Fractional order SEIR model with generalized incidence rate

Muhammad Altaf Khan; Sajjad Ullah; Saif Ullah; Muhammad Farhan; Muhammad Altaf Khan; Sajjad Ullah; Saif Ullah; Muhammad Farhan

doi:10.3934/math.2020182

AIMS Mathematics

2020, Volume 5, Issue 4: 2843-2857. doi: 10.3934/math.2020182

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Research article

Fractional order SEIR 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

Received: 07 November 2019 Accepted: 09 March 2020 Published: 18 March 2020
MSC : 34A08, 37N30

The incidence rate function describes the mechanism of a disease transmission and has a key role in mathematical epidemiology. In the present paper, we develop a fractional SEIR epidemic model in the Caputo sense with generalized incidence function. Initially, we present the existence and positivity of the Caputo SEIR epidemic model and calculate the basic reproduction number. Further, we investigate the model equilibria and prove the detail stability analysis of the model. Finally, the numerical simulations are provided for various values of fractional order α and different incidence rates. From the numerical simulations we conclude that the order of the fractional derivative plays a significant role to provides more insights about the disease dynamics.
- Incidence rate function,
- SEIR model,
- Caputo fractional derivative,
- stability analysis
Citation: Muhammad Altaf Khan, Sajjad Ullah, Saif Ullah, Muhammad Farhan. Fractional order SEIR model with generalized incidence rate[J]. AIMS Mathematics, 2020, 5(4): 2843-2857. doi: 10.3934/math.2020182

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Abstract

The incidence rate function describes the mechanism of a disease transmission and has a key role in mathematical epidemiology. In the present paper, we develop a fractional SEIR epidemic model in the Caputo sense with generalized incidence function. Initially, we present the existence and positivity of the Caputo SEIR epidemic model and calculate the basic reproduction number. Further, we investigate the model equilibria and prove the detail stability analysis of the model. Finally, the numerical simulations are provided for various values of fractional order α and different incidence rates. From the numerical simulations we conclude that the order of the fractional derivative plays a significant role to provides more insights about the disease dynamics.

References

[1]	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.
[2]	S. Ullah, M. A. Khan, and J. F. Gomez-Aguilar, Mathematical formulation of hepatitis B virus with optimal control analysis, Optim. Cont. Appl. Meth., 40 (2019), 1-17. doi: 10.1002/oca.2459
[3]	F. B. Agusto and M. A. Khan, Optimal control strategies for dengue transmission in Pakistan, Math. Biosci., 305 (2018), 102-121. doi: 10.1016/j.mbs.2018.09.007
[4]	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.
[5]	F. Rao, P. S. Mandal and Y. Kang, Complicated endemics of an SIRS model with a generalized incidence under preventive vaccination and treatment controls, Appl. Math. Model., 67 (2019), 38-61. doi: 10.1016/j.apm.2018.10.016
[6]	W. O. Kermack, A. G. MCKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118
[7]	V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41-61.
[8]	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. doi: 10.3934/mbe.2004.1.57
[9]	A. Denes, G. Rost, Global stability for SIR and SIRS models with nonlinear incidence and removal terms via dulac functions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1101-1117. doi: 10.3934/dcdsb.2016.21.1101
[10]	D. P. Gao, N. J. Huang, S. M. Kang, et al. Global stability analysis of an SVEIR epidemic model with general incidence rate, Bound. Value Probl., 2018 (2018), 42.
[11]	D. P. Ahokpossi, A. Atangana and D. P. Vermeulen, Modelling groundwater fractal flow with fractional differentiation via Mittag-Leffler law, Eur. Phy. J. Plus, 132 (2017), 165.
[12]	S. Ullah, M. A. Khan and M. Farooq, A fractional model for the dynamics of TB virus, Cha. Solit. Frac., 116 (2018), 63-71. doi: 10.1016/j.chaos.2018.09.001
[13]	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. doi: 10.1016/j.physa.2017.10.038
[14]	Y. Khan, N. Faraz, A. Yildirim, et al. Fractional variational iteration method for fractional initialboundary value problems arising in the application of nonlinear science, Comput. Math. Appl., 62 (2011), 2273-2278. doi: 10.1016/j.camwa.2011.07.014
[15]	Y. Khan, Q. Wu, N. Faraz, et al. A new fractional analytical approach via a modified RiemannLiouville derivative, Appl. Math. Lett., 25 (2012), 1340-1346. doi: 10.1016/j.aml.2011.11.041
[16]	Y. Khan, N. Faraz, S. KUMARA, et al. A coupling method of homotopy perturbation and Laplace transformation for fractional models, U.P.B. Sci. Bull. Series A, 74 (2012), 57-68.
[17]	Y. X. Jun, D. Baleanu, Y. Khan, et al. Local Fractional Variational Iteration Method for Diffusion and Wave Equations on Cantor Sets, Rom. J. Phys., 59 (2014), 36-48.
[18]	I. Podlubny, Fractional Differential Equations, Academic Press, California, USA, 1999.
[19]	M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 73-85.
[20]	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. doi: 10.2298/TSCI160111018A
[21]	M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, et al. Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model, Phy. Rev. E, 95 (2017), 022409.
[22]	A. Mouaouine, A. Boukhouima, K. Hattaf, et al. A fractional order SIR epidemic model with nonlinear incidence rate, Adv. Diff. Eq., 2018 (2018), 1-9. doi: 10.1186/s13662-017-1452-3
[23]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, CRC, 1993.
[24]	H. Delavari, D. Baleanu and J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dyn., 67 (2012), 2433-2439. doi: 10.1007/s11071-011-0157-5
[25]	C. V. D. Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 75-85. doi: 10.1016/j.cnsns.2014.12.013
[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. doi: 10.1016/j.cam.2016.01.044
[27]	Z. M. Odibat and N. T. Shawagfeh, Generalized Taylors formula, Appl. Math. Comput., 186 (2007), 286-293.
[28]	W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709-726. doi: 10.1016/j.jmaa.2006.10.040
[29]	J. J. Wang, J. Z. Zhang and Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Anal-Real, 11 (2010), 2390-2402. doi: 10.1016/j.nonrwa.2009.07.012
[30]	X. Liu and L. Yang, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Anal-Real, 13 (2012), 2671-2679. doi: 10.1016/j.nonrwa.2012.03.010
[31]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-341. doi: 10.2307/3866
[32]	D. L. DeAngelis, R. A. Goldstein and R. V. ONeill, A model for tropic interaction, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298
[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. doi: 10.1016/j.physa.2017.08.043

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