The effective reproduction number is a key figure to monitor the course of the COVID-19 pandemic. In this study we consider a retrospective modelling approach for estimating the effective reproduction number based on death counts during the first year of the pandemic in Germany. The proposed Bayesian hierarchical model incorporates splines to estimate reproduction numbers flexibly over time while adjusting for varying effective infection fatality rates. The approach also provides estimates of dark figures regarding undetected infections over time. Results for Germany illustrate that estimated reproduction numbers based on death counts are often similar to classical estimates based on confirmed cases. However, considering death counts proves to be more robust against shifts in testing policies: during the second wave of infections, classical estimation of the reproduction number suggests a flattening/ decreasing trend of infections following the "lockdown light" in November 2020, while our results indicate that true numbers of infections continued to rise until the "second lockdown" in December 2020. This observation is associated with more stringent testing criteria introduced concurrently with the "lockdown light", which is reflected in subsequently increasing dark figures of infections estimated by our model. These findings illustrate that the retrospective viewpoint can provide additional insights regarding the course of the pandemic. In light of progressive vaccinations, shifting the focus from modelling confirmed cases to reported deaths with the possibility to incorporate effective infection fatality rates might be of increasing relevance for the future surveillance of the pandemic.
Heidelberg University（海德堡大学）；Chinese Academy of Medical Sciences and Peking Union Medical College（中国医学科学院与北京协和医学院）；University Hospital Bonn（波恩大学医院）；University Hospital Heidelberg（海德堡大学医院）；Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）；Africa Health Research Institute（非洲健康研究所）
Mathematical models in epidemiology are an indispensable tool to determine the dynamics and important characteristics of infectious diseases. Apart from their scientific merit, these models are often used to inform political decisions and intervention measures during an ongoing outbreak. However, reliably inferring the dynamics of ongoing outbreaks by connecting complex models to real data is still hard and requires either laborious manual parameter fitting or expensive optimization methods which have to be repeated from scratch for every application of a given model. In this work, we address …
The infection fatality rate (IFR) of the Coronavirus Disease 2019 (COVID-19) is one of the most discussed figures in the context of this pandemic. Using German COVID-19 surveillance data and age-group specific IFR estimates from multiple international studies, this work investigates time-dependent variations in effective IFR over the course of the pandemic. Three different methods for estimating (effective) IFRs are presented: (a) population-averaged IFRs based on the assumption that the infection risk is independent of age and time, (b) effective IFRs based on the assumption that the age dist…
We develop a fully automatic Bayesian Lasso via variational inference. This is a scalable procedure for approximating the posterior distribution. Special attention is driven to the knot selection in regression spline. In order to carry through our proposal, a full automatic variational Bayesian Lasso, a Jefferey's prior is proposed for the hyperparameters and a decision theoretical approach is introduced to decide if a knot is selected or not. Extensive simulation studies were developed to ensure the effectiveness of the proposed algorithms. The performance of the algorithms were also tested i…
The COVID-19 pandemic has caused severe public health consequences in the United States. In this study, we use a hierarchical Bayesian model to estimate the age-specific COVID-19 attributable deaths over time in the United States. The model is specified by a novel non-parametric spatial approach, a low-rank Gaussian Process (GP) projected by regularised B-splines. We show that this projection defines a new GP with attractive smoothness and computational efficiency properties, derive its kernel function, and discuss the penalty terms induced by the projected GP. Simulation analyses and benchmar…
We analyse the temporal and regional structure in mortality rates related to COVID-19 infections. We relate the fatality date of each deceased patient to the corresponding day of registration of the infection, leading to a nowcasting model which allows us to estimate the number of present-day infections that will, at a later date, prove to be fatal. The numbers are broken down to the district level in Germany. Given that death counts generally provide more reliable information on the spread of the disease compared to infection counts, which inevitably depend on testing strategy and capacity, t…
In many applications that involve the inference of an unknown smooth function, the inference of its derivatives will often be just as important as that of the function itself. To make joint inferences of the function and its derivatives, a class of Gaussian processes called $p^{\text{th}}$ order Integrated Wiener's Process (IWP), is considered. Methods for constructing a finite element (FEM) approximation of an IWP exist but have focused only on the order $p = 2$ case which does not allow appropriate inference for derivatives, and their computational feasibility relies on additional approximat…
