Time-continuous and time-discrete SIR models revisited: theory and applications

Advances in Difference Equations

Benjamin Wacker¹  Jan Schlüter² 

[1] Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077, Göttingen, Germany;Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077, Göttingen, Germany;Institute for Dynamics of Complex Fluids, Faculty of Physics, Georg-August-University of Göttingen, Friedrich-Hund-Platz 1, D-37077, Göttingen, Germany;

关键词: COVID-19;    Difference equations;    Existence and uniqueness;    Mathematical epidemiology;    Numerical analysis;    Nonlinear ordinary differential equations;    SIR model;    Well-posedness;   

DOI  :  10.1186/s13662-020-02995-1

来源: Springer

【 摘 要 】

Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. As our main goal, we establish an implicit time-discrete SIR (susceptible people–infectious people–recovered people) model. For this purpose, we first introduce its continuous variant with time-varying transmission and recovery rates and, as our first contribution, discuss thoroughly its properties. With respect to these results, we develop different possible time-discrete SIR models, we derive our implicit time-discrete SIR model in contrast to many other works which mainly investigate explicit time-discrete schemes and, as our main contribution, show unique solvability and further desirable properties compared to its continuous version. We thoroughly show that many of the desired properties of the time-continuous case are still valid in the time-discrete implicit case. Especially, we prove an upper error bound for our time-discrete implicit numerical scheme. Finally, we apply our proposed time-discrete SIR model to currently available data regarding the spread of COVID-19 in Germany and Iran.

【 授权许可】

CC BY