Fractional derivative analysis of Asthma with the effect of environmental factors

Year 2024, Volume: 42 Issue: 1, 177 - 188, 27.02.2024

Rashid Jan Şuayip Yüzbaşı Attaullah Attaullah Muhammad Jawad Muhammad Jawad Asif Jan

Abstract

It is observed that the exposure to environmental factors such as indoors and outdoors air pollution, cigarette smoke, and allergens are highly related to asthma attacks. It is also re-ported that limited exposure to asthma riggers, cure due to medicine, the attacks of asthma can be minimized. In this paper, we formulate the dynamics of asthma with smoking and environmental factors classes in the fractional Caputo-Fabrizio (CF) framework to visualize its dynamical behaviour. We delineate the important properties of the CF derivative for the analysis of our model. The model is then analyzed for the basic properties and the uniqueness and existence of the hypothesized asthma system are investigated via the theory of fixed point. Furthermore, a novel numerical scheme is presented for the solution of our fractional system to illustrate the time series of asthma model. The dynamical behaviour of our asthma model is then highlighted numerically to show the impact of fractional-order ϑ on the system and to visualize the role of input factors on the dynamics of asthma disease.

Keywords

Asthma Disease, Environmental Factors, Mathematical Model, Numerical Method, Behaviour

References

• REFERENCES
• [1] Ghosh M. Industrial pollution and Asthma: A mathematical model. J Biol Syst 2000;8:347371. [CrossRef]
• [2] Polosa R, Thomson NC. Smoking and asthma: dangerous liaisons. Eur Respir J 2013;41:716726. [CrossRef]
• [3] Strachan DP. The role of environmental factors in asthma. Br Med Bull 2000;56:865882. [CrossRef]
• [4] Driscoll AJ, Arshad SH, Bont L, Brunwasser SM, Cherian T, Englund JA, et al. Does respiratory syncytial virus lower respiratory illness in early life cause recurrent wheeze of early childhood and asthma? Critical review of the evidence and guidance for future studies from a World Health Organization-sponsored meeting. Vaccine 2020;38:24352448. [CrossRef]
• [5] Canadian Medical Association. Secondhand cigarette smoke worsens symptoms in children with asthma. Section on Allergy, Canadian Paediatric Society. CMAJ 1986;135:321323.
• [6] Collishaw NE, Kirkbride J, Wigle DT. Tobacco smoke in the workplace: an occupational health hazard. Can Med Assoc J 1984;131:11991204.
• [7] Dockery DW, Pope CA, Xu X, Spengler JD, Ware JH, Fay ME, et al. An association between air pollution and mortality in six US cities. N Eng J Med 1993;329:17531759. [CrossRef]
• [8] Utell MJ, Looney RJ. Environmentally induced asthma. Toxicol Lett 1995;82:4753. [CrossRef]
• [9] Jan R, Xiao Y. Effect of partial immunity on transmission dynamics of dengue disease with optimal control. Math Methods Appl Sci 2019;42:19671983. [CrossRef]
• [10] Secer A, Onder I, Ozışık M. Sinc-Galerkin method for solving system of singular perturbed reaction-diffusion problems. Sigma J Eng Nat Sci 2021;39:203212. [CrossRef]
• [11] Sisman S, Merdan M. Global stability of Susceptible Diabetes Complication (SDC) model in discrete time. Sigma J Eng Nat Sci 2021;39:290312. [CrossRef]
• [12] Ergene B, Yalçın B. A finite element study on modal analysis of lightweight pipes. Sigma J Eng Nat Sci 2021;39:268278. [CrossRef]
• [13] Akbari N, Gholinia M, Gholinia S, Dabbaghian S, Ganji DD. Analyatical and numerical study of hydrodynamic nano fluid flow in a tow-dimensional semi-poropus channel with transverse magnetic field. Sigma J Eng Nat Sci 2018;36:587608.
• [14] Jan R, Xiao Y. Effect of pulse vaccination on dynamics of dengue with periodic transmission functions. Adv Differ Equ 2019;2019:17. [CrossRef]
