Analysis of COVID-19 epidemic with intervention impacts by a fractional operator
This study introduces an innovative fractional methodology for analyzing the dynamics of COVID-19 outbreak, examining the impact of intervention strategies like lockdown, quarantine, and isolation on disease transmission. The analysis incorporates the Caputo fractional derivative to grasp long-term memory effects and non-local behavior in the advancement of the infection. Emphasis is placed on assessing the boundedness and non-negativity of the solutions. Additionally, the Lipschitz and Banach contraction theorem are utilized to validate the existence and uniqueness of the solution. We determine the basic reproduction number associated with the model utilizing the next generation matrix technique. Subsequently, by employing the normalized sensitivity index, we perform a sensitivity analysis of the basic reproduction number to effectively identify the controlling parameters of the model. To validate our theoretical findings, numerical simulations are conducted for various fractional order values, utilizing a two-step Lagrange interpolation technique. Furthermore, the numerical algorithms of the model are represented graphically to illustrate the effectiveness of the proposed methodology and to analyze the effect of arbitrary order derivatives on disease dynamics.
[1] Baca¨er, N. (2011). Mckendrick and kermack on epidemic modelling (1926-1927). A Short History of Mathematical Population Dynamics, 89-96. ht tps://doi.org/10.1007/978-0-85729-115-8-16
[2] Worldometer. Coronavirus incubation period. Avaialable from:https://www.worldometers.i nfo/coronavirus/coronavirus-incubation-p eriod/.
[3] Li, M. T., Sun, G. Q., Zhang, J., Zhao, Y., Pei, X., Li, L., & Jin, Z. (2020). Analysis of covid-19 transmission in shanxi province with discrete time imported cases. Mathematical Biosciences and Engineering, 17(4), 3710. https://doi.org/10 .3934/mbe.2020208
[4] Eikenberry, S. E., Mancuso, M., Iboi, E., Phan, T., Eikenberry, K., Kuang, Y., & Gumel, A. B. (2020). To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the covid-19 pandemic. Infectious Disease Modelling, 5, 293-308. https://doi.or g/10.1016/j.idm.2020.04.001
[5] Sarkar, K., Khajanchi, S., & Nieto, J. J.(2020). Modeling and forecasting the COVID-19 pandemic in India. Chaos, Solitons & Fractals, 139, 110049. https://doi.org/10.1016/j.chao s.2020.110049
[6] Podlubny, I. (1999). Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 198.
[7] Sadki, M., Danane, J., & Allali, K. (2023). Hepatitis C virus fractional-order model: mathematical analysis. Modeling Earth Systems and Environment, 9(2), 1695-1707. https: //doi.org/10.1007/s40808-022-01582-5
[8] Kumawat, S., Bhatter, S., Suthar, D. L., Purohit, S. D., & Jangid, K. (2022). Numerical modeling on age-based study of coronavirus transmission. Applied Mathematics in Science and Engineering , 30(1), 609-634. https://doi.org/10.1080/27 690911.2022.2116435
[9] 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. http s://doi.org/10.1140/epjp/s13360-020-008 19-5
[10] Asamoah, J. K. K., Okyere, E., Yankson, E., Opoku, A. A., Adom-Konadu, A., Acheampong, E., & Arthur, Y. D. (2022). Non-fractional and fractional mathematical analysis and simulations for Q fever. Chaos, Solitons & Fractals, 156,111821. https://doi.org/10.1016/j.chaos. 2022.111821
[11] Naik, P. A., Owolabi, K. M., Yavuz, M., & Zu, J. (2020). Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons & Fractals, 140, 110272. https: //doi.org/10.1016/j.chaos.2020.110272
[12] Karaagac, B., Owolabi, K. M., & Nisar, K. S. (2020). Analysis and dynamics of illicit drug use described by fractional derivative with Mittag-Leffler kernel. CMC-Comput Mater Cont, 65(3), 1905-1924. https://doi.org/10.32604/c mc.2020.011623
[13] Bhatter, S., Jangid, K., & Purohit, S. D.(2022). Fractionalized mathematical models for drug diffusion. Chaos, Solitons & Fractals, 165,112810. https://doi.org/10.1016/j.chaos.20 22.112810
[14] Nazir, G., Zeb, A., Shah, K., Saeed, T., Khan, R. A., & Khan, S. I. U. (2021). Study of COVID-19 mathematical model of fractional order via modified Euler method. Alexandria Engineering Journal, 60(6), 5287-5296. https: //doi.org/10.1016/j.aej.2021.04.032
