N-Step-Ahead Optimal Control of a Compartmental Model of COVID-19

Douglas Martins1, Amit Bhaya1, Fernando Pazos2
1Department of Electrical Engineering, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
2Technology and Administration Department, National University of Avellaneda, Buenos Aires, Argentina

Tóm tắt

This paper uses a compartmental model that accounts for some of the main features of the COVID-19 pandemic. Assuming a control that represents the aggregated intensity of non pharmaceutical interventions, such as lockdown in varying degrees and the use of masks and social distancing, this text proposes an N-step-ahead optimal control (NSAOC) method that is easy to calculate and provides a guideline for implementation. The compartmental model is extended to account for vaccination, and the N-step-ahead optimal control is calculated for this case as well. The proposed control is robust to parameter variation in all model parameters, when they are assumed to be normally distributed about nominal values. In addition, the proposed NSAOC is shown to compare favorably with a recently proposed PID-like controller.

Tài liệu tham khảo

Acuña-Zegarra, M., Diaz Infante, S., Baca Carrasco, D., et al. (2021). COVID-19 optimal vaccination policies: A modeling study on efficacy, natural and vaccine-induced immunity responses. Mathematical Biosciences, 337(108), 614. https://doi.org/10.1016/j.mbs.2021.108614 Alleman, T., Torfs, E., & Nopens, I. (2020). COVID-19: From model prediction to model predictive control. https://doi.org/10.13140/RG.2.2.11772.00648 Almeida, L., Bliman, P. A., Nadin, G., et al. (2021). Final size and convergence rate for an epidemic in heterogeneous population. Mathematical Models and Methods in Applied Sciences, 31(5), 1021–1055. https://doi.org/10.1142/S0218202521500251 Ames, A. D., Molnár, T. G., Singletary, A. W., et al. (2020). Safety-critical control of active interventions for COVID-19 mitigation. IEEE Access, 8, 188454–188474. https://doi.org/10.1109/ACCESS.2020.3029558 Angulo, M. T., Castaños, F., Moreno-Morton, R., et al. (2020). A simple criterion to design optimal non-pharmaceutical interventions for epidemic outbreaks. Journal of the Royal Society Interface, 18, 20200803. https://doi.org/10.1101/2020.05.19.20107268 Armaou, A., Katch, B., Russo, L., et al. (2022). Designing social distancing policies for the COVID-19 pandemic: A probabilistic model predictive control approach. Mathematical Biosciences and Engineering, 19(9), 8804–8832. https://doi.org/10.3934/mbe.2022409 Bezanson, J., Edelman, A., & Karpinski, S. et al. (2014). Julia: A fresh approach to numerical computing. https://doi.org/10.48550/ARXIV.1411.1607, URL arxiv:1411.1607 Bin, M., Cheung, P. Y. K., Crisostomi, E., et al. (2021). Post-lockdown abatement of COVID-19 by fast periodic switching. PLOS Computational Biology, 17(1), 1–34. https://doi.org/10.1371/journal.pcbi.1008604 Bliman, P. A., & Duprez, M. (2020). How best can finite-time social distancing reduce epidemic final size? Journal of Theoretical Biology, 511(110), 557. https://doi.org/10.1016/j.jtbi.2020.110557 Bliman, P. A., Duprez, M., Privat, Y., et al. (2021). Optimal immunity control and final size minimization by social distancing for the sir epidemic model. Journal of Optimization Theory and Applications, 189, 408–436. https://doi.org/10.1007/s10957-021-01830-1 Camacho, E. F., & Bordons, C. (1999). Model Predictive Control. London: Springer. Canon, M. D., Cullum, C. D., Jr., & Polak, E. (1970). Theory of optimal control and mathematical programming. New York: McGraw-Hill. Carcione, J., Santos, J., Bagaini, C., et