Abstract
Monkeypox virus is a viral zoonotic disease that mostly found in a tropical area of West and central Africa. The disease outbreak occurred in 2022 affected 106 countries from all the six World Health Organization geographic regions and the outbreak accounts for the highest and unprecedented human-to-human transmission. In this paper, a new deterministic mathematical model incorporating post-exposure vaccination, pre-exposure vaccination and isolation is proposed to describe the human-to-human transmission dynamics of monkeypox virus. The analysis of the model shows that it exhibit two equilibria, which are disease free equilibrium and endemic equilibrium. However, the disease free equilibrium is proved to be both locally and globally asymptotically stable whenever the effective reproduction number is less than unity, while the endemic equilibrium is locally and globally asymptotically stable whenever the effective reproduction number is greater than unity. Some of the parameter values of the model are estimated using Nigeria 2022 monkeypox data, while some are obtained from the literature of monkeypox virus mathematical models. Optimal control system is also developed with three control strategies (public awareness campaign, post-exposure vaccination and isolation) to determine the best combination that can immensely decline the transmission or eradicate the disease. We proved that the optimal control indeed exist and we established the optimality conditions through the application of Pontryagin’s principle. The simulation result shows that post-exposure vaccination alone has rare impact in decreasing the spread of monkeypox virus while combination of two strategies (pre-exposure vaccination and isolation or post-exposure vaccination and isolation) have higher impact. In addition, the optimal control simulations show that combining public awareness campaign, post-exposure vaccination and isolation is the best to eradicate monkeypox virus in the society for insufficient vaccine that can cover the large susceptible population.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig9_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig10_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig11_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig12_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig13_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig14_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig15_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig16_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig17_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig18_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig19_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig20_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig21_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40808-023-01920-1/MediaObjects/40808_2023_1920_Fig22_HTML.png)
Similar content being viewed by others
Availability of data and materials
All data used in this research can be found within the manuscript.
References
Ackora-Prah J, Okyere S, Bonyah E, Adebanji A. O, Boateng Y (2023) Optimal control model of human-to-human transmission of monkeypox virus. F1000 Res 12(326):326. https://doi.org/10.12688/f1000research.130276.1
Al Qurashi M, Rashid S, Alshehri AM, Jarad F, Safdar F (2023) New numerical dynamics of the fractional monkeypox virus model transmission pertaining to nonsingular kernels. Math Biosci Eng 20(1):402–436. https://doi.org/10.3934/mbe.2023019
Bankuru SV, Kossol S, Hou W, Mahmoudi P, Rychtář J, Taylor D (2020) A game-theoretic model of Monkeypox to assess vaccination strategies. PeerJ 8:e9272. https://doi.org/10.7717/peerj.9272
Bhunu CP, Mushayabasa S (2011) Modelling the transmission dynamics of pox-like infections
Castillo-Chavez C, Song B (2004) Dynamical model of tuberculosis and their applications. Math Biosci 1:361–404. https://doi.org/10.3934/mbe.2004.1.361
DeJesus EX, Kaufman C (1987) Routh–Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. Phys Rev A 35(12):5288. https://doi.org/10.1103/PhysRevA.35.5288
Diekmann O, Heesterbeek JAP, Metz JA (1990) On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382. https://doi.org/10.1007/BF00178324
Diekmann O, Heesterbeek JAP, Roberts MG (2010) The construction of next-generation matrices for compartmental epidemic models. J R Soc Interface 7(47):873–885. https://doi.org/10.1098/rsif.2009.0386
Grant R, Nguyen LBL, Breban R (2020) Modelling human-to-human transmission of monkeypox. Bull World Health Organ 98(9):638. https://doi.org/10.2471/BLT.19.242347
