We study the datafor the cumulative as well as daily number of cases in the Covid-19 outbreak in China. The cumulative data can be fit to an empirical form obtained from a Susceptible-Infected-Removed (SIR) model studied on an Euclidean network previously. Plotting the number of cases against the distance from the epicenter for both China and Italy, we find an approximate power law variation with an exponent $\sim 1.85$ showing strongly that the spatial dependence plays a key role, a factor included in the model. We report here that the SIR model on the Eucledean network can reproduce with a high accuracy the data for China for given parameter values, and can also predict when the epidemic, at least locally, can be expected to be over.
Calibration of a SIR (Susceptibles-Infected-Recovered) model with official international data for the COVID-19 pandemics provides a good example of the difficulties inherent the solution of inverse problems. Inverse modeling is set up in a framework of discrete inverse problems, which explicitly considers the role and the relevance of data. Together with a physical vision of the model, the present work addresses numerically the issue of parameters calibration in SIR models, it discusses the uncertainties in the data provided by international authorities, how they influence the reliability of c…
The present paper introduces a data-driven framework for describing the time-varying nature of an SIRD model in the context of COVID-19. By embedding a rolling regression in a mixed integer bilevel nonlinear programming problem, our aim is to provide the research community with a model that reproduces accurately the observed changes in the number of infected, recovered, and death cases, while providing information about the time dependency of the parameters that govern the SIRD model. We propose this optimization model and a genetic algorithm to tackle its solution. Moreover, we test this algo…
This paper shows that the generalized logistic distribution model is derived from the well-known compartment model, consisting of susceptible, infected and recovered compartments, abbreviated as the SIR model, under certain conditions. In the SIR model, there are uncertainties in predicting the final values for the number of infected population and the infectious parameter. However, by utilizing the information obtained from the generalized logistic distribution model, we can perform the SIR numerical computation more stably and more accurately. Applications to severe acute respiratory syndrom…
In our previous paper, the logistic curve of the removed number was derived from SIR and SEIR models in the case of the small basic reproduction number. In this paper, we derive various logistic curves of the removed, unsusceptible and infectious numbers respectively from SIS and SIR models in the case of small and large basic reproduction numbers.
We discuss the generation of various reproduction ratios or numbers that are very useful to monitor an ongoing epidemic like Covid-19 and examine the effects of intervention measures. A detailed SEIR algorithm is described for their computation, with applications given to the current Covid-19 outbreaks in a number of countries (Argentina, Brazil, France, Italy, Mexico, Spain, UK and USA). The corresponding matlab script, complete and ready to use, is provided for free downloading.
The COVID-19 disease has forced countries to make a considerable collaborative effort between scientists and governments to provide indicators to suitable follow-up the pandemic's consequences. Mathematical modeling plays a crucial role in quantifying indicators describing diverse aspects of the pandemic. Consequently, this work aims to develop a clear, efficient, and reproducible methodology for parameter optimization, whose implementation is illustrated using data from three representative regions from Chile and a suitable generalized SIR model together with a fitted positivity rate. Our res…
Mathematical models of infectious diseases exhibit robust dynamics such as stable endemic or a disease-free equilibrium, or convergence of the solutions to periodic epidemic waves. The present work shows that the accuracy of such dynamics can be significantly improved by incorporating both local and global dynamics of the infection in disease models. To demonstrate improved accuracies, we extended a standard Susceptible-Infected-Recovered (SIR) model by incorporating global dynamics of the COVID-19 pandemic. The extended SIR model assumes three possibilities for the susceptible individuals tra…
Using the classical Susceptible-Infected-Recovered epidemiological model, an analytical formula is derived for the number of beds occupied by Covid-19 patients. The analytical curve is fitted to data in Belgium, France, New York City and Switzerland, with a correlation coefficient exceeding 98.8%, suggesting that finer models are unnecessary with such macroscopic data. The fitting is used to extract estimates of the doubling time in the ascending phase of the epidemic, the mean recovery time and, for those who require medical intervention, the mean hospitalization time. Large variations can be…
We present a modified age-structured SIR model based on known patterns of social contact and distancing measures within Washington, USA. We find that population age-distribution has a significant effect on disease spread and mortality rate, and contribute to the efficacy of age-specific contact and treatment measures. We consider the effect of relaxing restrictions across less vulnerable age-brackets, comparing results across selected groups of varying population parameters. Moreover, we analyze the mitigating effects of vaccinations and examine the effectiveness of age-targeted distributions.…
In this paper we develop a predictive model for the spread of COVID-19 infection at a provincial (i.e. EU NUTS-3) level in Italy by using official data from the Italian Ministry of Health integrated with data extracted from daily official press conferences of regional authorities and from local newspaper websites. This integration is mainly concerned with COVID-19 cause specific death data which are not available at NUTS-3 level from open official data data channels. An adjusted time-dependent SIRD model is used to predict the behavior of the epidemic, specifically the number of susceptible, i…
A simplified method to compute $R_t$, the Effective Reproduction Number, is presented. The method relates the value of $R_t$ to the estimation of the doubling time performed with a local exponential fit. The condition $R_t = 1$ corresponds to a growth rate equal to zero or equivalently an infinite doubling time. Different assumptions on the probability distribution of the generation time are considered. A simple analytical solution is presented in case the generation time follows a gamma distribution.
