Numerical Simulation of Fractional Order Dengue Disease with Incubation Period of Virus

Fractional Order Dengue Disease

Authors

  • Zain Ul Abadin Zafar Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan.Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan
  • M. Mushtaq Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan
  • Kashif Rehan Department of Mathematics, University of Engineering and Technology, KSK Campus, Lahore, Pakistan
  • M. Rafiq Faculty of Engineering, University of Central Punjab, Lahore, Pakistan

Keywords:

Dengue virus, incubation period, stability, fraction order numerical modeling

Abstract

Nowadays, numerical models have great importance in epidemiology. These helps us to understand the transmission dynamics of infectious diseases in a very comprehensive manner. In disease epidemiology, vector-host models are important because many diseases are spreading through vectors. Mosquitoes are vectors of dengue disease as these spread the disease in a population. The infectious vectors infect the hosts while infectious hosts infect to vectors .Two main groups of dengue patients are septic and contagious. The susceptible mosquitoes can get dengue infection from infectious humans but not from infected ones. Humans can be categorized into Susceptible, infected, infectious and recovered ones while mosquitoes are susceptible, infected and infectious. Susceptible individual can transfer dengue infection from diseased mosquitoes only. The transmission dynamics of “Fractional order dengue fever” with incubation period of virus has been analyzed in this paper. Using standard methods for analyzing a system, the stability of equilibrium points of the model has been determined. Finally, numerical simulation has been performed for the same problem for different values of discretization parameter ‘h’.

References

Yıldırım, A. & Y. Cherruault. Analytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method. Kybernetes 38(9): 1566-1575 (2009).

Makinde, O.D. Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Applied Mathematics & Computation 184(2): 842-848 (2007).

Arafa, A.A.M., S.Z. Rida & M. Khalil. Fractional modeling dynamics of HIV and 4 T-cells during primary Infection. Nonlinear Biomedical Physics 6: 1-7 (2012).

Hethcote, H.W. The mathematics for infectious diseases, Siam Review 42(4): 599-653 (2000).

Biazar, J. Solution of the epidemic model by adomian decomposition method. Applied Mathematics & Computation 173(2):1101–1106 (2006).

Busenberg, S. & P. Driessche. Analysis of a disease transmission model in a population with varying size. Journal of Mathematical Biology 28:257- 270 (1990).

El-Sayed, A.M.A, S.Z. Rida & A.A.M. Arafa. On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber. International Journal of Nonlinear Science 7: 485-492 (2009).

Hassan, H.N. & M.A. El-Tawil. A new technique of using homotopy analysis method for solving high-order non-linear differential equations. Mathematical Methods in Applied Science 34:728–742(2011).

Liao, S.J. A kind of approximate solution technique which does not depend upon small parameters: a special Example. International Journal of Non-Linear Mechanics 30:371–380 (1995).

Liao, S.J. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton (2003).

Zibaei, S. & M. Namjoo. A nonstandard finite differnce scheme for solving fractional-order Model of HIV-1 infection of CD4+ T-cells. Iranian Journal of Mathematical Chemistry 6(2): 169-184(2015).

Ahmad, E., A.M. Atial, El-Sayed & H.A.A. El-Saka. On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems. Physics Letters A 358(1): 1-4 (2006).

Mickens, R.E. Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numerical Methods of partial Differential Equations 23(3): 672-691 (2007).

Matignon, D. Stability result on fractional differential equations with applications to control processing to control processing, In: Computational Engineering in Systems Applications(Conference), p. 963–968 (1996).

Zafar, Z., K. Rehan, M. Mushtaq & M. Rafiq. Numerical modelling for nonlinear biochemical reaction networks. Iranian Journal of Mathematical Chemistry (Accepted for publication).

Pooseh, S., H.S. Rodrigues, & D.F.M. Torres. Fractional derivatives in dengue epidemics. doi:10.1063/1.3636838 (2011).

Jan, R., & A. Jan. MSGDTM for solution of fractional order dengue disease model. International Journal of Science and Research 6(3): 1140-1144 (2017).

Al-Sulami, H., M. El-Shahed, J.J. Nieto, & W. Shammakh. On fractional order dengue epidemic model. Mathematical Problems in Engineering, http://dx.doi.org/10.1155/2014/456537 (2014).

Metzler, R., & J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports 339:1-77 (2000).

West, B.J., P. Grigolini, & R. Metzler. Nonnenmacher, TF: Fractional diffusion and Le’vy stable processes. Physics Review E. 55(1): 99-106 (1997).

Golmankhaneh, A.K., R. Arefi, & D. Baleanu. Synchronization in a nonidentical fractional order of a proposed modified system. Journal of Vibration and Control, 21(6): 1154-1161 (2015).

Baleanu, D., A.K. Golmankhaneh, A.K. Golmankhaneh, & R.R. Nigmatullin. Newtonian Law with memory. Nonlinear Dynamics 60: 81-86 (2010).

Agila, A., D. Baleanu, R. Eid, & B. Irfanoglu. Applications of the extended fractional euler- lagrange equations model to freely oscillating dynamical systems. Romanian Journal of Physics 61(3-4): 350-359 (2016).

Baleanu, D & O.G. Mustafa. On the global existence of solutions to a class of fractional differential equations. Computers & Mathematics with Applications 59: 1835-1841 (2010).

He, J, Li, Z, Wang, Q: A new fractional derivative and its application to explanation of polar bear basis. Journal of King Saud University-Science 28:190-192 (2016).

Oliveira, E.C., & J.A.T. Machado. A review of definitions for fractional derivatives and Integral. Mathematical Problems in Engineering (2014). http://dx.doi.org/10.1155/2014/238459.

Caputo, M, & M. Fabrizio. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiations and Applications 1(2):73-85 (2015).

Atangana, A., & D. Baleanu. New fractional derivatives with non-local and non-singular kernel. Thermal Science 20(2):763-769 (2016).

Zafar, Z., K. Rehan, & M. Mushtaq. Fractional-order scheme for bovine babesiosis disease and tick populations. advances in difference equations. p. 86 (2017). http://dx.doi.org/10.1186/s13662-017-1133-2.

Zafar, Z., K. Rehan, M. Mushtaq, & M. Rafiq. Numerical Treatment for nonlinear Brusselator Chemical Model. Journal of Difference Equations and Applications 23(3): 521-538 (2017).

Zafar, Z., K. Rehan, & M. Mushtaq. HIV/AIDS epidemic fractional-order model. Journal of Difference Equations and Applications 23(7): 1298-1315 (2017).

Published

2021-04-23

How to Cite

Zafar, Z. U. A. ., Mushtaq, M. ., Rehan, K. ., & Rafiq, M. . (2021). Numerical Simulation of Fractional Order Dengue Disease with Incubation Period of Virus: Fractional Order Dengue Disease. Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 54(3), 277–296. Retrieved from http://ppaspk.org/index.php/PPAS-A/article/view/233

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