Investigation of nonstandard finite difference for fractional order Covid-19 model

Abstract

This article examines a mathematical model of the Covid-19 type. We demonstrate how the population is impacted by immigration, protection, the mortality, exposure, curing, and interactions between sick and healthy individuals. There are five classifications in our model: exposed, susceptible, infected, quarantined, and recovered. The model is subjected to numerical and fractional analysis in this instance. The numerical analysis is performed using the fractional order non-standard finite difference (NSFD) scheme. The Grunwald-Letnikov numerical approximation technique is used for fractional analysis. The findings are evaluated by simulations using the Matlab tool.

Keywords

• Caputo fractional derivative
• Covid 19 model
• Fractional order nonstandard finite difference
• Grünwald-Letnikov

References

1. [1] Abbey, H., “An examination of the Reed Frost theory of epidemics”, Human Biology, 24(3): 201, (1952).
2. [2] Kermack, W.O., McKendrick, A.G., “A contribution to the mathematical theory of epidemics: Proceedings of the royal society of london”, Series A, Containing papers of a mathematical and physical character, 115 (772):700-721, (1927).
3. [3] Hethcote, H.W., “The mathematics of infectious diseases”. SIAM review, 42 (4):599-653, (2000).
4. [4] Anderson, R.M., “The population dynamics of infectious diseases: theory and applications”, Springer, (2013).
5. [5] Brauer, F., Castillo-Chavez, C., “Mathematical models in population biology and epidemiology “(Vol. 2, p. 508). New York: Springer, (2012).
6. [6] Murray, J.D., “Mathematical biology: I. An introduction (Vol. 17)”, Springer Science & Business Media, (2007).
7. [7] Oliveira, G., “Refined compartmental models, asymptomatic carriers and COVID-19”, arXiv preprint arXiv: 2004. 14780, (2020).
8. [8] Abou-Ismail, A., “Compartmental Models of the COVID-19 pandemic for physicians and physician scientists”, SN Comprehensive Clinical Medicine, 2: 852-858, (2020).

