A nonstandard finite difference scheme for the SVICDR model to predict COVID-19 dynamics

Sarah Treibert; Helmut Brunner; Matthias Ehrhardt; Sarah Treibert; Helmut Brunner; Matthias Ehrhardt

doi:10.3934/mbe.2022056

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 2: 1213-1238. doi: 10.3934/mbe.2022056

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A nonstandard finite difference scheme for the SVICDR model to predict COVID-19 dynamics

1.
Chair of Applied Mathematics and Numerical Analysis, School of Mathematics and Natural Sciences, University of Wuppertal, Wuppertal 42119, Germany
2.
Chair of Health Economics, Faculty of Management and Economics, University of Wuppertal, Wuppertal 42119, Germany

Academic Editor: Salihu S. Musa

Received: 15 September 2021 Accepted: 27 October 2021 Published: 01 December 2021

In the context of 2019 coronavirus disease (COVID-19), considerable attention has been paid to mathematical models for predicting country- or region-specific future pandemic developments. In this work, we developed an SVICDR model that includes a susceptible, an all-or-nothing vaccinated, an infected, an intensive care, a deceased, and a recovered compartment. It is based on the susceptible-infectious-recovered (SIR) model of Kermack and McKendrick, which is based on ordinary differential equations (ODEs). The main objective is to show the impact of parameter boundary modifications on the predicted incidence rate, taking into account recent data on Germany in the pandemic, an exponential increasing vaccination rate in the considered time window and trigonometric contact and quarantine rate functions. For the numerical solution of the ODE systems a model-specific non-standard finite difference (NSFD) scheme is designed, that preserves the positivity of solutions and yields the correct asymptotic behaviour.
- nonstandard finite difference scheme,
- COVID-19,
- SARS-CoV-2,
- compartment models,
- epidemiology
Citation: Sarah Treibert, Helmut Brunner, Matthias Ehrhardt. A nonstandard finite difference scheme for the SVICDR model to predict COVID-19 dynamics[J]. Mathematical Biosciences and Engineering, 2022, 19(2): 1213-1238. doi: 10.3934/mbe.2022056

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Abstract

In the context of 2019 coronavirus disease (COVID-19), considerable attention has been paid to mathematical models for predicting country- or region-specific future pandemic developments. In this work, we developed an SVICDR model that includes a susceptible, an all-or-nothing vaccinated, an infected, an intensive care, a deceased, and a recovered compartment. It is based on the susceptible-infectious-recovered (SIR) model of Kermack and McKendrick, which is based on ordinary differential equations (ODEs). The main objective is to show the impact of parameter boundary modifications on the predicted incidence rate, taking into account recent data on Germany in the pandemic, an exponential increasing vaccination rate in the considered time window and trigonometric contact and quarantine rate functions. For the numerical solution of the ODE systems a model-specific non-standard finite difference (NSFD) scheme is designed, that preserves the positivity of solutions and yields the correct asymptotic behaviour.

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