Maymun Çiçeğinin Modellemesi: Sosyal Mesafe, Karantina ve Aşılama ile Salgının Yayılımı

Abstract

Bu çalışmada, zoonotik bir viral hastalık olan maymun çiçeğinin yayılımını simüle etmek için hem insan hem de kemirgen popülasyonlarını içeren yenilikçi bir matematiksel model sunduk. Önceki modellerden farklı olarak, enfekte insanlar için bir karantina bölmesi, hassas bireyler için bir sosyal mesafe bölmesi ve doğrudan iyileşen bölmeye geçişle birlikte aşılama sürecini ekleyerek maymun çiçeği yayılımını kontrol etmeye yönelik daha kapsamlı bir çerçeve sağladık. Ardından, bu üç kontrol önleminin hastalığın bulaşmasını azaltmadaki etkinliğini karşılaştırdık.Modelin dinamiklerini incelemek için öncelikle modelin çözümünün, tek, pozitif ve sınırlı olduğunu gösterdik. Daha sonra önerilen model için temel üreme sayısını (R_0) hesapladık. Anahtar parametreleri belirlemek için bir duyarlılık analizi gerçekleştirdik ve bu parametrelerin R_0 üzerindeki etkilerini değerlendirdik. Ayrıca, hastalığın yok olduğu veya endemik kaldığı koşulları belirlemek için hem hastalıksız denge noktasının hem de endemik denge noktasının kararlılığını analiz ettik. Bulgularımız, maymun çiçeği dinamiklerini etkileyen temel parametrelerin aşılama, sosyal mesafe ve karantina olduğunu ortaya koydu. Duyarlılık analizi, iyileşme oranlarının ve kuluçka sürelerinin salgın eğrisi üzerindeki kritik rolünü vurguladı. Bunun yanında, insan-insan bulaşmasının (β_hh) kemirgen-insan bulaşmasından ( β_rh) daha büyük bir etkiye sahip olduğunu gösterdik. Kontrolleri eş zamalı olarak birlikte uyguladığımızda, özellikle kontrollerin parametre değerleri belirli seviyelere ulaşıldığında enfekte vakaları büyük ölçüde azaltmaktadır.

Keywords

• Mpox
• duyarlılık analizi
• bulaşıcı hastalık
• temel üreme sayısı
• kararlılık analizi

Modeling Monkeypox: Spread of Outbreak with Social Distancing, Quarantine and Vaccination

Abstract

In this paper, we introduced a novel mathematical model to simulate the spread of the zoonotic viral disease monkeypox, incorporating both human and rodent populations to capture the disease dynamics. Unlike previous models, we included a quarantine compartment for infected humans, a social distancing compartment for susceptible individuals, and vaccination with direct transmission to the recovered compartment, offering a more comprehensive framework for controlling the spread of monkeypox. We then compared the effectiveness of these three control measures in reducing disease transmission. To investigate the dynamics of the model, we first demonstrated that it has a unique, positive, and bounded solution. Next, we calculated the basic reproduction number, R_0 for the proposed model. A sensitivity analysis is then conducted to identify key parameters, followed by an assessment of their effects on R_0. Additionally, we analyzed the local stability of both the disease-free and endemic equilibrium points to identify the conditions under which the disease dies out or remains endemic. We first showed in stability analysis section that these three control parameters play important roles in stability of equlibrium points. After that our findings in sensitivity analysis indicated the critical role of recovery rates and incubation periods in shaping the outbreak trajectory. Besides them, our analysis of R_0 in 3-D plots showed that the human-to-human transmission (β_hh) has about 3 times greater impact than rodent-to-human transmission (β_rh) on R_0. Finally, we presented simulations to show single and combined effects of the control parameters: quarantine, social distancing and vaccination on the transmission of monkeypox virus.

Keywords

• Mpox
• sensitivity analysis
• infectious disease
• basic reproduction number
• stability analysis.

Ethical Statement

The study is complied with research and publication ethics.

References

1. P. R. Reed et al., "Monkeypox virus transmission and pathogenesis," Journal of Infectious Diseases, 2004.
2. Z. Jezek et al., "Human monkeypox," American Journal of Tropical Medicine and Hygiene, vol. 38, no. 2, pp. 3–13, 1988.
3. S. S. Bhullar et al., "Emerging zoonotic viral diseases: Insights into monkeypox and its management," Medical Research Reviews, 2022.
4. M. O. Sefiu, "Modeling the dynamics of rodent-human transmission of monkeypox using an SIQR-SEI framework," Journal of Mathematical Epidemiology, vol. 12, no. 3, pp. 256–271, 2024. https://doi.org/ 10.38088/jise.1344860
5. P. Emeka, M. Ounorah, F. Eguda, and B. Babangida, "Mathematical model for monkeypox virus transmission dynamics," Epidemiology, vol. 8, no. 3, p. 1000348, 2018, doi: 10.4172/2161-1165.1000348.
6. S. Somma, N. Akinwande, and U. Chado, "A mathematical model of monkeypox virus transmission dynamics," Ife Journal of Science, vol. 21, no. 1, pp. 195–204, 2019.
7. Q. Huang, Y. Sun, M. Jia, M. Jiang, Y. Xu, L. Feng, and W. Yang, "An effectiveness study of vaccination and quarantine combination strategies for containing mpox transmission on simulated college campuses," Infectious Disease Modelling, vol. 9, no. 3, pp. 805–815, 2024, doi: 10.1016/j.idm.2024.04.004.
8. M. S. Ullah and K. M. A. Kabir, "Behavioral game of quarantine during the monkeypox epidemic: Analysis of deterministic and fractional order approach," Heliyon, vol. 10, no. 5, p. e26998, 2024, doi: 10.1016/j.heliyon.2024.e26998.

