### Analysis of SIRC model for influenza A with Caputo-Fabrizio derivative

Document Type : Research Paper

Authors

1 Laboratory of Advanced Materials, Department of Mathematics, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000, Annaba, Algeria

2 Department of Mathematics, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000, Annaba, Algeria

3 Laboratory of Embedded Systems (LASE), Department of Computer Science, Badji Mokhtar-Annaba University, P.O. Box 12, 23000, Annaba, Algeria

Abstract

In this manuscript, we study the fractional-order SIRC epidemiological model for influenza A in the human population in the Caputo-Fabrizio sense. The existence and uniqueness of the solution of the proposed problem are established using fixed point theory. The local stability of both disease-free equilibrium and endemic equilibrium points is investigated. Using the three-step fractional Adams-Bashforth scheme, an iterative solution of our system is generated. In the numerical simulation, many plots are given for different values of the fractional order to check the stability of equilibrium points. Also, the effect of varying some parameters of the model was presented. Furthermore, we compared our numerical solutions with those using Caputo fractional derivative model via graphical representations. The obtained results show the efficiency and accuracy of our approach.

Keywords

[1] E. Ahmed, A.M.A. El-Sayed and H.A.A. El-Saka, On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R¨ossler, Chua and Chen systems, Phys. Lett. A 358 (2006), no. 1, 1–4.
[2] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2016), no. 2, 763–769.
[3] E.J. Chow, M.A. Rolfes, A. O’Halloran, N.B. Alden, E.J. Anderson, N.M. Bennett, L. Billing, E. Dufort, P.D. Kirley, A. George, L. Irizarry, S. Kim, R. Lynfield, P. Ryan, W. Schaffner, H.K. Talbot, A. Thomas, K. YouseyHindes, C. Reed and S. Garg, Respiratory and nonrespiratory diagnoses associated with influenza in hospitalized adults, JAMA network open 3 (2020), no. 3, 201323-201323.
[4] R. Casagrandi, L. Bolzoni, S. A. Levin and V. Andreasen, The SIRC model and influenza A, Math. Biosci. 200 (2006), 152–169.
[5] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1 (2015), no. 2, 73–85.
[6] M. Caputo and M. Fabrizio, On the singular kernels for fractional derivatives. Some applications to partial differential equations, Progr. Fract. Differ. Appl. 7 (2021), 79–82.
[7] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Springer, Heidelberg, 2010.
[8] K. Diethelm and A.D. Freed, The Frac PECE subroutine for the numerical solution of differential equations of fractional order, Forsch. wissenschaft. Rechnen 1999 (1998), 57–71.
[9] P. Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48.
[10] M.A. Dokuyucu, E. Celik, H. Bulut and H.M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, Eur. Phys. J. Plus 133 (2018), no. 3, 1–6.
[11] M. El-Shahed and A. Alsaedi, The fractional SIRC model and Influenza A, Math. Probl. Eng. 2011 (2011), 1-9.
[12] J.F. Gomez-Aguilar, J.J. Rosales-Garcia, J.J. Bernal-Alvarado, T. Cordova-Fraga and R. Guzman-Cabrera, Fractional mechanical oscillators, Rev. Mex. Fis. 58 (2012), no. 4, 348–352.
[13] G. Gonzalez-Parra, A.J. Arenas and B.M. Chen-Charpentier, A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1), Math. Methods Appl. Sci. 37 (2014), no. 15, 2218–2226.
[14] K. Hattaf and N. Yousfi, Mathematical model of influenza AH1N1 infection, Adv, Stud. Biol. 1 (2009), no. 8, 383–390.
[15] M. Higazy and M. A. Alyami, New Caputo-Fabrizio fractional order SEIASqEqHR model for COVID-19 epidemic transmission with genetic algorithm based control strategy, Alexandria Engin. J. 59 (2020), no. 6, 4719–4736.
[16] R. Hilfer, Applications of fractional calculus in physics, World Scientific World Publishing Company, Singapore, 2000.
