Optimal control the prevalence of horizontally transmitted infectious diseases in the community

Document Type : Original Paper

Authors

1 Department of Mathematical, Payam Noor University, Tehran- Iran

2 Department of Statistics, Payam Noor University, Tehran- Iran

Abstract

‎In this paper, we propose a three-component mathematical model, including Suspected-Infected-Recovered individuals (SIR), under the control of maple vaccination, for infectious diseases. In such, infectious disease can be transmitted through contact with an infected person (horizontal transmission). Vaccination of suspected population will reduce the horizontal transmission of patients in the community. The mathematical model has two disease-free and endemic equilibrium points. The basic reproduction rate of the model, the existence and local asymptotic stability of these two equilibrium points are investigated. By using Pontriagin's minimum principle, we have investigated the conditions of reducing the suspected and infected population and increasing the recovered population due to the use of vaccination in the community. Numerical simulations to the optimal control problem show that control measures can lead to a decrease in the number of suspected population and an increase in recovered population ‎a‎nd it prevents the spread of the disease and becoming into an epidemic.

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References

  1. A. S. Ahmad, S. Owyed, A. H. A. Aty, E. E. Mahmoud, K. Shah and H. Alrabaiahg , Mathematical analysis of COVID-19 via new mathematical model, Chaos, Solitons and Fractals. 143 (2021) 110585.
    2.R. Akbari, A. V. Kamyad, A. A. Heydari and A. Heydari, The analysis of a disease-free equilibrium of hepatitis B model, Sahand Communications in Mathematical Analysis, 3(2) (2016) 1–11.
    3. R. Akbari, A. V. Kamyad, A. A. Heydari and A. Heydari, Stability analysis of the transmission dynamics of an HBV model, Int. J. Industrial Mathematics. 8(2) (2016) 11 pages.
    4. R. M. Anderson and R. M. May, Infectious diseases of humans dynamics and control, Oxford University Press, Oxford, 1991.
    5. E. A. Bakare, A. Nwagwo and E. Danso-Addo, Optimal control analysis of an SIR epidemic model with constant recruitment. International Journal of Applied Mathematical Research. 3(3) (2014) 273–285.
    6. S. Bhattacharyya and S. Ghosh, Optimal control of vertically transmitted disease, Computational and Mathematical Methods in Medicine, 11 (2010) 369–387.
    7.G. Birkhoff and G. C. C. Rota, Ordinary differential equations, John Wiley and Sons, New York, 1989.

8.S. Boccaletti, G. Bianconi and R. Criado, The structure and dynamics of multilayer networks, Phys. Rep. 554 (2014) 1–122.
9. O. Diekmann and J. A. P. Heesterbeek , Mathematical epidemiology of infective diseases: model building, Analysis and Interpretation, Wiley, New York, 2000.
10.A. V. Kamyad, R. Akbari, A. A. Heydari and A. Heydari, Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus, Computational and Mathematical Methods in Medicine. 2014 http://dx.doi.org/10.1155/2014/475451.
11. T. K. Kar and A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems. 104 (2011) 127–135.
12. M. J. Keeling and P. Rohani, Modeling infectious diseases in humans and animals, Princeton University Press, Princeton, NJ, 2007.
13. M. K. Libbus and L. M. Phillips, Public health management of perinatal hepatitis B virus, Public Health Nursing. 26(4) (2009) 353–361.
14. C. C. McCluskey and P. Van Den Driessche , Global analysis of two tuberculosis models, J. Dyn. Differ. Equ. 16(1) (2004) 139–166.
15. J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mt. J. Math. 20 (1990) 857–872.
16. S. Thornley , C. Bullen and M. Roberts, Hepatitis B in a high prevalence New Zealand population: a mathematical model applied to infection control policy, Journal ofTheoretical Biology. 254(3) (2008) 599–603.
17. L. Wang and X. Li, Spatial epidemiology of net worked metapopulation: an overview, Chin. Sci. Bull. 59 (2014) 3511–3522.
18.L. Wang, Z. Wang, Y. Zhang and X. Li, How human location-specific contact patterns impact spatial transmission between populations, Sci Rep. 3 (2013) 1468.
19.reading of diseases in complex networks, Physica A. 392 (2013) 1577–1585.
20. J. Zhang, The geometric theory and bifurcation problem of ordinary differential equations, Peking University Press, Beijing, 1987.
21. G. Zhang and Q. Sun, Noise-induced enhancement of network reciprocity in social dilemmas, Chaos, Solitons Fractals. 51 (2003) 31–35.
[2 C. Xia, Z. Wang and J. Sanz, Effects of delayed recovery and nonuniform transmission on the sp2] H. Zhang, Z. Wu and X. Xu, Impacts of subsidy policies on vaccination decisions in contact networks, Phys. Rev. E. 88 (2013) 012813.

 

Journal of Advanced Mathematical Modeling
Volume 13, Issue 2 - Serial Number 30
September 2023
Pages 284-296

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History

  • Receive Date: 15 January 2023
  • Revise Date: 22 August 2023
  • Accept Date: 30 August 2023

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