Modelling Influenza A disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates

Year 2023, , 58 - 73, 31.03.2023

Fırat Evirgen , Esmehan Uçar Sümeyra Uçar Necati Özdemir

https://doi.org/10.53391/mmnsa.1274004

Cited By: 16

Abstract

The objective of this manuscript is to present a novel approach to modeling influenza A disease dynamics by incorporating the Caputo-Fabrizio (CF) fractional derivative operator into the model. Particularly distinct contact rates between exposed and infected individuals are taken into account in the model under study, and the fractional derivative concept is explored with respect to this component. We demonstrate the existence and uniqueness of the solution and obtain the series solution for all compartments using the Laplace transform method. The reproduction number of the Influenza A model, which was created to show the effectiveness of different contact rates, was obtained and examined in detail in this sense. To validate our approach, we applied the predictor-corrector method in the sense of the Caputo-Fabrizio fractional derivative and demonstrate the effectiveness of the fractional derivative in accurately predicting disease dynamics. Our findings suggest that the use of the Caputo-Fabrizio fractional derivative can provide valuable insights into the mechanisms underlying influenza A disease and enhance the accuracy of disease models.

Keywords

Fractional differential equations, fixed point theory, Caputo-Fabrizio derivative, influenza

References

• Wu, Y., Wu, Y., Tefsen, B., Shi, Y. and Gao, G.F. Bat-derived influenza-like viruses H17N10 and H18N11. Trends in Microbiology, 22(4), 183-191, (2014).
• Kilbourne, E.D. Influenza pandemics of the 20th century. Emerging Infectious Diseases, 12(1), 9-14, (2006).
• Uçar, E., Ozdemir, N. and Altun, E. Fractional order model of immune cells influenced by cancer cells. Mathematical Modeling Natural Phenomea, 14(3), 12, (2019).
• Özköse, F., Şenel, M.T. and Habbireeh, R. Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83, (2021).
• Uçar, E., Özdemir, N. and Altun, E. Qualitative analysis and numerical simulations of new model describing cancer. Journal of Computational and Applied Mathematics, 422, 114899, (2023).
• Qureshi, S., Yusuf, A., Shaikh, A.A. and Inc, M. Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data. Physica A: Statistical Mechanics and its Applications, 534, 122149, (2019).
• Ahmad, S., Qiu, D. and ur Rahman, M. Dynamics of a fractional-order COVID-19 model under the nonsingular kernel of Caputo-Fabrizio operator. Mathematical Modelling and Numerical Simulation with Applications, 2(4), 228-243, (2022).
• Hamou, A.A., Rasul, R.R.Q., Hammouch, Z. and Özdemir, N. Analysis and dynamics of a mathematical model to predict unreported cases of COVID-19 epidemic in Morocco. Computational and Applied Mathematics, 41, 289, (2022).
• Okundalaye, O.O., Othman, W.A.M. and Oke, A.S. Toward an efficient approximate analytical solution for 4-compartment COVID-19 fractional mathematical model. Journal of Computational and Applied Mathematics, 416, 114506, (2022).
• Koca, I, Bulut, H. and Akçetin, E. A different approach for behavior of fractional plant virüs model. Journal of Nonlinear Sciences and Applications, 15(3), 186-202, (2022).
• Uçar, S., Ozdemir, N., Koca, I. and Altun, E. Novel analysis of the fractional glucose–insulin regulatory system with non-singular kernel derivative. The European Physical Journal Plus, 135(5), 1-18, (2020).
• Naik, P.A., Eskandari, Z., Yavuz, M. and Zu, J. Complex dynamics of a discrete-time BazykinBerezovskaya prey-predator model with a strong Allee effect. Journal of Computational and Applied Mathematics, 413, 114401, (2022).
• Evirgen, F. Transmission of Nipah virus dynamics under Caputo fractional derivative. Journal of Computational and Applied Mathematics, 418, 114654, (2023).
• Olumide, O.O., Othman, W.A.M. and Ozdemir, N. Efficient solution of fractional-order SIR epidemic model of childhood diseases with optimal homotopy asymptotic method. IEEE Access, 10, 9395-9405, (2022).
• Uçar, S. Analysis of hepatitis B disease with fractal-fractional Caputo derivative using real data from Turkey. Journal of Computational and Applied Mathematics, 419, 114692, (2023).
• Tajadodi, H., Jafari, H. and Ncube, M.N. Genocchi polynomials as a tool for solving a class of fractional optimal control problems. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12(2), 160–168, (2022).
• Chatterjee, A. and Pal, S. A predator-prey model for the optimal control of fish harvesting through the imposition of a tax. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 13(1), 68–80, (2023).
