An epidemiological compartmental model with automated parameter estimation and forecasting of the spread of COVID-19 with analysis of data from Germany and Brazil
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. From this estimated time series we obtained a time-dependent contagion rate, which reflects a lower or higher adherence to social distancing by the involved populations. By also analysing the data on daily deaths, we obtained the daily lethality and recovery rates. We then integrate the equations of motion of the model using these time-dependent parameters. We validate our epidemiological model by fitting the official data of confirmed, recovered, death, and active cases due to the pandemic with the theoretical predictions. We obtained very good fits of the data with this method. The automated procedure developed here could be used for basically any population with a minimum of extra work. Finally, we also propose and validate a forecasting method based on Markov chains for the evolution of the epidemiological data for up to two weeks.
Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an effic…
The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation (ODE) models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of a…
We propose a compartmental mathematical model for the spread of the COVID-19 disease, showing its usefulness with respect to the pandemic in Portugal, from the first recorded case in the country till the end of the three states of emergency. New results include the compartmental model, described by a system of seven ordinary differential equations; proof of positivity and boundedness of solutions; investigation of equilibrium points and their stability analysis; computation of the basic reproduction number; and numerical simulations with official real data from the Portuguese health authoritie…
Nowcasting and forecasting of epidemic spreading rely on incidence series of reported cases to derive the fundamental epidemiological parameters for a given pathogen. Two relevant drawbacks for predictions are the unknown fractions of undocumented cases and levels of nonpharmacological interventions, which span highly heterogeneously across different places and times. We describe a simple data-driven approach using a compartmental model including asymptomatic and presymptomatic contagions that allows to estimate both the level of undocumented infections and the value of effective reproductive …
We developed a mathematical model to describe the transmission of new coronavirus in the São Paulo State, Brazil. The model divided a community in subpopulations comprised by young and elder persons, in order to take into account higher risk of fatality among elder persons with severe CoViD-19. From data collected in the São Paulo State, we estimated the transmission and additional mortality rates, from which we calculated the basic reproduction number R0. From estimated parameters, estimation of the deaths due to CoViD-19 was three times lower than those found in literature. Considering isola…
Knowing COVID-19 epidemiological distributions, such as the time from patient admission to death, is directly relevant to effective primary and secondary care planning, and moreover, the mathematical modelling of the pandemic generally. We determine epidemiological distributions for patients hospitalised with COVID-19 using a large dataset ($N=21{,}000-157{,}000$) from the Brazilian Sistema de Informação de Vigilância Epidemiológica da Gripe database. A joint Bayesian subnational model with partial pooling is used to simultaneously describe the 26 states and one federal district of Brazil, and…
In [Ecological Complexity 44 (2020) Art. 100885, DOI: 10.1016/j.ecocom.2020.100885] a continuous-time compartmental mathematical model for the spread of the Coronavirus disease 2019 (COVID-19) is presented with Portugal as case study, from 2 March to 4 May 2020, and the local stability of the Disease Free Equilibrium (DFE) is analysed. Here, we propose an analogous discrete-time model and, using a suitable Lyapunov function, we prove the global stability of the DFE point. Using COVID-19 real data, we show, through numerical simulations, the consistence of the obtained theoretical results.
We present a compartmental meta-population model for the spread of Covid-19 in India. Our model simulates populations at a district or state level using an epidemiological model that is appropriate to Covid-19. Different districts are connected by a transportation matrix developed using available census data. We introduce uncertainties in the testing rates into the model that takes into account the disparate responses of the different states to the epidemic and also factors in the state of the public healthcare system. Our model allows us to generate qualitative projections of Covid-19 spread …
The outbreak of Covid-19 led the world to an unprecedent health and economical crisis. In an attempt to responde to this emergency researchers worldwide are intensively studying the Covid-19 pandemic dynamics. In this work, a SIRSi compartmental model is proposed, which is a modification of the known classical SIR model. The proposed SIRSi model considers differences in the immunization within a population, and the possibility of unreported or asymptomatic cases. The model is adjusted to three major cities of São Paulo State, in Brazil, namely, São Paulo, Santos and Campinas, providing estimat…
The outbreak of COVID-19 in 2020 has led to a surge in the interest in the mathematical modeling of infectious diseases. Disease transmission may be modeled as compartmental models, in which the population under study is divided into compartments and has assumptions about the nature and time rate of transfer from one compartment to another. Usually, they are composed of a system of ordinary differential equations (ODE's) in time. A class of such models considers the Susceptible, Exposed, Infected, Recovered, and Deceased populations, the SEIRD model. However, these models do not always account…
