An investigation into the optimal control of the horizontal and vertical incidence of communicable infectious diseases in society

Document Type : Research Article

Author

Department of Mathematics, Payame Noor Unvierstiy, Tehran, Irann

Abstract

This article aims at proposing and developing a three-component mathematical model for susceptible, infected and recovered $(SIR)$ population, under the control of vaccination of the susceptible population and drug therapy (antivirus) of the infected population (patient) in case of an infectious disease. The infectious disease under study can be transmitted through direct contact with an infected person (horizontal transmission) and from parent to child (vertical transmission). We investigate the basic reproduction number of the mathematical model, the existence and local asymptotic stability of both the disease free and endemic equilibrium. Using Pontryagin's minimum principle, we investigate the conditions of reducing the susceptible and infected population and increasing the recovered population based on the use of these two controllers in society. A numerical simulation of the optimal control problem shows, using  both controllers is much more effective and leads to a rapid increase in the recovered population and prevents the disease from spreading and becoming an epidemic in
the society.

Keywords

Main Subjects

References

[1] A.S. Ahmad, S. Owyed, A.H. Abdel-Aty, E.E. Mahmoud, K. Shah, H. Alrabaiahg, Mathematical analysis of COVID-19 via new mathematical model, Chaos Solit. 143 (2021) 110585.
[2] R. Akbari, A.V. Kamyad, A.A. Heydari, A. Heydari, The analysis of a disease-free equilibrium of hepatitis B model, Sahand Commun. Math. Anal. 3 (2016) 1–11.
[3] R. Akbari, A.V. Kamyad, A.A. Heydari, A. Heydari, Stability analysis of the transmission dynamics of an HBV model, Int. J. Ind. Math. 8 (2016) 119–129.
[4] R.M. Anderson, R.M. May, Infectious Diseases of Humans Dynamics and Control, Oxford University Press, 1991.
[5] E.A. Bakar, A. Nwagwo, E. Danso-Addo, Optimal control analysis of an SIR epidemic model with constant recruitment, Int. J. Appl. Math. Res. 3 (2014) 273–285.
[6] S. Bhattacharyyaa, S. Ghosh, Optimal control of vertically transmitted disease, Comput. Math. Methods Med. 11 (2010) 369–387.
[7] G. Birkhoff, G.C.C. Rota, Ordinary Differential Equations, Ginn, Boston, 1982.
[8] S. Boccaletti, G. Bianconi, R. Criado, C.I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep. 544 (2014) 1–122.
[9] O. Diekmann, J.A.P. Heesterbeek, Mathematical Epidemiology of Infective Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000.
[10] P.V.D. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental systems of disease transmission, Math. Biosci. 180 (2002) 29–48.
[11] W. Fleming, R. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.
[12] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000) 599–653.
[13] S. Janaa, P. Haldarb, T.K. Kar, Optimal control and stability analysis of an epidemic model with population dispersal, Chaos, Solitons & Fractals 83 (2016) 67–81.
[14] A.V. Kamyad, R. Akbari, A.A. Heydari, A. Heydari, Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B Virus, Comput. Math. Methods Med. 2014, Article ID 475451.
[15] T.K. Kar, A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosyst. 104 (2011) 127–135.
[16] M.J. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2007.
[17] C.C. McCluskey, P.V.D. Driessche, Global analysis of two tuberculosis models, J. Dyn. Differ. Equ. 16 (2004) 139–166.
[18] J.S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mt. J. Math. 20 (1990) 857-872.
[19] R. Shi, Y. Li, C. Wang, Stability analysis and optimal control of a fractional-order model for African swine fever, Virus Res. 288 (2020) 198111.
[20] L. Wang, X. Li, Spatial epidemiology of networked metapopulation: an overview, Chin. Sci. Bull. 59(28) (2014) 3511–3522.
[21] L. Wang, Z. Wang, Y. Zhang, X. Li, How human location-specific contact patterns impact spatial transmission between populations, Sci. Rep. 3 (2013) 1468.
[22] C. Xia, Z. Wang, J. Sanz, Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks, Physica A. 392 (2013) 1577-1585.
[23] J. Zhang, The Geometric Theory and Bifurcation Problem of Ordinary Differential Equations, Peking University Press, Beijing, 1987.
[24] G. Zhang, Q. Sun, L. Wang, Noise-induced enhancement of network reciprocity in social dilemmas, Chaos, Solitons & Fractals 51 (2013) 31–35.
[25] S. Zhang, Y. Zhou, The analysis and application of an HBV model, Appl. Math. Model. 36 (2012) 1302–1312.
Journal of Mathematical Modeling
Volume 11, Issue 4
December 2023
Pages 605-616

Files

Share

How to cite

Statistics