Optimal control of an epidemiological Covid-19 model with state constraint

Elisa Paparelli; Roberto Giambó; Helmut Maurer

doi:10.3934/dcdsb.2024095

Article Contents

2025, Volume 30, Issue 2: 422-448. Doi: 10.3934/dcdsb.2024095

This issue Previous Article The Hartman-Grobman theorem for mean hyperbolic systems Next Article Minimization with differential inequality and equality constraints applied to complete Lyapunov functions

Optimal control of an epidemiological Covid-19 model with state constraint

1.
Department of Mathematical Sciences "G. L. Lagrange", Dipartimento di Eccellenza 2018-2022, Italy
2.
Mathematics Division, School of Science and Technology, University of Camerino, Italy
3.
INFN, Sezione di Perugia, Italy
4.
Universität Münster, Institut für Analysis und Numerik, Münster, Germany

^*Corresponding author: Elisa Paparelli
^*Corresponding author: Elisa Paparelli

Received: December 2023

Revised: May 2024

Early access: July 15, 2024

Published online: July 15, 2024

The first author is supported by [insert grant information here].

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Abstract

The outbreak of the Covid-19 pandemic has forced governments to impose restrictions on the individual liberty of people. Such containment measures have considerably reduced the number of infections but have also caused substantial damage. In this context the following main issue arises: which policy is the best to contain fatalities and economic losses complying with the intensive care units capacity? This issue is investigated through the study of an optimal control problem based on a SEAIRD epidemic model referring to Covid-19. A state constraint is imposed on the number of infected individuals in order to maintain the infectious level under the health-facilities capacity threshold. The challenge is to find a control function that minimizes the total cost which represents a trade-off between economic losses and human deaths. After showing the existence of an optimal solution, the necessary optimality conditions provided by Pontryagin Minimum Principle are derived. Numerical solutions are obtained by discretizing the optimal control problem and applying nonlinear optimization methods. Various scenarios with different initial conditions representing different degrees of infection are studied and the solutions are compared.The COVID-19 control problem treated here may also serve as a prototypical example for solving an epidemiological control model with state constraints.

Keywords:

Mathematics Subject Classification: Primary: 49N90, 49K15, 49M37; Secondary: 91-08.

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Figure 1. Covid-19 spread under control $ u\equiv u_0 = 0.001 $, according to equations (2)-(7) for the three initial conditions (17)-(18)-(19)

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Figure 2. Uncontrolled Covid-19 spread according to SEAIRD system (2)-(7). Infected population $ I(t) $ reaches a peak of almost $ 0.09 $ after $ 114 $ days

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Figure 3. Case 1: Optimal solution for $ T = 70 $ without state constraint $ I(t) \le 0.005 $. (A) control $ u(t) $, (B) asymptomatic $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I (t) $

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Figure 4. Case 1: Solution for terminal time $ T = 70 $ with state constraint $ I(t) \leq 0.005 $. (A) control $ u(t) $. (B) asymptomatics $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 5. Case 1: Solution for $ T = 200 $ with state constraint $ I(t) \leq 0.005 $. (A) control $ u(t) $ with boundary interval $ [56.2, 178] $. (B) asymptomatics $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 6. Case 2: Solution for $ T = 70 $ without state constraint $ I(t) \le I_{max} = 0.005 $. (A) control $ u(t) $. (B) asymptomatics $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 7. Case 2: Solution for $ T = 70 $ with state constraint $ I(t) \le I_{max} = 0.005 $. (A) control $ u(t) $ with boundary interval $ [43.2, 48] $. (B) asymptomatics $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 8. Case 2: Solution for $ T = 200 $ with state constraint $ I(t) \le 0.005 $. (A) control $ u(t) $ with boundary interval $ [42.5, 178] $. (B) asymptomatics $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 9. Case 3: solution for $ T = 70 $ without state constraint $ I(t) \le I_{max} = 0.005 $. (A) control $ u(t) $. (B) asymptomatics $ A(t) $, infected $ A(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 10. Case 3: Solution for $ T = 70 $ with state constraint $ I(t) \le I_{max} = 0.005 $. (A) control $ u(t) $ with boundary interval $ [28.5, 48] $. (B) asymptomatics $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 11. Case 3: Solution for $ T = 200 $ with state constraint $ I(t) \le 0.005 $. (A) control $ u(t) $ with boundary interval $ [28.4, 178] $. (B) asymptomatics $ A(t) $, infected $ I(t) $ and deaths $ D(t) $. (C) adjoint variable $ \lambda_I(t) $

