Classical SIR Model Research Articles

Protection Strategy against an Epidemic Disease on Edge-Weighted Graphs Applied to a COVID-19 Case.

Simple SummaryInfectious diseases have been part of human history. Countless epidemics have produced high mortality rates in vulnerable populations. With the understanding of the spread of these types of diseases, population groups have been able to adapt and better cope with infections. Given the COVID-19 pandemic, one of the strategies used is the modeling of infectious diseases with the aim of establishing protection measures for people and stopping the spread of the epidemic. Our study evaluates protection strategies through infectious disease modeling with COVID-19 data in a commune in Chile. The results of the simulations indicate that the model generates important protection for the population by recognizing the super-propagating people (bridge nodes). This type of protection can be key in the fight against COVID-19.Among the diverse and important applications that networks currently have is the modeling of infectious diseases. Immunization, or the process of protecting nodes in the network, plays a key role in stopping diseases from spreading. Hence the importance of having tools or strategies that allow the solving of this challenge. In this paper, we evaluate the effectiveness of the DIL-W ranking in immunizing nodes in an edge-weighted network with 3866 nodes and 6,841,470 edges. The network is obtained from a real database and the spread of COVID-19 was modeled with the classic SIR model. We apply the protection to the network, according to the importance ranking list produced by DIL-W, considering different protection budgets. Furthermore, we consider three different values for ; in this way, we compare how the protection performs according to the value of .

Modeling analytics in COVID-19: prediction, prevention, control, and evaluation

The outbreak of COVID-19 has attracted attention from all around the world. Governments and institutions have adopted ways to fight COVID-19, but its prevalence is still strong. The SIR model has important reference value for the novel coronavirus epidemic, offering both preventive measures and the ability to predict future trends. Based on an analysis of the classical epidemiological SIR model along with key parameters, this paper aims to analyze the patterns of COVID-19, to discuss potential anti-COVID-19 measures, and to explain why we need to conduct appropriate measures against COVID-19. The use of the SIR model can play an important role in public health emergencies. Among the parameters of the SIR model, the contact ratio and the reproduction ratio are the factors that have the potential to mitigate the consequences of COVID-19. Anti-COVID-19 measures include wearing a mask, washing one’s hands, keeping social distance, and staying at home if possible.

Kinetic models of distribution and testing of epidemic diseases in an isolated contingent

Generalization of the classical SIR model of infection propagation are considered. The generalization of the models is carried out in two directions: 1) accounting for testing (monitoring), which is carried out in practice in all countries, by expanding SIR model by including an observation model; 2) accounting for the lack of reliable information about the state of the population is modeled by random processes (randomization), the statistics of which are determined by Kalman–Bucy estimation algorithms. The resulting model is mathematically more correct and stable, which makes it possible to obtain more reliable estimates of infection processes. The estimation model obtained by extension is described by equations corresponding to the regularization method for solving incorrect problems. The resulting system of equations simultaneously with the state estimation allows us to find the estimation error. To assess the infection process, when the number of recovered and deceased people is generally small compared to the number of susceptible to infection and infected people, a model is used that generalizes the Lotka-Volterra model, the natural course of the epidemic process. On the basis of the obtained model, the solution of reference problems is considered. A solution is obtained for the optimal estimate and its error at the initial stage of infection, when it can be assumed that the observed number of infected people increases linearly as a function of time, and the estimation error at the initial time is large relative to the number of really infected people. Obtaining reliable observations is the basis for making effective decisions to combat the epidemic. The developed model reflects the uncertainty that exists in practice in assessing the level of the epidemic condition of the population. The reference problem is considered, when the estimate of the number of infected people is a function that varies with time, although in reality the number of infected people remains constant over time (stationary state). Thus, the model describes the effect of pseudo epidemic, which can exist in an amount with a constant total number of all groups.

Mathematical modeling of COVID-19 in India and its states with optimal control.

A pandemic is an epidemic spread over a huge geographical area. COVID-19 is 5hbox {th} such pandemic documented after 1918 flu pandemic. In this work, we frame a mathematical epidemic model taking inspiration from the classic SIR model and develop a compartmental model with ten compartments to study the coronavirus dynamics in India and three of its most affected states, namely, Maharashtra, Karnataka, and Tamil Nadu, with inclusion of factors related to face mask efficacy, contact tracing, and testing along with quarantine and isolation. We fit the developed model and estimate optimum values of disease transmission rate, detection rate of undetected asymptomatic, and the same of undetected symptomatic. A sensitivity analysis is presented stressing on the importance of higher face mask usage, rapid testing, and contact tracing for curbing the disease spread. An optimal control analysis is performed with two control parameters to study the increase and decrease of the infected population with and without control. This study suggests that improved and rapid testing will help in identifying more infectives, thereby contributing in the decline of disease transmission rate. Optimal control analysis results on stressing on the importance of abiding by strict usage of face mask and social distancing for drastic decrease in number of infections. Time series behaviour of the symptomatic, asymptomatic, and hospitalized population is studied for a range of parameters, resulting in thorough understanding of significance of different parameters.

Superspreading of airborne pathogens in a heterogeneous world

Epidemics are regularly associated with reports of superspreading: single individuals infecting many others. How do we determine if such events are due to people inherently being biological superspreaders or simply due to random chance? We present an analytically solvable model for airborne diseases which reveal the spreading statistics of epidemics in socio-spatial heterogeneous spaces and provide a baseline to which data may be compared. In contrast to classical SIR models, we explicitly model social events where airborne pathogen transmission allows a single individual to infect many simultaneously, a key feature that generates distinctive output statistics. We find that diseases that have a short duration of high infectiousness can give extreme statistics such as 20% infecting more than 80%, depending on the socio-spatial heterogeneity. Quantifying this by a distribution over sizes of social gatherings, tracking data of social proximity for university students suggest that this can be a approximated by a power law. Finally, we study mitigation efforts applied to our model. We find that the effect of banning large gatherings works equally well for diseases with any duration of infectiousness, but depends strongly on socio-spatial heterogeneity.

Optimal lockdown policy for vaccination during COVID-19 pandemic

As the COVID-19 spreads across the world, many nations impose lockdown measures at the early stage of the pandemic to prevent the spread of the disease. Controversy surrounds the lockdown as it is a choice between economic freedom and public health. The ultimate solution to a pandemic is to vaccinate a massive population to achieve herd immunity. However, the whole vaccination programme is a long and complicated process. The virus and the vaccine will coexist for quite a long time. How to gradually ease the lockdown based on vaccination progress is an important question, as both economic and epidemiological issues are involved. In this paper, we extend the classic SIR model to find optimal decision to balance between economy and public health in the process of vaccination rollout. The model provides an approach of vaccine value estimation. Our results provide scientific suggestion for policymakers to make important decisions on how to gradually relax the strength for the lockdown over the entire vaccination cycle.

An SIR-type epidemiological model that integrates social distancing as a dynamic law based on point prevalence and socio-behavioral factors

Modeling human behavior within mathematical models of infectious diseases is a key component to understand and control disease spread. We present a mathematical compartmental model of Susceptible–Infectious–Removed to compare the infected curves given by four different functional forms describing the transmission rate. These depend on the distance that individuals keep on average to others in their daily lives. We assume that this distance varies according to the balance between two opposite thrives: the self-protecting reaction of individuals upon the presence of disease to increase social distancing and their necessity to return to a culturally dependent natural social distance that occurs in the absence of disease. We present simulations to compare results for different society types on point prevalence, the peak size of a first epidemic outbreak and the time of occurrence of that peak, for four different transmission rate functional forms and parameters of interest related to distancing behavior, such as: the reaction velocity of a society to change social distance during an epidemic. We observe the vulnerability to disease spread of close contact societies, and also show that certain social distancing behavior may provoke a small peak of a first epidemic outbreak, but at the expense of it occurring early after the epidemic onset, observing differences in this regard between society types. We also discuss the appearance of temporal oscillations of the four different transmission rates, their differences, and how this oscillatory behavior is impacted through social distancing; breaking the unimodality of the actives-curve produced by the classical SIR-model.

Small-world effects in a modified epidemiological model with mutation and permanent immune mechanism.

