Mathematical modeling and optimal control strategy for the monkeypox epidemic

In this study, we propose a discrete time mathematical model (SEIQR) that describes the dynamics of monkeypox within a human population.  The studied population is divided into five compartments: susceptible ($S$), exposed ($E$), infected ($I$), quarantined ($Q$), and recovered ($R$).  Also, we propose an optimal strategy to fight against the spread of this epidemic.  In this sense we use three controls which represent: 1) the awarness of vulnerable people through the media, civil society and education; 2) the quarantine of infected persons at home or, if required, in hospital; 3) encouraging of vaccination of susceptible persons.  To characterize these optimal controls, we apply the Pontryagin's maximum principle.  The optimality system is solved numerically using Matlab.  Therefore, the obtained results confirm the effectiveness of the proposed optimization approach.

  1. Durski K. N., McCollum A. M., Nakazawa Y., Petersen B. W., et al.  Emergence of Monkeypox – West and Central Africa, 1970–2017.  Morbidity and Mortality Weekly Report.  67 (10), 306 (2018).
  2. World Health Organization (WHO). Mpox (monkeypox). (2022).
  3. Yan X., Zou Y. Optimal and sub-optimal quarantine and isolation control in SARS epidemics.  Mathematical and Computer Modelling.  47 (1–2), 235–245 (2008).
  4. Likos A. M., Sammons S. A., Olson V. A., Frace A. M., Li Y., Olsen-Rasmussen M., Davidson W., et al.  A tale of two clades: monkeypox viruses.  Journal of General Virology.  86 (10), 2661–2672 (2005).
  5. Yinka-Ogunleye A., Aruna O., Ogoina D., Aworabhi N., Eteng W., Badaru S., Mohammed A., et al.  Reemergence of Human Monkeypox in Nigeria, 2017.  Emerging Infectious Diseases.  24 (6), 1149 (2018).
  6. Centers for Disease Control and Prevention (CDC).  Multistate outbreak of monkeypox – Illinois, Indiana, and Wisconsin, 2003.  Morbidity and Mortality Weekly Report.  52 (23), 537–540 (2003).
  7. Sklenovská N., Van Ranst M.  Emergence of monkeypox as the most important orthopoxvirus infection in humans.  Frontiers in Public Health.  6, 241 (2018).
  8. Eteng W. E., Mandra A., Doty J., Yinka-Ogunleye A., Aruna S., Reynolds M. G., McCollum A. M., Davidson W., Wilkins K., et al.  Notes from the field: responding to an outbreak of monkeypox using the one health approach –- Nigeria, 2017–2018.  Morbidity and Mortality Weekly Report.  67 (37), 1040 (2018).
  9. Organisation panaméricaine de la Santé.  VIIIe réunion ad hoc du Groupe consultatif technique (GCT) de l'OPS sur les maladies évitables par la vaccination. Briefing technique sur l'épidémie multi-pays de variole du singe, 31 mai 2022 (virtual).
  10. Grant R., Nguyen L. B., Breban R.  Modelling human-to-human transmission of monkeypox.  Bulletin of the World Health Organization.  98 (9), 638 (2020).
  11. Weinstein R. A., Nalca A., Rimoin A. W., Bavari S., Whitehouse C. A.  Reemergence of monkeypox: prevalence, diagnostics, and countermeasures.  Clinical Infectious Diseases.  41 (12), 1765–1771 (2005).
  12. Jezek Z., Szczeniowski M., Paluku K. M., Mutombo M., Grab B.  Human monkeypox: confusion with chickenpox.  Acta Tropica.  45 (4), 297–307 (1988).
  13. Jezek Z., Marennikova S. S., Mutumbo M., Nakano J. H., Paluku K. M., Szczeniowski M.  Human monkeypox: a study of 2,510 contacts of 214 patients.  Journal of Infectious Diseases.  154 (4), 551–555 (1986).
  14. Lai C.-C., Hsu C.-K., Yen M.-Y., Lee P.-I., Ko W.-C., Hsueh P.-R.  Monkeypox: an emerging global threat during the COVID-19 pandemic.  Journal of Microbiology, Immunology and Infection.  55 (5), 787–794 (2022).
  15. Samaranayake L., Anil S.  The monkeypox outbreak and implications for dental practice.  International Dental Journal.  72 (5), 589–596 (2022).
