Malaria is an infectious disease caused by Plasmodium parasite and it is transmitted among humans through bites of female Anopheles mosquitoes. In this paper, a new deterministic mathematical model for the endemic malaria disease transmission that incorporates imperfect quarantine and optimal control is proposed. Impact of various intervention strategies in the community with varying population at time t are analyzed using mathematical techniques. Further, the model is analyzed using stability theory of differential equations and the basic reproduction number is obtained from the largest eigenvalue of the next-generation matrix. Conditions for local and global stability of disease free, local stability of endemic equilibria and bifurcations are determined in terms of the basic reproduction number. The Center manifold theory is used to analyze the bifurcation of the model. It is shown that the model exhibit both a backward and a forward bifurcation. Reducing the biting rate of the quarantined people is advice able to minimize the spread of endemic malaria disease. The optimal control is designed by applying Pontryagins’s Maximum Principle (PMP) with four control strategies namely, insecticide treated nets, screening, treatment and indoor residual spray. The best strategy to control endemic malaria disease is the combination that incorporated all four control strategies.
| Published in | Mathematical Modelling and Applications (Volume 6, Issue 2) |
| DOI | 10.11648/j.mma.20210602.12 |
| Page(s) | 29-55 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Endemic Malaria, Imperfect Quarantine, Reproduction Number, Stability, Bifurcation, Optimal Control
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APA Style
Dereje Gutema Edossa, Alemu Geleta Wedajo, Purnachandra Rao Koya. (2021). Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control. Mathematical Modelling and Applications, 6(2), 29-55. https://doi.org/10.11648/j.mma.20210602.12
ACS Style
Dereje Gutema Edossa; Alemu Geleta Wedajo; Purnachandra Rao Koya. Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control. Math. Model. Appl. 2021, 6(2), 29-55. doi: 10.11648/j.mma.20210602.12
AMA Style
Dereje Gutema Edossa, Alemu Geleta Wedajo, Purnachandra Rao Koya. Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control. Math Model Appl. 2021;6(2):29-55. doi: 10.11648/j.mma.20210602.12
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Download@article{10.11648/j.mma.20210602.12,
author = {Dereje Gutema Edossa and Alemu Geleta Wedajo and Purnachandra Rao Koya},
title = {Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control},
journal = {Mathematical Modelling and Applications},
volume = {6},
number = {2},
pages = {29-55},
doi = {10.11648/j.mma.20210602.12},
url = {https://doi.org/10.11648/j.mma.20210602.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20210602.12},
abstract = {Malaria is an infectious disease caused by Plasmodium parasite and it is transmitted among humans through bites of female Anopheles mosquitoes. In this paper, a new deterministic mathematical model for the endemic malaria disease transmission that incorporates imperfect quarantine and optimal control is proposed. Impact of various intervention strategies in the community with varying population at time t are analyzed using mathematical techniques. Further, the model is analyzed using stability theory of differential equations and the basic reproduction number is obtained from the largest eigenvalue of the next-generation matrix. Conditions for local and global stability of disease free, local stability of endemic equilibria and bifurcations are determined in terms of the basic reproduction number. The Center manifold theory is used to analyze the bifurcation of the model. It is shown that the model exhibit both a backward and a forward bifurcation. Reducing the biting rate of the quarantined people is advice able to minimize the spread of endemic malaria disease. The optimal control is designed by applying Pontryagins’s Maximum Principle (PMP) with four control strategies namely, insecticide treated nets, screening, treatment and indoor residual spray. The best strategy to control endemic malaria disease is the combination that incorporated all four control strategies.},
year = {2021}
}
TY - JOUR T1 - Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control AU - Dereje Gutema Edossa AU - Alemu Geleta Wedajo AU - Purnachandra Rao Koya Y1 - 2021/07/16 PY - 2021 N1 - https://doi.org/10.11648/j.mma.20210602.12 DO - 10.11648/j.mma.20210602.12 T2 - Mathematical Modelling and Applications JF - Mathematical Modelling and Applications JO - Mathematical Modelling and Applications SP - 29 EP - 55 PB - Science Publishing Group SN - 2575-1794 UR - https://doi.org/10.11648/j.mma.20210602.12 AB - Malaria is an infectious disease caused by Plasmodium parasite and it is transmitted among humans through bites of female Anopheles mosquitoes. In this paper, a new deterministic mathematical model for the endemic malaria disease transmission that incorporates imperfect quarantine and optimal control is proposed. Impact of various intervention strategies in the community with varying population at time t are analyzed using mathematical techniques. Further, the model is analyzed using stability theory of differential equations and the basic reproduction number is obtained from the largest eigenvalue of the next-generation matrix. Conditions for local and global stability of disease free, local stability of endemic equilibria and bifurcations are determined in terms of the basic reproduction number. The Center manifold theory is used to analyze the bifurcation of the model. It is shown that the model exhibit both a backward and a forward bifurcation. Reducing the biting rate of the quarantined people is advice able to minimize the spread of endemic malaria disease. The optimal control is designed by applying Pontryagins’s Maximum Principle (PMP) with four control strategies namely, insecticide treated nets, screening, treatment and indoor residual spray. The best strategy to control endemic malaria disease is the combination that incorporated all four control strategies. VL - 6 IS - 2 ER -
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