Abstract
The main aim to conduct this research study is to show the advantage of using fractal-fractional differential operator over other differential operators in epidemiology. In this regard, an MSEIR mathematical model with fractal-fractional differential operator is being taken to study the qualitative analysis of the evolution of the chickenpox disease spread among the children of schools based in Schenzhen city of China in 2013. Balancing the dimensions of every differential equation by carrying the appropriate power on each dimensional quantity we prove the existence and uniqueness of the solutions of the said model using Schaefer’s theorem and fixed-point theory. Further, Ulam–Hyers stability of the model is shown. Comparing the results of our model with those of previously existed models with integer-order and other fractional-order derivatives, we show, using the real data of 25 weeks, that results of our model have come out with significant amount of accuracy, i.e., the fractal-fractional differential operator follows the disease evolution process more accurately than other operators. Numerical simulations are carried out using an efficient numerical technique and a MATLAB routine called ’fmincon’ is used to optimize the values of fractal and fractional orders as well as the transmission rate.
Similar content being viewed by others
Data availability
The author confirms that all data analysed during this study are included in the reference number [4].
References
DeWitte, S.N., Wood, J.W.: Selectivity of Black Death mortality with respect to preexisting health. Proc. Natl. Acad. Sci. 105(5), 1436–41 (2008)
Qureshi, S., Atangana, A.: Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data. Chaos Solitons Fractals 136, 109812 (2020)
Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)
Qureshi, S., Yusuf, A.: Modeling chickenpox disease with fractional derivatives: from Caputo to Atangana–Baleanu. Chaos Solitons Fractals 122, 111–118 (2019)
Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R Soc. Lond. Ser. A Contain. Pap. Math. Phys. Character 115(772), 700–721 (1927)
Reid, A.H., Taubenberger, J.K.: The origin of the 1918 pandemic influenza virus: a continuing enigma. J. Gen. Virol. 84(9), 2285–2292 (2003)
Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M., Rohani, P.: Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9(4), 467–484 (2006)
Tang, X., Zhao, S., Chiu, A.P., Ma, H., Xie, X., Mei, S., Kong, D., Qin, Y., Chen, Z., Wang, X., He, D.: Modelling the transmission and control strategies of varicella among school children in Shenzhen, China. PloS One 12(5), e0177514 (2017)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, vol. 198. Elsevier, Amsterdam (1998)
Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)
Muslim, M.: Existence and approximation of solutions to fractional differential equations. Math. Comput. Model. 49(5–6), 1164–1172 (2009)
Abbas, S., Banerjee, M., Momani, S.: Dynamical analysis of fractional-order modified logistic model. Comput. Math. Appl. 62(3), 1098–1104 (2011)
Flores-Tlacuahuac, A., Biegler, L.T.: Optimization of fractional order dynamic chemical processing systems. Ind. Eng. Chem. Res. 53(13), 5110–5127 (2014)
Caputo, M., Fabrizio, M.: Applications of new time and spatial fractional derivatives with exponential kernels. Progr. Fract. Differ. Appl. 2(2), 1–11 (2016)
Kumar, D., Singh, J., Baleanu, D.: A new analysis of the Forsberg–Whitham equation pertaining to a fractional derivative with Mittag–Leffler-type kernel. Eur. Phys. J. Plus 133(2), 1–10 (2018)
Khan, S.A., Shah, K., Zaman, G., Jarad, F.: Existence theory and numerical solutions to smoking model under Caputo–Fabrizio fractional derivative. Chaos Interdiscip. J. Nonlinear Sci. 29(1), 013128 (2019)
