Classical SIR Model Research Articles

Universal features of epidemic models under social distancing guidelines

Social distancing as a form of nonpharmaceutical intervention has been enacted in many countries as a form of mitigating the spread of COVID-19. There has been a large interest in mathematical modeling to aid in the prediction of both the total infected population and virus-related deaths, as well as to aid government agencies in decision making. As the virus continues to spread, there are both economic and sociological incentives to minimize time spent with strict distancing mandates enforced, and/or to adopt periodically relaxed distancing protocols, which allow for scheduled economic activity. The main objective of this study is to reduce the disease burden in a population, here measured as the peak of the infected population, while simultaneously minimizing the length of time the population is socially distanced, utilizing both a single period of social distancing as well as periodic relaxation. We derive a linear relationship among the optimal start time and duration of a single interval of social distancing from an approximation of the classic epidemic SIR model. Furthermore, we see a sharp phase transition region in start times for a single pulse of distancing, where the peak of the infected population changes rapidly; notably, this transition occurs well before one would intuitively expect. By numerical investigation of more sophisticated epidemiological models designed specifically to describe the COVID-19 pandemic, we see that all share remarkably similar dynamic characteristics when contact rates are subject to periodic or one-shot changes, and hence lead us to conclude that these features are universal in epidemic models. On the other hand, the nonlinearity of epidemic models leads to non-monotone behavior of the peak of infected population under periodic relaxation of social distancing policies. This observation led us to hypothesize that an additional single interval social distancing at a proper time can significantly decrease the infected peak of periodic policies, and we verified this improvement numerically. While synchronous quarantine and social distancing mandates across populations effectively minimize the spread of an epidemic over the world, relaxation decisions should not be enacted at the same time for different populations.

STRATEGIES TO DEAL WITH EPIDEMICS USING A SYSTEM DYNAMICS MODEL: A COVID-19 STUDY CASE

This paper presents a System Dynamics Model to verify well-known strategies to deal with the COVID-19 outbreak, based on the classical epidemiological SIR model. Three main strategies were tested, considering the idealized main goal of public policies, to save lives. In addition, the authorities' reaction time on implementing actions to control the pandemic is analyzed. From the tested strategies, between quarantine extension, vertical quarantine, and identification of infected persons, the most promising is to identify infected persons and the least suitable is vertical quarantine. Also, the simulations indicated that tightening quarantine at any time tends to save lives, but late action by authorities can bring overpressure on a country's health care system, which has also direct impact on the lethality of the epidemic.

Dynamic Adjustment of SIR Model with the Social Permissiveness: An Actual Measure of the Infection Rate

The COVID-19 pandemic has globally impacted the behavior of the social patterns affecting the disease’s contagiousness. This effect deviates the classical SIR model from reproducing the data of COVID-19 in most countries. This study incorporates a non-constant permissiveness function to the SIR model. The resulting model is computationally solved to obtain a likely permissiveness time-function. To solve the adjusted model, a technique based on a proportional-integral controller is applied. The resulting models are compared with previous results obtained by a manual iterative adjusting method.

A Two-wave Epidemiological model of COVID-19 outbreaks using MS-Excel®

The emergence of the coronavirus SARS-CoV-2 has raised a global issue and a pandemic disease outbreak, COVID-19, was declared by the World Health Organization on March 12th, 2020. Health authority advisors and governments need to quickly manage and deal with growing epidemiological data on a daily basis. In this work, current available data from reported cases and deaths were analyzed and treated. Lethality has been calculated by finding linearization of death cases against reported ones, using a time-delayed data transposition. A two-wave statistical model, 2WM, based on the superposition of normal distributions was used to fit current data and to estimate the evolution of infections and deaths, using Microsoft® Excel. A gamma distribution was used as a risk function to estimate death probability from patient admission to reported death. Evolution of fatality cases over time can then be estimated from the model with reasonable accuracy. Data from South Korea, China, Australia, Germany, Italy, Spain, United States, United Kingdom and Brazil were used. Constant lethality can be determined from the initial stage of the pandemic wave, even for places with low testing. Values ranged from 1.7% to 15.3%. The two-wave model can be fine-tuned to properly adjust the data. The second wave pattern was estimated according to the first wave parameter. The accuracy for estimating COVID-19 evolution was compared to the classic SIR model with good agreement. According to the model, approximately 10,000,000 cases and 860,000 deaths will be recorded worldwide.

Social contacts, epidemic spreading and health system. Mathematical modeling and applications to COVID-19 infection.

