Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives

Herick Laiton Kayange; Estomih Shedrack Massawe; Daniel Oluwole Makinde; Lathika Sunil Immanuel

doi:doi:10.11648/j.ijssam.20200504.12

| Peer-Reviewed

Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives

Herick Laiton Kayange, Estomih Shedrack Massawe, Daniel Oluwole Makinde, Lathika Sunil Immanuel

Published in International Journal of Systems Science and Applied Mathematics (Volume 5, Issue 4)

Received: 30 October 2020 Accepted: 11 November 2020 Published: 23 November 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, a non-linear mathematical model for the dynamics of Ebola virus diseases is formulated and analysed. The model has five classes namely susceptible human, exposed human, infected human, treated human and recovered human. Invariant region and positivity solution of the model are determined. Local stability analyses of disease free Equilibrium and endemic equilibrium are examined. The disease free equilibrium analysis is determined using Routh-Hurwitz criteria, whereby it is found to be locally stable if the reproduction number is less than one. Two control measures: control measure due to quarantine of exposed and susceptible individuals and control measure due to efficacy of treatment drug, used for treating Ebola virus disease Ebola victim are incorporated to the Ebola virus disease model. The control problem is then analysed in order to determine the optimal control. Numerical simulations for the model in the presence of control measures are finally performed. The results show that in the presence of optimal control, the Ebola virus disease can be eliminated in the Society. Furthermore, to minimize infections of Ebola virus disease, quarantine centres with skilled manpower must be prepared in advance so as to accommodate the significant number of exposed and susceptible individuals, in order to avoid further transmission in other areas out of quarantine centres. Also tracing of exposed and infected individuals must be efficiently done in order to quarantine the affected population and educate people on the transmission of the disease, symptoms and prevention measures in order to minimize human to human transmissions. Investing more on researches on new drugs which are effective in treating the Ebola virus disease victim is inevitable.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 5, Issue 4)
DOI	10.11648/j.ijssam.20200504.12
Page(s)	43-53
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), 2020. Published by Science Publishing Group

Keywords

Ebola, Control, Quarantine, Treatment, Infectives

References

[1]	Ustun C and Ozguller O 2004, Ebola: As an infectious Disease and as a potential Bioterrorism agent, Turk J med sci. 35(2005) 1-4
[2]	World Health Organisation (WHO), 2014, Ebola and Marburg Virus Disease Epidemics; Preparedness, alert, control and Evaluation. WHO Press; Geneva, Switzerland, www.paho.org/hq/Index.Php? option=com_docman&task=doc_view&gid=2416&Itemid
[3]	Birmingham K and Cooney S, 2002, Ebola: Small but real progress (News feature) Nature med, 8(4), 313
[4]	Sriram, 2014. An Overview on Ebola Virus Disease (EVD) or Ebola hemorrhagic Fever (EHF), International Journal of Allied Medical Sciences and clinical Research (IJAMSCR), 269 -278
[5]	Mupapa K, Massamba M, Kibadi K, Kuvula K, Bwaka A, Kipasa M, Colebunders R and Muyembe JJ, 2001. Treatment of Ebola hemorrhagic fever with blood transfusions from convalescent patients. The Journal of infectious diseases association of America 1999;179 (supp 1); S 18-23
[6]	Ki M, 2014. What do we really fear? The epidemiological characteristic of Ebola Virus and our preparedness. Korean Society of Epidemiology. Volume 36, Article ID: e2014014
[7]	Rodriguez HS, Monteiro MT and Torres DF, 2014. Optimal Control and Numerical Software: an Overview. Nova Science Publishers, New York, 93-110
[8]	Gaff H and Schaefer E, 2009. Optimal Control applied to Valuation and Treatment Strategies for Various Epidemiological models, pubmed. Jul; 6(3): 469-492
[9]	Gumel AB and Lenhart S, 2010. Modelling Paradigms and analysis of Disease Transmission Models, American Mathematical Society, New York, United States of America
[10]	Massawe LN (2015 Modelling Infectiology and Optimal Control of Dengue Epidemic, Applied and Computational Mathematics 2015; 4(3): 181-191
[11]	Chowell G, Hengartner NW, Castillo-Chavez C, Fenimore PW, Hyman JM: The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda. J Theor Biol. 2004, 229: 119-126. 10.1016/j.jtbi.2004.03.006
[12]	DiekMann O, Heesterbeek, JAP and Roberts MG, 2010, The construction of next generation matrices for compartmental epidemic models, J. R. Soc Interface, 7(47):873-885
[13]	Diekmann O, Heesterbeek JAP and Metz, JAJ, on the definition and computation of the basic Reproduction ratio, In models for infectious diseases in heterogeneous population, Journal of Mathematics Biology, 28(1990):365-382
[14]	Brauer F, Driessche VP and Wu J, 2008, mathematical epidemiology, Springer –verlag Berlin Heidelberg, Heidelberg, Germany
[15]	Cai L and Li Z, 2010, analysis of a simple vector-host epidemic model with direct transmission, xinyang university, Henan, China
[16]	Lenhart S and Workman JT, (2007). Optimal control, Applied to Biological Models, Chapman & hall/CRC, London, United Kingdom
[17]	Makinde OD and Okusun KO, 2013, Optimal control analysis of Malaria in the presence of Non-Linear incidence rate, Appl. Comp. Math, 12(1); 20-32
[18]	Rivers CM, Lofgren ET, Marathe M, Eubank S and Lewis BL, 2014. Modelling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia. Plos current outbreaks, fd 38 dd 8507856540bobe3fcd78f5ccf

