45

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Optimal control of a two-group malaria transmission model with vaccination

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Optimal control of a two-group malaria transmission model with vaccination

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