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Figure 12. Case 3: Optimal and approximative solution for $ T = 200 $ on 6 segments of $ [0, T] $. (A) optimal control $ u(t) $ and approximative control $ u_a(t) $. (B) infected $ I(t) $ and approximative infected $ I_a(t) $

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Table 1. Parameter values

Parameter	Value	Definition
$ \beta $	$ 0.25 $	The infection rate
$ s $	$ 0.1 $	Proportion of infected people who develop symptoms
$ n $	$ 0.00003 $	The natural death rate
$ k $	$ 0.2 $	The latency period after infection
$ \epsilon $	$ 2/3 $	The fraction of asymptomatic
$ \gamma $	$ 0.14 $	The recovery period
$ \delta $	$ 0.0028 $	The death rate due to Covid-19
$ r $	$ 0.04 $	The economic discount rate
$ \sigma $	$ 2 $	A parameter that expresses the GDP losses on production
$ \theta $	$ 1/3 $	Elasticity parameter that expresses the dependence
		of GDP change on infection spread
$ a $	$ 7833.11 $	The social cost of human deaths due to Covid-19
$ I_{max} $	$ 0.005 $	Intensive Care Units availability

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Table 2. Numerical results for the 3 scenarios of initial conditions (42), (43), (44)

$ X_0 $	$ T $	$ I \le I_{max} $	$ \min \, u(t) $	$ F $	$ I(T) $	$ A(T) $	$ D(T) $	$ \nu_T $
$ 55 $	$ 70 $	✘	$ 0.62 $	$ 3.6674 $	$ 0.0208 $	$ 0.01 $	$ 0.001 $	$ 0 $
$ 55 $	$ 70 $	√	$ 0.51 $	$ 3.8915 $	$ 0.005 $	$ 2.55 \cdot 10^{-3} $	$ 6.6 \cdot 10^{-4} $	$ 60 $
$ 55 $	$ 200 $	√	$ 0.53 $	$ 4.4084 $	$ 0.005 $	$ 2.55 \cdot 10^{-3} $	$ 2.5 \cdot 10^{-3} $	$ 0.3 $
$ 65 $	$ 70 $	✘	$ 0.59 $	$ 5.6448 $	$ 0.0209 $	$ 0.01 $	$ 0.001 $	$ 0 $
$ 65 $	$ 70 $	√	$ 0.51 $	$ 5.8888 $	$ 0.005 $	$ 2.55 \cdot 10^{-3} $	$ 8.5 \cdot 10^{-4} $	$ 59.2 $
$ 65 $	$ 200 $	√	$ 0.53 $	$ 6.3967 $	$ 0.005 $	$ 2.55 \cdot 10^{-3} $	$ 2.7 \cdot 10^{-3} $	$ 0.3 $
$ 70 $	$ 70 $	✘	$ 0.58 $	$ 7.0261 $	$ 0.0209 $	$ 0.0106 $	$ 0.0015 $	$ 0 $
$ 70 $	$ 70 $	√	$ 0.51 $	$ 7.2997 $	$ 0.005 $	$ 2.55 \cdot 10^{-3} $	$ 0.001 $	$ 59 $
$ 70 $	$ 200 $	√	$ 0.52 $	$ 7.8012 $	$ 0.005 $	$ 2.55 \cdot 10^{-3} $	$ 2.8\cdot 10^{-3} $	$ 0.3 $

