A review of optimal control strategies to inhibit contagious infectious diseases

[6] Balderrama, R., Peressutti, J., Pinasco, J. P., Vazquez, F., and Vega, C. S. D. L., 2022. Optimal
control for a SIR epidemic model with limited quarantine. Scientific Reports, 12(1), 12583. doi:
10.1038/s41598­022­16619­z
[7] Banerjee, R. and Biswas, R. K., 2020. Fractional optimal Control of Compartmental SIR model of
COVID­19: Shoing the Impact of Effective Vaccination. IFAC PapersOnLine, 55(1), PP. 616­622.
doi: https://doi.org/10.1016/j.ifacol.2022.04.101
[8] Barro, M., Guiro, A. and Ouedraogo, D., 2018. Optimal control of a SIR epidemic model with general incidence function and a time delays. CUBO A Mathematical Journal, 20(2), PP. 53­66. doi:
https://doi.org/10.4067/S0719­06462018000200053
[9] Benyah, F. and Yusuf, T. T., 2012. ,Optimal control of vaccination and treatment for an SIR epidemiological Model. World Journal of Modelling and Simulation, 8(3), PP., 194­204.
[10] Bhattacharyya, S. and Ghosh, S., 2010. Optimal control of vertically transmitted disease. Computational and Mathematical Methods in Medicine, 11 pp. 369­387. doi:
https://doi.org/10.1155/2010/520830
[11] Birkhoff, G. and Rota, G. C., 1989. Ordinary Differential Equations. 4th ed. John Wiley & Sons, New York.
[12] Bliman, P.A., Duprez, M., Privat, Y. and Vauchelet, N., 2021. Optimal Immunity Control and Final Size Minimization by Social Distancing for the SIR Epidemic Model. Journal of Optimization Theory and Applications, 189(2), PP., 408–436. doi: https://doi.org/10.1007/s10957­021­01830­1
[13] Bolzoni, L., Bonacini, E., Soresina, C., and Groppi, M., 2017. Time­optimal control strategies in SIR epidemic models. Mathematical biosciences, 292, PP.86­96. doi:
https://doi.org/10.1016/j.mbs.2017.07.011
[14] Bolzoni, L., Bonacini, E., Della Marca, R. and Groppi, M., 2019. Optimal control of epidemic size and duration with limited resources. Mathematical biosciences, 315, 108232. doi:
https://doi.org/10.1016/j.mbs.2019.108232
[15] Bolzoni, L., Della Marca, R., and Groppi, M., 2021. On the optimal control of SIR model with Erlangdistributed infectious period: isolation strategies. Journal of Mathematical Biology, 83, PP.1­21. doi: https://doi.org/10.1007/s00285­021­01668­1
[16] Dong, S., Xu, L., Zhong­Zhou Lan, Y, A., Xiao, D. and Gao, B., 2022. Application of a time­delay
SIR model with vaccination in COVID­19 prediction and its optimal control strategy, (2022). doi:
https://doi.org/10.1007/s11071­023­08308­x
[17] Elhia, M., Rachik, M. and Benlahmar, E., 2013. Optimal Control of an SIR Model with Delay in
State and Control Variables. ISRN Biomathematics, Volume 2013, Article ID 403549, 7 pages. doi:
https://doi.org/10.1155/2013/403549
[18] Farrington, C.P., 2003. On vaccine efficacy and reproduction numbers. Math. Biosci. 185(1), pp.
89­109. doi: https://doi.org/10.1016/S0025­5564(03)00061­0
[19] Fleming, W. and Rishel, R., 1975. Deterministic and Stochastic Optimal Control. Springer­Verlag, New York. doi: https://doi.org/10.1007/978­1­4612­6380­7

[20] Gani, S. R. and Halawar, S. V., 2017. Deterministic and Stochastic Optimal Control Analysis of an SIR Epidemic model. Global Journal of Pure and Applied Mathematics. 13(9), (2017), pp. 5761­5778.
[21] Grigorieva, E., Khailov, E. and Korobeinikov, A., 2016. Optimal Control for a SIR Epidemic model with Nonlinear Incidence Rate. Study in Applied Mathematics, 141(3), PP. 353­398.