We propose a physical theory underlying the temporal evolution of competing virus variants that relies on the existence of (quasi) fixed points capturing the large time scale invariance of the dynamics. To motivate our result we first modify the time-honoured compartmental models of the SIR type to account for the existence of competing variants and then show how their evolution can be naturally re-phrased in terms of flow equations ending at quasi fixed points. As the natural next step we employ (near) scale invariance to organise the time evolution of the competing variants within the effect…
An epidemiological compartmental model was used to simulate social distancing strategies to contain the COVID-19 pandemic and prevent a second wave in Sao Paulo, Brazil. Optimization using genetic algorithm was used to determine the optimal solutions. Our results suggest the best-case strategy for Sao Paulo is to maintain or increase the current magnitude of social distancing for at least 60 more days and increase the current levels of personal protection behaviors by a minimum of 10% (e.g., wearing facemasks, proper hand hygiene and avoid agglomeration). Followed by a long-term oscillatory le…
The subset selection problem of linear algebra is applied to identify independent patterns of COVID-19 evolution within Brazil. The data consist of a set of mortality curves in states of Brazil. A subset of the most independent curves is selected by using a functional version of the QR matrix decomposition technique with column pivoting. The selected subset is used next as a basis to represent the remaining curves filtering out any data redundancy. For each independent curve, an associated epidemiological region of influence is defined. The results show two main independent curves with a simil…
IMT School for Advanced Studies Lucca（卢cca高等研究院）；TREE-TOWER S.R.L.（TREE-TOWER有限责任公司）
Computational models for the simulation of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) epidemic evolution would be extremely useful to support authorities in designing healthcare policies and lockdown measures to contain its impact on public health and economy. In Italy, the devised forecasts have been mostly based on a pure data-driven approach, by fitting and extrapolating open data on the epidemic evolution collected by the Italian Civil Protection Center. In this respect, SIR epidemiological models, which start from the description of the nonlinear interactions between…
Epidemiological models for the spread of pathogens in a population are usually only able to describe a single pathogen. This makes their application unrealistic in cases where multiple pathogens with similar symptoms are spreading concurrently within the same population. Here we describe a method which makes possible the application of multiple single-strain models under minimal conditions. As such, our method provides a bridge between theoretical models of epidemiology and data-driven approaches for modeling of influenza and other similar viruses. Our model extends the Susceptible-Infected-Re…
In the present article we introduce an epidemiological model for the investigation of the spread of epidemics caused by viruses. The model is applied specifically to COVID-19, the disease caused by the SARS-Cov-2 virus (aka "novel coronavirus"). The SIR (Susceptible - Infectious - Recovered) model is used as a basis for studying the evolution of the epidemic. Nevertheless, we have modified some of the model hypotheses in order to obtain an estimate of the contamination free of overestimated predictions. This extended model is then applied to the case of the recent advance of the epidemic in Br…
Assessing the performance of compartmental and renewal models for learning $R_{t}$ using spatially heterogeneous epidemic simulations on real geographies
In this paper, we estimate epidemiological data of the COVID-19 pandemic in Hungary using only the daily number of hospitalized patients, and applying well-known techniques from systems and control theory. We use a previously published and validated compartmental model for the description of epidemic spread. Exploiting the fact that an important subsystem of the model is linear, first we compute the number of latent infected persons in time. Then an estimate can be given for the number of people in other compartments. From these data, it is possible to track the time dependent reproduction num…
In this paper we model the spreading of the SARS-CoV-2 in Mexico by introducing a new stochastic approximation constructed from first principles, structured on the basis of a Latent-Infectious- (Recovered or Deceased) (LI(RD)) compartmental approximation, where the number of new infected individuals caused by a single infectious individual per unit time (a day), is a random variable of a Poisson distribution and whose parameter is modulated through a weight-like time-dependent function. The weight function serves to introduce a time dependence to the average number of new infections and as we …
We collected individualized (unidentifiable) and aggregated openly available data from various sources related to suspected/confirmed SARS-CoV-2 infections, vaccinations, non-pharmaceutical government interventions, human mobility, and levels of population inequality in Brazil. In addition, a data structure allowing real-time data collection, curation, integration, and extract-transform-load processes for different objectives was developed. The granularity of this dataset (state- and municipality-wide) enables its application to individualized and ecological epidemiological studies, statistica…