Widespread opinions and discussion exist regarding the efficiency of social distancing after crucial spread of the SARS-CoV-2 virus during the actual Covid-19 pandemic. While Germany has released a federal law that prohibits any type of direct contact for more than 2 people other countries including the US released curfews. People are now wondering whether these measures are helpful to stop or hamper the Covid-19 pandemic and to limit the spread of the new corona virus. A quantitative statement on this question depends on many parameters that are difficult to grasp mathematically and cannot th…
Currently, due to the COVID-19 pandemic the public life in most European countries stopped almost completely due to measures against the spread of the virus. Efforts to limit the number of new infections are threatened by the advent of new variants of the SARS-COV-2 virus, most prominent the B.1.1.7 strain with higher infectivity. In this article we consider a basic two-strain SIR model to explain the spread of those variants in Germany on small time scales. For a linearized version of the model we calculate relevant variables like the time of minimal infections or the dynamics of the share of…
We generalise the epidemic Renormalisation Group framework while connecting it to a SIR model with time-dependent coefficients. We then confront the model with COVID-19 in Denmark, Germany, Italy and France and show that the approach works rather well in reproducing the data. We also show that a better understanding of the time dependence of the recovery rate would require extending the model to take into account the number of deaths whenever these are over 15% of the total number of infected cases.
We are currently facing a highly critical case of a world-wide pandemic. The novel coronavirus (SARS-CoV-2, a.k.a. COVID-19) has proved to be extremely contagious and the original outbreak from Asia has now spread to all continents. This situation will fruitfully profit from the study in regards of the spread of the virus, assessing effective countermeasures to weight the impact of the adopted strategies. The standard Susceptible-Infectious-Recovered (SIR) model is a very successful and widely used mathematical model for predicting the spread of an epidemic. We adopt the SIR model on a random …
Diffusion comes anytime and everywhere. If there is a gradient or a potential difference of a quantity a diffusion process happens and this ends if an equilibrium is reached only. The concentration of a species maybe such quantity, or the voltage. An electric currant will be driven by a voltage difference for example. In this COVID-19 pandemic one observes both regions with low incidence and other ones with high incidence. The local different people density could be a reason for that. In populous areas like big cities or congested urban areas higher COVID-19 incidences could be observed than i…
This is work in progress. We make it accessible hoping that people might find the idea useful. We propose a discrete, recursive 5-compartment model for the spread of epidemics, which we call {\em SEPIR-model}. Under mild assumptions which typically are fulfilled for the Covid-19 pandemic it can be used to reproduce the development of an epidemic from a small number of parameters closely related to the data. We demonstrate this at the development in Germany and Switzerland. It also allows model predictions assuming nearly constant reproduction numbers. Thus it might be a useful tool for sheddin…
Vaccines are the most powerful pharmaceutical tool to combat the COVID-19 pandemic. While the majority (about 65%) of the German population were fully vaccinated, incidence started growing exponentially in October 2021 with about 41% of recorded new cases aged twelve or above being symptomatic breakthrough infections, presumably also contributing to the dynamics. At the time, it (i) remains elusive how significant this contribution is and (ii) whether targeted non-pharmaceutical interventions (NPIs) may stop the amplification of the ongoing crisis. Here, we estimate that about 67%-76% of all n…
A plethora of prediction models of SARS-CoV-2 pandemic were proposed in the past. Prediction performances not only depend on the structure and features of the model, but also on its parametrization. Official databases are often biased due to lag in reporting of cases, changing testing policy or incompleteness of data. Moreover, model parametrization is time-dependent e.g. due to changing age-structures, new emerging virus variants, non-pharmaceutical interventions and ongoing vaccination programs. To cover these aspects, we develop a principled approach to parametrize SIR-type epidemiologic mo…
We have proposed a new form of growth rate for population ecology. Generally, the growth rate is dependent on the size of the population at that particular epoch. We have introduced an alternative time-dependent form of growth rate. This form satisfies essential conditions to represent population growth and can be an alternative form for growth models to analyze population dynamics. We have employed the generalized Richards model as a guideline to compare our results. Further, we have applied our model in the case of epidemics. To check the efficacy of our model, we have verified the 2003 SARS…