Cite This Article

Plain Text BibTeX RIS

APA Style

Herick Laiton Kayange, Estomih Shedrack Massawe, Daniel Oluwole Makinde, Lathika Sunil Immanuel. (2020). Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives. International Journal of Systems Science and Applied Mathematics, 5(4), 43-53. https://doi.org/10.11648/j.ijssam.20200504.12

Copy | Download

ACS Style

Herick Laiton Kayange; Estomih Shedrack Massawe; Daniel Oluwole Makinde; Lathika Sunil Immanuel. Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives. Int. J. Syst. Sci. Appl. Math. 2020, 5(4), 43-53. doi: 10.11648/j.ijssam.20200504.12

Copy | Download

AMA Style

Herick Laiton Kayange, Estomih Shedrack Massawe, Daniel Oluwole Makinde, Lathika Sunil Immanuel. Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives. Int J Syst Sci Appl Math. 2020;5(4):43-53. doi: 10.11648/j.ijssam.20200504.12

Copy | Download

@article{10.11648/j.ijssam.20200504.12,
  author = {Herick Laiton Kayange and Estomih Shedrack Massawe and Daniel Oluwole Makinde and Lathika Sunil Immanuel},
  title = {Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {5},
  number = {4},
  pages = {43-53},
  doi = {10.11648/j.ijssam.20200504.12},
  url = {https://doi.org/10.11648/j.ijssam.20200504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20200504.12},
  abstract = {In this paper, a non-linear mathematical model for the dynamics of Ebola virus diseases is formulated and analysed. The model has five classes namely susceptible human, exposed human, infected human, treated human and recovered human. Invariant region and positivity solution of the model are determined. Local stability analyses of disease free Equilibrium and endemic equilibrium are examined. The disease free equilibrium analysis is determined using Routh-Hurwitz criteria, whereby it is found to be locally stable if the reproduction number is less than one. Two control measures: control measure due to quarantine of exposed and susceptible individuals and control measure due to efficacy of treatment drug, used for treating Ebola virus disease Ebola victim are incorporated to the Ebola virus disease model. The control problem is then analysed in order to determine the optimal control. Numerical simulations for the model in the presence of control measures are finally performed. The results show that in the presence of optimal control, the Ebola virus disease can be eliminated in the Society. Furthermore, to minimize infections of Ebola virus disease, quarantine centres with skilled manpower must be prepared in advance so as to accommodate the significant number of exposed and susceptible individuals, in order to avoid further transmission in other areas out of quarantine centres. Also tracing of exposed and infected individuals must be efficiently done in order to quarantine the affected population and educate people on the transmission of the disease, symptoms and prevention measures in order to minimize human to human transmissions. Investing more on researches on new drugs which are effective in treating the Ebola virus disease victim is inevitable.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives
AU  - Herick Laiton Kayange
AU  - Estomih Shedrack Massawe
AU  - Daniel Oluwole Makinde
AU  - Lathika Sunil Immanuel
Y1  - 2020/11/23
PY  - 2020
N1  - https://doi.org/10.11648/j.ijssam.20200504.12
DO  - 10.11648/j.ijssam.20200504.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 43
EP  - 53
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20200504.12
AB  - In this paper, a non-linear mathematical model for the dynamics of Ebola virus diseases is formulated and analysed. The model has five classes namely susceptible human, exposed human, infected human, treated human and recovered human. Invariant region and positivity solution of the model are determined. Local stability analyses of disease free Equilibrium and endemic equilibrium are examined. The disease free equilibrium analysis is determined using Routh-Hurwitz criteria, whereby it is found to be locally stable if the reproduction number is less than one. Two control measures: control measure due to quarantine of exposed and susceptible individuals and control measure due to efficacy of treatment drug, used for treating Ebola virus disease Ebola victim are incorporated to the Ebola virus disease model. The control problem is then analysed in order to determine the optimal control. Numerical simulations for the model in the presence of control measures are finally performed. The results show that in the presence of optimal control, the Ebola virus disease can be eliminated in the Society. Furthermore, to minimize infections of Ebola virus disease, quarantine centres with skilled manpower must be prepared in advance so as to accommodate the significant number of exposed and susceptible individuals, in order to avoid further transmission in other areas out of quarantine centres. Also tracing of exposed and infected individuals must be efficiently done in order to quarantine the affected population and educate people on the transmission of the disease, symptoms and prevention measures in order to minimize human to human transmissions. Investing more on researches on new drugs which are effective in treating the Ebola virus disease victim is inevitable.
VL  - 5
IS  - 4
ER  -