As the interactions between people increases, the impending menace of COVID-19 outbreaks materialize, and there is an inclination to apply lockdowns. In this context, it is essential to have easy-to-use indicators for people to use as a reference. The basic reproduction number of confirmed positives, $R_t$, fulfill such a role. This document proposes a data-driven approach to nowcast $R_t$ based on previous observations' statistical behavior. As more information arrives, the method naturally becomes more precise about the final count of confirmed positives. Our method's strength is that it is …
Disease transmission is studied through disciplines like epidemiology, applied mathematics, and statistics. Mathematical simulation models for transmission have implications in solving public and personal health challenges. The SIR model uses a compartmental approach including dynamic and nonlinear behavior of transmission through three factors: susceptible, infected, and removed (recovered and deceased) individuals. Using the Lambert W Function, we propose a framework to study solutions of the SIR model. This demonstrates the applications of COVID-19 transmission data to model the spread of a…
In this paper, we study the effectiveness of the modelling approach on the pandemic due to the spreading of the novel COVID-19 disease and develop a susceptible-infected-removed (SIR) model that provides a theoretical framework to investigate its spread within a community. Here, the model is based upon the well-known susceptible-infected-removed (SIR) model with the difference that a total population is not defined or kept constant per se and the number of susceptible individuals does not decline monotonically. To the contrary, as we show herein, it can be increased in surge periods! In partic…
Since November 6$^{\mathrm{th}}$, 2020, Italian regions have been classified according to four levels, corresponding to specific risk scenarios, for which specific restrictive measures have been foreseen. By analyzing the time evolution of the reproduction number $R_t$, we estimate how much different restrictive measures affect $R_t$, and we quantify the combined effect of the diffusion of virus variants and the beginning of the vaccination campaign upon the $R_t$ trend. We also compute the time delay between implementation of restrictive measures and the resulting effects. Three different mod…
This article describes a simple Susceptible Infected Recovered (SIR) model fitting with COVID-19 data for the month of march 2020 in New York (NY) state. The model is a classical SIR, but is non-autonomous; the rate of susceptible people becoming infected is adjusted over time in order to fit the available data. The death rate is also secondarily adjusted. Our fitting is made under the assumption that due to limiting number of tests, a large part of the infected population has not been tested positive. In the last part, we extend the model to take into account the daily fluxes between New Jers…
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…
We introduce an epidemic spreading model on a network using concepts from percolation theory. The model is motivated by discussing the standard SIR model, with extensions to describe effects of lockdowns within a population. The underlying ideas and behavior of the lattice model, implemented using the same lockdown scheme as for the SIR scheme, are discussed in detail and illustrated with extensive simulations. A comparison between both models is presented for the case of COVID-19 data from the USA. Both fits to the empirical data are very good, but some differences emerge between the two appr…
Policy-makers require data-driven tools to assess the spread of COVID-19 and inform the public of their risk of infection on an ongoing basis. We propose a rigorous hybrid model-and-data-driven approach to risk scoring based on a time-varying SIR epidemic model that ultimately yields a simplified color-coded risk level for each community. The risk score $Γ_t$ that we propose is proportional to the probability of someone currently healthy getting infected in the next 24 hours. We show how this risk score can be estimated using another useful metric of infection spread, $R_t$, the time-varying a…
In this paper, we present a model to predict the spread of the Covid-19 epidemic and apply it to the specific case of Italy. We started from a simple Susceptible, Infected, Recovered (SIR) model and we added the condition that, after a certain time, the basic reproduction number $R_0$ exponentially decays in time, as empirically suggested by world data. Using this model, we were able to reproduce the real behavior of the epidemic with an average error of 5\%. Moreover, we illustrate possible future scenarios, associated to different intervals of $R_0$. This model has been used since the beginn…
Timely estimation of the current value for COVID-19 reproduction factor $R$ has become a key aim of efforts to inform management strategies. $R$ is an important metric used by policy-makers in setting mitigation levels and is also important for accurate modelling of epidemic progression. This brief paper introduces a method for estimating $R$ from biased case testing data. Using testing data, rather than hospitalisation or death data, provides a much earlier metric along the symptomatic progression scale. This can be hugely important when fighting the exponential nature of an epidemic. We deve…
