Solution of a dynamical memory effect COVID-19 infection system with leaky vaccination efficacy by non-singular kernel fractional derivatives

Biplab Dhar; Praveen Kumar Gupta; Mohammad Sajid; Biplab Dhar; Praveen Kumar Gupta; Mohammad Sajid

doi:10.3934/mbe.2022201

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 4341-4367. doi: 10.3934/mbe.2022201

Previous Article Next Article

Research article Special Issues

Solution of a dynamical memory effect COVID-19 infection system with leaky vaccination efficacy by non-singular kernel fractional derivatives

1.
Department of Mathematics-SoPS, DIT University, Uttarakhand 248009, India
2.
Department of Mathematics, NIT Silchar, Assam 788010, India
3.
Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah 51452, Saudi Arabia

Academic Editor: Masaki Ogura

Received: 10 January 2022 Revised: 09 February 2022 Accepted: 14 February 2022 Published: 28 February 2022

In this paper, the recent trends of COVID-19 infection spread have been studied to explore the advantages of leaky vaccination dynamics in SEVR (Susceptible Effected Vaccinated Recovered) compartmental model with the help of Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in the Caputo sense (ABC) non-singular kernel fractional derivative operators with memory effect within the model to show possible long–term approaches of the infection along with limited defensive vaccine efficacy that can be designed numerically over the closed interval ranging from 0 to 1. One of the main goals is to provide a stepping information about the usefulness of the aforementioned non-singular kernel fractional approaches for a lenient case as well as a critical case in COVID-19 infection spread. Another is to investigate the effect of death rate on state variables. The estimation of death rate for state variables with suitable vaccine efficacy has a significant role in the stability of state variables in terms of basic reproduction number that is derived using next generation matrix method, and order of the fractional derivative. For non-integral orders the pandemic modeling sense viz, CF and ABC, has been compared thoroughly. Graphical presentations together with numerical results have proposed that the methodology is powerful and accurate which can provide new speculations for COVID-19 dynamical systems.
- vaccine efficacy,
- non-singular kernel fractional derivative,
- basic reproduction number,
- stability analysis,
- Banach fixed-point theorem
Citation: Biplab Dhar, Praveen Kumar Gupta, Mohammad Sajid. Solution of a dynamical memory effect COVID-19 infection system with leaky vaccination efficacy by non-singular kernel fractional derivatives[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 4341-4367. doi: 10.3934/mbe.2022201

Related Papers:

Abstract

In this paper, the recent trends of COVID-19 infection spread have been studied to explore the advantages of leaky vaccination dynamics in SEVR (Susceptible Effected Vaccinated Recovered) compartmental model with the help of Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in the Caputo sense (ABC) non-singular kernel fractional derivative operators with memory effect within the model to show possible long–term approaches of the infection along with limited defensive vaccine efficacy that can be designed numerically over the closed interval ranging from 0 to 1. One of the main goals is to provide a stepping information about the usefulness of the aforementioned non-singular kernel fractional approaches for a lenient case as well as a critical case in COVID-19 infection spread. Another is to investigate the effect of death rate on state variables. The estimation of death rate for state variables with suitable vaccine efficacy has a significant role in the stability of state variables in terms of basic reproduction number that is derived using next generation matrix method, and order of the fractional derivative. For non-integral orders the pandemic modeling sense viz, CF and ABC, has been compared thoroughly. Graphical presentations together with numerical results have proposed that the methodology is powerful and accurate which can provide new speculations for COVID-19 dynamical systems.