Nouara, A., Amara, A., Kaslik, E., Etemad, S., Rezapour, S., Martinez, F., Kaabar, M.K.: A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type. Adv. Differ. Equ. 2021, 1–28 (2021)
Karthikeyan, P., Venkatachalam, K., Abbas, S.: Existence results for fractional impulsive integro-differential equations with integral conditions of Katugampola type. Acta Math. Univ. Comen. 90(4), 421–436 (2021)
Chandra, S., Abbas, S.: Fractal dimensions of mixed Katugampola fractional integral associated with vector valued functions. Chaos Solitons Fractals 164, 112648 (2022)
Abbas, S., Tyagi, S., Kumar, P., Ertürk, V.S., Momani, S.: Stability and bifurcation analysis of a fractional-order model of cell-to-cell spread of HIV-1 with a discrete time delay. Math. Methods Appl. Sci. 45, 7081–95 (2022)
Atangana, A., Khan, M.A.: Analysis of fractional global differential equations with power law. AIMS Math. 8(10), 24699–24725 (2023)
Qureshi, S., Abro, K.A., Gómez-Aguilar, J.F.: On the numerical study of fractional and non-fractional model of nonlinear duffing oscillator: a comparison of integer and non-integer order approaches. Int. J. Model. Simul. 43(4), 362–375 (2023)
Ullah, S., Khan, M.A., Farooq, M.: A fractional model for the dynamics of TB virus. Chaos Solitons Fractals. 116, 63–71 (2018)
Khan, M.A., Ullah, S., Farooq, M.: A new fractional model for tuberculosis with relapse via Atangana–Baleanu derivative. Chaos Solitons Fractals 116, 227–238 (2018)
Shaikh, A.S., Nisar, K.S.: Transmission dynamics of fractional order typhoid fever model using Caputo–Fabrizio operator. Chaos Solitons Fractals 128, 355–365 (2019)
Ahmad, S., Ullah, A., Al-Mdallal, Q.M., Khan, H., Shah, K., Khan, A.: Fractional order mathematical modeling of covid-19 transmission. Chaos Solitons Fractals 139, 110256 (2020)
Shahram, R., Hakimeh, M., Amin, J.: A new mathematical model for zika virus transmission. Adv. Differ. Equ. 2020(1), 1–15 (2020)
Jahanshahi, H., Munoz-Pacheco, J.M., Bekiros, S., Alotaibi, N.D.: A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19. Chaos Solitons Fractals 143, 110632 (2021)
Rahman, M.U., Arfan, M., Shah, Z., Kumam, P., Shutaywi, M.: Nonlinear fractional mathematical model of tuberculosis (TB) disease with incomplete treatment under Atangana–Baleanu derivative. Alex. Eng. J. 60(3), 2845–2856 (2021)
Kumar, V., Malik, M.: Existence, stability and controllability results of fractional dynamic system on time scales with application to population dynamics. Int. J. Nonlinear Sci. Numer. Simul. 22(6), 741–766 (2021)
Singh, H.K., Pandey, D.N.: Stability analysis of a fractional-order delay dynamical model on oncolytic virotherapy. Math. Methods Appl. Sci. 44(2), 1377–1393 (2021)
Khan, H., Ahmad, F., Tunç, O., Idrees, M.: On fractal-fractional Covid-19 mathematical model. Chaos Solitons Fractals 157, 111937 (2022)
Abro KA, Yıldırı A: Fractional treatment of vibration equation through modern analogy of fractional differentiations using integral transforms. Iran. J. Sci. Technol. Trans. A Sci. 43(5), 2307–2314 (2019)
Wang, Y, An, J-Y: Amplitude-frequency relationship to a fractional duffing oscillator arising in microphysics and tsunami motion. J. Low Freq. Noise Vib. Active Control 38(3–4), 1008–1012 (2019)
Abro, K.A.: A fractional and analytic investigation of thermo-diffusion process on free convection flow: an application to surface modification technology. Eur. Phys. J. Plus 135(1), 1–14 (2020)
Lohana, B., Abro, K.A., Shaikh, A.W.: Thermodynamical analysis of heat transfer of gravity-driven fluid flow via fractional treatment: an analytical study. J. Therm. Anal. Calorim. 144(1), 155–165 (2021)
Zhang, X., Wang, X., Pandey, M.D., Sørensen, J.D.: An effective approach for high-dimensional reliability analysis of train-bridge vibration systems via the fractional moment. Mech. Syst. Signal Process. 151, 107344 (2021)