Several analytical models have been used in this work to describe the evolution of death cases arising from coronavirus (COVID-19). The Death or `D' model is a simplified version of the SIR (susceptible-infected-recovered) model, which assumes no recovery over time, and allows for the transmission-dynamics equations to be solved analytically. The D-model can be extended to describe various focuses of infection, which may account for the original pandemic (D1), the lockdown (D2) and other effects (Dn). The evolution of the COVID-19 pandemic in several countries (China, Spain, Italy, France, UK,…
We here propose to model active and cumulative cases data from COVID-19 by a continuous effective model based on a modified diffusion equation under Lifshitz scaling with a dynamic diffusion coefficient. The proposed model is rich enough to capture different aspects of a complex virus diffusion as humanity has been recently facing. The model being continuous it is bound to be solved analytically and/or numerically. So, we investigate two possible models where the diffusion coefficient associated with possible types of contamination are captured by some specific profiles. The active cases curve…
This work evaluates the severity of COVID-19 Omicron variant in terms of the case-fatality-rates (CFR) with respective uncertainty intervals via a simultaneous fitting of confirmed cases and deaths in the USA, India, Brazil, France, Germany and United Kingdom. The CFRs were calculated combining Monte Carlo simulations and analytical methods based on Gompertz functions under the framework of the Least Square Method and assuming that the deaths can be described by a convolution of the confirmed cases with a common gamma function to describe the case to death period. Linear backgrounds both for c…
In this work, we adapt the epidemiological SIR model to study the evolution of the dissemination of COVID-19 in Germany and Brazil (nationally, in the State of Paraiba, and in the City of Campina Grande). We prove the well posedness and the continuous dependence of the model dynamics on its parameters. We also propose a simple probabilistic method for the evolution of the active cases that is instrumental for the automatic estimation of parameters of the epidemiological model. We obtained statistical estimates of the active cases based the probabilistic method and on the confirmed cases data. …
Background: The COVID-19 pandemic has had a profound impact on health, everyday life and economics around the world. An important complication that can arise in connection with a COVID-19 infection is acute kidney injury. A recent observational cohort study of COVID-19 patients treated at multiple sites of a tertiary care center in Berlin, Germany identified risk factors for the development of (severe) acute kidney injury. Since inferring results from a single study can be tricky, we validate these findings and potentially adjust results by including external information from other studies on …
We study the impact on the epidemiological dynamics of a class of restrictive measures that are aimed at reducing the number of contacts of individuals who have a higher risk of being infected with a transmittable disease. Such measures are currently either implemented or at least discussed in numerous countries worldwide to ward off a potential new wave of COVID-19 across Europe. They come in the form of Health Passes (HP), which grant full access to public life only to individuals with a certificate that proves that they have either been fully vaccinated, have recovered from a previous infec…
In order to slow the spread of the CoViD-19 pandemic, governments around the world have enacted a wide set of policies limiting the transmission of the disease. Initially, these focused on non-pharmaceutical interventions; more recently, vaccinations and large-scale rapid testing have started to play a major role. The objective of this study is to explain the quantitative effects of these policies on determining the course of the pandemic, allowing for factors like seasonality or virus strains with different transmission profiles. To do so, the study develops an agent-based simulation model, w…
A great variety of complex physical, natural and artificial systems are governed by statistical distributions, which often follow a standard exponential function in the bulk, while their tail obeys the Pareto power law. The recently introduced $κ$-statistics framework predicts distribution functions with this feature. A growing number of applications in different fields of investigation are beginning to prove the relevance and effectiveness of $κ$-statistics in fitting empirical data. In this paper, we use $κ$-statistics to formulate a statistical approach for epidemiological analysis. We vali…
Viral outbreaks, such as the current COVID-19 pandemic, are commonly described by compartmental models by means of ordinary differential equation (ODE) systems. The parameter values of these ODE models are typically unknown and need to be estimated based on accessible data. In order to describe realistic pandemic scenarios with strongly varying situations, these model parameters need to be assumed as time-dependent. While parameter estimation for the typical case of time-constant parameters does not pose larger issues, the determination of time-dependent parameters, e.g.~the transition rates o…