9. [9] Zafar, Z.A., Nigar A., Zaman, G., Thounthong, P., Tunc C., “Analysis and numerical simulations of fractional order Vallis system”, Alexandria Engineering Journal, 59: 2591–2605, (2020).
10. [10] Scherer, R., Kalla, S.L., Tangc, Y., Huang, J., “The Grünwald–Letnikov method for fractional differential equations”, Computers and Mathematics with Applications, 62: 902–917, (2011).
11. [11] Jacobs, B.A., “A New Grünwald-Letnikov derivative derived from a second-order scheme”, Abstract and Applied Analysis, 952057: 9, (2015).
12. [12] Chen, Y., Liu F., Yu Q., Li T., “Review of fractional epidemic models”, Applied Mathematical Modelling, 97: 281–307, (2021).
13. [13] Hanert, E., Schumacher, E., “Front dynamics in fractional-order epidemic modes”, Journal of Theoretical Biology, 279: 9–16, (2011).
14. [14] Podlubny, I., “Fractional Differential Equations”, Academic Press (1998).
15. [15] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., “Theory and Applications of Fractional Differential Equations”, Elsevier Science, USA, (2006).
16. [16] Caputo, M., Fabrizio, M.A., “New definition of fractional derivative without singular kernel”, Progress in Fractional Differentiation & Applications, 1(2): 73-85, (2015).
17. [17] Ouazi, S-ld., El Khomssi M., “Mathematical approaches to controlling COVID-19: optimal control and financial benefits”, Mathematical Modelling and Numerical Simulation with Applications, 4(1): 1-36, (2024).
18. [18] Joshi, H., Yavuz M., “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): 468, (2023).
19. [19] Kamrujjaman, M., Sinje, S.S., Nandi, T.R., Islam, F., Rahman, M.A., Akhi, A.A., Tasnim, F., Alam, M.S., “The impact of the COVID-19 pandemic on education in Bangladesh and its mitigation”, Bulletin of Biomathematics, 2(1): 57-84, (2024).
20. [20] Joshi, H., Yavuz M., Townley, S., Jha, B.K., “Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate”, Physica Scripta, 98(4): 045216, (2023).
21. [21] Atede, A.O., Omame, A., Inyama, S.C.A., “Fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data”, Bulletin of Biomathematics, 1(1): 78-110, (2023).
22. [22] Yavuz, M., Haydar, W.Y.A., “A new mathematical modelling and parameter estimation of COVID-19: a case study in Iraq”, AIMS Bioengineering, 9(4): 420-446, (2022).
23. [23] Ahmad, S., Qiu, D., Rahman, M.ur., “Dynamics of a fractional-order COVID-19 model under the nonsingular kernel of Caputo-Fabrizio operator”, Mathematical Modelling and Numerical Simulation with Applications, 2(4): 228-243, (2022).
24. [24] Evirgen, F., Özköse, F., Yavuz, M., Özdemir, N., “Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks”, AIMS Bioengineering, 10(3): 218-239, (2023).
25. [25] Özköse, F., “Long-term side effects: A mathematical modeling of COVID-19 and stroke with real data”, Fractal and Fractional, 7(10): 719, (2023).
26. [26] Özköse, F., Habbireeh, R., Şenel, M.T., “A novel fractional order model of SARS-CoV-2 and Cholera disease with real data”, Journal of Computational and Applied Mathematics, 423: 114969, (2023).
27. [27] Özköse, F., Yavuz, M., “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, (2022).
28. [28] Özköse, F., Yavuz, M., Şenel, M.T., Habbireeh, R., “Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom”, Chaos, Solitons and Fractals, 157: 111954, (2022).
29. [29] Mickens, R.E., “Application of Nonstandard Finite Difference Scheme”s, World Scientific Publishing Co. Pte. Ltd., (2000).
30. [30] Mickens, R.E.,” Nonstandard Finite Difference Models of Differential Equations”, World Scientific, (1994).
31. [31] Mickens, R.E., “Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition”, Numerical Methods Partial Differential Equations, 23: 672–691, (2007).
32. [32] Mickens, R.E., “Nonstandard Finite Difference Schemes for Differential Equations”, Journal of Difference Equations and Applications, 8(9): 823–847, (2002).
33. [33] Patidar, K.C., “On the Use of Nonstandard Finite Difference Methods”, Journal of Difference Equations and Applications, 11 (8): 735–758, (2005).
34. [34] Patidar, K.C., “Nonstandard finite difference methods: recent trends and further developments”, Journal of Difference Equations and Applications, 22 (6): 817–849, (2016).
35. [35] Baleanu, D., Zibaei, S., Namjoo, M., Jajarmi, A., “A Nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system”, Advances in Difference Equations, 308(1): 19, (2021).
36. [36] Dang, Q.A., Hoang, M.T., “Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models”, Journal of Difference Equations and Applications, 24 (1): 15–47, (2018).
37. [37] Dang, Q.A., Hoang, M.T., “Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model”, International Journal of Dynamics and Control, 8 (3): 772–778, (2020).
38. [38] Kocabıyık, M., Özdoğan, N., Ongun, M.Y., “Nonstandard finite difference scheme for a computer virus model”, Journal of Innovative Science and Engineering, 4 (2): 96–108, (2020).
39. [39] Özdoğan, N., Ongun, M.Y.,” Dynamical behaviours of a discretized model with Michaelis-Menten harvesting rate”, Journal of Universal Mathematics, 5 (2): 159–176, (2022).
40. [40] Kocabıyık, M., Ongun, M.Y., “Construction a distributed order smoking model and its nonstandard finite difference discretization”, AIMS Mathematics, 7 (3): 4636—4654, (2021).
41. [41] Ongun, M.Y., Arslan, D., “Explicit and implicit schemes for fractional–order hantavirus model”, Iranian Journal of Numerical Analysis and Optimization, 8 (2): 75–94, (2018).
42. [42] Abioye, A.I., Peter, O.J., Ogunseye, H.A., Oguntolu F.A., Oshinubi, K., Ibrahim A.A., Khan I., “Mathematical model of Covid-19 in Nigeria with optimal control”. Result in Physics, 28:104598, (2021).
43. [43] Anguelov R., Lubuma J.S., Mahudu S., “Qualitatively stable finite difference schemes for advection-reaction equations”, Journal of Computational and Applied Mathematics, 158: 19–30, (2003).
44. [44] Anguelov, R., Lubuma, JS., “Contributions to the mathematics of the nonstandard finite difference method and applications”, Numer. Methods Partial Differential Equations 17(5): 518–543, (2001).
45. [45] Arenasa, J.A., Parrab, G.G., Chen-Charpentier, B.M., “Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order”, Mathematics and Computers in Simulation, 121: 48–63, (2011).