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

Year 2022, Volume: 2 Issue: 2, 59 - 72, 30.06.2022
https://doi.org/10.53391/mmnsa.2022.006

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.

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).
Year 2022, Volume: 2 Issue: 2, 59 - 72, 30.06.2022
https://doi.org/10.53391/mmnsa.2022.006

Abstract

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).
There are 56 citations in total.

Details

Primary Language English
Subjects Bioinformatics and Computational Biology, Applied Mathematics
Journal Section Research Articles
Authors

Anwarud Din This is me 0000-0003-0463-0360

Muhammad Zainul Abidin This is me 0000-0001-8567-0676

Publication Date June 30, 2022
Submission Date March 22, 2022
Published in Issue Year 2022 Volume: 2 Issue: 2

Cite

APA Din, A., & Abidin, M. Z. (2022). Analysis of fractional-order vaccinated Hepatitis-B epidemic model with Mittag-Leffler kernels. Mathematical Modelling and Numerical Simulation With Applications, 2(2), 59-72. https://doi.org/10.53391/mmnsa.2022.006

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