Modelling of Malaria Transmission Using Delay Differential Equation

Kipkirui Mibei; Kirui Wesley; Adicka Daniel

doi:doi:10.11648/j.mma.20200503.15

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Modelling of Malaria Transmission Using Delay Differential Equation

Kipkirui Mibei, Kirui Wesley, Adicka Daniel

Published in Mathematical Modelling and Applications (Volume 5, Issue 3)

Received: 10 May 2020 Accepted: 14 July 2020 Published: 4 August 2020

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Abstract

Malaria is one of the major causes of deaths and ill health in endemic regions of sub-Saharan Africa and beyond despite efforts made to prevent and control its spread. Epidemiological models on how malaria is spread have made a substantial contribution on the understanding of disease changing aspects. Previous researchers have used Susceptible –Exposed-Infectious-Recovered (SEIR) model to explain how malaria is spread using ordinary differential equations. In this paper we develop mathematical SEIR model to define the dynamics of the spread of malaria using Delay differential equations with four control measures such as long lasting treated insecticides bed nets, intermittent preventive treatment of malaria in pregnant women (IPTP), intermittent preventive treatment of malaria in infancy (IPTI) and indoor residual spraying. The model is analyzed and reproduction number derived using next generation matrix method and its stability is checked by Jacobean matrix. Positivity of solutions and boundedness of the model is proved. We show that the disease free equilibrium is locally asymptotically stable if R₀<1 (R₀ – reproduction number) and is unstable if R₀>1. Numerical simulation shows that, with proper treatment and control measures put in place the disease is controlled.

Published in	Mathematical Modelling and Applications (Volume 5, Issue 3)
DOI	10.11648/j.mma.20200503.15
Page(s)	167-175
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

Stability, Basic Reproduction Number, Delay Differential Equations

References

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[2]	WHO (2016). World malaria report. Retrieved from http://www.who.int/malaria/publications.
[3]	Diekmann o, H. J. (2000). Mathematical epidemiology of infectious diseases. model builing, analysis and interpretation, 503-522.
[4]	Tumwiine, J. e. (2014). A mathematical model for transmission and spread of drug sensitive and resistant malaria strains within a human population. ID 636973.
[5]	Sunita, D. e. (2017). A SEIR model for malaria with infective immigrant. International Journal of Recent Research Vol 11, No 2, pg 155-160.
[6]	Abay, A. (2015). Mathematical modelling of endemic malaria transmission. America Journal of Applied Mathematics, 36-46.
[7]	Jessicca, M. e., & etal. (2018). Qualitative analysis of a mathematical model applied to malaria transmission in Tumaco.
[8]	Oduro, F. a. (2012). Transmission dynamics of malaria in Ghana. Journal of mathematics research, 4:6.
[9]	Mojeeb, K. A. (2017). A Simple SEIR mathematical model of malaria transmission. Asian Research Journalof Mathematics.
[10]	Ephraim Agyingi, M. N. (2016). The dynamics of multiple species and strains of malaria. School of mathematical sciences, Rochester Institute of technology vol 1, 29-40.
[11]	Prince Harvim (2014), An epidemiological model of malaria transmission in Ghana. Thesis paper Kwame Nkurumah University and Technology.
[12]	Nisha Budhar, S. D. (2017). Stability analysis of Human- mosquito model of malaria. International journal of mathematical and computational sciences Vol 11.
[13]	Kbenesh, B. e. (2009). 0ptimal control of vector-borne diseases: treatment and prevention. Discrete continuous dynamical system B11 (3), 587-611.
[14]	https./dph.georgia.gov.(2014). Malaria Georgia department of public health.
[15]	Flores J. D. Math-735. Mathematical modelling. 4.5 Routh-Hurwitz criteria. Department of mathematical sciences. The University of south Dakota, jFlores@usd.edu.

