Classical SEIR Model Research Articles

Impact of trans-regional travel-related testing on epidemic spreading

The convenient transport network enables people to travel quickly and frequently across regions, but it also brings serious challenges to the emergency management of infectious diseases. This work extends the classical SEIR model by incorporating trans-regional population movements and contact infections during travel. It also includes cross-regional population testing and treatment measures to curb the spread of infectious diseases across regions. According to these characteristics, a Net-SEIHR model is established and its diffusion properties are deduced. Further, the cooperative allocation problem of limited detection resources and its control strategy are discussed. Numerical experiments show that the control measures can reduce the speed of trans-regional transmission of infectious diseases and the peak of treatment. In the case of limited testing resources, it is necessary to strengthen the detection of people in the inflow regions.

Study on intentional control strategy of COVID-19

With the ongoing evolution of the novel coronavirus pathogen and continuous improvements in our social environment, the mortality rate of COVID-19 is on a decline. In response to this, we introduce an adaptive control strategy known as intentional control, which offers cost-efficiency and superior control effectiveness. The classical SEIR model faces limitations in accurately representing close contacts and sub-close contacts and fails to distinguish their varying levels of infectivity. To address this, our study modifies the classical model by incorporating close contact (E) and a sub-close contact (E2) while reworking the infectious mechanism. Once the model is formulated, we employ various statistical methods to identify crucial parameters, including R2, adjusted R2, and standard deviation. For disease control, we implement an intentional control program with four distinct grades. We develop and apply a scheme in MATLAB for our proposed model, generating diverse simulation results based on realistic parameter values for discussion. Additionally, we explore a range of strategy combinations to differentiate their effectiveness under various social conditions, aiming to identify an optimal approach. Comparing the intentional control strategy to random control, our findings consistently demonstrate the superiority of intentional control across all scenarios. Furthermore, the results indicate that our approach better aligns with the characteristics of the novel coronavirus, characterized by an “extremely low fatality rate and strong infectivity,” while offering detailed insights into the transmission dynamics among different compartments.

Port congestion under the COVID-19 pandemic: The simulation-based countermeasures

As the hub connecting global production and consumption, ports play a pivotal role in the global maritime supply chain system. The outbreak of the COVID-19 pandemic and its rapid global spread have caused significant port congestion, resulting in skyrocketing freight rates and posing a serious threat to the smooth operation of international trade. Based on the impact of the COVID-19 pandemic, we have developed a novel container port congestion evaluation model by integrating the classical SEIR model, using the system dynamics (SD) theory. In this study, we simulate the impact of different port congestion countermeasures using data from Ningbo Zhoushan Port. The results demonstrate that implementing smart regulations at a high level of epidemic prevention and control is the most efficient approach to swiftly eliminate port congestion. In addition, it is worth mentioning that in the scenarios with a low level of epidemic prevention and control, increasing port investment to optimize smart ports can yield even better effects. Furthermore, we have discussed the impact of factors such as the ratio of infected people, the number of contacted people, and epidemic prevention and control capacity on port congestion index. Hence, we propose that the implementation of advanced smart regulation systems can enhance port operational efficiency and effectively alleviate port congestion. Additionally, amidst the normalization of the COVID-19 pandemic, which has yet to disappear, it is crucial to maintain a vigilant approach to prevention and control to ensure the seamless operation of the port industry.

Modelling the COVID-19 pandemic: Focusing on the case of Greece

The SARS-CoV-2 infection (COVID-19) pandemic created an unprecedented chain of events at a global scale, with European counties initially following individual pathways on the confrontation of the global healthcare crisis, before organizing coordinated public vaccination campaigns, when proper vaccines became available. In the meantime, the viral infection outbreaks were determined by the inability of the immune system to retain a long-lasting protection as well as the appearance of SARS-CoV-2 variants with differential transmissibility and virulence. How do these different parameters regulate the domestic impact of the viral epidemic outbreak? We developed two versions of a mathematical model, an original and a revised one, able to capture multiple factors affecting the epidemic dynamics. We tested the original one on five European countries with different characteristics, and the revised one in one of them, Greece. For the development of the model, we used a modified version of the classical SEIR model, introducing various parameters related to the estimated epidemiology of the pathogen, governmental and societal responses, and the concept of quarantine. We estimated the temporal trajectories of the identified and overall active cases for Cyprus, Germany, Greece, Italy and Sweden, for the first 250 days. Finally, using the revised model, we estimated the temporal trajectories of the identified and overall active cases for Greece, for the duration of the 1230 days (until June 2023). As shown by the model, small initial numbers of exposed individuals are enough to threaten a large percentage of the population. This created an important political dilemma in most countries. Force the virus to extinction with extremely long and restrictive measures or merely delay its spread and aim for herd immunity. Most countries chose the former, which enabled the healthcare systems to absorb the societal pressure, caused by the increased numbers of patients, requiring hospitalization and intensive care.