[15] Carvalho, A. R., Pinto, C. M., & Tavares, J. N. (2019). Maintenance of the latent reservoir by pyroptosis and superinfection in a fractional order HIV transmission model. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3), 69-75. https: //doi.org/10.11121/ijocta.01.2019.00643
[16] Koca, I. (2018). Analysis of rubella disease model with non-local and non-singular fractional derivatives. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 17-25. https: //doi.org/10.11121/ijocta.01.2018.00532
[17] Bhatter, S., Jangid, K., Abidemi, A., Owolabi, K. M., & Purohit, S. D. (2023). A new fractional mathematical model to study the impact of vaccination on COVID-19 outbreaks. Decision Analytics Journal, 6, 100156. https://doi.or g/10.1016/j.dajour.2022.100156
[18] Owolabi, K. M., & Pindza, E. (2022). A nonlinear epidemic model for tuberculosis with Caputo operator and fixed point theory. Healthcare Analytics, 2, 100111. https://doi.org/10.1 016/j.health.2022.100111
[19] Naik, P. A., Owolabi, K. M., Zu, J., & Naik, M. U. D. (2021). Modeling the transmission dynamics of COVID-19 pandemic in Caputo type fractional derivative. Journal of Multiscale Modelling, 12(03), 2150006. https://doi.org/ 10.1142/S1756973721500062
[20] Nisar, K. S., Ahmad, S., Ullah, A., Shah, K., Alrabaiah, H., & Arfan, M. (2021). Mathematical analysis of SIRD model of COVID-19 with Caputo fractional derivative based on real data. Results in Physics, 21, 103772. https://doi.or g/10.1016/j.rinp.2020.103772
[21] Yıldız, T. A. (2019). A comparison of some control strategies for a non-integer order tuberculosis model. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3), 21-30. https: //doi.org/10.11121/ijocta.01.2019.00657
[22] Bugalia, S., Bajiya, V. P., Tripathi, J. P., Li, M. T., & Sun, G. Q. (2020). Mathematical modeling of COVID-19 transmission: the roles of intervention strategies and lockdown. Math. Biosci. Eng, 17(5), 5961-5986. https://doi.or g/10.3934/mbe.2020318
[23] Menaria, N., Purohit, S. D., & Parmar, R. K.(2016). On a new class of integrals involving generalized Mittag-Leffler function. Surveys in Mathematics and its Applications, 11, 1-9.
[24] Wang, J. L., & Li, H. F. (2011). Surpassing the fractional derivative: Concept of the memory-dependent derivative. Computers & Mathematics with Applications, 62(3), 1562-1567. https://doi.org/10.1016/j.camwa.2011.04. 028
[25] O(¨)zk¨ose, F. (2024). Modeling of psoriasis by considering drug influence: A mathematical approach with memory trace. Computers in Biology and Medicine, 168, 107791. https://do i.org/10.1016/j.compbiomed.2023.107791
[26] O(¨)zk¨ose, F. (2023). Long-term side effects: a mathematical modeling of COVID-19 and stroke with real data. Fractal and Fractional, 7(10), 719. https://doi.org/10.3390/fractalfract7100 719
[27] O(¨)zk¨ose, F., Habbireeh, R., & S¸enel, 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. https://doi.org/10.1016/j.cam.20 22.114969
[28] Odibat, Z. M., & Shawagfeh, N. T. (2007). Generalized Taylor’s formula. Applied Mathematics and Computation, 186(1), 286-293. https://doi.org/10.1016/j.amc.2006.07.10 2
[29] Lin, W. (2007). Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications, 332(1), 709-726. https: //doi.org/10.1016/j.jmaa.2006.10.040
[30] Van den Driessche, P., & Watmough, J.(2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1-2), 29-48. https://doi.org/10.1016/S0 025-5564(02)00108-6
[31] Martcheva, M. (2015). An introduction to mathematical epidemiology, Springer, 61. https: //doi.org/10.1007/978-1-4899-7612-3_1
[32] Chitnis, N., Hyman, J. M., & Cushing, J. M.(2008). Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology, 70, 1272-1296. https:// doi.org/10.1007/s11538-008-9299-0
[33] Mishra, A. M., Purohit, S. D., Owolabi, K. M., & Sharma, Y. D. (2020). A nonlinear epidemiological model considering asymptotic and quarantine classes for SARS CoV-2 virus. Chaos, Solitons & Fractals, 138, 109953 https: //doi.org/10.1016/j.chaos.2020.109953
[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., Ford, N. J., & Freed, A. D. (2004). Detailed error analysis for a fractional Adams method. Numerical Algorithms, 36, 31-52. https: //doi.org/10.1023/B:NUMA.0000027736.8507 8.be
[36] Atangana, A., & Owolabi, K. M. (2018). New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena, 13(1), 3. https://doi.org/10.1 051/mmnp/2018010