Lockdown and social distancing, as well as testing and contact tracing, are the main measures assumed by the governments to control and limit the spread of COVID-19 infection. In reason of that, special attention was recently paid by the scientific community to the mathematical modeling of infection spreading by including in classical models the effects of the distribution of contacts between individuals. Among other approaches, the coupling of the classical SIR model with a statistical study of the distribution of social contacts among the population, led some of the present authors to build a Social SIR model, able to accurately follow the effect of the decrease in contacts resulting from the lockdown measures adopted in various European countries in the first phase of the epidemic. The Social SIR has been recently tested and improved through a fruitful collaboration with the Health Protection Agency (ATS) of the province of Pavia (Italy), that made it possible to have at disposal all the relevant data relative to the spreading of COVID-19 infection in the province (half a million of people), starting from February 2020. The statistical analysis of the data was relevant to fit at best the parameters of the mathematical model, and to make short-term predictions of the spreading evolution in order to optimize the response of the local health system.

Mathematical Modelling of Lockdown Policy for COVID-19

In this paper, we extend the classic SIR model to find an optimal lockdown policy to balance between the economy and people’s health during the outbreak of COVID-19. In our model, we intend to solve a two phases optimisation problem: policymakers control the lockdown rate to maximise the overall welfare of the society; people in different health statuses take different decisions on their working hours and consumption to maximise their utility. We develop a novel method to estimate parameters for the model through various additional sources of data. We use the Cournot equilibrium to model people’s behaviour. The analysis of simulation results provides scientific suggestions for policymakers to make critical decisions on when to start the lockdown and how strong it should be during the whole period of the outbreak.

Epidemic Dynamics via Wavelet Theory and Machine Learning with Applications to Covid-19.

Simple SummaryUsing tools from both mathematics (especially wavelet theory) and computer science (machine learning), we present a general new method for modelling the evolution of epidemics which is not restricted to human populations. A crucial novel feature of our approach is that it significantly takes into account that an epidemic may take place in certain types of waves which cannot only be of a global as well as local nature, but can also occur at multiple different times and locations. In the particular case of the current Covid-19 pandemic, based on recent figures from the Johns Hopkins database we apply our model to France, Germany, Italy, the Czech Republic, as well as the US federal states New York and Florida, and compare it and its predictions to established as well as other recently developed forecasting methods and techniques.We introduce the concept of epidemic-fitted wavelets which comprise, in particular, as special cases the number of infectious individuals at time t in classical SIR models and their derivatives. We present a novel method for modelling epidemic dynamics by a model selection method using wavelet theory and, for its applications, machine learning-based curve fitting techniques. Our universal models are functions that are finite linear combinations of epidemic-fitted wavelets. We apply our method by modelling and forecasting, based on the Johns Hopkins University dataset, the spread of the current Covid-19 (SARS-CoV-2) epidemic in France, Germany, Italy and the Czech Republic, as well as in the US federal states New York and Florida.

Propagation of Epidemics Along Lines with Fast Diffusion.

It has long been known that epidemics can travel along communication lines, such as roads. In the current COVID-19 epidemic, it has been observed that major roads have enhanced its propagation in Italy. We propose a new simple model of propagation of epidemics which exhibits this effect and allows for a quantitative analysis. The model consists of a classical SIR model with diffusion, to which an additional compartment is added, formed by the infected individuals travelling on a line of fast diffusion. The line and the domain interact by constant exchanges of populations. A classical transformation allows us to reduce the proposed model to a system analogous to one we had previously introduced Berestycki et al. (J Math Biol 66:743–766, 2013) to describe the enhancement of biological invasions by lines of fast diffusion. We establish the existence of a minimal spreading speed, and we show that it may be quite large, even when the basic reproduction number R_0 is close to 1. We also prove here further qualitative features of the final state, showing the influence of the line.

On Equilibria Stability in an Epidemiological SIR Model with Recovery-dependent Infection Rate

We consider an epidemiological SIR model with an infection rate depending on the recovered population. We establish sufficient conditions for existence, uniqueness, and stability (local and global) of endemic equilibria and consider also the stability of the disease-free equilibrium. We show that, in contrast with classical SIR models, a system with a recovery-dependent infection rate can have multiple endemic stable equilibria (multistability) and multiple stable and unstable saddle points of equilibria. We establish conditions for the occurrence of these phenomena and illustrate the results with some examples.

Estimation of SIR model\u2019s parameters of COVID-19 in Algeria

BackgroundThe coronavirus disease 2019 (COVID-19) is reported in Algeria on February 25th, 2020. Since then, the number is still increasing leading to a total number of 36,699 cases and 1333 deaths on August 12th, 2020. Thus, comprehension of the epidemic curve is very important to predict its evolution and subsequently adapt the best prevention strategies. In this way, the current study was conducted to estimate the parameters of the classical SIR model and to predict the peak of the COVID-19 epidemic in Algeria using data from February 25th, 2020 to August 12th, 2020.ResultsResults showed that the peak of the epidemic will be reached on September 8th, 2021 and the total infected persons will exceed 800,000 cases at the end of the epidemic. Also, more than 15 million persons will be susceptible. The reproduction number (R0) is estimated at 1.23254.ConclusionThese results may be helpful for the Algerian authorities to adapt their strategies and may be taken into consideration in the future phase of discontainment.

