Fractional Mathematical Modelling of The Spread of Rotavirus Disease

Yıl 2023, Cilt: 39 Sayı: 2, 253 - 270, 31.08.2023

Fatma Özköse

Öz

In this study, rotavirus disease is examined from a different perspective. In this situation, many variables are used to construct a fractional mathematical model. The model is employed to determine how the disease's transmission will affect susceptible, infected, and recovered individuals. The implications of the fractional derivative on the stability and dynamic behaviour of solutions are examined using the formulation of the Caputo fractional operator. The existence and uniqueness, positivity and boundedness of the solution are next examined. Findings include equilibrium points and stability requirements.
Numerical simulations are used to examine the system's dynamic behaviour. With the use of these simulations, it is possible to study how susceptible, infected, and recovered people change over time by giving fractional values to ϑ. This highlights the advantages of using fractional differential equations. Then it is seen how changing some parameters causes changes in susceptible, infected and recovered individuals.

Anahtar Kelimeler

Fractional order derivative, Stability, Numerical solutions, Numerical simulation, Existence-uniqueness

Kaynakça

• [1] Parashar, U. D., Hummelman, E. G., Bresee, J. S., Miller, M. A., & Glass, R. I. 2003. Global illness and deaths caused by rotavirus disease in children. Emerging infectious diseases, 9(5), 565.
• [2] Clark, H. F., Glass, R. I., & Offitt, P. A. 1999. Rotavirus vaccines. In plotkin SA, Orenstein WA (eds). Vaccines.
• [3] Collins, P. L., McIntosh, K., & Chanock, R. M. 1996. Respiratory syncytial virus In: Fields BN, Knipe DM, Howley PM, et al.(eds) Fields Virology.
• [4] Parashar, U. D., Gibson, C. J., Bresee, J. S., & Glass, R. I. 2006. Rotavirus and severe childhood diarrhea. Emerging infectious diseases, 12(2), 304.
• [5] Liu, L., Johnson, H. L., Cousens, S., Perin, J., Scott, S., Lawn, J. E., ... & Black, R. E. 2012. Global, regional, and national causes of child mortality: an updated systematic analysis for 2010 with time trends since 2000. The lancet, 379(9832), 2151-2161.
• [6] Bozdayi, G., Dogan, B., Dalgic, B., Bostanci, I., Sari, S., Battaloglu, N. O., ... & Ahmed, K. 2008. Diversity of human rotavirus G9 among children in Turkey. Journal of medical virology, 80(4), 733-740.
• [7] Kurugol, Z., Geylani, S., Karaca, Y., Umay, F., Erensoy, S., Vardar, F., ... & Ozkinay, C. 2003. Rotavirus gastroenteritis among children under five years of age in Izmir, Turkey. Turkish Journal of Pediatrics, 45(4), 290-294.
• [8] Cataloluk, O., Iturriza, M., & Gray, J. 2005. Molecular characterization of rotaviruses circulating in the population in Turkey. Epidemiology & Infection, 133(4), 673-678.
• [9] Ceyhan, M., Alhan, E., Salman, N., Kurugol, Z., Yildirim, I., Celik, U., ... & Pawinski, R. 2009. Multicenter prospective study on the burden of rotavirus gastroenteritis in Turkey, 2005–2006: a hospital-based study. The Journal of infectious diseases, 200(Supplement_1), S234-S238.
• [10] Kurugöl, Z., & Salman, N. 2008. Rotavirus infeksiyonları ve aşılar. Ankem Dergisi, 22(3), 160-170.
• [11] Santos, N., & Hoshino, Y. 2005. Global distribution of rotavirus serotypes/genotypes and its implication for the development and implementation of an effective rotavirus vaccine. Reviews in medical virology, 15(1), 29-56.
• [12] Mızrakçı, S. 2022. Rotavirus konulu yayınlara global bakış. Black Sea Journal of Health Science, 5(2), 239-244.
• [13] Aydın, E., Aydin, N., & Perçin Renders, D. U. Y. G. U. 2022. Evaluation of the Effect of Acute Gastroenteritis Factors on Laboratory Parameters in Pediatric Patients. Flora Infeksıyon Hastalıkları Ve Klınık Mıkrobıyolojı Dergısı, 27(1).
• [14] Dalir, M., & Bashour, M. 2010. Applications of fractional calculus. Applied Mathematical Sciences, 4(21), 1021-1032.
• [15] Loverro, A. 2004. Fractional calculus: history, definitions and applications for the engineer. Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering, 1-28.