[22] Hackbusch, W. K., 1978. A numerical Method for solving parabolic equations with opposite orientations. Computing, 20 , PP.229­240. doi: 10.1007/BF02251947
[23] Hethcote, H.W., 2000. The mathematics of infectious diseases. SIAM Rev, 42(4): 599. doi:
https://doi.org/10.1137/S0036144500371907
[24] Huang, C., Wang, Y., Li, X., Ren, L., Zhao, J., Hu, Y., Zhang, L., Fan, G., Xu, J. and Gu, X., 2020. et
al. Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet, 395,
PP.497­506. doi: https://doi.org/10.1016/S0140­6736(20)30183­5
[25] Hussain, T., Ozair, M., Ali, F., ur Rehman, S., Assiri, T. A., and Mahmoud, E. E., 2021. Sensitivity
analysis and optimal control of COVID­19 dynamics based on SEIQR model. Results in Physics, 22,
103956. doi: https://doi.org/10.1016/j.rinp.2021.103956
[26] Jahangiri, K. and Tabibi, H., 2003. Disaster management: designing a new
model for effective planning in bioterrorism. Payesh, 2(3), PP.205­214. doi:
https://www.researchgate.net/publication/303667867
[27] Kang, Y. H., Zaman, G. and . Jung, I. H, 2009. Optimal treatment of an SIR epidemic model with time delay. BioSystems, 98(1), PP. 43­50. doi: https://doi.org/10.1016/j.biosystems.2009.05.006
[28] Kar, T. K. and Batabyal, A., 2011. Stability analysis and optimal control of an
SIR epidemic model with vaccination. Biosystems, 104(2–3), PP. 127­135. doi:
https://doi.org/10.1016/j.biosystems.2011.02.001
[29] Kennedy, J. and Eberhart, R. C., 1995. Particle swarm optimization. In: IEEE International Conference on Neural Networks, IEEE. pp. 1942–1948. doi: 10.1109/ICNN.1995.488968
[30] Kermack, W. O., and McKendrick, A. G., 1927. A contribution to the mathematical theory
of epidemics. Proceedings of the Royal Society of London Series A, 115(772), pp.700–721. doi:
https://doi.org/10.1098/rspa.1927.0118
[31] Ketcheson, D. I., Matteo Parsani, and LeVeque, R. J., 2013. High­order Wave Propagation Algorithms for Hyperbolic Systems. SIAM Journal on Scientific Computing, 35(1), PP. 351­377. doi:
https://doi.org/10.1137/110830320
[32] Ketcheson, D. I., 2021. Optimal control of an SIR epidemic through finite­time non­pharmaceutical intervention. Journal of Mathematical Biology, doi: https://doi.org/10.1007/s00285­021­01628­9
[33] Kierzenka, J. and Shampine, L. F., 2001. A BVP solver based on residual control and the MATLAB PSE. ACM Trans Math Softw, 27(3), PP. 299­316. doi: https://doi.org/10.1145/502800.502801
[34] Kmeta, T. and Kmetova, M., Be´zier, 2019. Curve parametrisation and echo state network methods for solving optimal control problems of SIR model. BioSystems,186, doi:
https://doi.org/10.1016/j.biosystems.2019.104029

[35] Laarabi, H., Rachik, M., El Kahlaoui, O. and Labriji, E., 2013. Optimal Vaccination Strategies of an SIR Epidemic Model with a Saturated Treatment. Universal Journal of Applied Mathematics, 1(3), pp.185­191. doi: 10.13189/ujam.2013.010305
[36] Lashari, A. A., 2016. Optimal Control of an SIR Epidemic Model with a Saturated Treatment. Applied Mathematics & Information Sciences, 10(1), PP. 185­191. doi: 10.18576/amis/100117
[37] Ledzewicz, U. and Schattler, H., On optimal singular controls for a general SIR­model with vaccination and treatment. Manuscript submitted to AIMS,PP. 981­990. doi: 10.3934/proc.2011.2011.981
[38] Li, Y., Ye, M. and Zhang, Q., 2019. Strong convergence of the partially truncated Euler­Maruyama scheme for a stochastic age­structured SIR epidemic model. Appl Math Comput, 362:124519. doi: https://doi.org/10.1016/j.amc.2019.06.033
[39] Lukes, D.L., 1982. Differential Equations: Classical to Controlled, Math. Sci. Eng. 162, Academic
Press, New York.
[40] Malik, T. and Sharomi, O., 2015. Optimal control in epidemiology. Article in Annals of Operations Research, 251, PP.55–71. doi: https://doi.org/10.1007/s10479­015­1834­4
[41] Mallapaty, S., 2020. Why does the coronavirus spread so easily between people? Nature, 579:183–183. doi: 10.1038/d41586­020­00660­x
[42] Mehrabian, A., and Lucas, C., 2006. A novel numerical optimization algorithm inspired from weed colonization. Ecol. Inform, 1 (4), PP.355–366. doi: https://doi.org/10.1016/j.ecoinf.2006.07.003
[43] Murray, J.D., 1993. Mathematical biology, mathematical biology. Berlin: Springer. doi:
https://link.springer.com/book/10.1007/978­3­662­08542­4
[44] Nepomuceno, E. G., Barbosa, A. M., Silva, M. X. and Perc, M., 2018. Individualbased modelling and control of bovine brucellosis. R Soc Open Sci, 5(5):180200. doi: https://doi.org/10.1098/rsos.180200
[45] Nepomuceno, E.G., Peixoto, M.L.C., Lacerda, M.J., Campanharo, A.S.L.O., Takahashi, R.H.C.,
Aguirre, L.A., 2021. Application of Optimal Control of Infectious Diseases in a Model‑Free Scenario.
SN Computer Science, 2(405). doi: https://doi.org/10.1007/s42979­021­00794­3
[46] Noupoue, Y.Y.Y., Tandogˇdu, Y. and Awadalla, M., 2019. On numerical techniques for solving
the fractional logistic differential equation. Advances in Difference Equations, 9(1), pp.1­13. doi:
10.1186/s13662­019­2055­y
[47] Ogunmiloroa, O. M., Fadugbab, S. E. and Ogunlade, T. O., 2018. Stability Analysis and Optimal
Control of Vaccination and Treatment of a SIR Epidemiological Deterministic Model with Relapse.
International Journal of Mathematical Modelling & Computations, 8(1), PP.39­ 51.
[48] Piret, J. and Boivin, G., 2021 Pandemics Throughout History, doi: 10.3389/fmicb.2020.631736.
frontiers in microbiology,
[49] Slama, H., Hussein, A., El­Bedwhey, N. A. and Selim, M. M., 2019. An approximate probabilistic
solution of a random SIR­type epidemiological model using RVT technique. Appl Math Comput, 361, PP. 144­156. doi: https://doi.org/10.1016/j.amc.2019.05.019
[50] Staneˇk, J., 2008. Kermack­McKendrick epidemics vaccinated. Kybernetika, 44(5), PP.705­714. doi: http://hdl.handle.net/10338.dmlcz/135883

[51] Tchoumi, S,Y., Diagne, M,L. and Rwezaura, H., 2021. Malaria and COVID­19 codynamics: A mathematical model and optimal control. Appl.Math. Modell, 99, pp.294. doi:
https://doi.org/10.1016/j.apm.2021.06.016
[52] Wickwire, K., 1977. Mathematical­models for control of pest and infectious­diseases ­a survey:
Theor Popul Biol., 11(2), PP.182­238. doi: https://doi.org/10.1016/0040­5809(77)90025­9
[53] Yang, J. and Wang, X., 2018. Threshold dynamics of an SIR model with nonlinear incidence rate and age­dependent susceptibility. Complexity, 2018:1,PP.1­16. doi: https://doi.org/10.1155/2018/9613807
[54] Yoshida, N. and Hara, T., 2007. Global stability of a delayed SIR epidemic model with
density dependent birth and death rates. J. Comput. Appl. Math, 201(2), PP.339–347. doi:
https://doi.org/10.1016/j.cam.2005.12.034
[55] Zakary, O., Rachik, M., and Elmouki, I., 2017. On the analysis of a multi­regions discrete
SIR epidemic model: an optimal control approach. Int. J. Dynam. Control, 5917–930. doi:
https://doi.org/10.1007/s40435­016­0233­2
[56] Zhang, Q., Tang, B., and Tang, S., 2018. Vaccination threshold size and backward bifurcation of
SIR model with state­dependent pulse control. Journal of theoretical biology, 455, PP.75­85. doi:
https://doi.org/10.1016/j.jtbi.2018.07.010
[57] Zhou, Y., Yang, K., Zhou, K., and Liang, Y., 2014. Optimal vaccination policies for an SIR model
with limited resources. Acta biotheoretica, 62, PP.171­181. doi: https://doi.org/10.1007/s10441­014­
9216­x