A spatiotemporal SIR epidemic model two-dimensional with problem of optimal control
Resumo
In this work, we are interested in studying a spatiotemporal two-dimensional SIR epidemic model, in the form of a system of partial differential equations (PDE). A distribution of a vaccine in the form of a control variable is considered to force immunity. The purpose is to characterize a control that minimizes the number of susceptible, infected individuals and the costs associated with vaccination over a finite space and time domain. In addition, the existence of the solution of the state system and the optimal control is proved. The characterization of the control is given in terms of state function and adjoint function. The numerical resolution of the state system shows the effectiveness of our control strategy.
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Referências
Medlock J, Kot M (2003) Spreading disease: integro-differential equations old and new. Math Biosci 184(2):201-222. https://doi.org/10.1016/S0025-5564(03)00041-5
Bailey N T J (1975) The mathematical theory of infectious diseases and its applications. Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP 13 6LE, London 3.
Anderson R M, May R M, Anderson B (1992) Infectious diseases of humans: dynamics and control, vol 28. Wiley Online Library, New York.
Hethcote H W (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599-653. https://doi.org/10.1137/S0036144500371907
Brauer F, Castillo-Chavez C (2001) Mathematical models in population biology and epidemiology, vol 40. Springer, New York. https://doi.org/10.1007/978-1-4757-3516-1
Keeling M J, Rohani P (2008) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton. https://doi.org/10.1515/9781400841035
Huppert A, Katriel G (2013) Mathematical modelling and prediction in infectious disease epidemiology. Clin Microbiol Infect 19(11):999-1005. https://doi.org/10.1111/1469-0691.12308
J. R. Beddington, "Mutual interference between parasites or predators and its effect on searching efficiency," Journal of Animal Ecology,pp.331-341, 1975. https://doi.org/10.2307/3866
D. L. DeAngelis, R. A. Goldsten,and R. Neill, "A modelfor trophic interaction," Ecology,vol.56,pp.881-892,1975. https://doi.org/10.2307/1936298
R. S. Cantrell andC.Cosner, "Onthe Dynamics of PredatorPrey Models with the Beddington-Deangelis Functional Response," Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp. 206-222, 2001. https://doi.org/10.1006/jmaa.2000.7343
P. H. Crowley and E. K. Martin, "Functional responses and interference within and between year classes of a dragonfly population," Journal of theNorth American BenthologicalSociety, vol.8,pp.211-221,1989. https://doi.org/10.2307/1467324
X. Zhou and J. Cui, "Global stability of the viral dynamics with crowley-martin functional response," Bulletin of the Korean Mathematical Society,vol.48,no.3,pp.555-574,2011. https://doi.org/10.4134/BKMS.2011.48.3.555
X. Q. Liu, S. M. Zhong, B. D. Tian, and F. X. Zheng,"Asymptotic properties of a stochastic predator-prey model with CrowleyMartin functional response," Journal of Applied Mathematics and Computing,vol.43,no.1-2,pp.479-490,2013. https://doi.org/10.1007/s12190-013-0674-0
Lotfi, El Mehdi and Maziane, Mehdi and Hattaf, Khalid and Yousfi, Noura Partial differential equations of an epidemic model with spatial diffusion. International Journal of Partial Differential Equations journal=International Journal of Partial Differential Equations, 2014. https://doi.org/10.1155/2014/186437
K. Hattaf, A. A. Lashari, Y. Louartassi, and N. Yousfi, "A delayed SIR epidemic model with general incidence rate," Electronic Journal of Qualitative Theory of Differential Equations,vol.3,pp. 1-9, 2013. https://doi.org/10.14232/ejqtde.2013.1.3
Viorel Barbu. Mathematical methods in optimization of differential systems, volume 310. Springer Science & Business Media, 2012.
Amnon Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44. Springer Science & Business Media, 2012.
Ioan I Vrabie. C0-semigroups and applications, volume 191 of north-holland mathematics studies, 2003.
Tim Clayton, Scott Duke-Sylvester, Louis J Gross, Suzanne Lenhart, and Leslie A Real. Optimal control of a rabies epidemic model with a birth pulse. Journal of biological dynamics, 4(1):43-58, 2010. https://doi.org/10.1080/17513750902935216
Joshua Kiddy K Asamoah, Francis T Oduro, Ebenezer Bonyah, and Baba Seidu. Modelling of rabies transmission dynamics using optimal control analysis. Journal of Applied Mathematics, 2017. https://doi.org/10.1155/2017/2451237
McAsey, Michael and Mou, Libin and Han, Weimin Convergence of the forward-backward sweep method in optimal control. Computational Optimization and Applications,53(1):207-226, Springer, 2012. https://doi.org/10.1007/s10589-011-9454-7
Haim Brezis, Philippe G Ciarlet, and Jacques Louis Lions. Analyse fonctionnelle: th'eorie et applications, volume 91. Dunod Paris, 1999.
Ioan I Vrabie. C0-semigroups and applications, volume 191 of north-holland mathematics studies, 2003.
Joel Smoller. Shock waves and reaction-diffusion equations, volume 258. Springer Science & Business Media, 2012.
Adil El-Alami Laaroussi, Mostafa Rachik,Mohamed Elhia, "An optimal control problem for a spatiotemporal SIR model", "International Journal of Dynamics and Control " https://doi.org/10.1007/s40435-016-0283-5
Ghazzali, R., Laaroussi, A. E., EL Bhih, A. et al. Int. J. Dynam. Control (2019) 7: 1021. https://doi.org/10.1007/s40435-019-00553-6
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