Differential order analysis and sensitivity analysis of a CoVID-19 infection system with memory effect

Mohammad Sajid; Biplab Dhar; Ahmed S. Almohaimeed; Mohammad Sajid; Biplab Dhar; Ahmed S. Almohaimeed

doi:10.3934/math.20221129

AIMS Mathematics

2022, Volume 7, Issue 12: 20594-20614. doi: 10.3934/math.20221129

Previous Article Next Article

Research article

Differential order analysis and sensitivity analysis of a CoVID-19 infection system with memory effect

1.
Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah 51452, Saudi Arabia
2.
Department of Mathematics-SoPS, DIT University, Dehradun 248009, Uttarakhand, India
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

Received: 28 July 2022 Revised: 04 September 2022 Accepted: 13 September 2022 Published: 21 September 2022
MSC : 26A33, 92-10, 92D25

The paper deals with numerical analysis of solutions for state variables of a CoVID-19 model in integer and fractional order. The solution analysis for the fractional order model is done by the new generalized Caputo-type fractional derivative and Predictor-Corrector methodology, and that for the integer order model is carried out by Multi-step Differential Transformation Method. We have performed sensitivity analysis of the basic reproduction number with the help of a normalized forward sensitivity index. The Arzelá-Ascoli theorem and Fixed point theorems with other important properties are used to establish a mathematical analysis of the existence and uniqueness criteria for the solution of the fractional order. The obtained outcomes are depicted with the help of diagrams, narrating the nature of the state variables. According to the results, the Predictor-Corrector methodology is favorably unequivocal for the fractional model and very simple in administration for the system of equations that are non-linear. The research done in this manuscript can assure the execution and relevance of the new generalized Caputo-type fractional operator for mathematical physics.
- Chebyshev norms,
- trapezoidal quadrature rule,
- Adams-Bashforth method,
- memory effect,
- Ms-DTM,
- Caputo-type fractional derivative
Citation: Mohammad Sajid, Biplab Dhar, Ahmed S. Almohaimeed. Differential order analysis and sensitivity analysis of a CoVID-19 infection system with memory effect[J]. AIMS Mathematics, 2022, 7(12): 20594-20614. doi: 10.3934/math.20221129

Related Papers:

Abstract

The paper deals with numerical analysis of solutions for state variables of a CoVID-19 model in integer and fractional order. The solution analysis for the fractional order model is done by the new generalized Caputo-type fractional derivative and Predictor-Corrector methodology, and that for the integer order model is carried out by Multi-step Differential Transformation Method. We have performed sensitivity analysis of the basic reproduction number with the help of a normalized forward sensitivity index. The Arzelá-Ascoli theorem and Fixed point theorems with other important properties are used to establish a mathematical analysis of the existence and uniqueness criteria for the solution of the fractional order. The obtained outcomes are depicted with the help of diagrams, narrating the nature of the state variables. According to the results, the Predictor-Corrector methodology is favorably unequivocal for the fractional model and very simple in administration for the system of equations that are non-linear. The research done in this manuscript can assure the execution and relevance of the new generalized Caputo-type fractional operator for mathematical physics.