• [16] Podlubny, 1999. I. Fractional differential equations, Academic Pres, New York.
• [17] Oldham, K.B.; Spanier, 1974. J. The fractional calculus, Academic Pres, New York.
• [18] Miller, K.S.; Ross, B. 1993. An introduction to the fractional calculus and fractional differential equations, Wiley, New York.
• [19] Hilfer, 2000. R. Applications of fractional calculus in physics, World Scientific, Singapore.
• [20] Öztürk, I., & Özköse, F. 2020. Stability analysis of fractional order mathematical model of tumorimmune system interaction. Chaos, Solitons & Fractals, 133, 109614.
• [21] Naik, P. A., Zu, J., & Owolabi, K. M. 2020. Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control. Chaos, Solitons & Fractals, 138, 109826.
• [22] Özköse, F., Şenel, M. T., & Habbireeh, R. 2021. Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83.
• [23] Özköse, F., Habbireeh, R., & Şenel, M. T. 2023. A novel fractional order model of SARS-CoV-2 and Cholera disease with real data. Journal of Computational and Applied Mathematics, 423, 114969.
• [24] Yavuz, M., Özköse, F., Susam, M., & Kalidass, M. 2023. A new modeling of fractional-order and sensitivity analysis for hepatitis-b disease with real data. Fractal and Fractional, 7(2), 165.
• [25] Sabbar, Y., Yavuz, M., & Özköse, F. 2022. Infection Eradication Criterion in a General Epidemic Model with Logistic Growth, Quarantine Strategy, MediaIntrusion, and Quadratic Perturbation. Mathematics, 10(22), 4213.
• [26] Sene, N. 2022. Theory and applications of new fractional-order chaotic system under Caputo operator. An International Journal of Optimization and Control, 12(1), 20-38.
• [27] Evirgen, F., Esmehan, U. Ç. A. R., Sümeyra, U. Ç. A. R., & Özdemir, N. 2023. Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 58-72.
• [28] Veeresha, P., Yavuz, M., & Baishya, C. 2021. A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(3), 52-67.
• [29] Odionyenma, U. B., Ikenna, N., & Bolaji, B. 2023. Analysis of a model to control the co-dynamics of Chlamydia and Gonorrhea using Caputo fractional derivative. Mathematical Modelling and Numerical Simulation with Applications, 3(2), 111-140.
• [30] Atede, A. O., Omame, A., & Inyama, S. C. 2023. A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data. Bulletin of Biomathematics, 1(1), 78-110.
• [31] NWAJERI�, U. K., ATEDE, A. O., PANLE, A. B., & EGEONU, K. U. 2023. Malaria and cholera co-dynamic model analysis furnished with fractional-order differential equations. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 33-57.
• [32] Özköse, F., Yavuz, M., Şenel, M. T., & Habbireeh, R. 2022. Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom. Chaos, Solitons & Fractals, 157, 111954.
• [33] Özköse, F., & Yavuz, M. 2022. Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey. Computers in biology and medicine, 141, 105044.
• [34] Diethelm, K., & Freed, A. D. 1998. The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forschung und wissenschaftliches Rechnen, 1999, 57-71.
• [35] Diethelm, K. 1997. An algorithm for the numerical solution of differential equations of fractional order. Electronic transactions on numerical analysis, 5(1), 1-6.
• [36] Garrappa, R. 2010. On linear stability of predictor–corrector algorithms for fractional differential equations. International Journal of Computer Mathematics, 87(10), 2281-2290.
• [37] Garrappa, R. 2018. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics, 6(2), 16.
• [38] Li C,Tao C. 2009.On the fractional adams method. Comput Math Appl,58(8):1573–88.
• [39] Naik, P. A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. 2020. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135, 1-42.
• [40] O� zköse, F., Yılmaz, S., Yavuz, M., O� ztürk, I�., Şenel, M. T., Bağcı, B. Ş., ... & O� nal, O� . 2022. A fractional modeling of tumor–immune system interaction related to lung cancer with real data. The European Physical Journal Plus, 137, 1-28.
• [41] Susam, M. 2022. Hepatit-B Hastalıgının Kesirli Mertebeden Matematiksel Modeli ve Türkiye’den Gerçek Veri ile Parametre Tahmini (Master's thesis, Necmettin Erbakan U� niversitesi Fen Bilimleri Enstitüsü).