Pandemic with mutation and permanent immune spreading in a small-world network described is studied by a modified SIR model, with consideration of mutation-immune mechanism. First, a novel mutation-immune model is proposed to modify the classical SIR model to simulate the transmission of mutable viruses that can be permanently immunized in small-world networks. Then, the influences of the size, coordination number and disorder parameter of the small-world network on the spread of the epidemic are analyzed in detail. Finally, the influences of mutation cycle and infection rate on epidemic transmission in small-world network are investigated further. The results show that the structure of the small-world network and the virus mutation cycle have an important impact on the spread of the epidemic. For viruses that can be permanently immunized, virus mutation is equivalent to making the immune cycle of human beings from infinite to finite. The dynamical behavior of the modified SIR epidemic model changes from an irregular, low-amplitude evolution at small disorder parameter to a spontaneous state of wide amplitude oscillations at large disorder parameter. Moreover, similar transition can also be found in increasing mutation cycle parameter. The maximum valid variation mutation decreases with the increase of disorder parameter and coordination number, but increase with respect to system size. In addition above, as the infection rate increases, the fraction of the infected increases and then decreases. As the mutation cycle increases, the time-average fraction of the infected and the infection rate corresponding to the maximum time-average fraction of the infected also decrease. As one conclusion, the results could give a deep understanding Pandemic with mutation and permanent immune spreading, from viewpoint of small-world network.

Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19.

The understanding of infectious diseases is a priority in the field of public health. This has generated the inclusion of several disciplines and tools that allow for analyzing the dissemination of infectious diseases. The aim of this manuscript is to model the spreading of a disease in a population that is registered in a database. From this database, we obtain an edge-weighted graph. The spreading was modeled with the classic SIR model. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics. Moreover, a deterministic approximation is provided. With database COVID-19 from a city in Chile, we analyzed our model with relationship variables between people. We obtained a graph with 3866 vertices and 6,841,470 edges. We fitted the curve of the real data and we have done some simulations on the obtained graph. Our model is adjusted to the spread of the disease. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics, in this case with real data of COVID-19. This valuable information allows us to also include/understand the networks of dissemination of epidemics diseases as well as the implementation of preventive measures of public health. These findings are important in COVID-19’s pandemic context.

On Assessing the Impact of Coronavirus Epidemic in Russia on Population Incomes

The paper discusses an approach to assessing the impact of the coronavirus epidemic in Russia on economic efficiency and, as a result, on the monetary income of the country's population. The approach used is based on the application of the methodology of mathematical modeling. Based on the analysis of statistical information, it is shown that there is a correlation between the dynamics of the average per capita income of the population and GDP. To assess the dynamics of GDP, a dynamic model of the impact of restrictive measures aimed at curbing the spread of the coronavirus epidemic on macroeconomic efficiency is constructed. The main hypothesis of the model is that the main factor affecting the efficiency of the economy is the productivity of workers who create GDP. In the constructed model, all employees are divided into three groups. The first group – ​workers whose activities were not affected by the coronavirus; the second group-workers whose productivity decreased due to the coronavirus; the third group-workers whose productivity fully or partially recovered after the easing of restrictive measures. As a result, the dynamics of GDP is determined by a system of three ordinary differential equations with parameters depended on the epidemiological situation. To assess the indicators that characterize the spread of infection and affect the parameters of the macroeconomic efficiency model, a discrete modification of the classical SIR-model of the epidemic with piecewise constant parameters is constructed. This model allowed us to estimate the dynamics of the average for the four day values of the basic reproductive numbers and other indicators of spread of infection through the use of official statistical information in the base period, and to perform scenario calculations for the development of the epidemic in Moscow and beyond until July 2021 Developed modification of the SIR model allows for its clarification with regard to the influence of vaccination on the dynamics of epidemiological process.