This paper is an exploratory study of two epidemiological questions on a worldwide basis. How fast is the disease spreading? Are the restrictions (especially mobility restrictions) for people bring the expected effect? To answer the first question, we propose a tool for estimating the reproduction number of epidemic (the number of secondary infections $R_t$) based on the SEIR model and compare it with an non-model $R_t$ estimation. To measure the $R_t$ of COVID-19 for different countries, real-time data on coronavirus daily cases of infections, recoveries, deaths are retrieved from the Center …
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…
Monitoring the evolution of the Covid19 pandemic constitutes a critical step in sanitary policy design. Yet, the assessment of the pandemic intensity within the pandemic period remains a challenging task because of the limited quality of data made available by public health authorities (missing data, outliers and pseudoseasonalities, notably), that calls for cumbersome and ad-hoc preprocessing (denoising) prior to estimation. Recently, the estimation of the reproduction number, a measure of the pandemic intensity, was formulated as an inverse problem, combining data-model fidelity and space-ti…
We proposed a Monte-Carlo method to estimate temporal reproduction number without complete information about symptom onsets of all cases. Province-level analysis demonstrated the huge success of Chinese control measures on COVID-19, that is, provinces' reproduction numbers quickly decrease to <1 by just one week after taking actions.
Initially emerged in the Chinese city Wuhan and subsequently spread almost worldwide causing a pandemic, the SARS-CoV-2 virus follows reasonably well the SIR (Susceptible-Infectious-Recovered) epidemic model on contact networks in the Chinese case. In this paper, we investigate the prediction accuracy of the SIR model on networks also for Italy. Specifically, the Italian regions are a metapopulation represented by network nodes and the network links are the interactions between those regions. Then, we modify the network-based SIR model in order to take into account the different lockdown measu…
Since the end of 2019, COVID-19 has significantly affected the lives of people around the world. Towards the end of 2020, several COVID-19 vaccine candidates with relatively high efficacy have been reported in the final phase of clinical trials. Vaccines have been considered as critical tools for opening up social and economic activities, thereby lessening the impact of this disease on the society. This paper presents a simulation of COVID-19 spread using modified Susceptible-Infected-Removed (SIR) model under vaccine intervention in several localities of Malaysia, i.e. those cities or states …
The dramatic outbreak of the coronavirus disease 2019 (COVID-19) pandemics and its ongoing progression boosted the scientific community's interest in epidemic modeling and forecasting. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive an explicit solution. The present work demonstrates that this is a non-trivial task. Notably, it is proven that the explicit solution of the model requires the introduction of a new transcendental special function, related to the Wright's Omega functi…
The recent COVID-19 pandemic highlighted the need of non-pharmaceutical interventions in the first stages of a pandemic. Among these, lockdown policies proved unavoidable yet extremely costly from an economic perspective. To better understand the tradeoffs between economic and epidemic costs of lockdown interventions, we here focus on a simple SIR epidemic model and study lockdowns as solutions to an optimal control problem. We first show numerically that the optimal lockdown policy exhibits a phase transition from suppression to mitigation as the time horizon grows, i.e., if the horizon is sh…
A great issue of discussion of an infectious disease is its basic reproduction number R0, which provides an estimation of the contagiousness of the disease. When R0 > 1, the disease spread will potentially lead to an outbreak, such that of the ongoing COVID-19 pandemics. During the evolution of an outbreak, various non-pharmaceutical interventions are employed, the impact of which is frequently assessed by the reduction that they introduce to the effective reproduction number Rt; reduction below 1 is an indication of eventual dying out of the disease spread. Motivated by the fact that R0 essen…
Near-real time estimations of the effective reproduction number are among the most important tools to track the progression of a pandemic and to inform policy makers and the general public. However, these estimations rely on reported case numbers, commonly recorded with significant biases. The epidemic outcome is strongly influenced by the dynamics of social contacts, which are neglected in conventional surveillance systems as their real-time observation is challenging. Here, we propose a concept using online and offline behavioral data, recording age-stratified contact matrices at a daily rat…
During the COVID-19 pandemic, governments faced difficulties in implementing mobility restriction measures, as no clear quantitative relationship between human mobility and infection spread in large cities is known. We developed a model that enables quantitative estimations of the infection risk for individual places and activities by using smartphone GPS data for the Tokyo metropolitan area. The effective reproduction number is directly calculated from the number of infectious social contacts defined by the square of the population density at each location. The difference in the infection rat…
The present article studies the extension of two deterministic models for describing the novel coronavirus pandemic crisis, the SIR model and the SEIR model. The models were studied and compared to real data in order to support the validity of each description and extract important information regarding the pandemic, such as the basic reproductive number R0, which might provide useful information concerning the rate of increase of the pandemic predicted by each model. We next proceed to making predictions and comparing more complex models derived from the SEIR model with the SIRD model, in ord…