  16. Yinka-Ogunleye A., Aruna O., Dalhat M., Ogoina D., McCollum A., Disu Y., Mamadu I., Akinpelu A., et al.  Outbreak of human monkeypox in Nigeria in 2017–18: a clinical and epidemiological report.  The Lancet Infectious Diseases.  19 (8), 872–879 (2019).
  17. Petersen E., Kantele A., Koopmans M., Asogun D., Yinka-Ogunleye A., Ihekweazu C., Zumla A.  Human monkeypox: epidemiologic and clinical characteristics, diagnosis, and prevention.  Infectious Disease Clinics of North America.  33 (4), 1027–1043 (2019).
  18. Thornhill J. P., Barkati S., Walmsley S., Rockstroh J., et al.  Monkeypox virus infection in humans across 16 countries — April – June 2022.  New England Journal of Medicine.  387 (8), 679–691 (2022).
  19. Foster S. O., Brink E. W., Hutchins D. L., Pifer J. M., Lourie B., Moser C. R., Cummings E. C., et al.  Human monkeypox.  Bulletin of the World Health Organization.  46 (5), 569–576 (1972).
  20. Ježek Z., Szczeniowski M., Paluku K. M., Mutombo M.  Human monkeypox: clinical features of 282 patients.  Journal of Infectious Diseases.  156 (2), 293–298 (1987).
  21. Cheema A. Y., Ogedegbe O. J., Munir M., Alugba G., Ojo T. K.  Monkeypox: A Review of Clinical Features, Diagnosis, and Treatment.  Cureus.  14 (7), e26756 (2022).
  22. Desenclos J.-C., De Valk H.  Emergent infectious diseases: importance for public health, epidemiology, promoting factors, and prevention.  Médecine et Maladies Infectieuses.  35 (2), 49–61 (2005).
  23. Artois M., Caron A., Leighton F. A., Bunn C., Vallat B.  La faune sauvage et les maladies émergentes.  Rev. sci. tech. Off. int. Epiz.  25 (3), 897–912 (2006).
  24. Vaughan A., Aarons E., Astbury J., Balasegaram S., Beadsworth M., Beck C. R., Chand M., O'Connor C., Dunning J., Ghebrehewet S., Harper N. et al.  Two cases of monkeypox imported to the United Kingdom, September 2018.  Eurosurveillance.  23 (38), 1800509 (2018).
  25. Uwishema O., Adekunbi O., Peñamante C. A., Bekele B. K., Khoury C., Mhanna M., Nicholas A., Adanur I., Dost B., Onyeaka H.  The burden of monkeypox virus amidst the Covid-19 pandemic in Africa: A double battle for Africa.  Annals of Medicine and Surgery.  80, 104197 (2022).
  26. Koenig K. L., Beÿ C. K., Marty A. M.  Monkeypox 2022 Identify-Isolate-Inform: A 3I Tool for frontline clinicians for a zoonosis with escalating human community transmission.  One Health.  15, 100410 (2022).
  27. Dashraath P., Nielsen-Saines K., Mattar C., Musso D., Tambyah P., Baud D.  Guidelines for pregnant individuals with monkeypox virus exposure.  The Lancet.  400 (10345), 21–22 (2022).
  28. George L. A., Regi A.  Monkey Pox: An Upcoming Threat in World; A Systematic Review.  International Journal of Research Publication and Reviews.  3 (7), 2202–2206 (2022).
  29. Mbala P. K., Huggins J. W., Riu-Rovira T., Ahuka S. M., Mulembakani P., Rimoin A. W., Martin J. W., Muyembe J. J.  Maternal and fetal outcomes among pregnant women with human monkeypox infection in the Democratic Republic of Congo.  Journal of Infectious Diseases.  216 (7), 824–828 (2017).
  30. Antinori A., Mazzotta V., Vita S., Carletti F., Tacconi D., Lapini L. E., D'Abramo A., Cicalini S., Lapa D., Pittalis S., Puro V.  Epidemiological, clinical and virological characteristics of four cases of monkeypox support transmission through sexual contact, Italy, May 2022.  Eurosurveillance.  27 (22), 2200421 (2022).
  31. Islam Q. T.  Monkeypox – A New Threat for the Globe!  Journal of Medicine.   23 (2), 104–105 (2022).
  32. Kermack W. O., McKendrick A. G.  A contribution to the mathematical theory of epidemics.  Proceedings of the Royal Society of London. Series A.  115 (772), 700–721 (1927).
  33. Kumar A., Srivastava P. K.  Vaccination and treatment as control interventions in an infectious disease model with their cost optimization.  Communications in Nonlinear Science and Numerical Simulation.  44, 334–343 (2017).