The two-step exponential decay reaction network: analysis of the solutions and relation to epidemiological SIR models with logistic and Gompertz type infection contact patterns

We mathematically analyze the solutions to the dynamical system induced by the two-step exponential (growth-)decay (2SED) reaction network involving three species and two rate parameters. We study the influence of the rate parameters on the shape of the solutions. We compare the latter to those of the classic Kermack–McKendrick epidemiological SIR model. We then discuss the similarities and differences between the 2SED and the SIR models from the perspective of chemical reaction network theory (CRNT), as well as from epidemiological modelling view-point. The CRNT approach suggests that the classical SIR model, based on the logistic reaction mechanism, describes well epidemic events related to diseases spreading via a ‘one-to-one’ contact pattern between individuals. On the other side, the 2SED model can be used to simulate epidemic data coming from non-communicable diseases. Our comparative analysis naturally suggests the formulation of a SIR-type model, which is situated between the classic SIR model and the 2SED model, such that the logistic ‘one-to-one’ contact mechanism is replaced by a catalytic (Gompertzian) one. The proposed G-SIR model can be considered as an intermediate step between the SIR and the 2SED models. We compare the shapes of the solutions to the three discussed models and formulate a hypothesis that relates the characteristics of the solution shapes to the model reaction mechanism, resp. to the contact patterns of the particular disease.

An adaptive social distancing SIR model for COVID-19 disease spreading and forecasting

Abstract Recently, various mathematical models have been proposed to model COVID-19 outbreak. These models are an effective tool to study the mechanisms of coronavirus spreading and to predict the future course of COVID-19 disease. They are also used to evaluate strategies to control this pandemic. Generally, SIR compartmental models are appropriate for understanding and predicting the dynamics of infectious diseases like COVID-19. The classical SIR model is initially introduced by Kermack and McKendrick (cf. (Anderson, R. M. 1991. “Discussion: the Kermack–McKendrick Epidemic Threshold Theorem.” Bulletin of Mathematical Biology 53 (1): 3–32; Kermack, W. O., and A. G. McKendrick. 1927. “A Contribution to the Mathematical Theory of Epidemics.” Proceedings of the Royal Society 115 (772): 700–21)) to describe the evolution of the susceptible, infected and recovered compartment. Focused on the impact of public policies designed to contain this pandemic, we develop a new nonlinear SIR epidemic problem modeling the spreading of coronavirus under the effect of a social distancing induced by the government measures to stop coronavirus spreading. To find the parameters adopted for each country (for e.g. Germany, Spain, Italy, France, Algeria and Morocco) we fit the proposed model with respect to the actual real data. We also evaluate the government measures in each country with respect to the evolution of the pandemic. Our numerical simulations can be used to provide an effective tool for predicting the spread of the disease.

Modeling and analysis of the spread of the COVID-19 pandemic using the classical SIR model

Modeling and analysis of the spread of the COVID-19 pandemic using the classical SIR model

Exactly solvable SIR models, their extensions and their application to sensitive pandemic forecasting.

The classic SIR model of epidemic dynamics is solved completely by quadratures, including a time integral transform expanded in a series of incomplete gamma functions. The model is also generalized to arbitrary time-dependent infection rates and solved explicitly when the control parameter depends on the accumulated infections at time t. Numerical results are presented by way of comparison. Autonomous and non-autonomous generalizations of SIR for interacting regions are also considered, including non-separability for two or more interacting regions. A reduction of simple SIR models to one variable leads us to a generalized logistic model, Richards model, which we use to fit Mexico’s COVID-19 data up to day number 134. Forecasting scenarios resulting from various fittings are discussed. A critique to the applicability of these models to current pandemic outbreaks in terms of robustness is provided. Finally, we obtain the bifurcation diagram for a discretized version of Richards model, displaying period doubling bifurcation to chaos.

A SIR-type model describing the successive waves of COVID-19.

It is well-known that the classical SIR model is unable to make accurate predictions on the course of illnesses such as COVID-19. In this paper, we show that the official data released by the authorities of several countries (Italy, Spain and The USA) regarding the expansion of COVID-19 are compatible with a non-autonomous SIR type model with vital dynamics and non-constant population, calibrated according to exponentially decaying infection and death rates. Using this calibration we construct a model whose outcomes for most relevant epidemiological paramenters, such as the number of active cases, cumulative deaths, daily new deaths and daily new cases (among others) fit available real data about the first and successive waves of COVID-19. In addition to this, we also provide predictions on the evolution of this pandemic in Italy and the USA in several plausible scenarios.

