Research Article
BibTex RIS Cite

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

Year 2025, Volume: 38 Issue: 2, 874 - 889, 01.06.2025

Mehmet Merdan , Pınar Açıkgöz

https://doi.org/10.35378/gujs.1456440

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] Abbey, H., “An examination of the Reed Frost theory of epidemics”, Human Biology, 24(3): 201, (1952).
  • [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] Hethcote, H.W., “The mathematics of infectious diseases”. SIAM review, 42 (4):599-653, (2000).
  • [4] Anderson, R.M., “The population dynamics of infectious diseases: theory and applications”, Springer, (2013).
  • [5] Brauer, F., Castillo-Chavez, C., “Mathematical models in population biology and epidemiology “(Vol. 2, p. 508). New York: Springer, (2012).
  • [6] Murray, J.D., “Mathematical biology: I. An introduction (Vol. 17)”, Springer Science & Business Media, (2007).
  • [7] Oliveira, G., “Refined compartmental models, asymptomatic carriers and COVID-19”, arXiv preprint arXiv: 2004. 14780, (2020).
  • [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] 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] 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] Jacobs, B.A., “A New Grünwald-Letnikov derivative derived from a second-order scheme”, Abstract and Applied Analysis, 952057: 9, (2015).
  • [12] Chen, Y., Liu F., Yu Q., Li T., “Review of fractional epidemic models”, Applied Mathematical Modelling, 97: 281–307, (2021).
  • [13] Hanert, E., Schumacher, E., “Front dynamics in fractional-order epidemic modes”, Journal of Theoretical Biology, 279: 9–16, (2011).
  • [14] Podlubny, I., “Fractional Differential Equations”, Academic Press (1998).
  • [15] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., “Theory and Applications of Fractional Differential Equations”, Elsevier Science, USA, (2006).
  • [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] Mickens, R.E., “Application of Nonstandard Finite Difference Scheme”s, World Scientific Publishing Co. Pte. Ltd., (2000).
  • [30] Mickens, R.E.,” Nonstandard Finite Difference Models of Differential Equations”, World Scientific, (1994).
  • [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] Mickens, R.E., “Nonstandard Finite Difference Schemes for Differential Equations”, Journal of Difference Equations and Applications, 8(9): 823–847, (2002).
  • [33] Patidar, K.C., “On the Use of Nonstandard Finite Difference Methods”, Journal of Difference Equations and Applications, 11 (8): 735–758, (2005).
  • [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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).

Year 2025, Volume: 38 Issue: 2, 874 - 889, 01.06.2025

Mehmet Merdan , Pınar Açıkgöz

https://doi.org/10.35378/gujs.1456440

Abstract

References

  • [1] Abbey, H., “An examination of the Reed Frost theory of epidemics”, Human Biology, 24(3): 201, (1952).
  • [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] Hethcote, H.W., “The mathematics of infectious diseases”. SIAM review, 42 (4):599-653, (2000).
  • [4] Anderson, R.M., “The population dynamics of infectious diseases: theory and applications”, Springer, (2013).
  • [5] Brauer, F., Castillo-Chavez, C., “Mathematical models in population biology and epidemiology “(Vol. 2, p. 508). New York: Springer, (2012).
  • [6] Murray, J.D., “Mathematical biology: I. An introduction (Vol. 17)”, Springer Science & Business Media, (2007).
  • [7] Oliveira, G., “Refined compartmental models, asymptomatic carriers and COVID-19”, arXiv preprint arXiv: 2004. 14780, (2020).
  • [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] 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] 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] Jacobs, B.A., “A New Grünwald-Letnikov derivative derived from a second-order scheme”, Abstract and Applied Analysis, 952057: 9, (2015).
  • [12] Chen, Y., Liu F., Yu Q., Li T., “Review of fractional epidemic models”, Applied Mathematical Modelling, 97: 281–307, (2021).
  • [13] Hanert, E., Schumacher, E., “Front dynamics in fractional-order epidemic modes”, Journal of Theoretical Biology, 279: 9–16, (2011).
  • [14] Podlubny, I., “Fractional Differential Equations”, Academic Press (1998).
  • [15] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., “Theory and Applications of Fractional Differential Equations”, Elsevier Science, USA, (2006).
  • [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] Mickens, R.E., “Application of Nonstandard Finite Difference Scheme”s, World Scientific Publishing Co. Pte. Ltd., (2000).
  • [30] Mickens, R.E.,” Nonstandard Finite Difference Models of Differential Equations”, World Scientific, (1994).
  • [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] Mickens, R.E., “Nonstandard Finite Difference Schemes for Differential Equations”, Journal of Difference Equations and Applications, 8(9): 823–847, (2002).
  • [33] Patidar, K.C., “On the Use of Nonstandard Finite Difference Methods”, Journal of Difference Equations and Applications, 11 (8): 735–758, (2005).
  • [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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).
There are 45 citations in total.

Details

Primary Language English
Subjects Biological Mathematics, Applied Mathematics (Other)
Journal Section Mathematics
Authors

Mehmet Merdan 0000-0002-8509-3044

Pınar Açıkgöz 0009-0006-7872-9135

Early Pub Date April 26, 2025
Publication Date June 1, 2025
Submission Date March 22, 2024
Acceptance Date January 15, 2025
Published in Issue Year 2025 Volume: 38 Issue: 2

Cite

APA Merdan, M., & Açıkgöz, P. (2025). Investigation of nonstandard finite difference for fractional order Covid-19 model. Gazi University Journal of Science, 38(2), 874-889. https://doi.org/10.35378/gujs.1456440
AMA Merdan M, Açıkgöz P. Investigation of nonstandard finite difference for fractional order Covid-19 model. Gazi University Journal of Science. June 2025;38(2):874-889. doi:10.35378/gujs.1456440
Chicago Merdan, Mehmet, and Pınar Açıkgöz. “Investigation of Nonstandard Finite Difference for Fractional Order Covid-19 Model”. Gazi University Journal of Science 38, no. 2 (June 2025): 874-89. https://doi.org/10.35378/gujs.1456440.
EndNote Merdan M, Açıkgöz P (June 1, 2025) Investigation of nonstandard finite difference for fractional order Covid-19 model. Gazi University Journal of Science 38 2 874–889.
IEEE M. Merdan and P. Açıkgöz, “Investigation of nonstandard finite difference for fractional order Covid-19 model”, Gazi University Journal of Science, vol. 38, no. 2, pp. 874–889, 2025, doi: 10.35378/gujs.1456440.
ISNAD Merdan, Mehmet - Açıkgöz, Pınar. “Investigation of Nonstandard Finite Difference for Fractional Order Covid-19 Model”. Gazi University Journal of Science 38/2 (June2025), 874-889. https://doi.org/10.35378/gujs.1456440.
JAMA Merdan M, Açıkgöz P. Investigation of nonstandard finite difference for fractional order Covid-19 model. Gazi University Journal of Science. 2025;38:874–889.
MLA Merdan, Mehmet and Pınar Açıkgöz. “Investigation of Nonstandard Finite Difference for Fractional Order Covid-19 Model”. Gazi University Journal of Science, vol. 38, no. 2, 2025, pp. 874-89, doi:10.35378/gujs.1456440.
Vancouver Merdan M, Açıkgöz P. Investigation of nonstandard finite difference for fractional order Covid-19 model. Gazi University Journal of Science. 2025;38(2):874-89.