Fractional order SIR model with generalized incidence rate

Muhammad Altaf Khan; Muhammad Ismail; Saif Ullah; Muhammad Farhan; Muhammad Altaf Khan; Muhammad Ismail; Saif Ullah; Muhammad Farhan

doi:10.3934/math.2020124

AIMS Mathematics

2020, Volume 5, Issue 3: 1856-1880. doi: 10.3934/math.2020124

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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

Received: 28 October 2019 Accepted: 07 February 2020 Published: 18 February 2020
MSC : 34A08, 37N30

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

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Abstract

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.

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