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APA Style

Kipkirui Mibei, Kirui Wesley, Adicka Daniel. (2020). Modelling of Malaria Transmission Using Delay Differential Equation. Mathematical Modelling and Applications, 5(3), 167-175. https://doi.org/10.11648/j.mma.20200503.15

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ACS Style

Kipkirui Mibei; Kirui Wesley; Adicka Daniel. Modelling of Malaria Transmission Using Delay Differential Equation. Math. Model. Appl. 2020, 5(3), 167-175. doi: 10.11648/j.mma.20200503.15

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AMA Style

Kipkirui Mibei, Kirui Wesley, Adicka Daniel. Modelling of Malaria Transmission Using Delay Differential Equation. Math Model Appl. 2020;5(3):167-175. doi: 10.11648/j.mma.20200503.15

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@article{10.11648/j.mma.20200503.15,
  author = {Kipkirui Mibei and Kirui Wesley and Adicka Daniel},
  title = {Modelling of Malaria Transmission Using Delay Differential Equation},
  journal = {Mathematical Modelling and Applications},
  volume = {5},
  number = {3},
  pages = {167-175},
  doi = {10.11648/j.mma.20200503.15},
  url = {https://doi.org/10.11648/j.mma.20200503.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200503.15},
  abstract = {Malaria is one of the major causes of deaths and ill health in endemic regions of sub-Saharan Africa and beyond despite efforts made to prevent and control its spread. Epidemiological models on how malaria is spread have made a substantial contribution on the understanding of disease changing aspects. Previous researchers have used Susceptible –Exposed-Infectious-Recovered (SEIR) model to explain how malaria is spread using ordinary differential equations. In this paper we develop mathematical SEIR model to define the dynamics of the spread of malaria using Delay differential equations with four control measures such as long lasting treated insecticides bed nets, intermittent preventive treatment of malaria in pregnant women (IPTP), intermittent preventive treatment of malaria in infancy (IPTI) and indoor residual spraying. The model is analyzed and reproduction number derived using next generation matrix method and its stability is checked by Jacobean matrix. Positivity of solutions and boundedness of the model is proved. We show that the disease free equilibrium is locally asymptotically stable if R00 – reproduction number) and is unstable if R0>1. Numerical simulation shows that, with proper treatment and control measures put in place the disease is controlled.},
 year = {2020}
}

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TY  - JOUR
T1  - Modelling of Malaria Transmission Using Delay Differential Equation
AU  - Kipkirui Mibei
AU  - Kirui Wesley
AU  - Adicka Daniel
Y1  - 2020/08/04
PY  - 2020
N1  - https://doi.org/10.11648/j.mma.20200503.15
DO  - 10.11648/j.mma.20200503.15
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 167
EP  - 175
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20200503.15
AB  - Malaria is one of the major causes of deaths and ill health in endemic regions of sub-Saharan Africa and beyond despite efforts made to prevent and control its spread. Epidemiological models on how malaria is spread have made a substantial contribution on the understanding of disease changing aspects. Previous researchers have used Susceptible –Exposed-Infectious-Recovered (SEIR) model to explain how malaria is spread using ordinary differential equations. In this paper we develop mathematical SEIR model to define the dynamics of the spread of malaria using Delay differential equations with four control measures such as long lasting treated insecticides bed nets, intermittent preventive treatment of malaria in pregnant women (IPTP), intermittent preventive treatment of malaria in infancy (IPTI) and indoor residual spraying. The model is analyzed and reproduction number derived using next generation matrix method and its stability is checked by Jacobean matrix. Positivity of solutions and boundedness of the model is proved. We show that the disease free equilibrium is locally asymptotically stable if R00 – reproduction number) and is unstable if R0>1. Numerical simulation shows that, with proper treatment and control measures put in place the disease is controlled.
VL  - 5
IS  - 3
ER  -

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Author Information

Kipkirui Mibei

Department of Mathematics, Faculty of Science and Technology, University of Kabianga, Kericho, Kenya
Kirui Wesley

Department of Mathematics, Faculty of Science South Eastern, Kenya University, Kitui, Kenya
Adicka Daniel