9. R. Alharbi, R. Jan, S. Alyobi, Y. Altayeb, and Z. Khan, "Mathematical modeling and stability analysis of the dynamics of monkeypox via fractional-calculus," Fractals, vol. 30, Art. no. 2240266, 2022, doi: 10.1142/S0218348X22402666.
10. T. D. Frank, "Mathematical analysis of four SEIR-type models for monkeypox outbreaks: Human-animal interactions," Mathematics, vol. 12, no. 20, p. 3215, 2024, doi: 10.3390/math12203215.
11. O. J. Peter, F. A. Oguntolu, M. M. Ojo, O. Abdulmumin, A. O. Oyeniyi, R. Jan, and I. Khan, "Fractional order mathematical model of monkeypox transmission dynamics," Physica Scripta, vol. 97, no. 8, p. 084005, 2022, doi: 10.1088/1402-4896/ac7ebc.
12. O. I. Idisi, T. T. Yusuf, E. Adeniyi, A. A. Onifade, Y. T. Oyebo, A. T. Samuel, and L. A. Kareem, "A new compartmentalized epidemic model to analytically study the impact of awareness on the control and mitigation of the monkeypox disease," Healthcare Analytics, vol. 4, 2023, Art. no. 100267, doi: 10.1016/j.health.2023.100267.
13. P. O. J. Peter, S. Kumar, N. Kumari, F. A. Oguntolu, K. Oshinubi, and R. Musa, "Transmission dynamics of Monkeypox virus: a mathematical modelling approach," Model Earth Syst Environ, vol. 8, pp. 3423–3434, 2022, doi: 10.1007/s40808-021-01313-2.
14. R. Sah et al., "Monkeypox and its possible sexual transmission: Where are we now with its evidence?," Pathogens, vol. 11, no. 8, p. 924, 2022.
15. P. van den Driessche and J. Watmough, "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission," Mathematical Biosciences, vol. 180, pp. 29–48, 2002, doi: 10.1016/S0025-5564(02)00108-6.
16. O. Diekmann, J. A. P. Heesterbeek, and J. A. Metz, "On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations," Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990.
17. O. I. Idisi and T. T. Yusuf, "A mathematical model for Lassa fever transmission dynamics with impacts of control measures: Analysis and simulation," European Journal of Mathematics and Statistics, vol. 2, no. 2, pp. 19–28, 2021, doi: 10.24018/ejmath.2021.2.2.17.
18. E. J. Routh, A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion. Cambridge, UK: Cambridge University Press, 1877.
19. A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt," Mathematische Annalen, vol. 46, no. 2, pp. 273–284, 1895, doi: 10.1007/BF01446812.
20. G. Chowell, H. Nishiura, and L. M. A. Bettencourt, "Comparative estimation of the reproduction number for pandemic influenza from daily case notification data," Journal of Theoretical Biology, vol. 364, pp. 149–159, 2014, doi: 10.1016/j.jtbi.2014.08.004.
21. I. H. Aslan, M. Demir, M. M. Wise, and S. Lenhart, "Modeling COVID-19: Forecasting and analyzing the dynamics of the outbreak in Hubei and Turkey," Mathematical Methods in the Applied Sciences, vol. 45, no. 10, pp. 6481–6494, 2022, doi: 10.1002/mma.8181.
22. M. Demir, I. H. Aslan, and S. Lenhart, "Analyzing the effect of restrictions on the COVID-19 outbreak for some US states," Theoretical Ecology, vol. 16, pp. 117–129, 2023, doi: 10.1007/s12080-023-00557-1.
23. S. Marino, I. B. Hogue, C. J. Ray, and D. E. Kirschner, "A methodology for performing global uncertainty and sensitivity analysis in systems biology," Journal of Theoretical Biology, vol. 254, no. 1, pp. 178–196, 2008.
24. H. Joshi, M. Yavuz, S. Townley, and B. K. Jha. "Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate". Physica Scripta 98.4 (2023): 045216.
25. H. Joshi. "Mechanistic insights of COVID-19 dynamics by considering the influence of neurodegeneration and memory trace", Physica Scripta 99.3 (2024): 035254
26. H. Joshi, and M. Yavuz. "Transition dynamics between a novel coinfection model of fractional-order for COVID-19 and tuberculosis via a treatment mechanism". The European Physical Journal Plus 138.5 (2023): 468.
27. H. Joshi, and B. K. Jha. Fractional reaction diffusion model for parkinson’s disease, Proceedings of the International Conference on ISMAC in Computational Vision and Bio-Engineering 2018 (ISMAC-CVB). Springer International Publishing, 2019.