References

[1]	S. K. Biswas, J. K. Ghosh, S. Sarkar, U. Ghosh, COVID-19 pandemic in India: a mathematical model study, Nonlinear Dyn., 102 (2020), 537–553. https://doi.org/10.1007/s11071-020-05958-z doi: 10.1007/s11071-020-05958-z
[2]	A. Anirudh, Mathematical modeling and the transmission dynamics in predicting the Covid-19-What next in combating the pandemic, Infect. Dis. Model., 5 (2020), 366–374. https://doi.org/10.1016/j.idm.2020.06.002 doi: 10.1016/j.idm.2020.06.002
[3]	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, Europe. PMC., 2020. https://doi.org/10.21203/rs.3.rs-50050/v1 doi: 10.21203/rs.3.rs-50050/v1
[4]	J. M. Ball, D. M. Mitchell, T. F. Gibbons, R. D. Parr, Rotavirus NSP4: a multifunctional viral enterotoxin, Viral. Immunol., 18 (2005), 27–40. https://doi.org/10.1089/vim.2005.18.27 doi: 10.1089/vim.2005.18.27
[5]	S. Gandon, M. J. Mackinnon, S. Nee, A. F. Read, Imperfect vaccines and the evolution of pathogen virulence, Nature, 414 (2001), 751–756. https://doi.org/10.1038/414751a doi: 10.1038/414751a
[6]	H. J. Larson, L. Z. Cooper, J. Eskola, S. L. Katz, S. Ratzan, Addressing the vaccine confidence gap, Lancet, 378 (2011), 526–535. https://doi.org/10.1016/S0140-6736(11)60678-8 doi: 10.1016/S0140-6736(11)60678-8
[7]	H. J. Larson, Politics and public trust shape vaccine risk perceptions, Nat. Hum. Behav., 2 (2018), 316–316. https://doi.org/10.1038/s41562-018-0331-6 doi: 10.1038/s41562-018-0331-6
[8]	B. Nyhan, J. Reifler, S. Richey, G. L. Freed, Effective messages in vaccine promotion: a randomized trial, Pediatrics, 133 (2014), e835–e842. https://doi.org/10.1542/peds.2013-2365 doi: 10.1542/peds.2013-2365
[9]	A. Rajput, M. Sajid, Tanvi, C. Shekhar, R. Aggarwal, Optimal control strategies on COVID-19 infection to bolster the efficacy of vaccination in India, Sci. Rep., 11 (2021), 20124. https://doi.org/10.1038/s41598-021-99088-0 doi: 10.1038/s41598-021-99088-0
[10]	Tanvi, M. Sajid, R. Aggarwal, A. Rajput, Assessing the impact of transmissibility on a cluster-based COVID-19 model in India, Int. J. Model. Simul. Sci. Comp., 12 (2021), 2141002. https://doi.org/10.1142/S1793962321410026 doi: 10.1142/S1793962321410026
[11]	Tanvi, R. Aggarwal, A. Rajput, M. Sajid, Modeling the optimal interventions to curtail the cluster based COVID-19 pandemic in India: Efficacy of prevention measures, Appl. Comput. Math., 20 (2021), 70–94.
[12]	S. Bagcchi, The world's largest COVID-19 vaccination campaign, Lancet Infect. Dis., 21 (2021), 323. https://doi.org/10.1016/S1473-3099(21)00081-5 doi: 10.1016/S1473-3099(21)00081-5
[13]	X. Chen, F. Fu, Imperfect vaccine and hysteresis, Proc. R. Soc. B, 286 (2019), 20182406. https://doi.org/10.1098/rspb.2018.2406 doi: 10.1098/rspb.2018.2406
[14]	B. Dhar, P. K. Gupta, Numerical solution of tumor-immune model including small molecule drug by multi-step differential transform method, Int. J. Adv. Trends Comput. Sci. Eng., 8 (2019), 1802–1807. https://doi.org/10.30534/ijatcse/2019/02852019 doi: 10.30534/ijatcse/2019/02852019
[15]	B. Dhar, P. K. Gupta, A numerical approach of tumor-immune model with B cells and monoclonal antibody drug by multi-step differential transformation method, Math. Methods Appl. Sci., 44 (2021), 4058–4070. https://doi.org/10.1002/mma.7009 doi: 10.1002/mma.7009
[16]	P. Liu, X. Liu, Dynamics of a tumor-immune model considering targeted chemotherapy, Chaos Solitons Fractals, 98 (2017), 7–13. https://doi.org/10.1016/j.chaos.2017.03.002 doi: 10.1016/j.chaos.2017.03.002