Department of Mathematics, Faculty of Science and Technology, University of Kabianga, Kericho, Kenya

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Plain Text BibTeX RIS

APA Style

Kipkirui Mibei, Kirui Wesley, Adicka Daniel. (2020). Modelling of Malaria Transmission Using Delay Differential Equation. Mathematical Modelling and Applications, 5(3), 167-175. https://doi.org/10.11648/j.mma.20200503.15

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ACS Style

Kipkirui Mibei; Kirui Wesley; Adicka Daniel. Modelling of Malaria Transmission Using Delay Differential Equation. Math. Model. Appl. 2020, 5(3), 167-175. doi: 10.11648/j.mma.20200503.15

Copy | Download

AMA Style

Kipkirui Mibei, Kirui Wesley, Adicka Daniel. Modelling of Malaria Transmission Using Delay Differential Equation. Math Model Appl. 2020;5(3):167-175. doi: 10.11648/j.mma.20200503.15

Copy | Download

@article{10.11648/j.mma.20200503.15,
  author = {Kipkirui Mibei and Kirui Wesley and Adicka Daniel},
  title = {Modelling of Malaria Transmission Using Delay Differential Equation},
  journal = {Mathematical Modelling and Applications},
  volume = {5},
  number = {3},
  pages = {167-175},
  doi = {10.11648/j.mma.20200503.15},
  url = {https://doi.org/10.11648/j.mma.20200503.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200503.15},
  abstract = {Malaria is one of the major causes of deaths and ill health in endemic regions of sub-Saharan Africa and beyond despite efforts made to prevent and control its spread. Epidemiological models on how malaria is spread have made a substantial contribution on the understanding of disease changing aspects. Previous researchers have used Susceptible –Exposed-Infectious-Recovered (SEIR) model to explain how malaria is spread using ordinary differential equations. In this paper we develop mathematical SEIR model to define the dynamics of the spread of malaria using Delay differential equations with four control measures such as long lasting treated insecticides bed nets, intermittent preventive treatment of malaria in pregnant women (IPTP), intermittent preventive treatment of malaria in infancy (IPTI) and indoor residual spraying. The model is analyzed and reproduction number derived using next generation matrix method and its stability is checked by Jacobean matrix. Positivity of solutions and boundedness of the model is proved. We show that the disease free equilibrium is locally asymptotically stable if R00 – reproduction number) and is unstable if R0>1. Numerical simulation shows that, with proper treatment and control measures put in place the disease is controlled.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Modelling of Malaria Transmission Using Delay Differential Equation
AU  - Kipkirui Mibei
AU  - Kirui Wesley
AU  - Adicka Daniel
Y1  - 2020/08/04
PY  - 2020
N1  - https://doi.org/10.11648/j.mma.20200503.15
DO  - 10.11648/j.mma.20200503.15
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 167
EP  - 175
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20200503.15
AB  - Malaria is one of the major causes of deaths and ill health in endemic regions of sub-Saharan Africa and beyond despite efforts made to prevent and control its spread. Epidemiological models on how malaria is spread have made a substantial contribution on the understanding of disease changing aspects. Previous researchers have used Susceptible –Exposed-Infectious-Recovered (SEIR) model to explain how malaria is spread using ordinary differential equations. In this paper we develop mathematical SEIR model to define the dynamics of the spread of malaria using Delay differential equations with four control measures such as long lasting treated insecticides bed nets, intermittent preventive treatment of malaria in pregnant women (IPTP), intermittent preventive treatment of malaria in infancy (IPTI) and indoor residual spraying. The model is analyzed and reproduction number derived using next generation matrix method and its stability is checked by Jacobean matrix. Positivity of solutions and boundedness of the model is proved. We show that the disease free equilibrium is locally asymptotically stable if R00 – reproduction number) and is unstable if R0>1. Numerical simulation shows that, with proper treatment and control measures put in place the disease is controlled.
VL  - 5
IS  - 3
ER  -

Copy | Download