• [15] Jan R, Khan MA, Gómez-Aguilar JF. Asymptomatic carriers in transmission dynamics of dengue with control interventions. Optim Control Appl Methods 2020;41:430447. [CrossRef]
• [16] Attaullah, Jan R, and Yüzbasi S. Dynamical behaviour of HIV Infection with the influence of variable source term through Galerkin method. Chaos Solit Fractals 2021;152:111429. [CrossRef]
• [17] Mahmoudi MR, Baleanu D, Band SS, Mosavi A. Factor analysis approach to classify COVID-19 datasets in several regions. Results Phys 2021;25:104071. [CrossRef]
• [18] Yüzbasi S. A numerical approach to solve the model for HIV infection of CD4+ T cells. Appl Math Model 2012;36:976352. [CrossRef]
• [19] Yüzbasi S, Ismailov N. A numerical method for the solutions of the HIV infection model of CD4+ T-cells. Int J Biomath 2017;10:1750098. [CrossRef]
• [20] Ahmed N, Rafiq M, Adel W, Rezazadeh H, Khan I, Nisar KS. Structure preserving numerical analysis of HIV and CD4+ T-cells reaction diffusion model in two space dimensions. Chaos Solit Fractals 2020;139:110307. [CrossRef]
• [21] Thirumalai S, Seshadri R, Yüzbasi S. On the solution of the Human Immunodeficiency Virus (HIV) infection model using spectral collocation method. Int J Biomath 2021;14:2050074. [CrossRef]
• [22] Yüzbasi S. An exponential collocation method for the solutions of the HIV infection model of CD4+ T cells. Int J Biomath 2016;9:1650036. [CrossRef]
• [23] Winkler T, Venegas JG, Harris RS. Mathematical modeling of ventilation defects in asthma. Drug Discov Today Dis Models 2015;15:38. [CrossRef]
• [24] Joseph GA, Balamuralitharan S. A Nonlinear differential equation model of Asthma effect of environmental pollution using LHAM. In Journal of Physics: Conference Series 2018; (Vol. 1000, No. 1, p. 012043). IOP Publishing. [CrossRef]
• [25] Rosenberg HF, Druey KM. Modeling asthma: Pitfalls, promises, and the road ahead. J Leukoc Biol 2018;104:4148. [CrossRef]
• [26] Burrowes KS, Doel T, Brightling C. Computational modeling of the obstructive lung diseases asthma and COPD. J Transl Med 2014;12:18. [CrossRef]
• [27] Sheffield PE, Knowlton K, Carr JL, Kinney PL. Modeling of regional climate change effects on ground-level ozone and childhood asthma. Am J Prev Med 2011;41:251257. [CrossRef]
• [28] Jang Y, Shin H, Lee MK, Kwon OS, Shin JS, Kim YI, Kim CW, Lee HR, Kim M. Antiviral activity of lambda-carrageenan against influenza viruses and severe acute respiratory syndrome coronavirus 2. Sci Rep 2021;11:12. [CrossRef]
• [29] Slowikowska M, Bajzert J, Miller J, Stefaniak T, Niedzwiedz A. The dynamics of circulating ımmune complexes in horses with severe equine asthma. Animals 2021;11:1001. [CrossRef]
• [30] Nápoles JEN, Guzmán PM, Lugo LM, Kashuri, A. The local generalized derivative and Mittag Leffler function. Sigma J Eng Nat Sci 2020;38:10071017.
• [31] Shah Z, Jan R, Kumam P, Deebani W, Shutaywi M. Fractional dynamics of HIV with source term for the supply of new CD4+ T-cells depending on the viral load via caputo-fabrizio derivative. Molecules 2021;26:1806. [CrossRef]
• [32] Srivastava HM, Jan R, Jan A, Deebani W, Shutaywi M. Fractional-calculus analysis of the transmission dynamics of the dengue infection. Chaos 2021;31:053130. [CrossRef]
• [33] Jan R, Jan A. MSGDTM for solution of fractional order dengue disease model. Int J Sci Res 2017;6:11401144.
• [34] Fatmawati F, Jan R, Khan MA, Khan Y, Ullah S. A new model of dengue fever in terms of fractional derivative. Math Biosci Eng 2020;17:52675287. [CrossRef]
• [35] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. 1st ed. Amsterdam: Elsevier; 1998.
• [36] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives, translated from the 1987 Russian original. Switzerland ; Philadelphia, Pa., USA : Gordon and Breach Science Publishers; 1993. [37] Qureshi S. Periodic dynamics of rubella epidemic under standard and fractional Caputo operator with real data from Pakistan. Math Comput Simul 2020;178:151165. [CrossRef]