A simulation of undiagnosed population and excess mortality during the COVID-19 pandemic

Whereas the extent of outbreak of COVID-19 is usually accessed via the number of reported cases and the number of patients succumbed to the disease, the officially recorded overall excess mortality numbers during the pandemic waves, which are significant and often followed the rise and fall of the pandemic waves, put a question mark on the above methodology. Gradually it has been recognized that estimating the size of the undiagnosed population (which includes asymptomatic cases and symptomatic cases but not reported) is also crucial. Here we used the classical mathematical SEIR model having an additional compartment, that is the undiagnosed group in addition to the susceptible, exposed, diagnosed, recovered and deceased groups, to link the undiagnosed COVID-19 cases to the reported excess mortality numbers and thereby try to know the actual size of the disease outbreak. The developed model wase successfully applied to relevant COVID-19 waves in USA (initial months of 2020), South Africa (mid of 2021) and Russia (2020–21) when a large discrepancy between the reported COVID-19 mortality and the overall excess mortality had been noticed.

MATHEMATICAL ANALYSIS OF SVEIQR MODEL FOR COVID-19

In this paper, we have proposed an SVEIQR compartmental model modifying the classical SEIR model with inclusion of vaccinated and quarantined classes to explain the COVID-19 outbreak mathematically. We have calibrated our model with the daily COVID-19 data reported by the WHO coronavirus dashboard. To observe the disease dynamics of COVID-19, a detailed stability analysis of the proposed SVEIQR model is carried out. Our results show that the disease free equilibrium (DFE) is stable if the basic reproduction number is less than unity and unstable otherwise. Moreover, endemic equilibrium (EE) is found to be stable when certain restrictions hold. The expression for effective reproduction number has been derived analytically and its value is calculated based on the reported cases. Sensitivity analysis of effective reproduction number is performed employing PRCCs and Latin hypercube scheme. We have compared short-term and long-term transmission dynamics of COVID-19 for India with different levels of vaccination and without control strategies. The impact of different degrees of control interventions is ascertained with the numerical simulation of the model.

Dynamics and Transmission of COVID-19 Epidemic in Malaysia: Atangana-Baleanu Fractional Model

The purpose of this artical is to specify the passage and mathematical model of COVID-19. This infectious disease COVID-19 originated in China and has affected the whole world. In the present study, we have considered, SEIR model to evaluate the spread of COVID-19 in Malaysia. In this article, we have explored the gesture of COVID-19 accurately by applying the newly advanced fractional operator of Atangana-Baleanu on the classical SEIR model. Based on (WHO) reported data in this study, we discussed the corona virus's behavior in Malaysia. We have taken the available infection cases from 1st May 2020 to 14th May 2020. We have matched our problem to Malaysia's actual data, which shows a firm agreement with the actual data. From the information collected from WHO, we have found the basic reproduction number . Moreover, stability analysis at DFE (Disease Free Equilibrium) and EE (Endemic Equilibrium) is accomplished for the given model to detect the model's gesture and transportation. The given fractional model has been solved numerically using MATLAB software. The influence of many parameters specifically has been shown graphically and discussed in detail. It is to be noted that in the present study, we have approximated the escalation of disease for the up-coming 300 days.

A Multistage Time-Delay Control Model for COVID-19 Transmission

With the transmission of the COVID-19 epidemic at home and abroad, this paper considers the spread process in China, improves the classic epidemic SEIR model, and establishes a multistage time-delay control model (MTCM) for COVID-19 transmission. The MTCM divides the spread of COVID-19 into three periods: the outbreak period, the control period and the steady period. The classical SEIR model, the improved SEQIR model and the SEQIR Ⅱ model correspond to the three periods. The classical SEIR model was adopted for the outbreak period and yielded results that were consistent with the observed early propagation of COVID-19 transmission. In the control period, adding isolation measures and a time delay to the MTCM and adjusting the rates yielded a better simulation effect. In the steady period, the focus of consideration is the number of new patients, population movement (in-migration and out-migration of the population) and patient classification (symptomatic and asymptomatic patients). The MCTM was used for simulation, and the comparison results revealed that the simulated data of the MCTM (improved SEQIR model) and the actual data are similar in the control period. The control policy of isolation measures is effective. New infections, population flow and patients with symptomatic or asymptomatic symptoms are more consistent with the steady period characteristics. The multi-stage time-delay control model for COVID-19 transmission provides theoretical methods and good prevention and control measures for future epidemic policy formulation.