Understanding how widely COVID-19 has spread is critical for examining the pandemic's progression. Despite efforts to carefully monitor the pandemic, the number of confirmed cases may underestimate the total number of infections. We introduce a demographic scaling model to estimate COVID-19 infections using an broadly applicable approach that is based on minimal data requirements: COVID-19 related deaths, infection fatality rates (IFRs), and life tables. As many countries lack reliable estimates of age-specific IFRs, we scale IFRs between countries using remaining life expectancy as a marker t…
This paper develops an individual-based stochastic network SIR model for the empirical analysis of the Covid-19 pandemic. It derives moment conditions for the number of infected and active cases for single as well as multigroup epidemic models. These moment conditions are used to investigate the identification and estimation of the transmission rates. The paper then proposes a method that jointly estimates the transmission rate and the magnitude of under-reporting of infected cases. Empirical evidence on six European countries matches the simulated outcomes once the under-reporting of infected…
One major bottleneck in the ongoing COVID-19 pandemic is the limited number of critical care beds. Due to the dynamic development of infections and the time lag between when patients are infected and when a proportion of them enters an intensive care unit (ICU), the need for future intensive care can easily be underestimated. To infer future ICU load from reported infections, we suggest a simple statistical model that (1) accounts for time lags and (2) allows for making predictions depending on different future growth of infections. We have evaluated our model for three regions, namely Berlin …
Since the beginning of the corona pandemic in March 2020, various parameters for describing the spread of the disease have been specified for Germany in addition to the daily infection figures (new infections and total infections), which are also used for political decisions. In addition to excess mortality and the weekly incidence, these include the doubling time $T_2$ and the reproduction number $R_t$. For the latter, various estimates can be found on the website of the Robert-Koch-Institute, see \cite{EstR:RKI}, which are calculated from the case numbers for all of Germany; local difference…
Background: Rapid testing for an infection is paramount during a pandemic to prevent continued viral spread and excess morbidity and mortality. This study aimed to determine whether alternative testing strategies based on sample pooling can increase the speed and throughput of screening for SARS-CoV-2. Methods: A mathematical modelling approach was chosen to simulate six different testing strategies based on key input parameters (infection rate, test characteristics, population size, testing capacity etc.). The situations in five countries (US, DE, UK, IT and SG) currently experiencing COVID-1…
This work provides an overview on deterministic and stochastic models that have previously been proposed by us to study the transmission dynamics of the Coronavirus Disease 2019 (COVID-19) in Europe and USA. Briefly, we describe realistic deterministic and stochastic models for the evolution of the COVID-19 pandemic, subject to the lockdown and quarantine measures, which take into account the time-delay for recovery or death processes. Realistic dynamic equations for the entire process have been derived by adopting the so-called "kinetic-type reactions approach". The lockdown and the quarantin…
We consider the pandemic spreading of COVID-19 for some selected countries after the outbreak of the coronavirus in Wuhan City, China. We estimated the infection rate and the initial infected individuals of COVID-19 by using the officially reported data at the early stage of the epidemic for the susceptible (S), infectable (I), quarantined (Q), and the cofirmed recovered (Rk) population model, so called SIQRk model. In the reported data we know the quarantined cases and the recovered cases. We can not know the recovered cases from the asymptomatic cases. In the SIQRk model we can estimated the…
An epidemiological model is developed for the spread of COVID-19 in South Africa. A variant of the classical compartmental SEIR model, called the SEIQRDP model, is used. As South Africa is still in the early phases of the global COVID-19 pandemic with the confirmed infectious cases not having peaked, the SEIQRDP model is first parameterized on data for Germany, Italy, and South Korea - countries for which the number of infectious cases are well past their peaks. Good fits are achieved with reasonable predictions of where the number of COVID-19 confirmed cases, deaths, and recovered cases will …
We investigate adaptive strategies to robustly and optimally control the COVID-19 pandemic via social distancing measures based on the example of Germany. Our goal is to minimize the number of fatalities over the course of two years without inducing excessive social costs. We consider a tailored model of the German COVID-19 outbreak with different parameter sets to design and validate our approach. Our analysis reveals that an open-loop optimal control policy can significantly decrease the number of fatalities when compared to simpler policies under the assumption of exact model knowledge. In …