• [38] Qureshi S. Effects of vaccination on measles dynamics under fractional conformable derivative with Liouville-Caputo operator. Eur Phys J Plus 2020;135:120. [CrossRef]
• [39] Qureshi S, Rangaig NA, Baleanu D. New numerical aspects of Caputo-Fabrizio fractional derivative operator. Mathematics 2019;7:374. [CrossRef]
• [40] Jan R, Khan MA, Kumam P, Thounthong P. Modeling the transmission of dengue infection through fractional derivatives. Chaos Solit Fractals 2019;127:189216. [CrossRef]
• [41] Qureshi S, Yusuf A. Fractional derivatives applied to MSEIR problems: Comparative study with real world data. Eur Phys J Plus 2019;134:13. [CrossRef]
• [42] Jan A, Jan R, Khan H, Zobaer MS, Shah R. Fractional-order dynamics of Rift Valley fever in ruminant host with vaccination. Commun Math Biol Neurosci 2020;2020:79.
• [43] Jan R, Khan H, Kumam P, Tchier F, Shah R, Bin Jebreen H. The investigation of the fractional-view dynamics of helmholtz equations within caputo operator. Comput Mater Contin 2021;68:31853201. [CrossRef]
• [44] Muresan CI, Birs IR, Dulf EH. Event-based ımplementation of fractional order IMC controllers for simple FOPDT processes. Mathematics 2020;8:1378. [CrossRef] [45] Goufo EF. A biomathematical view on the fractional dynamics of cellulose degradation. Fract Calc Appl Anal 2015;18:554564. [CrossRef]
• [46] Yusuf A, Qureshi S, Shah SF. Mathematical analysis for an autonomous financial dynamical system via classical and modern fractional operators. Chaos Solit Fractals 2020;132:109552. [CrossRef]
• [47] Sweilam NH, Al-Mekhlafi SM, Baleanu D. A hybrid stochastic fractional order Coronavirus (2019-nCov) mathematical model. Chaos Solit Fractals 2021;145:110762. [CrossRef]
• [48] Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 2015;1:13. [CrossRef]
• [49] Losada J, Nieto JJ. Properties of a new fractional derivative without singular kernel. Progr Fract Differ Appl 2015;1:8792.
• [50] Attaullah, Sohaib M. Mathematical modeling and numerical simulation of HIV infection model. Results Appl Math 2020;7:100118. [CrossRef]
• [51] Attaullah, Jan R, Jabeen A. Solution of the HIV infection model with full logistic proliferation and variable source term using Galerkin scheme. Matrix Sci Math 2020;4:3743.
• [52] Attaullah, Zeeshan, Tufail Khan M, Alyobi S, Yassen MF, Prathumwan D. A computational approach to a model for HIV and the immune system ınteraction. Axioms 2022;11:578. [CrossRef]
• [53] Attaullah RD, Khurshaid A, Alyobi S, Yassen MF, Prathumwan D. Computational framework of the SVIR epidemic model with a non-linear saturation incidence rate. Axioms 2022;11:651. [CrossRef]
• [54] Attaullah RD, Weera W. Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate. J AIMS Math 2022;6:1129211310. [CrossRef]
• [55] Attaullah RD, Alyobi S, Yassen MF. A study on the transmission and dynamical behavior of an HIV/AIDS epidemic model with a cure rate. AIMS Math 2022;7:1750717528. [CrossRef]
• [56] Attaullah RD, Yassen MF, Alyobi S, Al-Duais FS, Weera W. On the comparative performance of fourth order Runge–Kutta and the Galerkin–Petrov time discretization methods for solving nonlinear ordinary differential equations with application to some mathematical models in epidemiology. AIMS Math 2023;8:36993729. [CrossRef]
• [57] Attaullah, Jawad M, Alyobi S, Yassen MF, Weera W. A higher order Galerkin time discretization scheme for the novel mathematical model of COVID-19. AIMS Math 2023;8:37633790. [CrossRef]
• [58] Attaullah, Yüzbaşı Ş, Alyobi S, Yassen MF, Weera W. A Higher-Order Galerkin Time Discretization and Numerical Comparisons for Two Models of HIV Infection. Comput Math Methods Med 2022;2022:3599827. [CrossRef]