A nonstandard finite difference scheme for a time-fractional model of Zika virus transmission

Maghnia Hamou Maamar; Matthias Ehrhardt; Louiza Tabharit; Maghnia Hamou Maamar; Matthias Ehrhardt; Louiza Tabharit

doi:10.3934/mbe.2024039

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 1: 924-962. doi: 10.3934/mbe.2024039

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A nonstandard finite difference scheme for a time-fractional model of Zika virus transmission

1.
Department of Mathematics and Computer Science, Abdelhamid Ibn Badis University, Algeria
2.
Chair of Applied and Computational Mathematics, University of Wuppertal, Germany

Academic Editor: Sen Pei

Received: 05 November 2023 Revised: 01 December 2023 Accepted: 05 December 2023 Published: 21 December 2023

In this work, we investigate the transmission dynamics of the Zika virus, considering both a compartmental model involving humans and mosquitoes and an extended model that introduces a non-human primate (monkey) as a second reservoir host. The novelty of our approach lies in the later generalization of the model using a fractional time derivative. The significance of this study is underscored by its contribution to understanding the complex dynamics of Zika virus transmission. Unlike previous studies, we incorporate a non-human primate reservoir host into the model, providing a more comprehensive representation of the disease spread. Our results reveal the importance of utilizing a nonstandard finite difference (NSFD) scheme to simulate the disease's dynamics accurately. This NSFD scheme ensures the positivity of the solution and captures the correct asymptotic behavior, addressing a crucial limitation of standard solvers like the Runge-Kutta Fehlberg method (ode45). The numerical simulations vividly demonstrate the advantages of our approach, particularly in terms of positivity preservation, offering a more reliable depiction of Zika virus transmission dynamics. From these findings, we draw the conclusion that considering a non-human primate reservoir host and employing an NSFD scheme significantly enhances the accuracy and reliability of modeling Zika virus transmission. Researchers and policymakers can use these insights to develop more effective strategies for disease control and prevention.
- nonstandard finite difference scheme,
- positivity,
- Zika virus,
- compartment models,
- time-fractional models,
- epidemiology,
- SEIR model,
- human-vector models
Citation: Maghnia Hamou Maamar, Matthias Ehrhardt, Louiza Tabharit. A nonstandard finite difference scheme for a time-fractional model of Zika virus transmission[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 924-962. doi: 10.3934/mbe.2024039

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Abstract

In this work, we investigate the transmission dynamics of the Zika virus, considering both a compartmental model involving humans and mosquitoes and an extended model that introduces a non-human primate (monkey) as a second reservoir host. The novelty of our approach lies in the later generalization of the model using a fractional time derivative. The significance of this study is underscored by its contribution to understanding the complex dynamics of Zika virus transmission. Unlike previous studies, we incorporate a non-human primate reservoir host into the model, providing a more comprehensive representation of the disease spread. Our results reveal the importance of utilizing a nonstandard finite difference (NSFD) scheme to simulate the disease's dynamics accurately. This NSFD scheme ensures the positivity of the solution and captures the correct asymptotic behavior, addressing a crucial limitation of standard solvers like the Runge-Kutta Fehlberg method (ode45). The numerical simulations vividly demonstrate the advantages of our approach, particularly in terms of positivity preservation, offering a more reliable depiction of Zika virus transmission dynamics. From these findings, we draw the conclusion that considering a non-human primate reservoir host and employing an NSFD scheme significantly enhances the accuracy and reliability of modeling Zika virus transmission. Researchers and policymakers can use these insights to develop more effective strategies for disease control and prevention.

References

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