AIMS Mathematics, 2020, 5(3): 1856-1880. doi: 10.3934/math.2020124.

Research article Special Issues

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Fractional order SIR model with generalized incidence rate

1 Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa
2 Department of mathematics, City university of Science and Information Technology, Peshawar, KP, Pakistan
3 Department of mathematics university of Peshawar, KP, Pakistan
4 Department of mathematics Abdul Wali Khan university, Mardan, KP, Pakistan

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

The purpose of this work is to present the dynamics of a fractional SIR model with generalized incidence rate using two differential derivatives, that are the Caputo and the AtanganaBaleanu. Firstly, we formulate the proposed model in Caputo sense and carried out the basic mathematical analysis such as positivity, basic reproduction number, local and global dynamics of the disease free and endemic case. The disease free equilibrium is locally and globally asymptotically stable when the basic reproduction number is less than 1. We also show that the SIR fractional model is locally and globally asymptotically stable at endemic state when the basic reproduction number greater than unity. Further, to analyze the dynamics of Caputo model we provide some numerical results by considering various incidence rates and with different fractional order parameters. Then, we reformulate the same model using the Atangana-Baleanu operator having nonsingular and non-local kernel. We prove the existence and uniqueness of the Atangana-Baleanu SIR model using PicardLindelof approach. Furthermore, we present an iterative scheme of the model and provide graphical results for various values of the fractional order α. From the graphical interpretation we conclude that the Atangana-Baleanu derivative is more prominent and provides biologically more feasible results than Caputo operator.
  Figure/Table
  Supplementary
  Article Metrics

Keywords SIR model; Caputo operator; Atangana-Baleanu operator; stability results; Lyapunov function; Picard-Lindelof theorem; numerical solution

Citation: Muhammad Altaf Khan, Muhammad Ismail, Saif Ullah, Muhammad Farhan. Fractional order SIR model with generalized incidence rate. AIMS Mathematics, 2020, 5(3): 1856-1880. doi: 10.3934/math.2020124

References

  • 1. S. Ullah, M. A. Khan, J. F. Gez-Aguilar, Mathematical formulation of hepatitis B virus with optimal control analysis, Optim. Contr. Appl. Meth., 40 (2019), 529-544.    
  • 2. E. Bonyah, M. A. Khan, K. O. Okosun, et al. A theoretical model for Zika virus transmission, PLoS ONE, 12 (2017), 1-26.
  • 3. S. Ullah, M. A. Khan, M. Farooq, et al. Modeling and analysis of Tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan, Math. Comput. Simulat., 165 (2019), 181-199.    
  • 4. D. P. Ahokpossi, A. Atangana and D. P. Vermeulen, Modelling groundwater fractal flow with fractional differentiation via Mittag effler law, Eur. Phy. J. Plus, 132 (2017), 165.
  • 5. M. A. Khan, Y. Khan and S. Islam, Complex dynamics of an SEIR epidemic model with saturated incidence rate and treatment, Phy. A., 493 (2018), 210-227.    
  • 6. M. Caputo and M. Fabricio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.
  • 7. A. Atangana and D. Baleanu, New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model, Therm. Sci., 20 (2016), 763-769.    
  • 8. M. A. Khan, S. Ullah, M. Farhan, The dynamics of Zika virus with Caputo fractional derivative, AIMS Math., 4 (2019), 134-146.    
  • 9. S. Ullah, M. A. Khan, M. Farooq, A new fractional model for tuberculosis with relapse via Atangana-Baleanu derivative, Chaos, Solitons and Fractals, 116 (2018), 227-238.    
  • 10. S. Qureshi, A. Yusuf, Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu, Chaos, Solitons and Fractals, 122 (2019), 111-118.    
  • 11. S. Qureshi, E. Bonyah, A. A. Shaikh, Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data, Phy. A, 535 (2019), 1-22.
  • 12. Z. Wang, Y. K. Xie, J. Lu, et al. Stability and bifurcation of a delayed generalized fractional-order prey redator model with interspecific competition, Appl. Math. Comput., 347 (2019), 360-369.
  • 13. X. Wang, Z. Wang, X. Huang, et al. Dynamic Analysis of a Delayed Fractional-Order SIR Model with Saturated Incidence and Treatment Functions, Int. J. Bifurcat. Chaos, 28 (2018), 1850180.
  • 14. X. Wang, Z. Wang, H. Shen, Dynamical analysis of a discrete-time SIS epidemic model on complex networks, Appl. Math. Lett., 94 (2019), 292-299.    
  • 15. X. Wang, Z. Wang, J. Xia, Stability and bifurcation control of a delayed fractional-order ecoepidemiological model with incommensurate orders, J. Franklin I., 356 (2019), 8278-8295.    
  • 16. M. A. Khan, K. Shah, Y. Khan, et al. Mathematical modeling approach to the transmission dynamics of pine wilt disease with saturated incidence rate, Int. J. Biomath., 11 (2018), 1850035.
  • 17. F. Rao, P. S. Mandal and Y. Kang, Complicated endemics of an SIRS model with a generalized incidence under preventive vaccination and treatment controls, App. Math. Mod., 67 (2019), 38-61.    
  • 18. M. A. Khan, M. Farhan, S. Islam, et al. Modeling the transmission dynamics of avian influenza with saturation and psychological effect, Discrete and Continuous Dynamical Systems-S, 12 (2019), 455-474.    
  • 19. W. O. Kermack, A. G. MCKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721.    
  • 20. V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41-61.
  • 21. A. Korobeinikov and P. K Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.    
  • 22. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1999.
  • 23. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, 1993.
  • 24. H. Delavari, D. Baleanu, J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dyn., 67 (2012), 2433-2439.    
  • 25. C. Vargas-De-Leon, Volterra-type Lyapunov functions for fractional order epidemic systems, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 75-85.    
  • 26. J. Li, Y. Yang, Y. Xiao, et al. A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence, J. Appl. Anal. Comput., 6 (2016), 38-46.
  • 27. J. J. Wang, J. Z. Zhang and Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Analysis: Real World Applications, 11 (2010), 2390-2402.    
  • 28. M. A. Khan, Q. Badshah, S. Islam, et al. Global dynamics of SEIRS epidemic model with non-linear generalized incidences and preventive vaccination, Adv. Diff. Equ., 2015 (2015), 88.
  • 29. M. A. Khan, Y. Khan and S. Islam, Complex dynamics of an SEIR epidemic model with saturated incidence rate and treatment, Phy. A., 493 (2018), 210-227.    
  • 30. X. Liu and L. Yang, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Analysis: Real World Applications, 13 (2012), 2671-2679.    
  • 31. J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, The Journal of Animal Ecology, (1975), 331-341.
  • 32. D. L. DeAngelis, R. A. Goldstein and R. V. Oeill, A model for tropic interaction, Ecology, 56 (1975), 881-892.    
  • 33. K. Hattaf, M. Mahrouf, J. Adnani, et al. Qualitative analysis of a stochastic epidemic model with specific functional response cand temporary immunity, Phys. A, 490 (2018), 591-600.    
  • 34. Z. M. Odibat and N. T. Shawagfeh, Generalized taylor's formula, Appl. Math. Comput., 186 (2007), 286-293.
  • 35. W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appli., 332 (2007), 709-726.    
  • 36. M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 444.

 

This article has been cited by

  • 1. Wenbin Yang, Xiaozhou Feng, Shuhui Liang, Xiaojuan Wang, Asymptotic Behavior Analysis of a Fractional-Order Tumor-Immune Interaction Model with Immunotherapy, Complexity, 2020, 2020, 1, 10.1155/2020/7062957

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved