Data assimilation is used to optimally fit a classical epidemiology model to the Johns Hopkins data of the Covid-19 pandemic. The optimisation is based on the confirmed cases and confirmed deaths. This is the only data available with reasonable accuracy. Infection and recovery rates can be infered from the model as well as the model parameters. The parameters can be linked with government actions or events like the end of the holiday season. Based on this numbers predictions for the future can be made and control targets specified. With other words: We look for a solution to a given model which fits the given data in an optimal sense. Having that solution, we have all parameters.
Since the start of the still ongoing COVID-19 pandemic, there have been many modeling efforts to assess several issues of importance to public health. In this work, we review the theory behind some important mathematical models that have been used to answer questions raised by the development of the pandemic. We start revisiting the basic properties of simple Kermack-McKendrick type models. Then, we discuss extensions of such models and important epidemiological quantities applied to investigate the role of heterogeneity in disease transmission e.g. mixing functions and superspreading events, …
In this article, we develop a data assimilation procedure to predict the evolution of epidemics with data uncertainty, with application to the Covid-19 pandemic. We construct a vademecum of solutions by solving the SIR epidemic model for a set of data neighboring the estimated real (or official) ones. A reduced basis is constructed from this vademecum through Proper Orthogonal Decomposition (POD). The reduced POD base is then applied to assimilate the pandemic data (infected, recovered, deceased) during the period in which data are known, by a least squares procedure. The fitted curves are the…
We consider the $SEIRS$ epidemiology model with such features of the COVID-19 outbreak as: abundance of unidentified infected individuals, limited time of immunity and a possibility of vaccination. Within a compartmental realization of this model, we found the disease-free and the endemic stationary states. They exist in their respective restricted regions of the le via linear stability analysis. The expression for the basic reproductive number is obtained as well. The positions and heights of a first peak for the fractions of infected individuals are obtained numerically and are fitted to sim…
Universidad de Sevilla（塞维利亚大学）；Universidad Loyola Andalucía（安达卢西亚洛约拉大学）
This document analyzes the role of data-driven methodologies in Covid-19 pandemic. We provide a SWOT analysis and a roadmap that goes from the access to data sources to the final decision-making step. We aim to review the available methodologies while anticipating the difficulties and challenges in the development of data-driven strategies to combat the Covid-19 pandemic. A 3M-analysis is presented: Monitoring, Modelling and Making decisions. The focus is on the potential of well-known datadriven schemes to address different challenges raised by the pandemic: i) monitoring and forecasting the …
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…
Epidemic models play a key role in understanding and responding to the emerging COVID-19 pandemic. Widely used compartmental models are static and are of limited use to evaluate intervention strategies with the emerging pandemic. Applying the technology of data assimilation, we propose a Bayesian updating approach for estimating epidemiological parameters using observable information for the purpose of assessing the impacts of different intervention strategies. We adopt a concise renewal model and propose new parameters by disentangling the reduction of instantaneous reproduction number Rt int…
We provide an overview of the methods that can be used for prediction under uncertainty and data fitting of dynamical systems, and of the fundamental challenges that arise in this context. The focus is on SIR-like models, that are being commonly used when attempting to predict the trend of the COVID-19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIR-like models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it non-trivial to use these models for meaningful predictions. Most of…
A variety of approaches using compartmental models have been used to study the COVID-19 pandemic and the usage of machine learning methods with these models has had particularly notable success. We present here an approach toward analyzing accessible data on Covid-19's U.S. development using a variation of the "Physics Informed Neural Networks" (PINN) which is capable of using the knowledge of the model to aid learning. We illustrate the challenges of using the standard PINN approach, then how with appropriate and novel modifications to the loss function the network can perform well even in ou…
In this paper, we introduce a new control-theoretic paradigm for mitigating the spread of a virus. To this end, our discrete-time controller, aims to reduce the number of new daily deaths, and consequently, the cumulative number of deaths. In contrast to much of the existing literature, we do not rely on a potentially complex virus transmission model whose equations must be customized to the "particulars" of the pandemic at hand. For new viruses such as Covid-19, the epidemiology driving the modelling process may not be well known and model estimation with limited data may be unreliable. With …
