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

Abstract

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.

Time-discrete SIR model for COVID-19 in Fiji

“…Various studies [3,8,9,[17][18][19][20][21] have mainly investigated continuous or explicit time-discrete schemes. In contrast, Wacker and Schlüter [22] have analysed the properties of the implicit timediscrete SIR model, including nonnegativity and boundedness of solution, global existence and uniqueness in time, monotonicity properties and error analysis. We employed the implicit timediscrete SIR model [22] for short-time prediction and to keep modelling as interpretable as possible.…”

“…The data includes the cumulative number of infected cases ( Î), the cumulative number of recovered cases ( R) and the cumulative number of death cases ( D) in Fiji. Following [22], we define R i = R i + D i and I i = I i − R i . We considered the COVID-19 data of Fiji for the second wave of the pandemic from 4 May 2021 (t = 1) to 9 December 2021 (t = 220) of which the data corresponding to time {t i } 160 i=1 is used for estimating the parameters of the discrete-time SIR model and the data for time {t i } 220…”

Gaussian Parameters Correlate with the Spread of COVID-19 Pandemic: The Italian Case

“…where α(t) and β(t) are defined as transmission (α(t)) and removing (i.e., death or recovery, β(t)) rates [33]. This system should be accompanied by suitable initial conditions S(0) > 0, I(0) > 0 and R(0) ≥ 0 [32].…”

Inverse problem for adaptive SIR model: Application to COVID-19 in Latin America

“…This model is used for predicting the number of reported and unreported cases for the COVID-19 epidemics in several countries. There is a number of recent publications on COVID-19 aiming to provide insight and understanding the trends of the disease ( Ajbar, Alqahtani, & Boumaza, 2021 ; Griette, Magal, & Seydi, 2020 ; Griette & Magal, 2021 ; Kucharski, Russell, & Diamond, 2020 ; Li et al., 2020 ; Lin et al., 2020 ; Lobo et al., 2020 ; Nishiura, Linton, & Akhmetzhanov, 2020 ; Pereira, Schimit, & Bezerra, 2021 ; Roosa et al., 2020 ; Shereen, Khan, Kazmi, Bashir, & Siddique, 2020 ; Wacker & Schlüter, 2020 ).…”