[17] L. Jodar, R.J. Villanueva, A.J. Arenas and G.C. Gonzalez, Nonstandard numerical methods for a mathematical model for influenza disease, Math. Comput. Simul. 79 (2008), no. 3, 622–633.
[18] M.M. Khader and M.M. Babatin, Numerical treatment for solving fractional SIRC model and influenza A, Comp. Appl. Math. 33 (2014), no. 3, 543-556.
[19] M.M. Khader, N.H. Sweilam, A.M. S. Mahdy and N.K. Abdel Moniem, Numerical simulation for the fractional SIRC model and Influenza A, Appl. Math. Inf. Sci. 8 (2014), no. 3, 1029–1036.
[20] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Elsevier, Amsterdam, 2006.
[21] P. Krishnapriya, M. Pitchaimani and T.M. Witten, Mathematical analysis of an influenza A epidemic model with discrete delay, J. Comput. Appl. Math. 324 (2017), 155–172.
[22] H. Li, J. Cheng, H-B. Li and S.-M. Zhong, Stability analysis of a fractional order linear system described by the Caputo-Fabrizio derivative, Math. 7 (2019), 200.
[23] H. Li and S. Guo, Dynamics of a SIRC epidemiological model, Electron. J. Differ. Equ. 2017 (2017), 1-18.
[24] L. Li, C. Sun and J. Jia, Optimal control of a delayed SIRC epidemic model with saturated incidence rate, Optim. Control. Appl. Met. 40 (2018), no. 2, 367–374.
[25] J. Losada and J.J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), no. 2, 87–92.
[26] J. Losada and J. . Nieto, Fractional integral associated to fractional derivatives with nonsingular kernels, Progr. Fract. Differ. Appl. 7 (2021), 137-143.
[27] R.L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl. 59 (2010), 1586–1593.
[28] R.L. Magin, Fractional calculus in bioengineering, Begell. House. Inc., Redding, CT, 2006.
[29] R.L. Magin, Fractional calculus in bioengineering, part 2. Crit, Rev. Biomed. Eng. 32 (2004), 105–193.
[30] R.L. Magin, Fractional calculus in bioengineering, part 3. Crit. Rev. Biomed. Eng. 32 (2004), 195–378.
[31] C. Milici, G. Draganescu and J.T. Machado, Introduction to fractional differential equations Nonlinear Systems and Complexity, Springer, New York, 2019.
[32] M. Moghadami, A narrative review of influenza: a seasonal and pandemic disease, Iran. J. Med. Sci. 42 (2017), 2–13.
[33] E.J. Moore, S. Sirisubtawee and S. Koonprasert, A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment, Adv. Differ. Equ. 2019 (2019), no. 1, 1–20.
[34] K.B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41 (2010), 9–12.
[35] K. M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo Fabrizio derivative, Chaos, Solitons and Fractals, 105 (2017), 111–119.
[36] P. Palese and J. F. Young, Variation of influenza A, B, and C viruses, Science 215 (1982), 1468–1474.
[37] W. Pan, T. Li and S. Ali, A fractional-order epidemic model for the simulation of outbreaks of Ebola, Adv. Differ. Equ. 2021 (2021), 161.
[38] V. Petrova and C. Russell, The evolution of seasonal influenza viruses, Nat. Rev. Microbiol 16 (2018), 47–60.
[39] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[40] S. Rezapour and H. Mohammadi, A study on the AH1N1/09 influenza transmission model with the fractional Caputo–Fabrizio derivative, Adv. Differ. Equ. 2020 (2020), no. 1, 488.
[41] A. Shaikh, A. Tassaddiq, K.S. Nisar and D. Baleanu, Analysis of differential equations involving Caputo-Fabrizio fractional operator and its applications to reaction-diffusion equations, Adv. Differ. Equ. 2019 (2019), no. 1, 1–14.
[42] S. Ullah, M.A. Khan, M. Farooq, Z. Hamouch and D. Baleanu, A fractional model for the dynamics of tuberculosis infection using Caputo–Fabrizio derivative, Discrete Contin. Dyn. Syst. Ser. S 13 (2020), no 3.
[43] M.Z. Ullah, A.K. Alzahrani and D. Baleanu, An efficient numerical technique for a new fractional tuberculosis model with nonsingular derivative operator, J. Taibah Univ. Sci. 13 (2019), no 1, 1147–1157.
[44] T. Zhang, T. Ding, N. Gao and Y. Song, Dynamical behavior of a stochastic SIRC model for influenza A, Symmetry 12 (2020), 745.
###### Volume 13, Issue 2July 2022Pages 1239-1259
• Receive Date: 29 October 2021
• Accept Date: 09 February 2022