Impacts of a Delayed and Slow-Paced Vaccination on Cases and Deaths During the COVID-19 Pandemic: A Modelling Study

Background: In Brazil, vaccination has always been cutting across party political and ideological lines, which have delayed its start and brought the whole process into disrepute. Such divergences put the immunisation of the population in the background and create additional hurdles beyond the pandemic, mistrust and scepticism over vaccines.Methods: We conduct a mathematical modelling study to analyse the impacts of late vaccination and with slowly increasing coverage, as well as how harmful it would be if part of the population refused to get vaccinated or missed the second dose. We analyse data from confirmed cases, deaths caused by COVID-19, and vaccination in the state of Rio de Janeiro in the period between March 10, 2020, and October 27, 2021. The classical SIR model is extended to consider the effect of vaccination (efficacy, interval between doses, and vaccination rate) and data sets are regularised using Gaussian Process Regression. The model parameter distributions are estimated using Bayesian inference, aiming to obtain credible intervals in the simulations.Findings: We estimate that if the start of vaccination had been 30 days earlier, combined with efforts to drive vaccination rates up, 31,657 (25,801–35,117) deaths could have been averted. Our results also indicate that the slow pace of vaccination and the low demand for the second dose could cause a resurgence of cases as early as 2022.Interpretation: The government’s inaction and lack of a strategic plan to fight the pandemic meant that vaccination started late, leading to thousands of deaths that could have been prevented. Even when reaching the expected vaccination coverage for the first dose, it is still challenging to increase adherence to the second dose and maintain a high vaccination rate to avoid new outbreaks. Funding Information: Carlos Chagas Filho Foundation for Supporting Research in the State of Rio de Janeiro (FAPERJ) and Brazilian National Council for Scientific and Technological Development (CNPq).Declaration of Interests: The authors declare no competing interests.

Global proprieties of a delayed epidemic model with partial susceptible protection.

In the case of an epidemic, the government (or population itself) can use protection for reducing the epidemic. This research investigates the global dynamics of a delayed epidemic model with partial susceptible protection. A threshold dynamics is obtained in terms of the basic reproduction number, where for R0<1 the infection will extinct from the population. But, for R0>1 it has been shown that the disease will persist, and the unique positive equilibrium is globally asymptotically stable. The principal purpose of this research is to determine a relation between the isolation rate and the basic reproduction number in such a way we can eliminate the infection from the population. Moreover, we will determine the minimal protection force to eliminate the infection for the population. A comparative analysis with the classical SIR model is provided. The results are supported by some numerical illustrations with their epidemiological relevance.

Homotopy Perturbation Method for Mathematical Modelling of Dengue Fever

Dengue fever is a viral mosquito-transmitted infection that has become a major international infection in recent years. The leading cause of disease and death in tropical and sub-tropical regions is a public health concern. Models from mathematical epidemiology, such as the classical SIR-model and its variants, are used to characterize the spread of Dengue in a given population. The mathematical modelling of Dengue Fever is formulated into a first-order nonlinear differential equation. Homotopy Perturbation Method approaches the analytical solution of the model (HPM), and also simulation results are identified. Finally, the analytical solutions, simulation results are compared, and satisfactory agreement is noted.

Re-examination of the impact of some non-pharmaceutical interventions and media coverage on the COVID-19 outbreak in Wuhan.

In this paper, based on the classic Kermack-McKendrick SIR model, we propose an ordinary differential equation model to re-examine the COVID-19 epidemics in Wuhan where this disease initially broke out. The focus is on the impact of all those major non-pharmaceutical interventions (NPIs) implemented by the local public healthy authorities and government during the epidemics. We use the data publicly available and the nonlinear least-squares solver lsqnonlin built in MATLAB to estimate the model parameters. Then we explore the impact of those NPIs, particularly the timings of these interventions, on the epidemics. The results can help people review the responses to the outbreak of the COVID-19 in Wuhan, while the proposed model also offers a framework for studying epidemics of COVID-19 and/or other similar diseases in other places, and accordingly helping people better prepare for possible future outbreaks of similar diseases.

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.