al. (2020). A simulation of a COVID-19 epidemic based on a deterministic SEIR model. Frontiers in Public Health, 8, 230. https://doi.org/10.3389/fpubh.2020.00230 Carli, R., Cavone, G., Epicoco, N., et al. (2020). Model predictive control to mitigate the COVID-19 outbreak in a multi-region scenario. Annual Reviews in Control, 50, 373–393. https://doi.org/10.1016/j.arcontrol.2020.09.005 Charpentier, A., Elie, R., Laurière, M., et al. (2020). COVID-19 pandemic control: Balancing detection policy and lockdown intervention under ICU sustainability. Mathematical Modelling of Natural Phenomena. https://doi.org/10.1051/mmnp/2020045 Di Lauro, F., Kiss, I. Z., Russ, D., et al. (2021). COVID-19 and flattening the curve: A feedback control perspective. IEEE Control Systems Letters, 5(4), 1435–1440. https://doi.org/10.1109/LCSYS.2020.3039322 Djidjou-Demasse, R., Michalakis, Y., & Choisy, M., et al. (2020). Optimal COVID-19 epidemic control until vaccine deployment. https://www.medrxiv.org/content/10.1101/2020.04.02.20049189v3 Dunning, I., Huchette, J., & Lubin, M. (2015). Jump: A modeling language for mathematical optimization. SIAM Review. https://doi.org/10.1137/15M1020575 Ferguson, N. M., Laydon, D., Nedjati-Gilani, G., et al. (2020). Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand. Technical report, Imperial College, London. https://doi.org/10.25561/77482 Giordano, G., Blanchini, F., Bruno, R., et al. (2020). Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nature Medicine, 26, 1–6. https://doi.org/10.1038/s41591-020-0883-7 Gondim, J. A., & Machado, L. (2020). Optimal quarantine strategies for the COVID-19 pandemic in a population with a discrete age structure. Chaos, Solitons and Fractals, 140, 110166. https://doi.org/10.1016/j.chaos.2020.110166 Isee (2021). Stella online-COVID model. https://exchange.iseesystems.com/models/player/isee/covid-19-model Ivorra, B., Ruiz Ferrández, M., Vela, M., et al. (2020). Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections: the case of China. Communications in Nonlinear Science and Numerical Simulation, 88, 105303. https://doi.org/10.1016/j.cnsns.2020.105303 Jankhonkhan, J., & Sawangtong, W. (2021). Model predictive control of COVID-19 pandemic with social isolation and vaccination policies in Thailand. Axioms. https://doi.org/10.3390/axioms10040274 Kar, T., & Batabyal, A. (2011). Stability analysis and optimal control of an sir epidemic model with vaccination. Bio Systems, 104, 127–35. https://doi.org/10.1016/j.biosystems.2011.02.001 Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society, 115, 700–721. https://doi.org/10.1098/rspa.1927.0118 Kirk, D. E. (1970). Optimal control theory: An introduction. Prentice-Hall. Köhler, J., Schwenkel, L., Koch, A., et al. (2020). Robust and optimal predictive control of the COVID-19 outbreak. Annual Reviews in Control, 51, 525–539. https://doi.org/10.1016/j.arcontrol.2020.11.002 Lenhart, S., & Workman, J. (2007). Optimal control applied to Biological models. Mathematical and computational biology series. Chapman & Hall/CRC Press. Lin, F., Muthuraman, K., & Lawley, M. (2010). An optimal control theory approach to non-pharmaceutical interventions. BMC Infectious Diseases. https://doi.org/10.1186/1471-2334-10-32 Maciejowski, J. (2002). Predictive control with constraints. Prentice-Hall. Mallela, A. (2020). Optimal control applied to a SEIR model of 2019-nCoV with social distancing. https://doi.org/10.1101/2020.04.10.20061069 Moore, S., & Okyere, E. (2020). Controlling the transmission dynamics of COVID-19. https://arxiv.org/pdf/2004.00443.pdf Morato, M. M., Bastos, S. B., Cajueiro, D. O., et