References

[1]	A. Ali, F. Alshammari, S. Islam, M. Khan, S. Ullah, Modeling and analysis of the dynamics of novel coronavirus (COVID-19) with Caputo fractional derivative, Results Phys., 20 (2021), 103669. http://dx.doi.org/10.1016/j.rinp.2020.103669 doi: 10.1016/j.rinp.2020.103669
[2]	S. Akindeinde, E. Okyere, A. Adewumi, R. Lebelo, O. Fabelurin, S. Moore, Caputo fractional-order SEIRP model for COVID-19 epidemic, Alex. Eng. J., 61 (2022), 829–845. http://dx.doi.org/10.1016/j.aej.2021.04.097 doi: 10.1016/j.aej.2021.04.097
[3]	I. Ahmed, G. Modu, A. Yusuf, P. Kumam, I. Yusuf, A mathematical model of Coronavirus disease (COVID-19) containing asymptomatic and symptomatic classes, Results Phys., 21 (2021), 103776. http://dx.doi.org/10.1016/j.rinp.2020.103776 doi: 10.1016/j.rinp.2020.103776
[4]	A. Anirudh, Mathematical modeling and the transmission dynamics in predicting the Covid-19-what next in combating the pandemic, Infectious Disease Modelling, 5 (2020), 366–374. http://dx.doi.org/10.1016/j.idm.2020.06.002 doi: 10.1016/j.idm.2020.06.002
[5]	L. Barros, M. Lopes, F. Pedro, E. Esmi, J. Santos, D. Sánchez, The memory effect on fractional calculus: an application in the spread of COVID-19, Comp. Appl. Math., 40 (2021), 72. http://dx.doi.org/10.1007/s40314-021-01456-z doi: 10.1007/s40314-021-01456-z
[6]	S. Biswas, J. Ghosh, S. Sarkar, U. Ghosh, COVID-19 pandemic in India: a mathematical model study, Nonlinear Dyn., 102 (2020), 537–553. http://dx.doi.org/10.1007/s11071-020-05958-z doi: 10.1007/s11071-020-05958-z
[7]	M. Caputo, M. Fabrizio, On the notion of fractional derivative and applications to the hysteresis phenomena, Meccanica, 52 (2017), 3043–3052. http://dx.doi.org/10.1007/s11012-017-0652-y doi: 10.1007/s11012-017-0652-y
[8]	B. Dhar, P. Gupta, A numerical approach of tumor-immune model with B cells and monoclonal antibody drug by multi-step differential transformation method, Math. Method. Appl. Sci., 44 (2021), 4058–4070. http://dx.doi.org/10.1002/mma.7009 doi: 10.1002/mma.7009
[9]	B. Dhar, P. Gupta, M. Sajid, Solution of a dynamical memory effect COVID-19 infection system with leaky vaccination efficacy by non-singular kernel fractional derivatives, Math. Biosci. Eng., 19 (2022), 4341–4367. http://dx.doi.org/10.3934/mbe.2022201 doi: 10.3934/mbe.2022201
[10]	V. Erturk, P. Kumar, Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives, Chaos Soliton. Fract., 139 (2020), 110280. http://dx.doi.org/10.1016/j.chaos.2020.110280 doi: 10.1016/j.chaos.2020.110280
[11]	Y. Feng, X. Yang, J. Liu, On overall behavior of Maxwell mechanical model by the combined Caputo fractional derivative, Chinese J. Phys., 66 (2020), 269–276. http://dx.doi.org/10.1016/j.cjph.2020.05.006 doi: 10.1016/j.cjph.2020.05.006
[12]	M. Islam, A. Peace, D. Medina, T. Oraby, Integer versus fractional order SEIR deterministic and stochastic models of measles, Int. J. Environ. Res. Public Health, 17 (2020), 2014. http://dx.doi.org/10.3390/ijerph17062014 doi: 10.3390/ijerph17062014
[13]	A. Jajarmi, D. Baleanu, A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos Soliton. Fract., 113 (2018), 221–229. http://dx.doi.org/10.1016/j.chaos.2018.06.009 doi: 10.1016/j.chaos.2018.06.009
[14]	U. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, arXiv: 1411.5229.
[15]	E. Kharazmi, M. Cai, X. Zheng, Z. Zhang, G. Lin, G. Karniadakis, Identifiability and predictability of integer-and fractional-order epidemiological models using physics-informed neural networks, Nat. Comput. Sci., 1 (2021), 744–753. http://dx.doi.org/10.1038/s43588-021-00158-0 doi: 10.1038/s43588-021-00158-0
[16]	K. Kozioł, R. Stanisławski, G. Bialic, Fractional-order sir epidemic model for transmission prediction of covid-19 disease, Appl. Sci., 10 (2020), 8316. http://dx.doi.org/10.3390/app10238316 doi: 10.3390/app10238316
[17]	C. Li, Y. Zhu, C. Qi, L. Liu, D. Zhang, X. Wang, et al., Epidemic dynamics of COVID-19 based on SEAIUHR model considering asymptomatic cases in Henan province, China, Research Square, in press. http://dx.doi.org/10.21203/rs.3.rs-50050/v1
[18]	C. Li, F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Opt., 34 (2013), 149–179. http://dx.doi.org/10.1080/01630563.2012.706673 doi: 10.1080/01630563.2012.706673
[19]	D. McNamara, About 80% of asymptomatic people with CoVID-19 develop symptom, Medscape Medical News, September 28, 2020.
[20]	P. Naik, K. Owolabi, J. Zu, M. Naik, Modeling the transmission dynamics of COVID-19 pandemic in Caputo type fractional derivative, J. Multiscale Model., 12 (2021), 2150006. http://dx.doi.org/10.1142/S1756973721500062 doi: 10.1142/S1756973721500062
[21]	P. Naik, J. Zu, M. Ghori, M. Naik, Modeling the effects of the contaminated environments on COVID-19 transmission in India, Results Phys., 29 (2021), 104774. http://dx.doi.org/10.1016/j.rinp.2021.104774 doi: 10.1016/j.rinp.2021.104774
[22]	Z. Odibat, D. Baleanu, Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives, Appl. Numer. Math., 156 (2020), 94–105. http://dx.doi.org/10.1016/j.apnum.2020.04.015 doi: 10.1016/j.apnum.2020.04.015
[23]	Z. Odibat, C. Bertelle, M. Aziz-Alaouni, G. Duchamp, A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. Math. Appl., 59 (2010), 1462–1472. http://dx.doi.org/10.1016/j.camwa.2009.11.005 doi: 10.1016/j.camwa.2009.11.005
[24]	O. Postavaru, S. Anton, A. Toma, COVID-19 pandemic and chaos theory, Math. Comput. Simulat., 181 (2021), 138–149. http://dx.doi.org/10.1016/j.matcom.2020.09.029 doi: 10.1016/j.matcom.2020.09.029
[25]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Amsterdam: Elsevier, 1999. http://dx.doi.org/10.1016/s0076-5392(99)x8001-5
[26]	S. Rosa, D. Torres, Parameter estimation, sensitivity analysis and optimal control of a periodic epidemic model with application to HRSV in Florida, Stat. Optim. Inf. Comput., 6 (2018), 139–149. http://dx.doi.org/10.19139/soic.v6i1.472 doi: 10.19139/soic.v6i1.472
[27]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. http://dx.doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[28]	S. Yadav, D. Kumar, J. Singh, D. Baleanu, Analysis and dynamics of fractional order Covid-19 model with memory effect, Results Phys., 24 (2021), 104017. http://dx.doi.org/10.1016/j.rinp.2021.104017 doi: 10.1016/j.rinp.2021.104017
[29]	M. Zamir, G. Zaman, A. Alshomrani, Sensitivity analysis and optimal control of anthroponotic cutaneous leishmania, PloS One, 11 (2016), 0160513. http://dx.doi.org/10.1371/journal.pone.0160513 doi: 10.1371/journal.pone.0160513

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)