[17]	D. Baleanu, B. Shiri, H. M. Srivastava, M. Al Qurashi, A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Adv. Differ. Equations, 2018 (2018), 1–23. https://doi.org/10.1186/s13662-018-1822-5 doi: 10.1186/s13662-018-1822-5
[18]	D. Baleanu, A. Jajarmi, S. S. Sajjadi, D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos Int. J. Nonlinear Sci. 29 (2019), 083127. https://doi.org/10.1063/1.5096159 doi: 10.1063/1.5096159
[19]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative, Chaos Solitons Fractals, 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
[20]	B. Shiri, D. Baleanu, System of fractional differential algebraic equations with applications, Chaos Solitons Fractals, 120 (2019), 203–212. https://doi.org/10.1016/j.chaos.2019.01.028 doi: 10.1016/j.chaos.2019.01.028
[21]	A. Dold, B. Eckmann, F. Takens, Lecture Notes in Mathematics, Springer, 1975.
[22]	R. Almeida, A. M. C. Brito da Cruz, N. Martins, M. T. T. Monteiro, An epidemiological MSEIR model described by the Caputo fractional derivative, Int. J. Dyn. Control, 7 (2019), 776–784. https://doi.org/10.1007/s40435-018-0492-1 doi: 10.1007/s40435-018-0492-1
[23]	G. Chowell, J. M. Hyman, L. M. A. Bettencourt, C. Castillo-Chavez, Mathematical and Statistical Estimation Approaches in Epidemiology, 1$^{st}$ edition, Springer, Dordrecht, 2009. https: //doi.org/10.1007/978-90-481-2313-1
[24]	O. Vasilyeva, T. Oraby, F. Lutscher, Aggregation and environmental transmission in chronic wasting disease, Math. Biosci. Eng., 12 (2015), 209. https://doi.org/10.3934/mbe.2015.12.209 doi: 10.3934/mbe.2015.12.209
[25]	F. Bozkurt, A. Yousef, T. Abdeljawad, A. Kalinli, Q. Al Mdallal, A fractional-order model of COVID-19 considering the fear effect of the media and social networks on the community, Chaos Solitons Fractals, 152 (2021), 111403. https://doi.org/10.1016/j.chaos.2021.111403 doi: 10.1016/j.chaos.2021.111403
[26]	A. Din, L. Yongjin, T. Khan, G. Zaman, Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China, Chaos Solitons Fractals, 141 (2020), 110286. https://doi.org/10.1016/j.chaos.2020.110286 doi: 10.1016/j.chaos.2020.110286
[27]	A. Din, T. Khan, Y. Li, H. Tahir, A. Khan, W. A. Khan, Mathematical analysis of dengue stochastic epidemic model, Results Phys., 20 (2021), 103719. https://doi.org/10.1016/j.rinp.2020.103719 doi: 10.1016/j.rinp.2020.103719
[28]	A. Din, Y. Li, F. M Khan, Z. U. Khan, P. Liu, On analysis of fractional order mathematical model of Hepatitis B using Atangana–Baleanu Caputo (ABC) derivative, Fractals, 30 (2022), 2240017. https://doi.org/10.1142/S0218348X22400175 doi: 10.1142/S0218348X22400175
[29]	A. Din, Y. Li, A. Yusuf, A. I. Ali, Caputo type fractional operator applied to Hepatitis B system, Fractals, 30 (2022), 2240023. https://dx.doi.org/10.1142/S0218348X22400230 doi: 10.1142/S0218348X22400230
[30]	A. Din, Y. Li, Lévy noise impact on a stochastic hepatitis B epidemic model under real statistical data and its fractal–fractional Atangana–Baleanu order model, Phys. Scr., 96 (2021), 124008. https://doi.org/10.1088/1402-4896/ac1c1a doi: 10.1088/1402-4896/ac1c1a
[31]	A. Din, The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function, Chaos Int. J. Nonlinear Sci., 31 (2021), 123101. https://doi.org/10.1063/5.0063050 doi: 10.1063/5.0063050
[32]	A. Din, F. M. Khan, Z. U. Khan, A. Yusuf, T. Munir, The mathematical study of climate change model under nonlocal fractional derivative, Partial Differ. Equations Appl. Math., 5 (2022), 100204. https://doi.org/10.1016/j.padiff.2021.100204 doi: 10.1016/j.padiff.2021.100204