This paper extends the canonical model of epidemiology, SIRD model, to allow for time varying parameters for real-time measurement of the stance of the COVID-19 pandemic. Time variation in model parameters is captured using the generalized autoregressive score modelling structure designed for the typically daily count data related to pandemic. The resulting specification permits a flexible yet parsimonious model structure with a very low computational cost. This is especially crucial at the onset of the pandemic when the data is scarce and the uncertainty is abundant. Full sample results show …
This survey analyses the role of data-driven methodologies for pandemic modelling and control. We provide a roadmap from the access to epidemiological data sources to the control of epidemic phenomena. We review the available methodologies and discuss the challenges in the development of data-driven strategies to combat the spreading of infectious diseases. Our aim is to bring together several different disciplines required to provide a holistic approach to epidemic analysis, such as data science, epidemiology, and systems-and-control theory. A 3M-analysis is presented, whose three pillars are…
We demonstrate the ability of statistical data assimilation to identify the measurements required for accurate state and parameter estimation in an epidemiological model for the novel coronavirus disease COVID-19. Our context is an effort to inform policy regarding social behavior, to mitigate strain on hospital capacity. The model unknowns are taken to be: the time-varying transmission rate, the fraction of exposed cases that require hospitalization, and the time-varying detection probabilities of new asymptomatic and symptomatic cases. In simulations, we obtain accurate estimates of undetect…
Compartmental models are an important quantitative tool in epidemiology, enabling us to forecast the course of a communicable disease. However, the model parameters, such as the infectivity rate of the disease, are riddled with uncertainties, which has motivated the development and use of stochastic compartmental models. Here, we first show that a common stochastic model, which treats the uncertainties as white noise, is fundamentally flawed since it erroneously implies that greater parameter uncertainties will lead to the eradication of the disease. Then, we present a principled modeling of t…
Rule-based models have been successfully used to represent different aspects of the COVID-19 pandemic, including age, testing, hospitalisation, lockdowns, immunity, infectivity, behaviour, mobility and vaccination of individuals. These rule-based approaches are motivated by chemical reaction rules which are traditionally solved numerically with the standard Gillespie algorithm proposed in the context of molecular dynamics. When applying reaction system type of approaches to epidemiology, generalisations of the Gillespie algorithm are required due to the time-dependency of the problems. In this…
Epidemiological models with constant parameters may not capture satisfactory infection patterns in the presence of pharmaceutical and non-pharmaceutical mitigation measures during a pandemic, since infectiousness is a function of time. In this paper, an Epidemiology-Informed Neural Network algorithm is introduced to learn the time-varying transmission rate for the COVID-19 pandemic in the presence of various mitigation scenarios. There are asymptomatic infectives, mostly unreported, and the proposed algorithm learns the proportion of the total infective individuals that are asymptomatic infect…
In a recent paper (arXiv:2003.12055), Singh and Adhikari present results of an analysis of a mathematical model of epidemiology based on which they argue that a $49$ day lockdown is required in India for containing the pandemic in India. We assert that as a model study with the stated assumptions, the analysis presented in the paper is perfectly valid, however, any serious comparison with real data and attempts at prediction from the model are highly problematic. The main point of the present paper is to convincingly establish this assertion while providing a warning that the results and analy…
In the case of SARS-CoV-2 pandemic management, wastewater-based epidemiology aims to derive information on the infection dynamics by monitoring virus concentrations in the wastewater. However, due to the intrinsic random fluctuations of the viral signal in the wastewater (due to e.g., dilution; transport and fate processes in sewer system; variation in the number of persons discharging; variations in virus excretion and water consumption per day) the subsequent prevalence analysis may result in misleading conclusions. It is thus helpful to apply data filtering techniques to reduce the noise in…
During the early stage of the COVID-19 pandemic, many countries implemented non-pharmaceutical interventions (NPIs) to control the transmission of SARS-CoV-2, the causative pathogen of COVID-19. Among those NPIs, quarantine measures were widely adopted and enforced through stay-at-home and shelter-in-place orders. Understanding the effectiveness of quarantine measures can inform decision-making and control planning during the ongoing COVID-19 pandemic and for future disease outbreaks. In this study, we use mathematical models to evaluate the impact of quarantine measures on COVID-19 spread in …
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 …
We are asking: why are the dynamics and control of Covid-19 most interesting for mathematicians and why are mathematicians urgently needed for controlling the pandemic? First we present our comments in a Bottom-up approach, i.e., following the events from their beginning as they evolved through time. They happened differently in different countries, and the main objective of the first part is to compare these evolutions in a few selected countries with each other. The second part of the article is not "country-oriented" but "problem-oriented". From a given problem we go Top-down to its solutio…