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References

[1]	Z. Abbasi, I. Zamani, A. H. A. Mehra, M. Shafieirad and A. Ibeas, Optimal control design of impulsive SQEIAR epidemic models with application to COVID-19, Chaos, Solitons and Fractals, 139 (2020), 110054. doi: 10.1016/j.chaos.2020.110054.
[2]	D. Acemoglu, V. Chernozhukov, I. Werning and M. Whinston, Optimal targeted lockdowns in a multi-group SIR model, American Economic Review: Insights, 3 (2021), 487-502. doi: 10.1257/aeri.20200590.
[3]	L. J. S. Allen, An introduction to stochastic epidemic models, Mathematical Epidemiology. Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1945 (2008), 81-130. doi: 10.1007/978-3-540-78911-6_3.
[4]	J. K. K. Asamoah, Z. Jin, G. Q. Sun, B. Seidu, E. Yankson, A. Abidemi, F. T. Oduro, S. E. Moore and E. Okyere, Sensitivity assessment and optimal economic evaluation of a new COVID-19 compartmental epidemic model with control interventions, Chaos, Solitons and Fractals, 146 (2021), 110885. doi: 10.1016/j.chaos.2021.110885.
[5]	A. Aspri, E. Beretta, A. Gandolfi and E. Wasmer, Mortality containment vs. economics opening: Optimal policies in a SEIARD model, Journal of Mathematical Economics, 93 (2021), 102490. doi: 10.1016/j.jmateco.2021.102490.
[6]	J. Bauer, D. Brüggmann, D. Klingelhöfer and et al., Access to intensive care in 14 European countries: a spatial analysis of intensive care need and capacity in the light of COVID-19, Intensive Care Med, 46 (2020), 2026-2034. doi: 10.1007/s00134-020-06229-6.
[7]	H. Behncke, Optimal control of deterministic epidemics, Optimal Control Applications and Methods, 21 (2000), 269-285. doi: 10.1002/oca.678.
[8]	M. H. A. Biswas, L. T. Paiva and M. R. De Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784. doi: 10.3934/mbe.2014.11.761.
[9]	G. Bonaccorsi, F. Pierri, M. Cinelli, A. Flori, A. Galeazzi, F. Porcelli, A. L. Schmidt, C. M. Valensise, A. Scala, W. Quattrociocchi and F. Pammolli, Economic and social consequences of human mobility restrictions under COVID-19, Proceedings of the National Academy of Sciences of the United States of America, 117 (2020), 15530-15535. doi: 10.1073/pnas.2007658117.
[10]	J. P. Caulkins, D. Grass, G. Feichtinger, R. F. Hartl, P. M. Kort, A. Prskawetz, A. SeidL and S. Wrzaczekd, The optimal lockdown intensity for COVID-19, Journal of Mathematical Economics, 93 (2021), 102489. doi: 10.1016/j.jmateco.2021.102489.
[11]	L. Cesari, Optimization Theory and Applications. Problems with Ordinary Differential Equations, Springer-Verlag, 1983.
[12]	A. Charpentier, R. Elie, M. Laurière and V. C. Tran, COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability, Mathematical Modelling of Natural Phenomena, 15 (2020), Article Number 57. doi: 10.1051/mmnp/2020045.
[13]	W. Choi and E. Shim, Optimal strategies for social distancing and testing to control COVID-19, Journal of Theoretical Biology, 512 (2021), 110568. doi: 10.1016/j.jtbi.2020.110568.
[14]	L. Cianfanelli, F. Parise, D. Acemoglu, G. Como and A. Ozdaglar, Lockdown interventions in SIR model: Is the reproduction number the right control variable?, 60th IEEE Conference on Decision and Control (CDC), 2021, 4254-4259. doi: 10.1109/CDC45484.2021.9682977.
[15]	M. R. De Pinho, I. Kornienko and H. Maurer, Optimal control of a SEIR model with mixed constraints and L1 cost, Lecture Notes in Electrical Engineering, LNEE, 321 (2015), 135-145. doi: 10.1007/978-3-319-10380-8_14.