Adding a reaction-restoration type transmission rate dynamic-law to the basic SEIR COVID-19 model.

The classical SEIR model, being an autonomous system of differential equations, has important limitations when representing a pandemic situation. Particularly, the geometric unimodal shape of the epidemic curve is not what is generally observed. This work introduces the βSEIR model, which adds to the classical SEIR model a differential law to model the variation in the transmission rate. It considers two opposite thrives generally found in a population: first, reaction to disease presence that may be linked to mitigation strategies, which tends to decrease transmission, and second, the urge to return to normal conditions that pulls to restore the initial value of the transmission rate. Our results open a wide spectrum of dynamic variabilities in the curve of new infected, which are justified by reaction and restoration thrives that affect disease transmission over time. Some of these dynamics have been observed in the existing COVID-19 disease data. In particular and to further exemplify the potential of the model proposed in this article, we show its capability of capturing the evolution of the number of new confirmed cases of Chile and Italy for several months after epidemic onset, while incorporating a reaction to disease presence with decreasing adherence to mitigation strategies, as well as a seasonal effect on the restoration of the initial transmissibility conditions.

Assessing vaccination priorities for different ages and age-specific vaccination strategies of COVID-19 using an SEIR modelling approach.

In the past year, the global epidemic situation is still not optimistic, showing a trend of continuous expansion. With the research and application of vaccines, there is an urgent need to develop some optimal vaccination strategies. How to make a reasonable vaccination strategy to determine the priority of vaccination under the limited vaccine resources to control the epidemic and reduce human casualties? We build a dynamic model with vaccination which is extended the classical SEIR model. By fitting the epidemic data of three countries—China, Brazil, Indonesia, we have evaluated age-specific vaccination strategy for the number of infections and deaths. Furthermore, we have evaluated the impact of age-specific vaccination strategies on the number of the basic reproduction number. At last, we also have evaluated the different age structure of the vaccination priority. It shows that giving priority to vaccination of young people can control the number of infections, while giving priority to vaccination of the elderly can greatly reduce the number of deaths in most cases. Furthermore, we have found that young people should be mainly vaccinated to reduce the number of infections. When the emphasis is on reducing the number of deaths, it is important to focus vaccination on the elderly. Simulations suggest that appropriate age-specific vaccination strategies can effectively control the epidemic, both in terms of the number of infections and deaths.

Seroprevalence and attainment of herd immunity against SARS CoV-2: A modelling study.

Objective:The present study aims to predict the likelihood of and likely time required to attain herd immunity against COVID-19 in New Delhi due to natural infection.Method:An ODE-based mathematical model was constructed by extending the classical SEIR model to predict the seroprevalence rate. We estimated the parameter values for Delhi using available data (reported cases and the seroprevalence rate) and used them for future prediction. Also, changes in the seroprevalence rate with different possibilities of reinfection were predicted.Results:Maximum seroprevalence rate obtained through our model is 31.65% and also a reduction in the seroprevalence rate was observed for the upcoming one month (month of January, 2021) due to the reduced transmission rate. After increasing the transmission rate to the value same as the third wave in New Delhi, we obtained a maximum value of 54.96%. This maximum value significantly decreased with the reduction in the reinfection possibilities. Also, a little impact of the duration of persistence of antibodies, 180 vs 105 days, was observed on the maximum seroprevalence.Conclusion:This modelling study suggests that natural infection alone, as gauged by serial sero-surveys, may not result in attainment of herd immunity in the state of Delhi.

SIMULATION OF PANDEMIC DEVELOPMENT PROCESSES

The study is devoted to the development of mathematical models for predicting the incidence of COVID-19 using simulation tools. Methods of modeling the spread of infectious diseases have been studied since the beginning of the twentieth century. In the classical SIR model, the entire population was divided into three parts: «susceptible – infected – recovered». In the 1980s, the intermediate state – «infected, but not yet infectious» was added to the classical model. This model is called SEIR. The spread of the epidemic in both models is described by a system of differential equations, which is solved using numerical methods.
 Significantly new simulation modelling approaches are necessary to forecast evolution of the coronavirus pandemic. This study based on the classical SEIR model («susceptible – infected but not yet infectious – infected – recovered») suggests several methods for developing models: with a long plateau, with two or more waves. The modern tool used for the simulation is Anylogic 8.

