Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis

Timothy Kiprono Yano; Jacob Bitok

doi:doi:10.11648/j.acm.20221105.13

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Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis

Timothy Kiprono Yano, Jacob Bitok

Published in Applied and Computational Mathematics (Volume 11, Issue 5)

Received: 24 September 2022 Accepted: 9 October 2022 Published: 17 October 2022

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Abstract

A normalized pneumonia mathematical model is formulated and analyzed to describe the transmission dynamics of pneumonia disease with a varying population size and in the presence of drug resistance threats. The main aim of the study is to formulate and analyze a pneumonia optimal control model that implements varied control strategies against antibiotic resistance threats and varying population size. The stability theory of differential equations and Pontryagin's Maximum Principle for an optimality system were employed to determine the crucial properties of the mathematical model. The basic reproduction number is determined using the Next generation matrix approach and the stability analysis for the disease-free and as well as for the endemic equilibrium are determined. The sensitivity indices of the effective reproduction number to the crucial parameter values are determined and ranked as per their impact on the transmission of pneumonia disease. We extend the model to an optimal control problem with four control strategies: disease prevention effort, treatment effort that minimize the sensitive and resistant strain and immunity control effort. The optimal control analysis of the adopted control efforts revealed that the combination of prevention and treatment, prevention and immunity control and a combination of all controls are the effective intervention strategies that result in a decrease in infections in the community. Numerical simulations are performed for a combination of other strategies and pertinent results were displayed graphically.

Published in	Applied and Computational Mathematics (Volume 11, Issue 5)
DOI	10.11648/j.acm.20221105.13
Page(s)	130-139
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

Streptococcus Pneumoniae, Effective Reproductive Number, Pontryagin's Maximum Principle, Optimal Control, Sensitivity Indices, Numerical Simulation, Optimal Control Analysis

References

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[3]	Fleming, W. H. and Rishel, R. W. (2012). Deterministic and stochastic optimal control, volume 1. Springer Science & Business Media.
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[8]	Kizito, M. and Tumwiine, J. (2018). A mathematical model of treatment and vaccination interventions of pneumococcal pneumonia infection dynamics. Journal of Applied Mathematics, 2018.
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[10]	Melegaro, A., Gay, N., and Medley, G. (2004). Estimating the transmission parameters of pneumococcal carriage in households. Epidemiology Infection, 132 (3): 433-441.
[11]	Okosun, K. O., Ouifki, R., and Marcus, N. (2011). Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems, 106 (23): 136-145.
[12]	Organization, W. H. et al. (2013a). Fact sheet on pneumonia. Weekly Epidemiological Record= Relevé épidémiologique hebdomadaire, 88 (11): 126-127.
[13]	Organization, W. H. et al. (2013b). The global view of campylobacteriosis: report of an expert consultation, Utrecht, Netherlands, 9-11 July 2012.
[14]	Otieno, M. J. O. J. and Paul, O. (2013). Mathematical model for pneumonia dynamics with carriers.
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[17]	Swai, M. C., Shaban, N., and Marijani, T. (2021). Optimal control in two strain pneumonia transmission dynamics. Journal of Applied Mathematics, 2021.
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[19]	Tilahun, G. T., Makinde, O. D., and Malonza, D. (2018). Co-dynamics of pneumonia and typhoid fever diseases with cost effective optimal control analysis. Applied Mathematics and Computation, 316: 438-459.
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APA Style

Timothy Kiprono Yano, Jacob Bitok. (2022). Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis. Applied and Computational Mathematics, 11(5), 130-139. https://doi.org/10.11648/j.acm.20221105.13

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ACS Style

Timothy Kiprono Yano; Jacob Bitok. Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis. Appl. Comput. Math. 2022, 11(5), 130-139. doi: 10.11648/j.acm.20221105.13

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AMA Style

Timothy Kiprono Yano, Jacob Bitok. Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis. Appl Comput Math. 2022;11(5):130-139. doi: 10.11648/j.acm.20221105.13

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@article{10.11648/j.acm.20221105.13,
  author = {Timothy Kiprono Yano and Jacob Bitok},
  title = {Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {5},
  pages = {130-139},
  doi = {10.11648/j.acm.20221105.13},
  url = {https://doi.org/10.11648/j.acm.20221105.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221105.13},
  abstract = {A normalized pneumonia mathematical model is formulated and analyzed to describe the transmission dynamics of pneumonia disease with a varying population size and in the presence of drug resistance threats. The main aim of the study is to formulate and analyze a pneumonia optimal control model that implements varied control strategies against antibiotic resistance threats and varying population size. The stability theory of differential equations and Pontryagin's Maximum Principle for an optimality system were employed to determine the crucial properties of the mathematical model. The basic reproduction number is determined using the Next generation matrix approach and the stability analysis for the disease-free and as well as for the endemic equilibrium are determined. The sensitivity indices of the effective reproduction number to the crucial parameter values are determined and ranked as per their impact on the transmission of pneumonia disease. We extend the model to an optimal control problem with four control strategies: disease prevention effort, treatment effort that minimize the sensitive and resistant strain and immunity control effort. The optimal control analysis of the adopted control efforts revealed that the combination of prevention and treatment, prevention and immunity control and a combination of all controls are the effective intervention strategies that result in a decrease in infections in the community. Numerical simulations are performed for a combination of other strategies and pertinent results were displayed graphically.},
 year = {2022}
}