Copy | Download

Author Information

Herick Laiton Kayange

Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania
Estomih Shedrack Massawe

College of Engineering and Technology, St. Joseph University in Tanzania, Dar es Salaam, Tanzania
Daniel Oluwole Makinde

Faculty of Military Science, Stellenbosch University, Stellenbosch, South Africa
Lathika Sunil Immanuel

College of Engineering and Technology, St. Joseph University in Tanzania, Dar es Salaam, Tanzania

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Herick Laiton Kayange, Estomih Shedrack Massawe, Daniel Oluwole Makinde, Lathika Sunil Immanuel. (2020). Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives. International Journal of Systems Science and Applied Mathematics, 5(4), 43-53. https://doi.org/10.11648/j.ijssam.20200504.12

Copy | Download

ACS Style

Herick Laiton Kayange; Estomih Shedrack Massawe; Daniel Oluwole Makinde; Lathika Sunil Immanuel. Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives. Int. J. Syst. Sci. Appl. Math. 2020, 5(4), 43-53. doi: 10.11648/j.ijssam.20200504.12

Copy | Download

AMA Style

Herick Laiton Kayange, Estomih Shedrack Massawe, Daniel Oluwole Makinde, Lathika Sunil Immanuel. Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives. Int J Syst Sci Appl Math. 2020;5(4):43-53. doi: 10.11648/j.ijssam.20200504.12

Copy | Download

@article{10.11648/j.ijssam.20200504.12,
  author = {Herick Laiton Kayange and Estomih Shedrack Massawe and Daniel Oluwole Makinde and Lathika Sunil Immanuel},
  title = {Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {5},
  number = {4},
  pages = {43-53},
  doi = {10.11648/j.ijssam.20200504.12},
  url = {https://doi.org/10.11648/j.ijssam.20200504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20200504.12},
  abstract = {In this paper, a non-linear mathematical model for the dynamics of Ebola virus diseases is formulated and analysed. The model has five classes namely susceptible human, exposed human, infected human, treated human and recovered human. Invariant region and positivity solution of the model are determined. Local stability analyses of disease free Equilibrium and endemic equilibrium are examined. The disease free equilibrium analysis is determined using Routh-Hurwitz criteria, whereby it is found to be locally stable if the reproduction number is less than one. Two control measures: control measure due to quarantine of exposed and susceptible individuals and control measure due to efficacy of treatment drug, used for treating Ebola virus disease Ebola victim are incorporated to the Ebola virus disease model. The control problem is then analysed in order to determine the optimal control. Numerical simulations for the model in the presence of control measures are finally performed. The results show that in the presence of optimal control, the Ebola virus disease can be eliminated in the Society. Furthermore, to minimize infections of Ebola virus disease, quarantine centres with skilled manpower must be prepared in advance so as to accommodate the significant number of exposed and susceptible individuals, in order to avoid further transmission in other areas out of quarantine centres. Also tracing of exposed and infected individuals must be efficiently done in order to quarantine the affected population and educate people on the transmission of the disease, symptoms and prevention measures in order to minimize human to human transmissions. Investing more on researches on new drugs which are effective in treating the Ebola virus disease victim is inevitable.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives
AU  - Herick Laiton Kayange
AU  - Estomih Shedrack Massawe
AU  - Daniel Oluwole Makinde
AU  - Lathika Sunil Immanuel
Y1  - 2020/11/23
PY  - 2020
N1  - https://doi.org/10.11648/j.ijssam.20200504.12
DO  - 10.11648/j.ijssam.20200504.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 43
EP  - 53
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20200504.12
AB  - In this paper, a non-linear mathematical model for the dynamics of Ebola virus diseases is formulated and analysed. The model has five classes namely susceptible human, exposed human, infected human, treated human and recovered human. Invariant region and positivity solution of the model are determined. Local stability analyses of disease free Equilibrium and endemic equilibrium are examined. The disease free equilibrium analysis is determined using Routh-Hurwitz criteria, whereby it is found to be locally stable if the reproduction number is less than one. Two control measures: control measure due to quarantine of exposed and susceptible individuals and control measure due to efficacy of treatment drug, used for treating Ebola virus disease Ebola victim are incorporated to the Ebola virus disease model. The control problem is then analysed in order to determine the optimal control. Numerical simulations for the model in the presence of control measures are finally performed. The results show that in the presence of optimal control, the Ebola virus disease can be eliminated in the Society. Furthermore, to minimize infections of Ebola virus disease, quarantine centres with skilled manpower must be prepared in advance so as to accommodate the significant number of exposed and susceptible individuals, in order to avoid further transmission in other areas out of quarantine centres. Also tracing of exposed and infected individuals must be efficiently done in order to quarantine the affected population and educate people on the transmission of the disease, symptoms and prevention measures in order to minimize human to human transmissions. Investing more on researches on new drugs which are effective in treating the Ebola virus disease victim is inevitable.
VL  - 5
IS  - 4
ER  -

Copy | Download