[16]	N. Donthu and A. Gustafsson, Effects of COVID-19 on business and research, Journal of Business Research, 117 (2020), 284-289. doi: 10.1016/j.jbusres.2020.06.008.
[17]	K. Fister, S. Lenhart and J. McNally, Optimizing chemotherapy in an HIV model, Electron. J. Differ. Equat, 1998 (1998), 1-12.
[18]	F. Fernández-Aranda, M. Casas, L. Claes, D. C. Bryan, A. Favaro, R. Granero, C. Gudiol, S. Jiménez-Murcia, A. Karwautz, D. Le Grange, J. M. Menchón, K. Tchanturia and J. Treasure, COVID-19 and implications for eating disorders, European Eating Disorders Review, 28 (2020), 239-245. doi: 10.1002/erv.2738.
[19]	S. Flaxman, S. Mishra, A. Gandy, H. J. T. Unwin, T. A. Mellan, H. Coupland and C. Whittaker, Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe, Nature, 584 (2020), 257-261. doi: 10.1038/s41586-020-2405-7.
[20]	R. Fourer, M. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Management Science, 36 (1990), 519-641. doi: 10.1287/mnsc.36.5.519.
[21]	A. Gandolfi, A. Aspri, E. Beretta, K. Jamshad and M. Jiang, A new threshold reveals the uncertainty about the effect of school opening on diffusion of Covid-19, Scientific Reports, 12 (2022), Article number: 3012. doi: 10.1038/s41598-022-06540-w.
[22]	M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L. Carraro, R. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, 117 (2020), 10484-10491. doi: 10.1073/pnas.2004978117.
[23]	G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860. doi: 10.1038/s41591-020-0883-7.
[24]	J. A. Gondim and L. Machado, Optimal quarantine strategies for the COVID-19 pandemic in a population with a discrete age structure, Chaos, Solitons and Fractals, 140 (2020), 110166. doi: 10.1016/j.chaos.2020.110166.
[25]	R. Hartl, S. Sethi and R. Vickson, A survey of the Maximum Principles for Optimal Control Problems with State Constraints, SIAM Review, 37 (1995), 181-218. doi: 10.1137/1037043.
[26]	H. W. Hethcote, Mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.
[27]	D. H. Jacobson, M. M. Lele and J. L. Speyer, New necessary conditions of optimality for control problems with state-variable inequality constraints, J. Math. Anal. Appl., 35 (1971), 255-284. doi: 10.1016/0022-247X(71)90219-8.
[28]	M. Kantner and T. Koprucki, Beyond just "flattening the curve": Optimal control of epidemics with purely non-pharmaceutical interventions, Journal of Mathematics in Industry, 10 (2020), Paper No. 23, 23 pp. doi: 10.1186/s13362-020-00091-3.
[29]	A. J. Kucharski, T. W. Russell, C. Diamond, Y. Liu, J. Edmunds and S. Funk et al., Early dynamics of transmission and control of COVID-19: A mathematical modelling study, The Lancet Infectious Diseases, 20 (2020), 553-558. doi: 10.1016/S1473-3099(20)30144-4.
[30]	S. Lenhart and J. Workman, Optimal Control Applied to Biological Models, Mathematical and Computational Biology Series, Chapman and Hall/CRC, 2007. doi: 10.1201/9781420011418.
[31]	G. B. Libotte, F. S. Lobato, G. M. Platt and A. J. Silva Neto, Determination of an optimal control strategy for vaccine administration in COVID-19 pandemic treatment, Computer Methods and Programs in Biomedicine, 196 (2020), 105664. doi: 10.1016/j.cmpb.2020.105664.
[32]	F. Lin, K. Muthuraman and M. Lawley, An optimal control theory approach to non-pharmaceutical interventions, BMC Infectious Diseases, 10 (2010), Article number: 32. doi: 10.1186/1471-2334-10-32.