We consider the SIR epidemiological model applied to the evolution of COVID-19 with two approaches. In the first place we fit a global SIR model, with time delay, and constant parameters throughout the outbreak, including the contagion rate. The contention measures are reflected on an effective reduced susceptible population $N_{eff}$. In the second approach we consider a time-dependent contagion rate that reflects the contention measures either through a step by step fitting process or by following an exponential decay. In this last model the population is considered the one of the country $N…
As of December 2020, the COVID-19 pandemic has infected over 75 million people, making it the deadliest pandemic in modern history. This study develops a novel compartmental epidemiological model specific to the SARS-CoV-2 virus and analyzes the effect of common preventative measures such as testing, quarantine, social distancing, and vaccination. By accounting for the most prevalent interventions that have been enacted to minimize the spread of the virus, the model establishes a paramount foundation for future mathematical modeling of COVID-19 and other modern pandemics. Specifically, the mod…
Compartmental equations are primary tools in disease spreading studies. Their predictions are accurate for large populations but disagree with empirical and simulated data for finite populations, where uncertainties become a relevant factor. Starting from the agent-based approach, we investigate the role of uncertainties and autocorrelation functions in SIS epidemic model, including their relationship with epidemiological variables. We find new differential equations that take uncertainties into account. The findings provide improved predictions to the SIS model and it can offer new insights f…
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 …
The COVID-19 pandemic has placed unprecedented demands on local public health units in Ontario, Canada, one of which was the need for in-house epidemiological-modelling capabilities. To address this need, Ontario Tech University and the Durham Region Health Department developed a native Windows desktop app that performs epidemiological modelling of infectious diseases. The app is an implementation of a multi-stratified compartmental epidemiological model that can accommodate multiple virus variants and levels of vaccination, as well as public health measures such as physical distancing, contac…
The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources, such as the amount of infectious individuals. In this paper we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest, such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interacti…
We model and forecast the early evolution of the COVID-19 pandemic in Brazil using Brazilian recent data from February 25, 2020 to March 30, 2020. This early period accounts for unawareness of the epidemiological characteristics of the disease in a new territory, sub-notification of the real numbers of infected people and the timely introduction of social distancing policies to flatten the spread of the disease. We use two variations of the SIR model and we include a parameter that comprises the effects of social distancing measures. Short and long term forecasts show that the social distancin…
The infections and fatalities due to SARS-CoV-2 virus for cases specific to India have been studied using a deterministic susceptible-exposed-infected-recovered-dead (SEIRD) compartmental model. One of the most significant epidemiological parameter, namely the effective reproduction number of the infection is extracted from the daily growth rate data of reported infections and it is included in the model with a time variation. We evaluate the effect of control interventions implemented till now and estimate the case numbers for infections and deaths averted by these restrictive measures. We fu…
Approaching epidemiological dynamics of COVID-19 with physics-informed neural networks
Frankfurt Institute for Advanced Studies（法兰克福高等研究院）；Xidian-FIAS International Joint Research Center（西安电子科技大学-法兰克福高等研究院国际联合研究中心）；Institut für Theoretische Physik, Goethe Universität Frankfurt（法兰克福大学理论物理研究所）；GSI Helmholtzzentrum für Schwerionenforschung GmbH（GSI亥姆霍兹重离子研究中心）；Xidian University（西安电子科技大学）
本文提出将嵌入 SIR 模型的物理信息神经网络（PINN）用于传染病动力学建模，通过在合成数据和德国 COVID-19 数据上的实验证明其能准确识别和预测疫情传播趋势，展示了机器学习与仓室模型结合研究流行病的高潜力。
The presence of a large number of infected individuals with few or no symptoms is an important epidemiological difficulty and the main mathematical feature of COVID-19. The A-SIR model, i.e. a SIR (Susceptible-Infected-Removed) model with a compartment for infected individuals with no symptoms or few symptoms was proposed by Giuseppe Gaeta, arXiv:2003.08720 [q-bio.PE] (2020). In this paper we investigate a slightly generalized version of the same model and propose a scheme for fitting the parameters of the model to real data using the time series only of the deceased individuals. The scheme is…
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…
The Zika virus has been found in individual cases but has not been confirmed as the cause of in the large number of cases of microcephaly in Brazil in 2015-6. Indeed, disparities between the incidence of Zika and microcephaly across geographic locations has led to questions about the virus's role. Here we consider whether the insecticide pyriproxyfen used in Brazilian drinking water might be the primary cause or a cofactor. Pyriproxifen is a juvenile hormone analog which has been shown to correspond in mammals to a number of fat soluble regulatory molecules including retinoic acid, a metabolit…
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 …
Over a year after the start of the COVID-19 epidemics, we are still facing the virus and it is hard to correctly predict its future spread over weeks to come, as well as the impacts of potential political interventions. Current epidemic models mainly fall in two approaches: compartmental models, divide the population in epidemiological classes and rely on the mathematical resolution of differential equations to give a macroscopic view of the epidemical dynamics, allowing to evaluate its spread a posteriori; agent-based models are computer models that give a microscopic view of the situation, s…