Effect of Quarantine Strategies in a Compartmental Model with Asymptomatic Groups.

We present an epidemiological model, which extend the classical SEIR model by accounting for the presence of asymptomatic individuals and the effect of isolation of infected individuals based on testing. Moreover, we introduce two types of home quarantine, namely gradual and abrupt one. We compute the equilibria of the new model and derive its reproduction number. Using numerical simulations we analyze the effect of quarantine and testing on the epidemic dynamic. Given a constraint that limits the maximal number of simultaneous active cases, we demonstrate that the isolation rate, which enforces this constraint, decreases with the increasing testing rate. Our simulations show that massive testing allows to control the infection spread using a much lower isolation rate than in the case of indiscriminate quarantining. Finally, based on the effective reproduction number we suggest a strategy to manage the epidemic. It consists in introducing abrupt quarantine as well as relaxing the quarantine in such a way that the epidemic remains under control and further waves do not occur. We analyze the sensitivity of the model dynamic to the quarantine size, timing and strength of the restrictions.

RLIM: a recursive and latent infection model for the prediction of US COVID-19 infections and turning points.

Initially found in Hubei, Wuhan, and identified as a novel virus of the coronavirus family by the WHO, COVID-19 has spread worldwide at exponential speed, causing millions of deaths and public fear. Currently, the USA, India, Brazil, and other parts of the world are experiencing a secondary wave of COVID-19. However, the medical, mathematical, and pharmaceutical aspects of its transmission, incubation, and recovery processes are still unclear. The classical susceptible–infected–recovered model has limitations in describing the dynamic behavior of COVID-19. Hence, it is necessary to introduce a recursive, latent model to predict the number of future COVID-19 infection cases in the USA. In this article, a dynamic recursive and latent infection model (RLIM) based on the classical SEIR model is proposed to predict the number of COVID-19 infections. Given COVID-19 infection and recovery data for a certain period, the RLIM is able to fit current values and produce an optimal set of parameters with a minimum error rate according to actual reported numbers. With these optimal parameters assigned, the RLIM model then becomes able to produce predictions of infection numbers within a certain period. To locate the turning point of COVID-19 transmission, an initial value for the secondary infection rate is given to the RLIM algorithm for calculation. RLIM will then calculate the secondary infection rates of a continuous time series with an iterative search strategy to speed up the convergence of the prediction outcomes and minimize the maximum square errors. Compared with other forecast algorithms, RLIM is able to adapt the COVID-19 infection curve faster and more accurately and, more importantly, provides a way to identify the turning point in virus transmission by searching for the equilibrium between recoveries and new infections. Simulations of four US states show that with the secondary infection rate omega initially set to 0.5 within the selected latent period of 14 days, RLIM is able to minimize this value at 0.07 and reach an equilibrium condition. A successful forecast is generated using New York state’s COVID-19 transmission, in which a turning point is predicted to emerge on January 31, 2021.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11071-021-06520-1.

Study of SEIRD Adaptive-Compartmental Model of COVID-19 Epidemic Spread in Russian Federation Using Optimization Methods

A systemic approach to the study of a new multi-parameter model of the COVID-19 pandemic spread is proposed, which has the ultimate goal of optimizing the manage parameters of the model. The approach consists of two main parts: 1) an adaptive-compartmental model of the epidemic spread, which is a generalization of the classical SEIR model, and 2) a module for adjusting the parameters of this model from the epidemic data using intelligent optimization methods. Data for testing the proposed approach using the pandemic spread in some regions of the Russian Federation were collected on a daily basis from open sources during the first 130 days of the epidemic, starting in March 2020. For this, a so-called data farm was developed and implemented on a local server (an automated system for collecting, storing and preprocessing data from heterogeneous sources, which, in combination with optimization methods, allows most accurately tune the parameters of the model, thus turning it into an intelligent system to support management decisions). Among all model parameters used, the most important are the rate of infection transmission, the government actions and the population reaction.

Adaptive-compartmental model of coronavirus epidemic and its optimization by the methods of artificial intelligence

In this work, we develop a systemic approach to the study of a new model of COVID-19 pandemic. The main goal is to minimize the pandemic damage to economy and society by defining the model optimal management parameters. Our approach consists of two main parts: 1) the adaptive-compartmental model of the epidemic (ACM-SEIR) – a generalization of the classical SEIR model and 2) the module to tune ACM-SEIR parameters using artificial intelligence methods (collection, storage and processing of big data from heterogeneous sources) that allow the most accurate adjustment of ACM-SEIR parameters turning it into an intelligent system for decision support called herein iACM-SEIR. We show that among iACM-SEIR parameters, the most important are individual economic, demographic and psychologic characteristics of society and the governmental actions.

