Tikhonov regularization for a spatiotemporal model of the human monkeypox outbreak

2023;
: pp. 875–888
https://doi.org/10.23939/mmc2023.03.875
Received: November 26, 2022
Revised: April 17, 2023
Accepted: April 18, 2023

Mathematical Modeling and Computing, Vol. 10, No. 3, pp. 875–888 (2023)

1
Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University of Casablanca
2
Laboratory of Analysis, Modelling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University of Casablanca
3
Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University of Casablanca
4
Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University of Casablanca

Monkeypox is a contagious disease caused by the monkeypox virus.  There is currently an outbreak of monkeypox in the U.S. and other countries where the virus is not usually seen.  We develop and analyze a deterministic mathematical model for the monkeypox virus by proposing a spatiotemporal model describing the dynamics of the virus between humans.  The existence, the positivity, and the boundedness of the solutions have been proved.  Moreover, with the help of the optimal control, we add two different controls (blocking of contact and treatment in the case of infection) to prevent the propagation of monkeypox between humans. Finally, we present brief comments and numerical simulations to illustrate our findings.  The results show that keeping diseased people apart from the general population minimizes the spread of disease.

  1. Pazy A.  Semigroups of Linear Operators and Applications to Partial Differential Equations.  Applied Mathematical Sciences. Vol. 44.  Springer, New York, USA(1983).
  2. Bankuru S. V., Kossol S., Hou W., Mahmoudi P., Rychtář J., Taylor D.  A game-theoretic model of monkeypox to assess vaccination strategies.  PeerJ.  8, e9272 (2020).
  3. Bhunu C., Garira W., Magombedze G.  Mathematical analysis of a two strain HIV/AIDS model with antiretroviral treatment.  Acta Biotheoretica.  57 (3), 361–381 (2009).
  4. Bhunu C. P., Mushayabasa S.  Modelling the transmission dynamics of pox-like infections.  IAENG International Journal of Applied Mathematics.  41 (2), 141–149 (2011).
  5. Henry D.  Geometric Theory of Semilinear Parabolic Equations.  Lecture Notes in Mathematics. Vol. 840.  Springer–Verlag, Berlin, New York (1993).
  6. Mahase E.  Monkeypox: what do we know about the outbreaks in Europe and North America?  BMJ.  377, o1274 (2022).
  7. Petersen E., Abubakar I., Ihekweazu C., Heymann D., Ntoumi F., Blumberg L., Asogun D., et al.  Monkeypox – enhancing public health preparedness for an emerging lethal human zoonotic epidemic threat in the wake of the smallpox post eradication era.  International Journal of Infectious Diseases.  78, 78–84 (2019).
  8. Beer E. M., Rao V. B.  A systematic review of the epidemiology of human monkeypox outbreaks and implications for outbreak strategy.  PLoS Neglected Tropical Diseases. 13 (10), e0007791 (2019).
  9. Bunge E. M., Hoet B., Chen L., Lienert F., Weidenthaler H., Baer L. R., et al.  The changing epidemiology of human monkeypox – A potential threat? A systematic review.  PLoS Neglected Tropical Diseases.  16 (2), e0010141 (2022).
  10. Webb G. F.  A reaction-diusion model for a deterministic diffusive epidemic.  Journal of Mathematical Analysis and Applications.  84 (1), 150–161 (1981).
  11. Grant R., Nguyen L.-B. L., Breban R.  Modelling human-to-human transmission of monkeypox.  Bulletin of the World Health Organization.  98 (9), 638 (2020).
  12. Smoller J.  Shock Waves and Reaction–Diffusion Equations.  Grundlehren der mathematischen Wissenschaften. Vol. 258.  Springer–Verlag, Berlin, Germany (1983).
  13. Simpson K., Heymann D., Brown C. S., Edmunds W. J., Elsgaard J., et al.  Human monkeypox – after 40 years, an unintended consequence of smallpox eradication.  Vaccine.  38 (33), 5077–5081 (2020).
  14. Kozlov M.  Monkeypox goes global: why scientists are on alert.  Nature.  606, 15–16 (2022).
  15. McAsey M., Mou L., Han W.  Convergence of the forward–backward sweep method in optimal control.  Computational Optimization and Applications.  53, 207–226 (2012).
  16. Protter M. H., Weinberger H. F.  Maximum Principles in Differential Equations.  Prentice Hall, Englewood Cliffs (1967).
  17. Mauldin M. R., McCollum A. M., Nakazawa Y. J., Mandra A., Whitehouse E. R., Davidson W., et al.  Exportation of monkeypox virus from the African continent.  Journal of Infectious Diseases.  225 (8), 1367–1376 (2022).
  18. Whitney M. L.  Theoretical and numerical study of Tikhonov's regularization and Morozovs discrepancy principle.  Thesis, Georgia State University (2009).
  19. Alikakos N. D.  An application of the invariance principle to reaction-diffusion equations.  Journal of Differential Equations.  33 (2), 201–225 (1979).
  20. Fisher R. A.  The wave of advantageous genes.  Annals of Eugenics.  7 (4), 353–369 (1937).
  21. Somma S. A., Akinwande N. I., Chado U. D.  A mathematical model of monkey pox virus transmission dynamics.  Ife Journal of Science.  21 (1), 195–204 (2019).
  22. Lenhart S., Workman J. T.  Optimal control applied to biological models.  Chapman and Hall/CRC, Boca Raton (2007).
  23. TeWinkel R. E.  Stability analysis for the equilibria of a monkeypox model.  Thesis and Dissertations, University of Wisconsin (2019).
  24. Usman S., Isa Adamu I.  Modeling the transmission dynamics of the monkeypox virus infection with treatment and vaccination interventions.  Journal of Applied Mathematics and Physics.  5 (12), 2335–2353 (2017).
  25. WHO, Monkeypox.  https://www.who.int/news-room/fact-sheets/detail/monkeypox (2022).
  26. WHO, World health organization.  Disease outbreak news; multicountry monkeypox outbreak in non-endemic countries.  https://www.who.int/emergencies/disease-outbreak-news/item/2022-DON385 (2022).
  27. Khaloufi I., Lafif M., Benfatah Y., Laarabi H., Bouyaghroumni J., Rachik M.  A continuous SIR mathematical model of the spread of infectious illnesses that takes human immunity into account.  Mathematical Modeling and Computing.  10 (1), 53–65 (2023).
  28. El Youssoufi 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).
  29. Adi Y. A., Irsalinda N., Wiraya A., Sugiyarto S., Rafsanjani Z. A.  An epidemic model with viral mutations and vaccine interventions.  Mathematical Modeling and Computing.  10 (2), 311–325 (2023).
  30. Kouidere A., Elhia M., Balatif O.  A spatiotemporal spread of COVID-19 pandemic with vaccination optimal control strategy: A case study in Morocco.  Mathematical Modeling and Computing.  10 (1), 171–185 (2023).