• Kaliraj, K., Viswanath, K.S., Logeswari, K. and Ravichandran, C. Analysis of Ffractional integro–differential equation with robin boundary conditions using topological degree method. International Journal of Applied and Computational Mathematics, 8(4), 176, (2022).
• Manjula, M., Kaliraj, K., Botmart, T., Nisar, K.S. and Ravichandran, C. Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses. AIMS Mathematics, 8(2), 4645-4665, (2023).
• Vijayaraj, V., Ravichandran, C., Sawangtong, P. and Nisar, K.S. Existence results of AtanganaBaleanu fractional integro-differential inclusions of Sobolev type. Alexandria Engineering Journal, 66, 249-255, (2023).
• Sene, N. Theory and applications of new fractional-order chaotic system under Caputo Operator. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12(1), 20-38, (2022).
• Koca, I. Modeling the heat flow equation with fractional-fractal differentiation. Chaos, Solitons & Fractals, 128, 83-91, (2019).
• Alexander, M.E., Bowman, C., Moghadas, S.M., Summers, R., Gumel, A.B. and Sahai, B.M. A vaccination model for transmission dynamics of influenza. SIAM Journal on Applied Dynamical Systems, 3(4), 503-524, (2004).
• Casagrandi, R., Bolzoni, L., Levin, S.A. and Andreasen, V. The SIRC model and influenza A. Mathematical Biosciences, 200(2), 152-169, (2006).
• Iwami, S., Takeuchi, Y. and Liu, X. Avian–human influenza epidemic model. Mathematical Biosciences, 207(1), 1-25, (2007).
• Tracht, S.M., Del Valle, S.Y. and Hyman, J.M. Mathematical modeling of the effectiveness of facemasks in reducing the spread of novel influenza A (H1N1). PloS One, 5(2), e9018, (2010).
• González-Parra, G., Arenas, A.J. and Chen-Charpentier, B.M. A fractional order epidemic model for the simulation of outbreaks of influenza A (H1N1). Mathematical Methods in the Applied Sciences, 37(15), 2218-2226, (2014).
• Khanh, N.H. Stability analysis of an influenza virus model with disease resistance. Journal of the Egyptian Mathematical Society, 24, 193-199, (2016).
• Jia, J. and Xiao, J. Stability analysis of a disease resistance SEIRS model with nonlinear incidence rate. Advances in Difference Equations, 75, (2018).
• Quirouette, C., Younis, N.P., Reddy, M.B. and Beauchemin, C.A. A mathematical model describing the localization and spread of influenza A virus infection within the human respiratory tract. PLoS Computational Biology, 16(4), e1007705, (2020).
• Srivastava, H.M., Saad, K.M., Gómez-Aguilar, J.F. and Almadiy, A.A. Some new mathematical models of the fractional-order system of human immune against IAV infection. Mathematical Biosciences and Engineering, 17(5), 4942-4969, (2020).
• Baba, I.A., Ahmad, H., Alsulami, M.D., Abualnaja, K.M. and Altanji, M. A mathematical model to study resistance and non-resistance strains of influenza. Results in Physics, 26, 104390, (2021).
• Ojo, M.M., Benson, T.O., Peter, O.J. and Goufo, E.F.D. Nonlinear optimal control strategies for a mathematical model of COVID-19 and influenza co-infection. Physica A: Statistical Mechanics and its Applications, 607, 128173, (2022).
• Etemad, S., Avci, I., Kumar, P., Baleanu, D. and Rezapour, S. Some novel mathematical analysis on the fractal–fractional model of the AH1N1/09 virus and its generalized Caputotype version. Chaos, Solitons & Fractals, 162, 112511, (2022).
• Derradji, L.S., Hamidane, N. and Aouchal, S. A fractional SEIRS model with disease resistance and nonlinear generalized incidence rate in Caputo–Fabrizio sense. Rendiconti del Circolo Matematico di Palermo Series 2, 72(1), 81-98, (2023).
• Sabir, Z., Said, S.B. and Al-Mdallal, Q. A fractional order numerical study for the influenza disease mathematical model. Alexandria Engineering Journal, 65, 615-626, (2023).
• Caputo, M. and Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1, 73-85, (2015).
• Atangana, A. and Talkahtani, B.S. Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel. Advances in Mechanical Engineering, 7, 1-6, (2015).
• Gomez-Aguilar, J.F., Rosales-García, J.J. and Bernal-Alvarado, J.J. Fractional mechanical oscillators. Revista Mexicana Física, 58, 348-352, (2012).
• Driessche, V.P. and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(2), 29-48, (2002).
• Diekmann, O., Heesterbeek, J.A.P. and Roberts, M.G. The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47), 873-885, (2010).
• Toh, Y.T., Phang, C. and Loh, J.R. New predictor-corrector scheme for solving nonlinear differential equations with Caputo-Fabrizio operator. Mathematical Methods in the Applied Sciences, 42, 175-185, (2019).