A Mathematical Analysis and Modelling of Hepatitis B Model with Non-Integer Time Fractional Derivative

Muhammad Farman, Aqeel Ahmad, M. Umer Saleem, Amina Hafeez


In this paper, we develop time fractional Hepatitis B model and analyze qualitatively. The Caputo fractional derivative operator of order \(\alpha\in (0,1]\) is employed to obtain the system of fractional differential equations. The stability and sensitivity analysis of fractional order model has been made and verify the non-negative unique solution. LADM was successfully used for solving different problems. Laplace transform method is a useful technique in different field of biological science, engineering and applied mathematics. The latest technique (LADM) is employed on the developed fractional order model for numerical solutions. Finally, numerical simulations are also established to investigate the influence of the system parameter on the spread of the disease and which show the effect of fractional parameter \(\alpha\) on our obtained solutions.


Epidemic model; Stability analysis; Fractional derivative; LADM

Full Text:



A. Ahmad, M. Farman, F. Yasin and M. O. Ahmad, Dynamical transmission and effect of smoking in society, Int. J. Adv. Appl. Sci. 5(2) (2018), 71 – 75, DOI: 10.21833/ijaas.2018.02.012.

A. Ahmad, M. Farman, F. Yasin and M. O. Ahmad, Dynamical transmission and effect of smoking in society, Int. J. Adv. Appl. Sci. 5(2) (2018), 71 – 75, DOI: 10.21833/ijaas.2018.02.012.

A. Ahmad, M. Farman, M. O Ahmad, N. Raza and Abdullah, Dynamical behavior of SIR epidemic model with non-integer time fractional derivatives: A mathematical analysis, Int. J. Adv. Appl. Sci. 5(1) (2018), 123 – 129, DOI: 10.21833/ijaas.2018.01.016.

F. Ashraf and M.O. Ahmad, Nonstandard finite difference scheme for control of measles epidemiology, Int. J. Adv. Appl. Sci. 6(3) (2019), 79 – 85, DOI: 10.21833/ijaas.2019.03.012.

F. Ashraf, A. Ahmad, M. U. Saleem, M. Farman and M. O. Ahmad, Dynamical behavior of HIV immunology model with non-integer time fractional derivatives, Int. J. Adv. Appl. Sci. 5(3) (2018), 39 – 45, DOI: 10.21833/ijaas.2018.03.006.

B. Buonomo and D. Lacitignola, On the dynamics of an SEIR epidemic model with a convex incidence rate, Ricerche Mat. 57 (2008), 261 – 281, DOI: 10.1007/s11587-008-0039-4.

P. Elango, The role of mathematics in biology, 5th International Symposium 2015 – IntSym 2015, Department of Mathematical Sciences, Faculty of Applied Sciences, South Eastern University of Sri Lanka (2015), http://ir.lib.seu.ac.lk/handle/123456789/1299.

F. Haq, K. Shah, G. Rahman and M. Shahzad, Numerical solution of fractional order smoking model via Laplace Adomian decomposition method, Alex. Eng. J. 1110-0168.(2017), DOI: 10.1016/j.aej.2017.02.015.

H. Jafari, C. M. Khalique and M. Nazari, Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion wave equations, Applied Mathematics Letters 24(11) (2011), 1799 – 1805, DOI: 10.1016/j.aml.2011.04.037.

A. V. Kamyad, R. Akbari, A. A. Heydari and A. Heydari, Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus, Computational and Mathematical Methods in Medicine 2014 (2014), Article ID 475451, 15 pages, DOI: 10.1155/2014/475451.

World Health Organization (WHO), Prevention and Control of Viral Hepatitis Infection, Framework for Global Action (2012), http://www.who.int/iris/handle/10665/130012.

T. Zhu, B. T. Korber, A. J. Nahmias, E. Hooper, P. M. Sharp and D. D. Ho, An African HIV-1 sequence and implications for the origin of the epidemic, Nature 391 (1998), 594 – 597, DOI: 10.1038/35400.

DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1154


  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905