This paper extends the canonical model of epidemiology, the SIRD model, to allow for time-varying parameters for real-time measurement and prediction of the trajectory of the Covid-19 pandemic. Time variation in model parameters is captured using the generalized autoregressive score modeling structure designed for the typical daily count data related to the pandemic. The resulting specification permits a flexible yet parsimonious model with a low computational cost. The model is extended to allow for unreported cases using a mixed-frequency setting. Results suggest that these cases' effects on…
In the recent COVID-19 pandemic, a wide range of epidemiological modelling approaches have been used to predict the effective reproduction number, R(t), and other COVID-19 related measures such as the daily rate of exponential growth, r(t). These candidate models use different modelling approaches or differing assumptions about spatial or age mixing, and some capture genuine uncertainty in scientific understanding of disease dynamics. Combining estimates using appropriate statistical methodology from multiple candidate models is important to better understand the variation of these outcome mea…
We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China.
Governments around the world have implemented preventive measures against the spread of the coronavirus disease (COVID-19). In this study, we consider a multivariate discrete-time Markov model to analyze the propagation of COVID-19 across 33 provincial regions in China. This approach enables us to evaluate the effect of mobility restriction policies on the spread of the disease. We use data on daily human mobility across regions and apply the Bayesian framework to estimate the proposed model. The results show that the spread of the disease in China was predominately driven by community transmi…
The non-population conserving SIR (SIR-NC) model to describe the spread of infections in a community is proposed and studied. Unlike the standard SIR model, SIR-NC does not assume population conservation. Although similar in form to the standard SIR, SIR-NC admits a closed form solution while allowing us to model mortality, and also provides different, and arguably a more realistic, interpretation of the model parameters. Numerical comparisons of this SIR-NC model with the standard, population conserving, SIR model are provided. Extensions to include imported infections, interacting communitie…
We present a very simple and transparent method to interpret time series of confirmed Covid-19 cases. Roughly speaking, the analysis of weekly new infections for each day is a tool for the definition and early detection of the turning point of the epidemic. In Italy, the growth of Covid-19 activity was overcome one week ago, Austria and Switzerland followed suit. This note emphasizes the crucial delay of two weeks between infection and listing of infection in a central database. Our estimates of the time course of Covid-19 activity and reproduction rates can reduce the information gap by sever…
The purpose of this work is to give a contribution to the understanding of the COVID-19 contagion in Italy. To this end, we developed a modified Susceptible-Infected-Recovered (SIR) model for the contagion, and we used official data of the pandemic up to March 30th, 2020 for identifying the parameters of this model. The non standard part of our approach resides in the fact that we considered as model parameters also the initial number of susceptible individuals, as well as the proportionality factor relating the detected number of positives with the actual (and unknown) number of infected indi…
We examine the age-structured SIR model, a variant of the classical Susceptible-Infected-Recovered (SIR) model of epidemic propagation, in the context of COVID-19. In doing so, we provide a theoretical basis for the model, perform an empirical validation, and discover the limitations of the model in approximating arbitrary epidemics. We first establish the differential equations defining the age-structured SIR model as the mean-field limits of a continuous-time Markov process that models epidemic spreading on a social network involving random, asynchronous interactions. We then show that, as t…
The estimate of the remaining time of an ongoing wave of epidemic spreading is a critical issue. Due to the variations of a wide range of parameters in an epidemic, for simple models such as Susceptible-Infected-Removed (SIR) model, it is difficult to estimate such a time scale. On the other hand, multidimensional data with a large set attributes are precisely what one can use in statistical learning algorithms to make predictions. Here we show, how the predictability of the SIR model changes with various parameters using a supervised learning algorithm. We then estimate the condition in which…
The study carries out predictive modeling based on publicly available COVID-19 data for the duration 01 April to 20 June 2020 pertaining to India and five of its most infected states: Maharashtra, Tamil Nadu, Delhi, Gujarat, and Rajasthan using susceptible, infected, recovered, and dead (SIRD) model. The basic reproduction number R0 is derived by exponential growth method using RStudio package R0. The differential equations reflecting SIRD model have been solved using Python 3.7.4 on Jupyter Notebook platform. For visualization, Python Matplotlib 3.2.1 package is used. The study offers insight…
