A spatiotemporal spread of COVID-19 pandemic with vaccination optimal control strategy: A case study in Morocco

On March 2, 2020, the Moroccan Ministry of Health announced the first case of COVID-19 in the city of Casablanca for a Moroccan tourist who came from Italy.  The SARS-COV-2 virus has spread throughout the Kingdom of Morocco.  In this paper, we study the spatiotemporal transmission of the COVID-19 virus in the Kingdom of Morocco.  By supporting a SI$_{\rm W}$IHR partial differential equation for the spread of the COVID-19 pandemic in Morocco as a case study.  Our main goal is to characterize the optimum order of controlling the spread of the COVID-19 pandemic by adopting a vaccination strategy, the aim of which is to reduce the number of susceptible and infected individuals without vaccination and to maximize the recovered individuals by reducing the cost of vaccination using one of the vaccines approved by the World Health Organization.  To do this, we proved the existence of a pair of control. It provides a description of the optimal controls in terms of state and auxiliary functions.  Finally, we provided numerical simulations of data related to the transmission of the COVID-19 pandemic.  Numerical results are presented to illustrate the effectiveness of the adopted approach.

  1. Coronavirus update (live), https://www.worldometers.info/coronavirus/#countries, accessed: 2020-11-28 (2020).
  2. Coronavirus update (live), https://www.worldometers.info/coronavirus/#countries, accessed: 2021-10-07 (2021).
  3. Elhia M., Chokri K., Alkama M.  Optimal control and free optimal time problem for a COVID-19 model with saturated vaccination function.  Communications in Mathematical Biology and Neuroscience.  2021, 35 (2021).
  4. Kouidere A., Khajji B., El Bhih A., Balatif O., Rachik M.  A mathematical modeling with optimal control strategy of transmission of COVID-19 pandemic virus.  Communications in Mathematical Biology and Neuroscience.  2020, 24 (2020).
  5. Kada D., Kouidere A., Balatif O., Rachik M., Labriji E. H.  Mathematical modeling of the spread of COVID-19 among different age groups in Morocco: Optimal control approach for intervention strategies.  Chaos, Solitons & Fractals.  141, 110437 (2020).
  6. Elhia M., Boujallal L., Alkama M., Balatif O., Rachik M.  Set-valued control approach applied to a COVID-19 model with screening and saturated treatment function.  Complexity. 2020, 9501028 (2020).
  7. Kouidere A., Youssoufi L. E., Ferjouchia H., Balatif O., Rachik M.  Optimal control of mathematical modeling of the spread of the COVID-19 pandemic with highlighting the negative impact of quarantine on diabetics people with cost-effectiveness.  Chaos, Solitons & Fractals.  145, 110777 (2021).
  8. Bentout S., Tridane A., Djilali S., Touaoula T. M.  Age-structured modeling of COVID-19 epidemic in the USA, UAE and Algeria.  Alexandria Engineering Journal.  60 (1), 401–411 (2020).
  9. Liu Z., Magal P., Seydi O., Webb G.  A COVID-19 epidemic model with latency period.  Infectious Disease Modelling.  5, 323–337 (2020).
  10. El Bhih A., Benfatah Y., Kouidere A., Rachik M.  A discrete mathematical modeling of transmission of COVID-19 pandemic using optimal control.  Communications in Mathematical Biology and Neuroscience.  2020, 75 (2020).
  11. Kouidere A., Kada D., Balatif O., Rachik M., Naim M.  Optimal control approach of a mathematical modeling with multiple delays of the negative impact of delays in applying preventive precautions against the spread of the COVID-19 pandemic with a case study of Brazil and cost-effectiveness.  Chaos, Solitons & Fractals.  142, 110438 (2020).
  12. Castilho C., Gondim J. A., Marchesin M., Sabeti M.  Assessing the efficiency of different control strategies for the COVID-19 epidemic.  Electronic Journal of Differential Equations. 2020 (64), 1–17 (2020).
  13. Khajji B., Kouidere A., Elhia M., Balatif O., Rachik M.  Fractional optimal control problem for an age-structured model of COVID-19 transmission.  Chaos, Solitons & Fractals.  143, 110625 (2021).
  14. Atangana A., Araz S. İ.  Modeling and forecasting the spread of Covid-19 with stochastic and deterministic approaches: Africa and Europe.  Advances in Difference Equations.  2021, 57 (2021).
  15. Peter O. J., Shaikh A. S., Ibrahim M. O., Nisar K. S., Baleanu D., Khan I., Abioye A. I.  Analysis and dynamics of fractional order mathematical model of COVID-19 in NIGERIA using Atangana–Baleanu operator. Computers, Materials & Continua.  66 (2), 1823–1848 (2020).
  16. Alshomrani A. S., Ullah M. Z., Baleanu D.  Caputo SIR model for COVID-19 under optimized fractional order.  Advances in Difference Equations.  2021, 185 (2021).
  17. Akgül A., Ahmed N., Raza A., Iqbal Z., Rafiq M., Baleanu D., Rehman M. A.-u.  New applications related to COVID-19.  Results in Physics.  20, 103663 (2021).
  18. Baleanu D., Jajarmi A., Mohammadi H., Rezapour S.  A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative.  Chaos, Solitons & Fractals.  134, 109705 (2020).
  19. Vrabie I. I.  Co-Semigroups and applications.  Elsevier (2003).
  20. Barbu V.  Mathematical Methods in Optimization of Differential Systems.  Mathematics and Its Applications. Vol. 310. Springer Dordrecht (1994).
  21. Smoller J.  Shock Waves and Reaction–Diffusion Equations.  Grundlehren der mathematischen Wissenschaften. Vol. 258.  Springer New York, NY (1994).
  22. Brezis H., Ciarlet P. G., Lions J. L.  Analyse fonctionnelle: théorie et applications. Vol. 91. Dunod Paris (1999).
  23. Laaroussi A. E.-A., Rachik M., Elhia M.  An optimal control problem for a spatiotemporal SIR model.  International Journal of Dynamics and Control.  6, 384–397 (2018).
  24. Laaroussi A. E. A., Rachik M.  On the regional control of a reaction–diffusion system SIR.  Bulletin of Mathematical Biology.  82, 5 (2020).
  25. Adnaoui K., El Alami Laaroussi A.  An optimal control for a two-dimensional spatiotemporal SEIR epidemic model.  International Journal of Differential Equations.  2020, 4749365 (2020).
  26. El Yousoufi L., Kouidere A., Kada D., Balatif O., Daouia A., Rachik M.  On stability analysis study and strategies for optimal control of a mathematical model of hepatitis HCV with the latent state.  Mathematical Modeling and Computing.  10 (1), 101–118 (2023).
  27. Kouidere A., Balatif O., Rachik M.  Cost-effectiveness of a mathematical modeling with optimal control approach of spread of COVID-19 pandemic: A case study in Peru.  Chaos, Solitons & Fractals: X.  10, 100090 (2023).
Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 171–185 (2023)