Arqub, O.A., El-Ajou, A.: Solution of the fractional epidemic model by homotopy analysis method. J. King Saud Univer. Sci. 25(1), 73–81 (2013)
Atangana, A., Bildik, N.: Approximate solution of tuberculosis disease population dynamics model. In: Abstract and Applied Analysis, Vol. 2013, pp. 1–8, Hindawi (2013)
Diethelm, K.: A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn. 71(4), 613–619 (2013)
Atangana, A., Alqahtani, R.T.: Modelling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative. Entropy 18(2), 40 (2016)
Aliyu, A.I., Inc, M., Yusuf, A., Baleanu, D.: A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana–Baleanu fractional derivatives. Chaos Solitons Fractals 116, 268–277 (2018)
Yadav, R.P., Verma, R.: A numerical simulation of fractional order mathematical modeling of COVID-19 disease in case of Wuhan China. Chaos Solitons Fractals 140, 110124 (2020)
Padder, A., Almutairi, L., Qureshi, S., Soomro, A., Afroz, A., Hincal, E., Tassaddiq, A.: Dynamical analysis of generalized tumor model with Caputo fractional-order derivative. Fractal Fract. 7(3), 258 (2023)
He, J.-H.: Fatalness of virus depends upon its cell fractal geometry. Chaos Solitons Fractals 38(5), 1390–1393 (2008)
Atangana, A.: Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos Solitons Fractals 102, 396–406 (2017)
Li, Z., Liu, Z., Khan, M.A.: Fractional investigation of bank data with fractal-fractional Caputo derivative. Chaos Solitons Fractals 131, 109528 (2020)
Wang, W., Khan, M.A.: Analysis and numerical simulation of fractional model of bank data with fractal-fractional Atangana–Baleanu derivative. J. Comput. Appl. Math. 369, 112646 (2020)
Fatmawati, Khan MA, Alfiniyah C, Alzahrani E et al: Analysis of dengue model with fractal-fractional Caputo–Fabrizio operator. Adv. Differ. Equ. 2020(1), 1–23 (2020)
Shah, K., Arfan, M., Mahariq, I., Ahmadian, A., Salahshour, S., Ferrara, M.: Fractal-fractional mathematical model addressing the situation of corona virus in Pakistan. Results Phys. 19, 103560 (2020)
Khan, M.A., et al.: The dynamics of dengue infection through fractal-fractional operator with real statistical data. Alex. Eng. J. 60(1), 321–336 (2021)
Awadalla, M., Qureshi, S., Soomro, A., Abuasbeh, K.: A novel three-step numerical solver for physical models under fractal behavior. Symmetry 15(2), 330 (2023)
Abro, K.A., Atangana, A., Gomez-Aguilar, J.F.: Optimal synchronization of fractal–fractional differentials on chaotic convection for newtonian and non-newtonian fluids. E. Phys. J. Spec. Top. 232(14), 2403–2414 (2023)
Qureshi, S., Atangana, A., Shaikh, A.A.: Strange chaotic attractors under fractal-fractional operators using newly proposed numerical methods. Eur. Phys. J. Plus 134(10), 523 (2019)
Chen, W.: Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals 28(4), 923–929 (2006)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer Science & Business Media, New York (2013)
Atangana, A., Araz, S.I.: New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications. Academic Press, Cambridge (2021)
Funding
Funding was provided by Council of Scientific and Industrial Research, India (Grant no. 9012-11-44).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
This work does not have any conflicts of interest. Further, the research work of the first author is financially supported by the Council of Scientific and Industrial Research, New Delhi, India.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Singh, H.K., Pandey, D.N. Analysis of the Chickenpox Disease Evolution in an MSEIR Model Using Fractal-Fractional Differential Operator. Differ Equ Dyn Syst (2024). https://doi.org/10.1007/s12591-024-00690-1
Accepted:
Published:
DOI: https://doi.org/10.1007/s12591-024-00690-1
Keywords
- Chickenpox
- Caputo derivative
- Atangana–Baleanu derivative
- Fractal-Fractional derivative
- Least-squares
- Fmincon