We performed a thorough sensitivity analysis of the herd immunity threshold for discrete-time SIR compartmental models with a static network structure. We find unexpectedly that these models violate classical intuition which holds that the herd immunity threshold should monotonically increase with the transmission parameter. We find the existence of bifurcations in the herd immunity threshold in the high transmission probability regime. The extent of these bifurcations is modulated by the graph heterogeneity, the recovery parameter, and the network size. In the limit of large, well-mixed netwo…
Currently, Brazil has one of the fastest increasing COVID-19 epidemics in the world, that has caused at least 94 thousand confirmed deaths until now. The city of Sao Paulo is particularly vulnerable because it is the most populous in the country. Analyzing the spatiotemporal dynamics of COVID-19 is important to help the urgent need to integrate better actions to face the pandemic. Thus, this study aimed to analyze the COVID-19 mortality, from March to July 2020, considering the spatio-time architectures, the socio-economic context of the population, and using a fine granular level, in the most…
An analysis of the COVID-19 epidemic is proposed on the basis of the epiMOX dashboard (publicly accessible at https://www.epimox.polimi.it) that deals with data of the epidemic trends and outbreaks in Italy from late February 2020. Our analysis provides an immediate appreciation of the past epidemic development, together with its current trends by fostering a deeper interpretation of available data through several critical epidemic indicators. In addition, we complement the epiMOX dashboard with a predictive tool based on an epidemiological compartmental model, named SUIHTER, for the forecast …
University of Michigan（密歇根大学）；Harbin Institute of Technology（哈尔滨工业大学）；University of Connecticut（康涅狄格大学）；Massachusetts Institute of Technology（麻省理工学院）
COVID-19 pandemic has had a disruptive and irreversible impact globally, yet traditional epidemiological modeling approaches such as the susceptible-infected-recovered (SIR) model have exhibited limited effectiveness in forecasting of the up-to-date pandemic situation. In this work, susceptible-vaccinated-exposed-infected-dead-recovered (SVEIDR) model and its variants -- aged and vaccination-structured SVEIDR models -- are introduced to encode the effect of social contact for different age groups and vaccination status. Then, we implement the physics-informed neural network (PiNN) on both simu…
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…
The prevalence of COVID-19 has been the most serious health challenge of the 21th century to date, concerning national health systems on a daily basis, since December 2019 when it appeared in Wuhan City. Nevertheless, most of the proposed mathematical methodologies aiming to describe the dynamics of an epidemic, rely on deterministic models that are not able to reflect the true nature of its spread. In this paper, we propose a SEIHCRDV model - an extension/improvement of the classic SIR compartmental model - which also takes into consideration the populations of exposed, hospitalized, admitted…
The ongoing COVID-19 pandemic highlights the essential role of mathematical models in understanding the spread of the virus along with a quantifiable and science-based prediction of the impact of various mitigation measures. Numerous types of models have been employed with various levels of success. This leads to the question of what kind of a mathematical model is most appropriate for a given situation. We consider two widely used types of models: equation-based models (such as standard compartmental epidemiological models) and agent-based models. We assess their performance by modeling the s…
Detections of mutations of the SARS-CoV-2 virus gave rise to new packages of interventions. Among them, international travel restrictions have been one of the fastest and most visible responses to limit the spread of the variants. While inducing large economic losses, the epidemiological consequences of such travel restrictions are highly uncertain. They may be poorly effective when the new highly transmissible strain of the virus already circulates in many regions. Assessing the effectiveness of travel bans is difficult given the paucity of data on daily cross-border mobility and on existing …
Addressed in this work is the performance of five popular algorithms, which aim at assessing the dissemination dynamics of the COVID-19 disease on the basis of the time series of new confirmed cases. The tests are based on simulated data, generated by means of a deterministic compartmental epidemiological model \cite{Matsinos2020a}, adapted herein to also include the possibility of the loss of immunity by the group of the recovered (or vaccinated) subjects. Assuming a simple temporal dependence of the effective reproduction number (the exact details are of no relevance as far as the conclusion…
In this paper, we present a method to forecast the spread of SARS-CoV-2 across regions with a focus on the role of mobility. Mobility has previously been shown to play a significant role in the spread of the virus, particularly between regions. Here, we investigate under which epidemiological circumstances incorporating mobility into transmission models yields improvements in the accuracy of forecasting, where we take the situation in the Netherlands during and after the first wave of transmission in 2020 as a case study. We assess the quality of forecasting on the detailed level of municipali…