[33]	H. Habenom, M. Aychluh, D. L. Suthar, Q. Al-Mdallal, S. D. Purohit, Modeling and analysis on the transmission of covid-19 Pandemic in Ethiopia, Alexandria Eng. J., 61 (2022), 5323–5342. https://doi.org/10.1016/j.aej.2021.10.054 doi: 10.1016/j.aej.2021.10.054
[34]	F. A. Rihan, Q. M. Al-Mdallal, H. J. AlSakaji, A. Hashish, A fractional-order epidemic model with time-delay and nonlinear incidence rate, Chaos Solitons Fractals, 126 (2019), 97–105. https://doi.org/10.1016/j.chaos.2019.05.039 doi: 10.1016/j.chaos.2019.05.039
[35]	F. A. Rihan, H. J. AlSakaji, Dynamics of a stochastic delay differential model for COVID-19 infection with asymptomatic infected and interacting people: Case study in the UAE, Results Phys., 28 (2021), 104658. https://doi.org/10.1016/j.rinp.2021.104658 doi: 10.1016/j.rinp.2021.104658
[36]	A. Shafiq, S. A. Lone, T. N. Sindhu, Y. El Khatib, Q. M. Al-Mdallal, T. Muhammad, A new modified Kies Fréchet distribution: Applications of mortality rate of Covid-19, Results Phys., 28 (2021), 104638.
[37]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 28 (2015), 1–13.
[38]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 87–92. https://doi.org/10.1002/bate.201590002 doi: 10.1002/bate.201590002
[39]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, preprint, arXiv: 1602.03408.
[40]	A. Jajarmi, D. Baleanu, A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos Solitons Fractals, 113 (2018), 221–229. https://doi.org/10.1016/j.chaos.2018.06.009 doi: 10.1016/j.chaos.2018.06.009
[41]	S. Abuasad, A. Yildirim, I. Hashim, S. A. Abdul Karim, J. F. d Gómez-Aguilar, Fractional multi-step differential transformed method for approximating a fractional stochastic sis epidemic model with imperfect vaccination, Int. J. Environ. Res. Public Health, 16 (2019), 973. https://doi.org/10.3390/ijerph16060973 doi: 10.3390/ijerph16060973
[42]	Q. Liu, D. Jiang, N. Shi, T. Hayat, A. Alsaedi, The threshold of a stochastic SIS epidemic model with imperfect vaccination, Math. Comput. Simul., 114 (2018), 78–90.
[43]	J. Keehner, L. E. Horton, M. A. Pfeffer, C. A. Longhurst, R. T. Schooley, J. S. Currier, et al., SARS-CoV-2 infection after vaccination in health care workers in California, N. Engl. J. Med., 384 (2021), 1774–1775.
[44]	R. Ganguly, I. K. Puri, Mathematical model for chemotherapeutic drug efficacy in arresting tumour growth based on the cancer stem cell hypothesis, Cell Proliferation, 40 (2007), 338–354. https://doi.org/10.1111/j.1365-2184.2007.00434.x doi: 10.1111/j.1365-2184.2007.00434.x
[45]	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. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[46]	V. S. Panwar, P. S. S. Uduman, J. F. Gómez-Aguilar, Mathematical modeling of coronavirus disease COVID-19 dynamics using CF and ABC non-singular fractional derivatives, Chaos Solitons Fractals, 145 (2021), 110757. https://doi.org/10.1016/j.chaos.2021.110757 doi: 10.1016/j.chaos.2021.110757
[47]	J. Hellewell, S. Abbott, A. Gimma, N. I. Bosse, C. I. Jarvis, T. W. Russell, et al., Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, Lancet Global Health, 8 (2020), e488–e496.
[48]	H. Gupta, S. Kumar, D. Yadav, O. P. Verma, T. K. Sharma, C. W. Ahn, et al., Data analytics and mathematical modeling for simulating the dynamics of COVID-19 epidemic—A case study of India, Electron., 10 (2021), 127.
[49]	A. Atangana, J. F. Gómez-Aguilar, Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus., 133 (2018), 166. https://doi.org/10.1140/epjp/i2018-12021-3 doi: 10.1140/epjp/i2018-12021-3

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)