  34. Sari R. A., Habibah U., Widodo A.  Optimal control on model of SARS disease spread with vaccination and treatment.  Journal of Experimental Life Science.  7 (2), 61–68 (2017).
  35. El Mansouri A., Labzai A., Belam M., Rachik M.  Mathematical modeling and optimal control strategy for the obesity epidemic.  Communications in Mathematical Biology and Neuroscience.  2022, 20 (2022).
  36. Essounaini A., Labzai A., Laarabi H., Rachik M.  Mathematical modeling and optimal control strategy for a discrete time model of COVID-19 variants.  Communications in Mathematical Biology and Neuroscience.  2022, 25 (2022).
  37. Khajji B., Kada D., Balatif O., Rachik M.  A multi-region discrete time mathematical modeling of the dynamics of Covid-19 virus propagation using optimal control.  Journal of Applied Mathematics and Computing.  64, 255–281 (2020).
  38. Labzai A., Kouidere A., Balatif O., Rachik M.  Stability analysis of mathematical model new corona virus (COVID-19) disease spread in population.  Communications in Mathematical Biology and Neuroscience.  2020, 41 (2020).
  39. Khajji B., Boujallal L., Elhia M., Balatif O., Rachik M.  A fractional-order model for drinking alcohol behaviour leading to road accidents and violence.  Mathematical Modeling and Computing.  9 (3), 501–518 (2022).
  40. Sadki M., Harroudi S., Allali K.  Dynamical analysis of an HCV model with cell-to-cell transmission and cure rate in the presence of adaptive immunity.  Mathematical Modeling and Computing.  9 (3), 579–593 (2022).
  41. Bounkaicha C., Allali K., Tabit Y., Danane J.  Global dynamic of spatio-temporal fractional order SEIR model.  Mathematical Modeling and Computing.  10 (2), 299–310 (2023).
  42. Elkaf M., Allali K.  Fractional derivative model for tumor cells and immune system competition.  Mathematical Modeling and Computing.  10 (2), 288-298 (2023).
  43. Peter O. J., Kumar S., Kumari N., Oguntolu F. A., et al.  Transmission dynamics of Monkeypox virus: a mathematical modelling approach.  Modeling Earth Systems and Environment.  8, 3423–3434 (2022).
  44. Yuan P., Tan Y., Yang L., Aruffo E., Ogden N. H., Bélair J., Heffernan J., Arino J., Watmough J., Carabin H., Zhu H.  Assessing transmission risks and control strategy for monkeypox as an emerging zoonosis in a metropolitan area.  Journal of Medical Virology.  95 (1), e28137 (2023).
  45. Usman S., Adamu I. I.  Modeling the transmission dynamics of the monkeypox virus infection with treatment and vaccination interventions.  Journal of Applied Mathematics and Physics.  5 (12), 2335 (2017).
  46. Hwang C. L., Fan L. T.  A discrete version of Pontryagin's maximum principle.  Operations Research.  15 (1), 139–146 (1967).
  47. Zeb A., Zaman G., Momani S.  Square-root dynamics of a giving up smoking model.  Applied Mathematical Modelling.  37 (7), 5326–5334 (2013).
  48. Zhang D. C., Shi B.  Oscillation and global asymptotic stability in a discrete epidemic model.  Journal of Mathematical Analysis and Applications.  278 (1), 194–202 (2003).
  49. Ding W., Hendon R., Cathey B., Lancaster E., Germick R.  Discrete time optimal control applied to pest control problems.  Involve, a Journal of Mathematics.  7 (4), 479–489 (2014).
  50. Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F.  The Mathematical Theory of Optimal Processes.  Wiley, New York (1962).
  51. Rafal M. D., Stevens W. F.  Discrete dynamic optimization applied to on-line optimal control.  AIChE Journal.  14 (1), 85–91 (1968).
  52. Von Magnus P., Andersen E. K., Petersen K. B., Birch-Andersen A.  A pox-like disease in cynomolgus monkeys.  Acta Pathologica Microbiologica Scandinavica.  46, 156–176 (1959).
  53. The Eagle Online. Breaking: Monkey pox: FG confirms only four cases. (2017).
  54. Odom M. R., Hendrickson R. C., Lefkowitz E. J.  Poxvirus protein evolution: Family wide assessment of possible horizontal gene transfer events.  Virus Research.  144 (1–2), 233–249 (2009).
Mathematical Modeling and Computing, Vol. 10, No. 3, pp. 944–955 (2023)