[33]	Z. Liu, P. Magal, O. Seydi and G. Webb, A COVID-19 epidemic model with latency period, Infectious Disease Modelling, 5 (2020), 323-337. doi: 10.1016/j.idm.2020.03.003.
[34]	X. Lü, H. Hui, F. Liu and Y. Bai, Stability and optimal control strategies for a novel epidemic model of COVID-19, Nonlinear Dynamics, 106 (2021), 1491-1507. doi: 10.1007/s11071-021-06524-x.
[35]	A. Mahajan, N. A. Sivadas and R. Solanki, An epidemic model SIPHERD and its application for prediction of the spread of COVID-19 infection in India, Chaos, Solitons and Fractals, 140 (2020), 110156. doi: 10.1016/j.chaos.2020.110156.
[36]	M. Mandal, S. Jana, S. K. Nandi, A. Khatua, S. Adak and T. Kar, A model based study on the dynamics of COVID-19: Prediction and control, Chaos, Solitons and Fractals, 136 (2020), 109889. doi: 10.1016/j.chaos.2020.109889.
[37]	C. Maringe, J. Spicer, M. Morris, A. Purushotham, E. Nolte, R. Sullivan, B. Rachet and A. Aggarwal, The impact of the COVID-19 pandemic on cancer deaths due to delays in diagnosis in England, UK: a national, population-based, modelling study, The Lancet Oncology, 21 (2020), 1023-1034. doi: 10.1016/S1470-2045(20)30388-0.
[38]	L. Matrajt, J. Eaton, T. Leung and E. R. Brown, Vaccine optimization for COVID-19: Who to vaccinate first?, Science Advances, 7 (2021), 1374. doi: 10.1126/sciadv.abf1374.
[39]	H. Maurer, On the Minimum Principle for Optimal Control Problems with State Constraints, Schriftenreihe des Rechenzentrums der Universität Münster Nr. 41, Münster, 1979.
[40]	M. McAsey, L. Mou and W. Han, Convergence of the forward-backward sweep method in optimal control, Comput Optim Appl, 53 (2012), 207-226. doi: 10.1007/s10589-011-9454-7.
[41]	L. Miclo, D. Spiro and J. Weinbull, Optimal epidemic suppression under an ICU constraint: An analytical solution, Journal of Mathematical Economics, 101 (2022), 102669. doi: 10.1016/j.jmateco.2022.102669.
[42]	D. Moreira and M. P. Da Costa, The impact of the Covid-19 pandemic in the precipitation of intimate partner violence, Int J Law Psychiatry, 71 (2020), 101606. doi: 10.1016/j.ijlp.2020.101606.
[43]	R. Morton and K. Wickwire, On the Optimal Control of a Deterministic Epidemic, Advances in Applied Probability, 6 (1974), 622-635. doi: 10.2307/1426183.
[44]	M. Natilli, A. Rossi, A. Trecroci, L. Cavaggioni, G. Merati and D. Formenti, The long-tail effect of the COVID-19 lockdown on Italians' quality of life, sleep and physical activity, Scientific Data, 9 (2022), Article number: 250. doi: 10.1038/s41597-022-01376-5.
[45]	F. Ndaïrou, I. Area, J. J. Nieto and D. F. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos, Solitons and Fractals, 135 (2020), 109846. doi: 10.1016/j.chaos.2020.109846.
[46]	R. M. Neilan and S. M. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics, 75 (2010), 67-81.
[47]	https://neos-server.org/neos/index.html
[48]	M. Nicola, Z. Alsafi, C. Sohrabi, A. Kerwan, A. Al-Jabir, C. Iosifidis, M. Agha and R. Agha, The socio-economic implications of the coronavirus pandemic (COVID-19): A review, Int J Surg, 78 (2020), 185-193. doi: 10.1016/j.ijsu.2020.04.018.
[49]	R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics, 87 (2015), 925-979. doi: 10.1103/RevModPhys.87.925.
[50]	T. A. Perkins and G. España, Optimal control of the COVID-19 pandemic with non-pharmaceutical interventions, Bulletin of Mathematical Biology, 82 (2020), Paper No. 118, 24 pp. doi: 10.1007/s11538-020-00795-y.
[51]	N. Perra, D. Balcan, B. Gonçalves and A. Vespignani, Towards a characterization of behavior-disease models, PLOS ONE, 6 (2011), e23084. doi: 10.1371/journal.pone.0023084.