Analyzing and forecasting COVID-19 pandemic in the Kingdom of Saudi Arabia using ARIMA and SIR models.

The novel coronavirus COVID‐19 is spreading all across the globe. By June 29, 2020, the World Health Organization announced that the number of cases worldwide had reached 9 994 206 and resulted in more than 499 024 deaths. The earliest case of COVID‐19 in the Kingdom of Saudi Arabia (KSA) was registered on March 2 in 2020. Since then, the number of infections as per the outcome of the tests increased gradually on a daily basis. The KSA has 182 493 cases, with 124 755 recoveries and 1551 deaths on June 29, 2020. There have been significant efforts to develop models that forecast the risks, parameters, and impacts of this epidemic. These models can aid in controlling and preventing the outbreak of these infections. In this regard, this article details the extent to which the infection cases, prevalence, and recovery rate of this pandemic are in the country and the predictions that can be made using the past and current data. The well‐known classical SIR model was applied to predict the highest number of cases that may be realized and the flattening of the curve afterward. On the other hand, the ARIMA model was used to predict the prevalence cases. Results of the SIR model indicate that the repatriation plan reduced the estimated reproduction number. The results further affirm that the containment technique used by Saudi Arabia to curb the spread of the disease was efficient. Moreover, using the results, close interaction between people, despite the current measures remains a great risk factor to the spread of the disease. This may force the government to take even more stringent measures. By validating the performance of the applied models, ARIMA proved to be a good forecasting method from current data. The past data and the forecasted data, as per the ARIMA model provided high correlation, showing that there were minimum errors.

Capacitated SIR Model with an Application to COVID-19

The classical SIR model and its variants have seen great success in understanding and predicting infectious diseases' spread. To better capture the COVID-19 outbreak, we extend the SIR model to incorporate the limited testing capacity and account for asymptomatic people. Specifically, based on the SIR model, we impose a testing capacity and differentiate the infected people into symptomatic and asymptomatic. In this capacitated SIR model, we show first- and second-order structural properties of one measure, the number of uninfected people, with respect to the testing capacity, degree of testing people without symptoms (or level of a hospital panic run), testing turnaround time, and contact tracing accuracy. In particular, we show that the total number of infected cases is concavely decreasing in the testing capacity; the policies to increase the testing capacity, and reduce the degree of a panic run or decrease the testing turnaround time, are complementary. We use the COVID-19 data to calibrate our model and point out its public policy implications.

City lockdown and nationwide intensive community screening are effective in controlling the COVID-19 epidemic: Analysis based on a modified SIR model.

BackgroundIn December 2019, an outbreak of COVID-19 epidemic occurred in Wuhan, China and infection spread rapidly around the world. To limit the rapid spread locally and nationwide, the Chinese government locked down Wuhan city on January 23 and began implementing nationwide intensive community screening on February 16.MethodTo assess the effectiveness of city lockdown and intensive community screening, we built a modified SIR model by introducing an α value into the classic SIR model. The α value represents the proportion of infected individuals who are not effectively isolated from susceptible individuals at a given time point.ResultsThe accuracy of the modified SIR model was validated using data from Guangdong and Zhejiang provinces. The lockdown of Wuhan city substantially reduced the α value for the rest of China excluding Hubei province, while only slightly reducing the α value for the city itself. Intensive community screening rapidly reduced the α value for Wuhan.ConclusionCity lockdown was efficient in controlling the spread of the epidemic from Wuhan to the rest of the country. Nationwide intensive community screening was extremely effective in suppressing the spread locally within Wuhan city. These experiences should urgently be shared with other countries to help halt the progressing worldwide pandemic.

System inference for the spatio-temporal evolution of infectious diseases: Michigan in the time of COVID-19.

We extend the classical SIR model of infectious disease spread to account for time dependence in the parameters, which also include diffusivities. The temporal dependence accounts for the changing characteristics of testing, quarantine and treatment protocols, while diffusivity incorporates a mobile population. This model has been applied to data on the evolution of the COVID-19 pandemic in the US state of Michigan. For system inference, we use recent advances; specifically our framework for Variational System Identification (Wang et al. in Comput Methods Appl Mech Eng 356:44–74, 2019; arXiv:2001.04816 [cs.CE]) as well as Bayesian machine learning methods.

Social distancing versus early detection and contacts tracing in epidemic management.

Different countries - and sometimes different regions within the same countries - have adopted different strategies in trying to contain the ongoing COVID-19 epidemic; these mix in variable parts social confinement, early detection and contact tracing. In this paper we discuss the different effects of these ingredients on the epidemic dynamics; the discussion is conducted with the help of two simple models, i.e. the classical SIR model and the recently introduced variant A-SIR (arXiv:2003.08720) which takes into account the presence of a large set of asymptomatic infectives.