Gunasekar T, Manikandan S, Govindan V, Ahmad J, Emam W, Al-Shbeil I (2023) Symmetry analyses of epidemiological model for monkeypox virus with Atangana–Baleanu fractional derivative. Symmetry 15(8):1605. https://doi.org/10.3390/sym15081605
Heimann B (1979) Fleming, WH/Rishel, RW, deterministic and stochastic optimal control, vol 49, 9th edn. Springer, New York-Heidelberg-Berlin, pp 494–494. https://doi.org/10.1002/zamm.19790590940 ((XIII, 222 S, DM 60, 60. Zeitschrift Angewandte Mathematik und Mechanik, 1975))
Kermack W. O, McKendrick A. G (1927) A contribution to the mathematical theory of epidemics. In: Proceedings of the royal society of London. Series A, Containing papers of a mathematical and physical character 115(772):700–721. https://doi.org/10.1098/rspa.1927.0118
Kifle ZS, Lemecha Obsu L (2023) Optimal control analysis of a COVID-19 model. Appl Math Sci Eng 31(1):2173188. https://doi.org/10.1080/27690911.2023.2173188
Lasalle JP (1976) The stability of dynamical systems. In: Regional conference series in applied mathematics, SIAM, Philadelphia
Liu B, Farid S, Ullah S, Altanji M, Nawaz R, Wondimagegnhu Teklu S (2023) Mathematical assessment of monkeypox disease with the impact of vaccination using a fractional epidemiological modeling approach. Sci Rep 13(1):13550. https://doi.org/10.1038/s41598-023-40745
Ngungu M, Addai E, Adeniji A, Adam UM, Oshinubi K (2023) Mathematical epidemiological modeling and analysis of monkeypox dynamism with non-pharmaceutical intervention using real data from United Kingdom. Front Public Health 11:1101436. https://doi.org/10.3389/fpubh.2023.1101436
Nigeria Centre for Disease control (2022) https://reliefweb.int/report/nigeria/update-monkeypox-mpx-nigeria-epi-week-52-december-26-2022-january-1-2023
Okyere S, Ackora-Prah J (2023) Modeling and analysis of monkeypox disease using fractional derivatives. Results Eng 17:100786. https://doi.org/10.1016/j.rineng.2022.100786
Peter OJ, Kumar S, Kumari N, Oguntolu FA, Oshinubi K, Musa R (2022) Transmission dynamics of Monkeypox virus: a mathematical modelling approach. Model Earth Syst Environ 1–12. https://doi.org/10.1007/s40808-021-01313-2
Peter OJ, Oguntolu FA, Ojo MM, Olayinka Oyeniyi A, Jan R, Khan I (2022) Fractional order mathematical model of monkeypox transmission dynamics. Phys Scripta 97(8):084005. https://doi.org/10.1088/1402-4896/ac7ebc
Somma SA, Akinwande NI, Chado UD (2019) A mathematical model of monkey pox virus transmission dynamics. Ife J Sci 21(1):195–204. https://doi.org/10.4314/ijs.v21i1.17
United Kingdom Health Security Agency. Recommendations for the use of pre- and post-exposure vaccination during a monkeypox incident.https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1100600/recommendations-for-pre-and-post-exposure-vaccination-during-a-monkeypox-incident-26-august-2022.pdf
Usman S, Adamu II (2017) Modeling the transmission dynamics of the monkeypox virus infection with treatment and vaccination interventions. J Appl Math Phys 5(12):2335. https://doi.org/10.4236/jamp.2017.512191
Vargas-De-León C, d’Onofrio A (2017) Global stability of infectious disease models with contact rate as a function of prevalence index. Math Biosci Eng 14(4):1019–1033. https://doi.org/10.3934/mbe.2017053
World Health Organization (2022a) Multi-country outbreak of monkeypox. External Situation Report 7. https://www.who.int/docs/default-source/coronaviruse/situation-reports/20221005_monkeypox_external_sitrep-7.pdf?sfvrsn=4c3b4c35_4 &download=true
World Health Organization (2022b) Multi-country outbreak of monkeypox. External Situation Report 10. https://www.who.int/docs/default-source/coronaviruse/situation-reports/20221116_monkeypox_external_sitrep-10_cleared.pdf?sfvrsn=c1c0b710_3 &download=true
World Health Organization (2022c) Strategic Preparedness, Readiness and Response Plan. MONKEYPOX. https://www.who.int/publications/m/item/monkeypox-strategic-preparedness--readiness--and-response-plan-(sprp)
World Health Organization (2022d) WHO emergency appeal: monkeypox, July 2022–June 2023.https://www.who.int/publications/m/item/who-emergency-appeal--monkeypox---july-2022---june-2023
Acknowledgements
The researchers would like to acknowledge Deanship of Scientific Research, Taif University for funding this work.
Funding
Deanship of Scientific Research, Taif University.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
There is no competing interest whatsoever in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ahmad, Y.U., Andrawus, J., Ado, A. et al. Mathematical modeling and analysis of human-to-human monkeypox virus transmission with post-exposure vaccination. Model. Earth Syst. Environ. 10, 2711–2731 (2024). https://doi.org/10.1007/s40808-023-01920-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40808-023-01920-1