SEIRS Mathematical Model for Malaria with Treatment

Alemu Geleta Wedajo; Purnachandra Rao Koya; Dereje Legesse Abaire

doi:doi:10.11648/j.mma.20200502.16

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SEIRS Mathematical Model for Malaria with Treatment

Alemu Geleta Wedajo, Purnachandra Rao Koya, Dereje Legesse Abaire

Published in Mathematical Modelling and Applications (Volume 5, Issue 2)

Received: 1 January 2020 Accepted: 5 May 2020 Published: 28 May 2020

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Abstract

In this paper a deterministic mathematical model for the spread of malaria in human and mosquito populations are presented. The model has a set of eight non – linear differential equations with five state variables for human and three for mosquito populations respectively. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, treatment and recovered or immune classes before coming back to the susceptible class. Susceptible mosquitoes can become infected when they bite infectious humans, and once infected they move through exposed and infectious class. However, mosquitoes once infected will never recover from the disease during their lifetime. That is, infected mosquitoes will remain infectious until they die. Formula for the basic reproduction number R₀ is established and used to determine whether the disease dies out or persists in the populations. It is shown that the disease – free equilibrium point is locally asymptotically stable using the magnitude of Eigen value and Routh – Hurwitz stability Criterion. Result and detailed discussion of the analysis as well as the simulation study is incorporated in the text of the paper lucidly.

Published in	Mathematical Modelling and Applications (Volume 5, Issue 2)
DOI	10.11648/j.mma.20200502.16
Page(s)	105-117
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.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Dynamics of Malaria, SEIRS Model, Treatment, Local Stability, Routh – Hurwitz Criterion, Reproduction Number, Simulation Study

References

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[2]	Diekmann O., Heesterbeek J. A. P., Metz J. A. J. (1990). "On the definition and computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations". J. Math. Biol. 28, 365–382 (doi: 10.1007/BF00178324).
[3]	FekaduTadege Kobe and Purnachandra Rao Koya (2015)."Controlling the Spread of Malaria Disease Using Intervention Strategies". Journal of Multidisciplinary Engineering Science and Technology (JMEST), ISSN: 3159-0040, Vol. 2 Issue 5.
[4]	G. A. Ngwa, W. S. Shu (2000)."A mathematical model for endemic malaria with variable human and mosquito populations". Mathematical and computer modeling, 32 (7), Pp 747-763.
[5]	Hartman T K, Rogerson S J, Fischer P R (2010). "The impact of maternal malaria on newborns". Annals of Tropical Paediatrics 30 (4).
[6]	Howitt P, Darzi A, Yang G Z, Ashra_an H, Atun R, Barlow J, Blake-more A, Bull A M, Car J, Conteh L, Cooke G S, Ford N, Gregson S A, Kerr K, King D, Kulendran M, Malkin R A, Majeed A, Matlin S, Mer-ri_eld R, Penfold H A, Reid S D, Smith P C, Stevens M. M., Templeton M R, Vincent C, Wilson E (2012). "Technologies for global health". The Lancet 380 (9840): 50735. doi: 10.1016/S0140-6736(12)61127-1. PMID 22857974.
[7]	Kajfasz P (2009). "Malaria prevention". International Maritime Health 60 (12): 6770. PMID 20205131.
[8]	Lozano R et al. (2012). "Global and regional mortality from 235 causes of death for 20 age groups in 1990 and 2010: A systematic analysis for the Global Burden of Disease Study 2010". Lancet 380 (9859): 2095128. doi: 10.1016/S0140-6736(12)61728-0. PMID 23245604.
[9]	Murray C J, Rosenfeld L C, Lim S S, Andrews K G, Foreman K J, Haring D, Fullman N, Naghavi M, Lozano R, Lopez AD (2012). "Global malaria mortality between 1980 and 2010: A systematic analysis". Lancet 379 (9814): 41331. doi: 10.1016/S0140-6736(12)60034-8. PMID 22305225.
[10]	N. Chitnis J. M. Cushing and J. M. Hyman (2006)."Bifurcation analysis of a mathematical model for malaria transmission". SIAM Journal on Applied Mathematics, 67 (1), 24-45.
[11]	Nadjm B, Behrens RH (2012). "Malaria: An update for physicians". Infectious Disease Clinics of North America 26 (2): 24359. doi: 10.1016/j.idc.2012.03.010. PMID 22632637.
[12]	Olupot – Olupot P, Maitland, K (2013). "Management of severe malaria: Results from recent trials". Advances in Experimental Medicine and Biology. Advances in Experimental Medicine and Biology 764: 24150. doi: 10.1007/978-1-4614-4726-9-20. ISBN 978-1-4614-4725-2.PMID 23654072.
[13]	SamwelOsekoNyachae, Johana K. Sigey, Jeconiah A. Okello, James M. Okwoyo& D. Theuri (2014), "A Study for the Spread of Malaria in Nyamira Town – Kenya", The SIJ Transactions on Computer Science Engineering & its Applications (CSEA), The Standard International Journals (The SIJ), Vol. 2, No. 3 (1), Pp. 53-60.
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APA Style

Alemu Geleta Wedajo, Purnachandra Rao Koya, Dereje Legesse Abaire. (2020). SEIRS Mathematical Model for Malaria with Treatment. Mathematical Modelling and Applications, 5(2), 105-117. https://doi.org/10.11648/j.mma.20200502.16

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ACS Style

Alemu Geleta Wedajo; Purnachandra Rao Koya; Dereje Legesse Abaire. SEIRS Mathematical Model for Malaria with Treatment. Math. Model. Appl. 2020, 5(2), 105-117. doi: 10.11648/j.mma.20200502.16

