Dynamical Behavior of Fractional Order SVIR Epidemic Model

Article Sidebar

PDF
Cite as:
Aqeel Ahmad, Nouman Javeed, Muhammad Farman, M.O. Ahmad, Amna Hafeez, Ali Raza, Dynamical Behavior of Fractional Order SVIR Epidemic Model, Int. J. Anal. Appl., 17 (2) (2019), 260-274.

Main Article Content

Aqeel Ahmad, Nouman Javeed, Muhammad Farman, M.O. Ahmad, Amna Hafeez, Ali Raza

Abstract

In this Paper, we proposed a fractional order SVIR epidemic model is incorporated to investigate its dynamical behavior in random environments. S, V, I, R are known as variables and these variables represent the number of susceptible, vaccinated, infected and recovered cells from viruses in the body. The Caputo fractional derivative operator of order α \in (0,1] is employed to obtain the system of fractional differential equations. The basic reproductive number is derived for a general viral production rate which determines the local stability of the infection free equilibrium. The stability and sensitivity analysis of fractional order has been made and verify the non-negative unique solution. The solution of the time fractional model has been procured by employing Laplace Adomian decomposition method (LADM) and the accuracy of the scheme is presented by convergence analysis. Finally numerical solutions are also established to investigate the influence of system parameter on the spread of disease and which show the effect of fractional parameter on our obtained solution.

Article Details

References

  1. J. Biazar, Solution of the epidemic model by Adomian decomposition method, Appl. Math. Comput. 173 (2006), 1101-1106.
  2. S. Busenberg, P. Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol. 28 (1990), 65-82.
  3. AE A.M.A. El-Sayed, S.Z. Rida, A.A.M. Arafa, On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, Int. J. Nonlinear Sci. 7 (2009), 485-495.
  4. O.D. Makinde, Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy, Appl. Math. Comput. 184 (2007), 842-848.
  5. A.A.M. Arafa, S.Z. Rida, M. Khalil, Fractional modeling dynamics of HIV and 4 T-cells during primary infection, Nonlinear Biomed. Phys. 6 (2012), 1-7.
  6. C.M. Kribs-Zaleta, Structured models for heterosexual disease transmission, Math. Biosci. 160 (1999), 83-108.
  7. B. Buonomo, D. Lacitignola, On the dynamics of an SEIR epidemic model with a convex incidence rate, Ricerche Mat. 57 (2008), 261-281.
  8. X. Liu, C. Wang, Bifurcation of a predator-prey model with disease in the prey, Nonlinear Dyn. 62 (2010), 841-850.
  9. Fazal Haq, Kamal Shah, Ghaus ur Rahman, Muhammad Shahzad. Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alex. Eng. J. 57 (2) (2018), 1061-1069
  10. J.D Murray: Mathematical Biology I. An Introduction,USA ,Springer , Verlag Berlin Heidelberg. (2002).
  11. A. C. Guyton and J. E. Hall, Text Book of Medical Physiology, Elsevier Saunders, St. Louis, edition 11 972 (2012).
  12. Erturk, VS, Zaman, G, Momani, S: A numeric analytic method for approximating a giving up smoking model containing fractional derivatives. Comput. Math. Appl. 64 (2012), 3068-3074.
  13. Zeb, A, Chohan, I, Zaman, G: The homotopy analysis method for approximating of giving up smoking model in fractional order. Appl. Math. 3 (2012), 914-919.
  14. Alkhudhari, Z, Al-Sheikh, S, Al-Tuwairqi, S: Global dynamics of a mathematical model on smoking. Appl. Math. 2014 (2014), Article ID 847075.
  15. R. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2004.
  16. J. Biazar, Solution of the epidemic model by Adomian decomposition method, Appl. Math. Comput. 173 (2006), 1101-1106.
  17. T. Khan, G. Zaman and M. I. Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis B, J. Biol. Dyn. 11 (1) (2017), 172-189.
  18. Aqeel Ahmad, Muhammad Farman, Faisal Yasin, M. O. Ahmad, Dynamical transmission and effect of smoking in society, Int. J. Adv. Appl. Sci. 5(2) (2018), 71-75
  19. Aqeel Ahmad, Muhammad Farman, M. O. Ahmad, Nauman Raza, M. 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
  20. Farah Ashraf, Aqeel Ahmad, Muhammad Umer Saleem, Muhammad Farman, 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