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TY  - JOUR
T1  - Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis
AU  - Timothy Kiprono Yano
AU  - Jacob Bitok
Y1  - 2022/10/17
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221105.13
DO  - 10.11648/j.acm.20221105.13
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JO  - Applied and Computational Mathematics
SP  - 130
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PB  - Science Publishing Group
SN  - 2328-5613
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AB  - A normalized pneumonia mathematical model is formulated and analyzed to describe the transmission dynamics of pneumonia disease with a varying population size and in the presence of drug resistance threats. The main aim of the study is to formulate and analyze a pneumonia optimal control model that implements varied control strategies against antibiotic resistance threats and varying population size. The stability theory of differential equations and Pontryagin's Maximum Principle for an optimality system were employed to determine the crucial properties of the mathematical model. The basic reproduction number is determined using the Next generation matrix approach and the stability analysis for the disease-free and as well as for the endemic equilibrium are determined. The sensitivity indices of the effective reproduction number to the crucial parameter values are determined and ranked as per their impact on the transmission of pneumonia disease. We extend the model to an optimal control problem with four control strategies: disease prevention effort, treatment effort that minimize the sensitive and resistant strain and immunity control effort. The optimal control analysis of the adopted control efforts revealed that the combination of prevention and treatment, prevention and immunity control and a combination of all controls are the effective intervention strategies that result in a decrease in infections in the community. Numerical simulations are performed for a combination of other strategies and pertinent results were displayed graphically.
VL  - 11
IS  - 5
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Author Information

Timothy Kiprono Yano

Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya
Jacob Bitok

Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya

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APA Style

Timothy Kiprono Yano, Jacob Bitok. (2022). Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis. Applied and Computational Mathematics, 11(5), 130-139. https://doi.org/10.11648/j.acm.20221105.13

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ACS Style

Timothy Kiprono Yano; Jacob Bitok. Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis. Appl. Comput. Math. 2022, 11(5), 130-139. doi: 10.11648/j.acm.20221105.13

Copy | Download

AMA Style

Timothy Kiprono Yano, Jacob Bitok. Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis. Appl Comput Math. 2022;11(5):130-139. doi: 10.11648/j.acm.20221105.13

Copy | Download

@article{10.11648/j.acm.20221105.13,
  author = {Timothy Kiprono Yano and Jacob Bitok},
  title = {Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {5},
  pages = {130-139},
  doi = {10.11648/j.acm.20221105.13},
  url = {https://doi.org/10.11648/j.acm.20221105.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221105.13},
  abstract = {A normalized pneumonia mathematical model is formulated and analyzed to describe the transmission dynamics of pneumonia disease with a varying population size and in the presence of drug resistance threats. The main aim of the study is to formulate and analyze a pneumonia optimal control model that implements varied control strategies against antibiotic resistance threats and varying population size. The stability theory of differential equations and Pontryagin's Maximum Principle for an optimality system were employed to determine the crucial properties of the mathematical model. The basic reproduction number is determined using the Next generation matrix approach and the stability analysis for the disease-free and as well as for the endemic equilibrium are determined. The sensitivity indices of the effective reproduction number to the crucial parameter values are determined and ranked as per their impact on the transmission of pneumonia disease. We extend the model to an optimal control problem with four control strategies: disease prevention effort, treatment effort that minimize the sensitive and resistant strain and immunity control effort. The optimal control analysis of the adopted control efforts revealed that the combination of prevention and treatment, prevention and immunity control and a combination of all controls are the effective intervention strategies that result in a decrease in infections in the community. Numerical simulations are performed for a combination of other strategies and pertinent results were displayed graphically.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Computational Modelling of Pneumonia Disease Transmission Dynamics with Optimal Control Analysis
AU  - Timothy Kiprono Yano
AU  - Jacob Bitok
Y1  - 2022/10/17
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221105.13
DO  - 10.11648/j.acm.20221105.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 130
EP  - 139
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221105.13
AB  - A normalized pneumonia mathematical model is formulated and analyzed to describe the transmission dynamics of pneumonia disease with a varying population size and in the presence of drug resistance threats. The main aim of the study is to formulate and analyze a pneumonia optimal control model that implements varied control strategies against antibiotic resistance threats and varying population size. The stability theory of differential equations and Pontryagin's Maximum Principle for an optimality system were employed to determine the crucial properties of the mathematical model. The basic reproduction number is determined using the Next generation matrix approach and the stability analysis for the disease-free and as well as for the endemic equilibrium are determined. The sensitivity indices of the effective reproduction number to the crucial parameter values are determined and ranked as per their impact on the transmission of pneumonia disease. We extend the model to an optimal control problem with four control strategies: disease prevention effort, treatment effort that minimize the sensitive and resistant strain and immunity control effort. The optimal control analysis of the adopted control efforts revealed that the combination of prevention and treatment, prevention and immunity control and a combination of all controls are the effective intervention strategies that result in a decrease in infections in the community. Numerical simulations are performed for a combination of other strategies and pertinent results were displayed graphically.
VL  - 11
IS  - 5
ER  -

Copy | Download