[52]	L. Pontryagin, G. Boltyanskii, R. Gamkrelidze and E. Mishchenko, Mathematical Theory of Optimal Processes, NewYork, 1964.
[53]	K. Prem, Y. Liu, T. W. Russell, A. J. Kucharski, R. M. Eggo, N. Davies and et al., The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: a modelling study, The Lancet Public Health, 5 (2020), e261-e270. doi: 10.1016/S2468-2667(20)30073-6.
[54]	A. Remuzzi and G. Remuzzi, COVID-19 and Italy: What next?, The Lancet, 395 (2020), 1225-1228. doi: 10.1016/S0140-6736(20)30627-9.
[55]	Q. Richard, S. Alizon, M. Choisy, M. T. Sofonea and R. Djidjou-Demasse, Age-structured non-pharmaceutical interventions for optimal control of COVID-19 epidemic, PLoS Computational Biology, 17 (2021), e1008776. doi: 10.1371/journal.pcbi.1008776.
[56]	I. M. Ross, A Primer on Pontryagin's Principle in Optimal Control, 2$^nd$ edition, Collegiate Publishers, San Francisco, 2015.
[57]	S. Saha and G. Samanta, Modelling the role of optimal social distancing on disease prevalence of COVID-19 epidemic, International Journal of Dynamics and Control, 9 (2021), 1053-1077. doi: 10.1007/s40435-020-00721-z.
[58]	H. Schättler and U. Ledzewicz, Optimal control for problems with a quadratic cost functional on the therapeutic agents, in Optimal Control for Mathematical Models of Cancer Therapies, New York, Springer New York, 42 (2015), 141-170. doi: 10.1007/978-1-4939-2972-6_4.
[59]	S. Sethi and P. Staats, Optimal control of some simple deterministic epidemic models, The Journal of the Operational Research Society, 29 (1978), 129-136. doi: 10.1057/jors.1978.27.
[60]	O. Sharomi and T. Malik, Optimal control in epidemiology, Annals of Operations Research, 251 (2017), 55-71. doi: 10.1007/s10479-015-1834-4.
[61]	C. Sohrabi, Z. Alsafi, N. O'Neill, M. Khan, A. Kerwan, A. Al-Jabir, C. Iosifidis and R. Agha, World Health Organization declares global emergency: A review of the 2019 novel coronavirus (COVID-19), International Journal of Surgery, 76 (2020), 71-76. doi: 10.1016/j.ijsu.2020.02.034.
[62]	R. Tandon, COVID-19 and mental health: Preserving humanity, maintaining sanity, and promoting health, Asian J Psychiatr, 51 (2020), 102256. doi: 10.1016/j.ajp.2020.102256.
[63]	S. Ullah and M. A. Khan, Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study, Chaos, Solitons and Fractals, 139 (2020), 110075. doi: 10.1016/j.chaos.2020.110075.
[64]	T. Usherwood, Z. LaJoie and V. Srivastava, A model and predictions for COVID-19 considering population behavior and vaccination, Scientific Reports, 11 (2021), 12051. doi: 10.1038/s41598-021-91514-7.
[65]	N. Vindegaard and M. Benros, COVID-19 pandemic and mental health consequences: Systematic review of the current evidence, Brain, Behavior, and Immunity, 89 (2020), 531-542. doi: 10.1016/j.bbi.2020.05.048.
[66]	R. Vinter, Optimal Control, Modern Birkäuser Classics, Birkäuser, 2010. doi: 10.1007/978-0-8176-8086-2.
[67]	A. Wächter and L. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.
[68]	W. Wang, J. Tang and F. Wei, Updated understanding of the outbreak of 2019 novel coronavirus (2019-nCoV) in Wuhan, China, Journal of Medical Virology, 92 (2020), 441-447. doi: 10.1002/jmv.25689.
[69]	A. Yousefpour, H. Jahanshahi and S. Bekiros, Optimal policies for control of the novel coronavirus disease (COVID-19) outbreak, Chaos, Solitons and Fractals, 136 (2020), 109883. doi: 10.1016/j.chaos.2020.109883.