On the nonstandard numerical discretization of SIR epidemic model with a saturated incidence rate and vaccination

Agus Suryanto; Isnani Darti; Agus Suryanto; Isnani Darti

doi:10.3934/math.2021010

AIMS Mathematics

2021, Volume 6, Issue 1: 141-155. doi: 10.3934/math.2021010

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Research article

On the nonstandard numerical discretization of SIR epidemic model with a saturated incidence rate and vaccination

Agus Suryanto ,
Isnani Darti ^,

Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia

Received: 29 July 2020 Accepted: 14 September 2020 Published: 30 September 2020
MSC : 92D30, 37N25, 39A30

Recently, Hoang and Egbelowo (Boletin de la Sociedad Matemàtica Mexicana, 2020) proposed a nonstandard finite difference scheme (NSFD) to get a discrete SIR epidemic model with saturated incidence rate and constant vaccination. The discrete model was derived by discretizing the right-hand sides of the system locally and the first order derivative is approximated by the generalized forward difference method but with a restrictive denominator function. Their analysis showed that the NSFD scheme is dynamically-consistent only for relatively small time-step sizes. In this paper, we propose and analyze an alternative NSFD scheme by applying nonlocal approximation and choosing the denominator function such that the proposed scheme preserves the boundedness of solutions. It is verified that the proposed discrete model is dynamically-consistent with the corresponding continuous model for all time-step size. The analytical results have been confirmed by some numerical simulations. We also show numerically that the proposed NSFD scheme is superior to the Euler method and the NSFD method proposed by Hoang and Egbelowo (2020).
- SIR epidemic model,
- saturated incidence rate,
- local and global stability analysis,
- Lyapunov function,
- dynamically-consistent discretization
Citation: Agus Suryanto, Isnani Darti. On the nonstandard numerical discretization of SIR epidemic model with a saturated incidence rate and vaccination[J]. AIMS Mathematics, 2021, 6(1): 141-155. doi: 10.3934/math.2021010

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Abstract

Recently, Hoang and Egbelowo (Boletin de la Sociedad Matemàtica Mexicana, 2020) proposed a nonstandard finite difference scheme (NSFD) to get a discrete SIR epidemic model with saturated incidence rate and constant vaccination. The discrete model was derived by discretizing the right-hand sides of the system locally and the first order derivative is approximated by the generalized forward difference method but with a restrictive denominator function. Their analysis showed that the NSFD scheme is dynamically-consistent only for relatively small time-step sizes. In this paper, we propose and analyze an alternative NSFD scheme by applying nonlocal approximation and choosing the denominator function such that the proposed scheme preserves the boundedness of solutions. It is verified that the proposed discrete model is dynamically-consistent with the corresponding continuous model for all time-step size. The analytical results have been confirmed by some numerical simulations. We also show numerically that the proposed NSFD scheme is superior to the Euler method and the NSFD method proposed by Hoang and Egbelowo (2020).

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