Dynamical Evolution of COVID-19 in Italy With an Evaluation of the Size of the Asymptomatic Infective Population.

The present work deals with an Ordinary Differential Equation (ODE) model specifically designed to describe the COVID-19 evolution in Italy. The model is particularised on the basis of National data about the infection status of the Italian population to obtain numerical solutions that effectively reproduce the real data. Our epidemic model is a classical SEIR model that incorporates two compartments of infected subpopulations, representing diagnosed and undiagnosed individuals respectively, and an additional quarantine compartment. Possible control actions representing social, political, and medical interventions are also included. The numerical results of the proposed model identification by least square fitting are analysed and commented with special emphasis on the estimation of the number of asymptomatic infective individuals. Our fitting results are in good agreement with the epidemiological data. Short and long-term predictions on the evolution of the disease are also given.

Application of a Generalized SEIR Model for COVID-19 in Algeria

The novel coronavirus disease 2019 (COVID-19) reported in Wuhan is continuing to impress the world by its fast spread and the number of affected persons attracting an unprecedented attention. In this article, the classical SEIR model and a generalized SEIR model called SEIRDP were applied to predict the evolution of COVID-19 in Algeria for a future period of 100 days using official reported data from early April to early August, 2020. Initial evaluation showed that the two models had a net correspondence with the reported data during this period for cumulative infected cases but the number of cumulative deaths was underestimated with the classical SEIR model. Model prediction with the SEIRDP concluded that the number of cumulative infected cases will increase in the next days reaching a number of about 60 k in middle November with a median of about 300 daily cases. Also, the number of estimated deaths will be around 2k. These results suggest that the COVID-19 is ongoing to infect more persons which may push national authorities to carefully act in the probable leaving of containment.

On an SE(Is)(Ih)AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic

In this paper, we study the nonnegativity and stability properties of the solutions of a newly proposed extended SEIR epidemic model, the so-called SE(Is)(Ih)AR epidemic model which might be of potential interest in the characterization and control of the COVID-19 pandemic evolution. The proposed model incorporates both asymptomatic infectious and hospitalized infectious subpopulations to the standard infectious subpopulation of the classical SEIR model. In parallel, it also incorporates feedback vaccination and antiviral treatment controls. The exposed subpopulation has three different transitions to the three kinds of infectious subpopulations under eventually different proportionality parameters. The existence of a unique disease-free equilibrium point and a unique endemic one is proved together with the calculation of their explicit components. Their local asymptotic stability properties and the attainability of the endemic equilibrium point are investigated based on the next generation matrix properties, the value of the basic reproduction number, and nonnegativity properties of the solution and its equilibrium states. The reproduction numbers in the presence of one or both controls is linked to the control-free reproduction number to emphasize that such a number decreases with the control gains. We also prove that, depending on the value of the basic reproduction number, only one of them is a global asymptotic attractor and that the solution has no limit cycles.

Optimal control problem for a tuberculosis model with multiple infectious compartments and time delays

In this paper, we formulate an optimal control problem based on a tuberculosis model with multiple infectious compartments and time delays. In order to have a more realistic model that allows highlighting the role of detection, loss to follow-up and treatment in TB transmission, we propose an extension of the classical SEIR model by dividing infectious patients in the compartment (I) into three categories: undiagnosed infected (I), diagnosed patients who are under treatment (T) and diagnosed patients who are lost to follow-up (L). We incorporate in our model delays representing the incubation period and the time needed for treatment. We also introduce three control variables in our delayed system which represent prevention, detection and the efforts that prevent the failure of treatment. The purpose of our control strategies is to minimize the number of infected individuals and the cost of intervention. The existence of the optimal controls is investigated, and a characterization of the three controls is given using the Pontryagin's maximum principle with delays. To solve numerically the optimality system with delays, we present an adapted iterative method based on the iterative Forward-Backward Sweep Method (FBSM). Numerical simulations performed using Matlab are also provided. They indicate that the prevention control is the most effective one. To the best of our knowledge, it is the first work to apply optimal control theory to a TB model which considers infectious patients diagnosis, loss to follow-up phenomenon and multiple time delays.