F.O.G. and Teleworking: Some Labor Economics of Covid-19

FOG (« fear of going back to work ») is a new acronym reflecting workers stress to become contamined by COVID-19. In response, firms have been offering protections, extending teleworking as a way to continue to work during the pandemics. Leveraging a classical epidemiologic SIR model, we study how pandemics such as COVID-19 affect labor market, when the labor productivity is tied to the value of interactions, and under wage negotiations. Despite relatively schematic, our modelling highlights that workers participation during pandemics is dependent on reservation wages, and that the final dynamics are also critically dependent on a mix of health and wealth factors such as age, work interactions, workers power, and productivity of interactions. In general, teleworking may be a way to restore work participation, even if teleworking may be less productive, to the extent that the productivity gap can be compensated by a much higher protection of workers.

Toward Ultrametric Modeling of the Epidemic Spread

An ultrametric model of epidemic spread of infections based on the classical SIR model is proposed. Ultrametrics on a set of individuals is introduced based on their hierarchical clustering relative to the average time of infection contact. The general equations of the ultrametric SIR model are written down and their particular implementation using the p-adic parametrization is presented. A numerical analysis of the p-adic SIR model and a comparison of its behavior with the classical SIR model are performed. The concept of hierarchical isolation and the scenario of its management in order to reduce the level of epidemic spread is considered.

Spread of Infection on Homogeneous Tree

This paper concerns a probabilistic model of spread of infection on a homogeneous tree. The model is related to stochastic SIR model on graphs. We derive an integral equation for the distribution of the waiting time for an individual to get infected. In a special case the integral equation is equivalent to the Bernoulli differential equation and implies the classic SIR model under an appropriate scaling.

Next-Generation Matrices and Basic Reproductive Numbers for All Phases of the Coronavirus Disease

During the early phase of COVID-19, the transmissibility of the coronavirus disease was estimated using the classical SIR and SEIR models. However, with the advent of some controlling measures in its informative stages, these classical compartmental models have been ameliorated to provide accurate insight of the coronavirus disease. The paper seeks to derive the basic reproductive formulas for these improved models using the matrix approach. These transmissibility equations detail the dynamics of the coronavirus disease for all phases of the pandemic; either the infected population is on lockdown or not; either infectious persons are quarantined or not; either a vaccination program has been rolled out or yet to be rolled out. With the availability of data, any of these transmissibility equations could be adopted to report on the endemicity of the coronavirus disease.

The Effect of Stay-at-Home Orders on COVID-19 Infections in the United States

Background: In March and April 2020, public health authorities in the United States acted to mitigate transmission of and hospitalizations from the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which causes coronavirus disease 2019 (COVID-19). These actions were not coordinated at the national level, which raises the question of what might have happened if they were. It also creates an opportunity to use spatial and temporal variation to measure their effect with greater accuracy. Methods: We combine publicly available data sources on the timing of stay-at-home orders and daily confirmed COVID-19 cases at the county level in the United States (N = 132,048). We then derive from the classic SIR model a two-way fixed-effects model and apply it to the data with controls for unmeasured differences between counties and over time. This enables us to estimate the effect of stay-at-home orders while accounting for local variation in factors like health systems and demographics, and temporal variation in national mitigation actions, access to tests, or exposure to media reports that could influence the course of the disease. Findings: Mean county-level daily growth in COVID-19 infections peaked at 17.2% just before stay-at-home orders were issued. Two way fixed-effects regression estimates suggest that orders were associated with a 3.8 percentage point (95% CI 0.7 to 8.6) reduction in the growth rate after one week and an 8.6 percentage point (3.0 to 14.1) reduction after two weeks. By day 22 the reduction (18.2 percentage points, 12.3 to 24.0) had surpassed the growth at the peak, indicating that growth had turned negative and the number of new daily infections was beginning to decline. A hypothetical national stay-at-home order issued on March 13, 2020 when a national emergency was declared might have reduced cumulative county infections by 62.3%, and might have helped to reverse exponential growth in the disease by April 5. Interpretation: Although stay-at-home orders impose great costs to society, delayed responses and piecemeal application of these orders generate similar costs without obtaining the full potential benefits suggested by this analysis. The results here suggest that a coordinated nationwide stay-at-home order may have reduced by hundreds of thousands the current number of infections and by thousands the total number of deaths from COVID-19. Future efforts in the United States and elsewhere to control pandemics should coordinate stay-at-home orders at the national level, especially for diseases for which local spread has already occurred and testing availability is delayed. Since stay-at-home orders reduce infection growth rates, early implementation when infection counts are still low would be most beneficial. Funding Statement: None. Declaration of Interests: The authors declare no competing interests.