Collocation method with flood-based metaheuristic optimizer for optimal control on a multi-strain COVID-19 model

Abstract

This paper describes a new and powerful way to solve optimal control problems (OCPs) on a multi-strain COVID-19 model for strategies related to vaccination and amplification. We call it the collocation method with a flood-based metaheuristic optimizer (FBMO). We use a collocation method with Laguerre polynomials and their derivative operational matrices to turn the OCP into a nonlinear programming (NLP) problem. To address the NLP, the research employs the FBMO to determine the control variables u_i for i = 1, 2, and 3, representing isolation, vaccination efficacy, and treatment enhancement, in conjunction with the state function of the multi-strain COVID-19 model. These strategies are executed within an SVI_cI_vR-type control model for COVID-19 in Morocco, designed to control the outbreak of multi-strain disease. The paper’s primary aim is to achieve a high-quality optimal solution for the given OCP, thereby contributing to the advancement of efficient strategies for managing the COVID-19 pandemic.

Keywords

Multi-strain / Amplification / Optimal control / Vaccination / Collocation method / Flood-based metaheuristic optimizer

Funding

This work has financial support of Farhangian University (Contract No. 500.17474.120).

Declaration of competing interest

The authors declare that they have no conflict of interest regarding the publication of this article.

References

[1] Chen L, He Y, Liu H, Shang Y, Guo G. Potential immune evasion of the severe acute respiratory syndrome coronavirus 2 Omicron variants. Front Immunol., 2024;15:1339660. https://doi.org/10.3389/fimmu.2024.1339660

[2] Willett BJ, Grove J, MacLean OA et al. SARS-CoV-2 Omicron is an immune escape variant with an altered cell entry pathway. Nat Microbiol., 2022; 7(8):1161-1179. https://doi.org/10.1038/s41564-022-01143-7

[3] Carabelli AM, Peacock TP, Thorne LG et al. SARS-CoV-2 variant biology: immune escape, transmission and fitness. Nat Rev Microbiol., 2023; 21(3):162-177. https://doi.org/10.1038/s41579-022-00841-7

[4] Id Ouaziz S, El Khomssi M. Mathematical approaches to controlling COVID-19: optimal control and financial benefits. Math Model Numer Simul Appl., 2024; 4(1):1-36. https://doi.org/10.53391/mmnsa.1373093

[5] Evirgen F, Özkoşe F, Yavuz M, Özdemir N. Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks. AIMS Bioeng., 2023; 10(3):218-239. https://doi.org/10.3934/bioeng.2023015

[6] Joshi H, Yavuz M, Townley S, Jha BK. Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate. Phys Scr., 2023; 98(4):045216. https://doi.org/10.1088/1402-4896/acbe7a

[7] Chen L, He Y, Liu H, Shang Y, Guo G. Potential immune evasion of the severe acute respiratory syndrome coronavirus 2 Omicron variants. Front Immunol., 2024;15:1339660. https://doi.org/10.3389/fimmu.2024.1339660

[8] Burgess DJ. Network effects of disease mutations. Nat Rev Genet., 2015; 16(6):317. https://doi.org/10.1038/nrg3957

[9] Ghostine R, Gharamti M, Hassrouny S, Hoteit I. An extended SEIR model with vaccination for forecasting the COVID-19 pandemic in Saudi Arabia using an ensemble Kalman filter. Mathematics., 2021; 9(6):636. https://doi.org/10.3390/math9060636

[10] Hussain T, Ozair M, Ali F, ur Rehman S, Assiri TA, Mahmoud EE. Sensitivity analysis and optimal control of COVID-19 dynamics based on SEIQR model. Results Phys., 2021;22:103956. https://doi.org/10.1016/j.rinp.2021.103956

[11] Olaniyi S, Obabiyi OS, Okosun KO, Oladipo AT, Adewale SO. Mathematical modelling and optimal cost-effective control of COVID-19 transmission dynamics. Eur Phys J Plus., 2020; 135(11):938. https://doi.org/10.1140/epjp/s13360-020-00954-z

[12] Abidemi A, Zainuddin ZM, Aziz NAB. Impact of control interventions on COVID-19 population dynamics in Malaysia: a mathematical study. Eur Phys J Plus., 2021; 136(2):237. https://doi.org/10.1140/epjp/s13360-021-01205-5

[13] Libotte GB, Lobato FS, Platt GM, Neto AJS. Determination of an optimal control strategy for vaccine administration in COVID-19 pandemic treatment. Comput Methods Programs Biomed., 2020;196:105664. https://doi.org/10.1016/j.cmpb.2020.105664

[14] Khan AA, Ullah S, Amin R. Optimal control analysis of COVID-19 vaccine epidemic model: a case study. Eur Phys J Plus., 2022; 137(1):156. https://doi.org/10.1140/epjp/s13360-022-02365-8

[15] Shen ZH, Chu YM, Khan MA, Muhammad S, Al-Hartomy OA, Higazy M. Mathematical modeling and optimal control of the COVID-19 dynamics. Results Phys., 2021;31:105028. https://doi.org/10.1016/j.rinp.2021.105028

