Uncertainty quantification in epidemiological models for the COVID-19 pandemic

Abstract

Mathematical modeling of epidemiological diseases using differential equations are of great importance in order to recognize the characteristics of the diseases and their outbreak. The procedure of modeling consists of two essential components: the first component is to solve the mathematical model numerically, the so-called forward modeling. The second component is to identify the unknown parameter values in the model, which is known as inverse modeling and leads to identifying the epidemiological model more precisely. The main goal of this paper is to develop the forward and inverse modeling of the coronavirus (COVID-19) pandemic using novel computational methodologies in order to accurately estimate and predict the pandemic. This leads to governmental decisions support in implementing effective protective measures and prevention of new outbreaks. To this end, we use the logistic equation and the SIR (susceptible-infected-removed) system of ordinary differential equations to model the spread of the COVID-19 pandemic. For the inverse modeling, we propose Bayesian inversion techniques, which are robust and reliable approaches, in order to estimate the unknown parameters of the epidemiological models. We deploy an adaptive Markov-chain Monte-Carlo (MCMC) algorithm for the estimation of a posteriori probability distribution and confidence intervals for the unknown model parameters as well as for the reproduction number. We perform our analyses on the publicly available data for Austria to estimate the main epidemiological model parameters and to study the effectiveness of the protective measures by the Austrian government. The estimated parameters and the analysis of fatalities provide useful information for decision makers and makes it possible to perform more realistic forecasts of future outbreaks. According to our Bayesian analysis for the logistic model, the growth rate and the carrying capacity are estimated respectively as 0.28 and . Moreover for the parameters of the SIR model, namely the transmission rate and recovery rate, we estimate 0.36 and 0.06, respectively. Additionally, we obtained an average infectious period of 17 days and a transmission period of 3 days for COVID-19 in Austria. We also estimate the reproduction number over time for Austria. This quantity is estimated around 3 on March 26, when the first recovery was reported. Then it decays to 1 at the beginning of April. Furthermore, we present a fatality analysis for COVID-19 in Austria, which is also of importance for governmental protective decision making. According to our analysis, the case fatality rate (CFR) is estimated as and a prediction of the number of fatalities for the coming 10 days is also presented. Additionally, the ICU bed usage in Austria indicates that around of the active infected individuals are critical cases and require ICU beds. Therefore, if Austrian governmental protective measures would not have taken place and for instance if the number of active infected cases would ...

A Modified SIRD Model to Study the Evolution of the COVID-19 Pandemic in Spain

“…The precision of the estimated parameters is determined by the theoretical reasoning of Sections 2.2 and 2.3, and by the good behavior of the model against the observed data. In this sense, some studies [20,21] using Bayesian techniques, show a similar behavior in the estimates.…”

“…In addition to the works already considered about SIRD models, interested readers can find a wide variety of models for the study of the COVID-19 pandemic in the scientific literature published in the last months: using Bayesian and stochastic techniques [20][21][22], including mobility [23], confinement and quarantine [15,24], fractional models [25], and logistic models [26], among others.…”

Framework for enhancing the estimation of model parameters for data with a high level of uncertainty

“…The remaining parameters of the model are assumed to be known biological parameters. In this case, the incubation period is [46], the proportion of symptomatic infected individuals is ρ = 0.6 [10], and both the recovery rate of infected and positively diagnosed individuals are defined as γ I = γ P = 1 / 16.7 day −1 [47].…”

SUIHTER : a new mathematical model for COVID-19. Application to the analysis of the second epidemic outbreak in Italy

“…Calibration of epidemiological models has already been performed in a Bayesian framework, following the pioneering paper by O'Neill & Roberts [21], for several infectious diseases [22][23][24]. In the case of the COVID-19 epidemic, Bayesian inference has been performed using simpler SIR [25,26], meta-community SEIR-like [2,4,13,27,28] and SEIAR [7] models, in the last case aiming at estimating nine parameters-including a dynamic, time-dependent contact rate β(t)-during the first outbreak of the COVID-19 epidemic. In addition to model calibration, our analysis also provides a numerical assessment of the predictive capability of the model, in forecasting with adequate advance notice the occurrence of a peak for the most relevant compartments.…”