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AMA Style

Alemu Geleta Wedajo, Purnachandra Rao Koya, Dereje Legesse Abaire. SEIRS Mathematical Model for Malaria with Treatment. Math Model Appl. 2020;5(2):105-117. doi: 10.11648/j.mma.20200502.16

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@article{10.11648/j.mma.20200502.16,
  author = {Alemu Geleta Wedajo and Purnachandra Rao Koya and Dereje Legesse Abaire},
  title = {SEIRS Mathematical Model for Malaria with Treatment},
  journal = {Mathematical Modelling and Applications},
  volume = {5},
  number = {2},
  pages = {105-117},
  doi = {10.11648/j.mma.20200502.16},
  url = {https://doi.org/10.11648/j.mma.20200502.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200502.16},
  abstract = {In this paper a deterministic mathematical model for the spread of malaria in human and mosquito populations are presented. The model has a set of eight non – linear differential equations with five state variables for human and three for mosquito populations respectively. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, treatment and recovered or immune classes before coming back to the susceptible class. Susceptible mosquitoes can become infected when they bite infectious humans, and once infected they move through exposed and infectious class. However, mosquitoes once infected will never recover from the disease during their lifetime. That is, infected mosquitoes will remain infectious until they die. Formula for the basic reproduction number R0 is established and used to determine whether the disease dies out or persists in the populations. It is shown that the disease – free equilibrium point is locally asymptotically stable using the magnitude of Eigen value and Routh – Hurwitz stability Criterion. Result and detailed discussion of the analysis as well as the simulation study is incorporated in the text of the paper lucidly.},
 year = {2020}
}

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TY - JOUR
T1 - SEIRS Mathematical Model for Malaria with Treatment
AU - Alemu Geleta Wedajo
AU - Purnachandra Rao Koya
AU - Dereje Legesse Abaire
Y1 - 2020/05/28
PY - 2020
N1 - https://doi.org/10.11648/j.mma.20200502.16
DO - 10.11648/j.mma.20200502.16
T2 - Mathematical Modelling and Applications
JF - Mathematical Modelling and Applications
JO - Mathematical Modelling and Applications
SP - 105
EP - 117
PB - Science Publishing Group
SN - 2575-1794
UR - https://doi.org/10.11648/j.mma.20200502.16
AB - In this paper a deterministic mathematical model for the spread of malaria in human and mosquito populations are presented. The model has a set of eight non – linear differential equations with five state variables for human and three for mosquito populations respectively. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, treatment and recovered or immune classes before coming back to the susceptible class. Susceptible mosquitoes can become infected when they bite infectious humans, and once infected they move through exposed and infectious class. However, mosquitoes once infected will never recover from the disease during their lifetime. That is, infected mosquitoes will remain infectious until they die. Formula for the basic reproduction number R0 is established and used to determine whether the disease dies out or persists in the populations. It is shown that the disease – free equilibrium point is locally asymptotically stable using the magnitude of Eigen value and Routh – Hurwitz stability Criterion. Result and detailed discussion of the analysis as well as the simulation study is incorporated in the text of the paper lucidly.
VL - 5
IS - 2
ER -

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Author Information

Alemu Geleta Wedajo

Department of Mathematics, Wollega University, Nekemte, Ethiopia
Purnachandra Rao Koya

Department of Mathematics, Wollega University, Nekemte, Ethiopia
Dereje Legesse Abaire

Department of Mathematics, Madda Walabu University, Bale, Robe, Ethiopia

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APA Style

Alemu Geleta Wedajo, Purnachandra Rao Koya, Dereje Legesse Abaire. (2020). SEIRS Mathematical Model for Malaria with Treatment. Mathematical Modelling and Applications, 5(2), 105-117. https://doi.org/10.11648/j.mma.20200502.16

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ACS Style

Alemu Geleta Wedajo; Purnachandra Rao Koya; Dereje Legesse Abaire. SEIRS Mathematical Model for Malaria with Treatment. Math. Model. Appl. 2020, 5(2), 105-117. doi: 10.11648/j.mma.20200502.16

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AMA Style

Alemu Geleta Wedajo, Purnachandra Rao Koya, Dereje Legesse Abaire. SEIRS Mathematical Model for Malaria with Treatment. Math Model Appl. 2020;5(2):105-117. doi: 10.11648/j.mma.20200502.16

Copy | Download

@article{10.11648/j.mma.20200502.16,
  author = {Alemu Geleta Wedajo and Purnachandra Rao Koya and Dereje Legesse Abaire},
  title = {SEIRS Mathematical Model for Malaria with Treatment},
  journal = {Mathematical Modelling and Applications},
  volume = {5},
  number = {2},
  pages = {105-117},
  doi = {10.11648/j.mma.20200502.16},
  url = {https://doi.org/10.11648/j.mma.20200502.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200502.16},
  abstract = {In this paper a deterministic mathematical model for the spread of malaria in human and mosquito populations are presented. The model has a set of eight non – linear differential equations with five state variables for human and three for mosquito populations respectively. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, treatment and recovered or immune classes before coming back to the susceptible class. Susceptible mosquitoes can become infected when they bite infectious humans, and once infected they move through exposed and infectious class. However, mosquitoes once infected will never recover from the disease during their lifetime. That is, infected mosquitoes will remain infectious until they die. Formula for the basic reproduction number R0 is established and used to determine whether the disease dies out or persists in the populations. It is shown that the disease – free equilibrium point is locally asymptotically stable using the magnitude of Eigen value and Routh – Hurwitz stability Criterion. Result and detailed discussion of the analysis as well as the simulation study is incorporated in the text of the paper lucidly.},
 year = {2020}
}

Copy | Download