[16] Olivares A, Staffetti E. Optimal control applied to vaccination and testing policies for COVID-19. Mathematics., 2021; 9(23):3100. https://doi.org/10.3390/math9233100

[17] Rajput A, Sajid M, Tanvi, Shekhar C, Aggarwal R. Optimal control strategies on COVID-19 infection to bolster the efficacy of vaccination in India. Sci Rep., 2021; 11(1):20124. https://doi.org/10.1038/s41598-021-99088-0

[18] Arruda EF, Das SS, Dias CM, Pastore DH. Modelling and optimal control of multi strain epidemics, with application to COVID-19. Plos One., 2021; 16(9), e0257512. https://doi.org/10.1371/journal.pone.0257512

[19] Elqaddaoui A, El Bhih A, Laarabi H, Abta A, Rachik M. Mathematical modeling and optimal control of multi-strain COVID-19 spread in discrete time. Front Appl Math Stat., 2024;10:1392628. https://doi.org/10.3389/fams.2024.1392628

[20] Wu H, Zhang L, Li HL, Teng Z. Stability analysis and optimal control on a multi-strain coinfection model with amplification and vaccination. Results Phys., 2023;50:106556. https://doi.org/10.1016/j.rinp.2023.106556

[21] Elqaddaoui A, El Bhih A, Laarabi H, Rachik M, Abta A. A stochastic optimal control strategy for multi-strain COVID-19 spread. Commun Math Biol Neurosci., 2023;2023:130.

[22] Khajji B, Boujallal L, Balatif O, Rachik M. Mathematical modelling and optimal control strategies of a multistrain COVID-19 spread. J Appl Math., 2022; 2022(1): 9071890. https://doi.org/10.1155/2022/9071890

[23] Venkatesh A, Ankamma Rao M, Vamsi DKK. A comprehensive study of optimal control model simulation for COVID-19 infection with respect to multiple variants. Commun Math Biol Neurosci., 2023;2023:75.

[24] Nocedal J, Wright SJ. Numerical Optimization. Springer. 2006.

[25] Luenberger DG, Ye Y. Linear and Nonlinear Programming, Addison-Wesley. 1984.

[26] Bazaraa MS, Sherali HD, Shetty CM. Nonlinear programming: Theory and Algorithms. John Wiley & Sons. 2006. https://doi.org/10.1002/0471787779

[27] Ghasemi M, Golalipour K, Zare M et al. Flood algorithm (FLA): an efficient inspired meta-heuristic for engineering optimization. J Supercomput., 2024; 80(15):22913-23017. https://doi.org/10.1007/s11227-024-06291-7

[28] Guo J, Zhao M, Yu J et al. EHPR: Learning evolutionary hierarchy perception representation based on quaternion for temporal knowledge graph completion. Inf Sci., 2025;688:121409. https://doi.org/10.1016/j.ins.2024.121409

[29] Yildiz BS. Enhancing the performance of a additive manufactured battery holder using a coupled artificial neural network with a hybrid flood algorithm and water wave algorithm. Mater Test., 2024; 66(10):1557-1563. https://doi.org/10.1515/mt-2024-0217

[30] Li T, Wu D, Zhou M, Liao Q, Peng Y. ADCV: Unsupervised depth completion employing adaptive depth-based cost volume. Digit Signal Process., 2024;155:104750. https://doi.org/10.1016/j.dsp.2024.104750

[31] Yan C, Yang K. FSM-YOLO: Apple leaf disease detection network based on adaptive feature capture and spatial context awareness. Digit Signal Process., 2024;155:104770. https://doi.org/10.1016/j.dsp.2024.104770

[32] Liu H, Xiao J, Yao Y et al. A multi-strategy improved northern goshawk optimization algorithm for optimizing engineering problems. Biomimetics., 2024; 9(9):561. https://doi.org/10.3390/biomimetics9090561

[33] COVID-19 coronavirus outbreak, 2022, https://www.worldometers.info/coronavirus/repro [Accessed 19, 2022].

[34] Morocco population, 2022, https://www.worldometers.info/world-population/morocco-population [Accessed 19, 2022].

[35] Total confirmed COVID-19 cases vs. deaths per million, 2022, https://ourworldindata.org/grapher/rate-confirmed-cases-vs-rate-confirmed-deaths, [Accessed 19, 2022].

[36] Hammoumi A., Qesmi R. Impact assessment of containment measure against COVID-19 spread in Morocco. Chaos Solitons & Fractals., 140 (2020) 110231. https://doi.org/10.1016/j.chaos.2020.110231

[37] Daily COVID-19 vaccine doses administered, 2022, [Accessed 19, 2022].

[38] Live updates: COVID-19 epidemic map, 2022, [Accessed 19, 2022].

[39] Rodrigues P, Silva CJ, Torres DFM. Cost-effectiveness analysis of optimal control measures for tuberculosis. Bull Math Biol., 2014; 76:2627-2645. https://doi.org/10.1007/s11538-014-0028-6

[40] Guo T, Nikan O, Avazzadeh Z, Qiu W. Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems. Comput Appl Math., 2022;41:236. https://doi.org/10.1007/s40314-022-01934-y