Analysis of fractional-order vaccinated Hepatitis-B epidemic model with Mittag-Leffler kernels

Year 2022, Volume: 2 Issue: 2, 59 - 72, 30.06.2022

Anwarud Din Muhammad Zainul Abidin

https://doi.org/10.53391/mmnsa.2022.006

Cited By: 12

Abstract

The current paper investigates a newly developed model for Hepatitis-B infection in sense of the Atangana-Baleanu Caputo (ABC) fractional-order derivative. The proposed technique classifies the population into five distinct categories, such as susceptible, acute infections, chronic infections, vaccinated, and immunized. We obtain the Ulam-Hyers type stability and a qualitative study of the corresponding solution by applying a well-known principle of fixed point theory. Furthermore, we establish the deterministic stability of the proposed model. For the approximation of the ABC fractional derivative, we use a newly proposed numerical method. The obtained results are numerically verified by MATLAB 2020a.

Keywords

Fractional calculus, fractional-order model, hepatitis-B disease, ABC derivative, fixed-point theorem, numerical simulation

References

• Wang, L., Liu, Z., & Zhang, X. Global dynamics of an SVEIR epidemic model with distributed delay and nonlinear incidence. Applied Mathematics and Computation, 284, 47–65, (2016).
• Din, A., & Li, Y. Mathematical analysis of a new nonlinear stochastic hepatitis B epidemic model with vaccination effect and a case study. The European Physical Journal Plus, 137(5), 1-24, (2022).
• Veeresha, P. A numerical approach to the coupled atmospheric ocean model using a fractional operator. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 1-10, (2021).
• Din, A., & Li, Y. Lévy noise impact on a stochastic hepatitis B epidemic model under real statistical data and its fractal–fractional Atangana–Baleanu order model. Physica Scripta, 96(12), 124008, (2021).
• Duan, X., Yuan, S., & Li, X. Global stability of an SVIR model with age of vaccination. Applied Mathematics and Computation, 226, 528–540, (2014).
• Din, A., Li, Y., & Liu, Q. Viral dynamics and control of hepatitis B virus (HBV) using an epidemic model. Alexandria Engineering Journal, 59(2), 667-679, (2020).
• Poland, G.A., & Jacobson, R.M. Prevention of hepatitis B with the hepatitis B vaccine. New England Journal of Medicine, 351(27), 2832-2838, (2004).
• McAleer, W.J., Buynak, E.B., Maigetter, R.Z., Wampler, D.E., Miller, W.J., & Hilleman, M.R. Human hepatitis B vaccine from recombinant yeast. Nature, 307(5947), 178-180, (1984).
• Din, A., & Li, Y. Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity. Physica Scripta, 96(7), 074005, (2021).
• Atangana, A. Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties. Physica A: Statistical Mechanics and its Applications, 505, 688-706, (2018).
• Veeresha, P., Prakasha, D.G., & Baleanu, D. An efficient technique for fractional coupled system arisen in magnetothermoelasticity with rotation using Mittag–Leffler kernel. Journal of Computational and Nonlinear Dynamics, 16(1), (2021).
• Hammouch, Z., Yavuz, M., & Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).
• Atangana, A., & Baleanu, D. New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Thermal Science, 20(2), 763-769, (2016).
• Jajarmi, A., & Baleanu, D. A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Frontiers in Physics, 8, 220, (2020).
• Shah, K., Khalil, H., & Khan, R.A. Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos, Solitons and Fractals, 77, 240-246, (2015).
• Din, A., Liu, P., & Cui, T. Stochastic stability and optimal control analysis for a tobacco smoking model. Applied and Computational Mathematics, 10(6), 163-185, (2021).
• Cui, T., Liu, P., & Din, A. Fractal–fractional and stochastic analysis of norovirus transmission epidemic model with vaccination effects. Scientific Reports, 11(1), 1-25, (2021).
• Li, X.P., Din, A., Zeb, A., Kumar, S., & Saeed, T. The impact of Lévy noise on a stochastic and fractal-fractional Atangana–Baleanu order hepatitis B model under real statistical data. Chaos, Solitons & Fractals, 154, 111623, (2021).
• Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).
• Chen, S.B., Soradi-Zeid, S., Jahanshahi, H., Alcaraz, R., Gómez-Aguilar, J.F., Bekiros, S., & Chu, Y.M. Optimal control of time-delay fractional equations via a joint application of radial basis functions and collocation method. Entropy, 22(11), 1213, (2020).
• Ikram, R., Khan, A., Zahri, M., Saeed, A., Yavuz, M., & Kumam, P. Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay. Computers in Biology and Medicine, 141, 105115, (2022).
• Joshi, H., & Jha, B.K. Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 84-94, (2021).
• Din, A., Shah, K., Seadawy, A., Alrabaiah, H., & Baleanu, D. On a new conceptual mathematical model dealing the current novel coronavirus-19 infectious disease. Results in Physics, 19, 103510, (2020).
• Gholami, M., Ghaziani, R.K., & Eskandari, Z. Three-dimensional fractional system with the stability condition and chaos control. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 41-47, (2022).