In the last decade, humanity has faced many different pandemics such as SARS, H1N1, and presently novel coronavirus (COVID-19). On one side, scientists are focusing on vaccinations, and on the other side, there is a need to propose models that can help us in understanding the spread of these pandemics as it can help governmental and other concerned agencies to be well prepared, especially from pandemics, which spreads faster like COVID-19. The main reason for some epidemic turning into pandemics is the connectivity among different regions of the world, which makes it easier to affect a wider g…
OBJECTIVES: Estimates of the basic reproduction number (R0) of COVID-19 vary across countries. This paper aims to characterise the spatial variability in R0 across the first six months of the global COVID-19 outbreak, and to explore social factors that impact R0 estimates at national and regional level. METHODS: We searched PubMed, LitCOVID and the WHO COVID-19 database from January to June 2020. Peer-reviewed English-language papers were included that provided R0 estimates. For each study, the value of the estimate, country under study and publication month were extracted. The median R0 value…
We study the increases of infections and deaths in Sweden caused by COVID-19 with several different models: Firstly an analytical susceptible-infected (SI) model and the standard susceptible-infected-recovered (SIR) model. Then within the SIR framework we study the susceptible-infected-deceased (SID) correlations. All models reproduce well the number of infected cases and give similar predictions. What causes us deep concern is the large number of deaths projected by the SI and SID models. Our analysis shows that, irrespective of the possible uncertainty of our model prediction, the next few d…
This paper deals with the problem of estimating variables in nonlinear models for the spread of disease and its application to the COVID-19 epidemic. First unconstrained methods are revisited and they are shown to correspond to the application of a linear filter followed by a nonlinear estimate of the effective reproduction number after a change-of-coordinates. Unconstrained methods often fail to keep the estimated variables within their physical range and can lead to unreliable estimates that require aggressively smoothing the raw data. In order to overcome these shortcomings a constrained es…
The current COVID-19 pandemic and subsequent lockdowns have highlighted the close and delicate relationship between a country's public health and economic health. Macroeconomic models that use preexisting epidemic models to calculate the impacts of a disease outbreak are therefore extremely useful for policymakers seeking to evaluate the best course of action in such a crisis. We develop an SIR model of the COVID-19 pandemic that explicitly considers herd immunity, behavior-dependent transmission rates, remote workers, and indirect externalities of lockdown. This model is presented as an exit …
We provide a probabilistic SIRD model for the COVID-19 pandemic in Italy, where we allow the infection, recovery and death rates to be random. In particular, the underlying random factor is driven by a fractional Brownian motion. Our model is simple and needs only some few parameters to be calibrated.
Epidemics like Covid-19 and Ebola have impacted people's lives significantly. The impact of mobility of people across the countries or states in the spread of epidemics has been significant. The spread of disease due to factors local to the population under consideration is termed the endogenous spread. The spread due to external factors like migration, mobility, etc. is called the exogenous spread. In this paper, we introduce the Exo-SIR model, an extension of the popular SIR model and a few variants of the model. The novelty in our model is that it captures both the exogenous and endogenous …
Background: Northern Italy was one of the most impacted areas by COVID. It is now widely assumed that the virus was silently spreading for at least 2 weeks before the first patient was identified. During this silent phase, and in the following weeks when the hospital system was overburdened, data collection was not performed in an accurate enough way to estimate an epidemic curve. With the aim of assessing both the dynamics of the introduction of the virus and the effectiveness of containment measures introduced, we try to reconstruct the epidemic curve using all cause mortality data. Methods:…
In order to analyze the effectiveness of three successive nationwide lockdown enforced in India, we present a data-driven analysis of four key parameters, reducing the transmission rate, restraining the growth rate, flattening the epidemic curve and improving the health care system. These were quantified by the consideration of four different metrics, namely, reproduction rate, growth rate, doubling time and death to recovery ratio. The incidence data of the COVID-19 (during the period of 2nd March 2020 to 31st May 2020) outbreak in India was analyzed for the best fit to the epidemic curve, ma…
This paper introduces a dynamic panel SIR (DP-SIR) model to investigate the impact of non-pharmaceutical interventions (NPIs) on the COVID-19 transmission dynamics with panel data from 9 countries across the globe. By constructing scenarios with different combinations of NPIs, our empirical findings suggest that countries may avoid the lockdown policy with imposing school closure, mask wearing and centralized quarantine to reach similar outcomes on controlling the COVID-19 infection. Our results also suggest that, as of April 4th, 2020, certain countries such as the U.S. and Singapore may requ…
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 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.