al. (2020). An optimal predictive control strategy for COVID-19 (SARS-CoV-2) social distancing policies in Brazil. Annual Reviews in Control, 50, 417–431. https://doi.org/10.1016/j.arcontrol.2020.07.001 Morato, M. M., Pataro, I., da Costa, M. A., et al. (2020). A parametrized nonlinear predictive control strategy for relaxing COVID-19 social distancing measures in Brazil. ISA Transactions, 124, 197–214. https://doi.org/10.1016/j.isatra.2020.12.012 Olivier, L., Botha, S., & Craig, I. K. (2020). Optimized lockdown strategies for curbing the spread of COVID-19: A South African case study. IEEE Access, 8, 205755–205765. https://doi.org/10.1109/ACCESS.2020.3037415 Parino, F., Zino, L., Calafiore, G. C., et al. (2021). A model predictive control approach to optimally devise a two-dose vaccination rollout: A case study on COVID-19 in Italy. International Journal of Robust and Nonlinear Control. https://doi.org/10.1002/rnc.5728 Pazos, F., & Felicioni, F. (2021). A control approach to the Covid-19 disease using a SEIHRD dynamical model. Complex Systems, 30(3), 323–346. https://doi.org/10.25088/ComplexSystems.30.3.323 Péni, T., Csutak, B., Szederkényi, G., et al. (2020). Nonlinear model predictive control with logic constraints for COVID-19 management. Nonlinear Dynamics, 102, 1965–1986. https://doi.org/10.1007/s11071-020-05980-1 Perkins, T. A., & España, G. (2020). Optimal control of the COVID-19 pandemic with non-pharmaceutical interventions. Bulletin of Mathematical Biology. https://doi.org/10.1007/s11538-020-00795-y Sadeghi, M., Greene, J. M., & Sontag, E. D. (2021). Universal features of epidemic models under social distancing guidelines. Annual Reviews in Control, 51, 426–440. https://doi.org/10.1016/j.arcontrol.2021.04.004 Shah, N. H., Suthar, A. H., & Jayswal, E. N. (2020). Control strategies to curtail transmission of COVID-19. International Journal of Mathematics and Mathematical Sciences. https://doi.org/10.1155/2020/2649514 Sharma, A., & Agarwal, B. (2021). A cyber-physical system approach for model based predictive control and modeling of COVID-19 in India. Journal of Interdisciplinary Mathematics, 24(1), 1–18. https://doi.org/10.1080/09720502.2020.1830479 Stewart, G., Heusden, K. V., & Dumont, G. (2020). How control theory can help us control COVID-19. IEEE Spectrum, 57(6), 26–29. https://doi.org/10.1109/MSPEC.2020.9099929 Tsay, C., Lejarza, F., Stadtherr, M., et al. (2020). Modeling, state estimation, and optimal control for the US COVID-19 outbreak. Scientific Reports, 10, 10711. https://doi.org/10.1038/s41598-020-67459-8 Ullah, S., & Khan, M. A. (2020). 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. https://doi.org/10.1016/j.chaos.2020.110075 Wächter, A. (2009). Short tutorial: Getting started with ipopt in 90 minutes. In: Naumann, U., Schenk, O., Simon, H.D., et al. (Eds.), Combinatorial scientific computing, dagstuhl seminar proceedings (DagSemProc) (Vol 9061, pp. 1–17). Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl. https://doi.org/10.4230/DagSemProc.09061.16, https://drops.dagstuhl.de/opus/volltexte/2009/2089 Watkins, N., Nowzari, C., & Pappas, G. (2019). Robust economic model predictive control of continuous-time epidemic processes. IEEE Transactions on Automatic Control, 65(3), 1116–1131. https://doi.org/10.1109/TAC.2019.2919136 Worldometers (2020). BenchmarkTools: Julia language package. https://github.com/JuliaCI/BenchmarkTools.jl Zamir, M., Shah, Z., Nadeem, F., et al. (2020). Non pharmaceutical interventions for optimal control of COVID-19. Computer Methods and Programs in Biomedicine, 196, 105642. https://doi.org/10.1016/j.cmpb.2020.105642