• Khan, A., Hussain, G., Inc, M., & Zaman, G. Existence, uniqueness, and stability of fractional hepatitis B epidemic model. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(10), 103104, (2020).
• Atangana, A., Akgül, A., & Owolabi, K.M. Analysis of fractal fractional differential equations. Alexandria Eng. J, 59(3), 1117–1134, (2020).
• Kumar, P., & Erturk, V.S. Dynamics of cholera disease by using two recent fractional numerical methods. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 102-111, (2021).
• Alzahrani, E.O., & Khan, M.A. Modeling the dynamics of Hepatitis E with optimal control. Chaos, Solitons & Fractals, 116, 287-301, (2018).
• Din, A., & Li, Y. The extinction and persistence of a stochastic model of drinking alcohol. Results in Physics, 28, 104649, (2021).
• Din, A., Li, Y., & Shah, M.A. The complex dynamics of hepatitis B infected individuals with optimal control. Journal of Systems Science and Complexity, 34(4), 1301-1323, (2021).
• Daşbaşı, B. Stability analysis of an incommensurate fractional-order SIR model. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 44-55, (2021).
• Naik, P.A., Eskandari, Z., Yavuz, M., & Zu, J. Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect. Journal of Computational and Applied Mathematics, 413, 114401, (2022).
• Özköse, F., Yavuz, M., Şenel, M. T., & Habbireeh, R. Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom. Chaos, Solitons & Fractals, 157, 111954, (2022).
• Rossikhin, Y.A., & Shitikova, M.V. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Applied Mechanics Reviews, 50(1), 15-67, (1997).
• Atangana, A., & Qureshi, S. Modeling attractors of chaotic dynamical systems with fractal–fractional operators. Chaos, Solitons & Fractals, 123, 320-337, (2019).
• Özköse, F., & Yavuz, M. Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey. Computers in Biology and Medicine, 141, 105044, (2022).
• Jena, R.M., Chakraverty, S., Yavuz, M., & Abdeljawad, T. A new modeling and existence–uniqueness analysis for Babesiosis disease of fractional order. Modern Physics Letters B, 35(30), 2150443, (2021).
• Din, A., Khan, A., & Baleanu, D. (2020). Stationary distribution and extinction of stochastic coronavirus (COVID-19) epidemic model. Chaos, Solitons & Fractals, 139, 110036, (2020).
• Din, A., Khan, T., Li, Y., Tahir, H., Khan, A., & Khan, W.A. Mathematical analysis of dengue stochastic epidemic model. Results in Physics, 20, 103719, (2021).
• Yavuz, M., Coşar, F.Ö., Günay, F., & Özdemir, F.N. A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign. Open Journal of Modelling and Simulation, 9(3), 299-321, (2021).
• Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A., & Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a Sars-Cov-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 56-66, (2021).
• Li, X.P., Al Bayatti, H., Din, A., & Zeb, A. A vigorous study of fractional order COVID-19 model via ABC derivatives. Results in Physics, 29, 104737, (2021).
• Sene, N. Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 13-25, (2022).
• Özköse, F., Şenel, M.T., & 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).
• Din, A., Li, Y., Khan, T., & Zaman, G. Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China. Chaos, Solitons & Fractals, 141, 110286, (2020).
• Atangana, A., & Koca, I. Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos, Solitons & Fractals, 89, 447-454, (2016).
• Din, A., Li, Y., Khan, T., Anwar, K., & Zaman, G. Stochastic dynamics of hepatitis B epidemics. Results in Physics, 20, 103730, (2021).
• Atangana, E., & Atangana, A. Facemasks simple but powerful weapons to protect against COVID-19 spread: Can they have sides effects?. Results in Physics, 19, 103425, (2020).
• Din, A., & Li, Y. Controlling heroin addiction via age-structured modeling. Advances in Difference Equations, 2020(1), 1-17, (2020).
• Atangana, A. Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?. Chaos, Solitons & Fractals, 136, 109860, (2020).
• Owolabi, K.M., Gómez-Aguilar, J.F., & Karaagac, B. Modelling, analysis and simulations of some chaotic systems using derivative with Mittag-Leffler kernel. Chaos, Solitons & Fractals, 125, 54-63, (2019).
• Toufik, M., & Atangana, A. New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. The European Physical Journal Plus, 132(10), 1-16, (2017).
• Owolabi, K.M., & Atangana, A. On the formulation of Adams–Bashforth scheme with Atangana–Baleanu–Caputo fractional derivative to model chaotic problems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29, 023111, (2019).
• Atangana, A., & Owolabi, K.M. New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena, 13(1), 3, (2018).
• Atangana, A., & Bonyah, E. Fractional stochastic modeling: new approach to capture more heterogeneity. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(1), 013118, (2019).
• Koca, I. Modelling the spread of ebola virus with Atangana–Baleanu fractional operators. The European Physical Journal Plus, 133, 100, (2018).