SEIR Model Research Articles

Modeling public opinion dissemination in a multilayer network with SEIR model based on real social networks

Due to the complexity of social relationships, Internet users are active in multiple social networks and receive or release information in various ways and channels. Therefore, the spread of public opinion is not limited to a closed single network. Considering the structural characteristics of WeChat, Weibo and Tiktok, we built a multi-layer network based on the complex network theory. Based on the environmental and personal factors, the calculation methods of transmission rate and immunity rate is proposed. The trends of information transmission in the multi-layer network is simulated through an improved SEIR model. Through simulation, the differences of information transmission between multi-layer networks and single-layer networks are explored, and the impacts of network structure, coupling strength between layers and interlayer connection mode on information transmission are studied. More than 76,000 tweets from 7 events on Weibo are collected to verify the effectiveness of the model and explore the dissemination trends of actual public opinions. Through the research, it is found that (1) the public opinion transmission efficiency of multi-layer network is higher, the scale is larger, and the time is longer. (2) Compared with Weibo and Tiktok, the information transmission in WeChat is more efficient and lasts longer, and promotes the information transmission in the other two networks. (3) With the increase of coupling strength between layers, the efficiency of information transmission becomes higher and the scale becomes larger. This study provides new ideas for the research of public opinion dissemination, and helps relevant departments to correctly guide public opinion.

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.

Dynamical analysis of a class of SEIR models through delayed strategies

In recent decades, the mathematical modeling of infectious diseases, real-world problems, non-linear dynamical complex systems, etc., has increased significantly. According to World Health Organization, tobacco use is the cause of about 22% of cancer deaths. Another 10% are due to obesity, poor diet, lack of physical activity, and excessive drinking of alcohol. Approximately 5%–10% of cancers are due to inherited genetic defects. The objective is to investigate the impact of time delays in implementing control measures on the epidemic dynamics. The classification of cell population has four compartments: susceptible cells (x), cancer-infected cells (y), virus-free cells (v), and immune cells (z). Our focus is to find the equilibria of the problem and their stability. The stability of the solutions is of two types: locally asymptotic and globally asymptotic. The Routh–Hurwitz criterion, Volterra-type Lyapunov function, and LaSalle’s invariance principle are used to verify the stability of solutions. The graphical behavior depicts the stable solutions to a real-world problem and supports the stability analysis of the problem. The findings contribute to the understanding of epidemic dynamics and provide valuable information for designing and implementing effective intervention strategies in public health systems.

Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach.

Compartment models are implemented to understand the dynamic of a system. To analyze the models, a numerical tool is required. This manuscript presents an alternative numerical tool for the SIR and SEIR models. The same idea could be applied to other compartment models. The result starts with transforming the SIR model to an equivalent differential equation. The Dirichlet series satisfying the differential equation leads to an alternative numerical method to obtain the model's solutions. The derived Dirichlet solution not only matches the numerical solution obtained by the fourth-order Runge-Kutta method (RK-4), but it also carries the long-run behavior of the system. The SIR solutions obtained by the RK-4 method, an approximated analytical solution, and the Dirichlet series approximants are graphically compared. The Dirichlet series approximants order 15 and the RK-4 method are almost perfectly matched with the mean square error less than 2 × 10-5. A specific Dirichlet series is considered in the case of the SEIR model. The process to obtain a numerical solution is done in the similar way. The graphical comparisons of the solutions achieved by the Dirichlet series approximants order 20 and the RK-4 method show that both methods produce almost the same solution. The mean square errors of the Dirichlet series approximants order 20 in this case are less than 1.2 × 10-4.

Implementation of Hidden Infections in the SEIR Model Analysis of the Omicron Variant Spread in Bulgaria

In this paper we study the impact of the hidden cases of Covid-19 spread in Bulgaria by means of the SEIR model, in the period December 15, 2021 – until March 2022. In particular, the numerical experiments based on the hidden cases till mid-January, provide a forecast of the peak of the infectious curve about February 8, 2022, and reaching the maximum of about 90 000 cases. The forecast shows that end of the epidemics (in Bulgaria) would be in mid-May, 2022. This forecast is compared to the previous forecasts based on the same data excluding hidden cases; they have shown that the peak will be about February 28, reaching about 75 000 cases and will vanish until end of May. As seen in Fig. 3, our forecast of the end of the particular wave of epidemics shows just two weeks difference with what happened in real life – the wave ended in the first week of June. It is curious, that though the peak of the curve in thehidden cases approach has shifted to February 8, the cumulative number of cases in both approaches is nearly the same. More details on the methods used in the present paper and the non-hidden cases study, are provided as well.

Stability analysis and optimal control of a fractional-order generalized SEIR model for the COVID-19 pandemic

In view of the spread of corona virus disease 2019 (COVID-19), this paper proposes a fractional-order generalized SEIR model. The non-negativity of the solution of the model is discussed. Based on the established threshold R0, the existence of the disease-free equilibrium and endemic equilibrium is analyzed. Then, sufficient conditions are established to ensure the local asymptotic stability of the equilibria. The parameters of the model are identified based on the statistical data of COVID-19 cases. Furthermore, the validity of the model for describing the COVID-19 outbreak is verified. Meanwhile, the accuracy of the relevant theoretical results are also verified. Considering the relevant strategies of COVID-19 prevention and control, the fractional optimal control problem (FOCP) is proposed. Numerical schemes for Riemann–Liouville (R–L) fractional-order adjoint system with transversal conditions is presented. Based on the relevant statistical data, the corresponding FOCP is numerically solved, and the control effect of the COVID-19 outbreak under the optimal control strategy is discussed.

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.

A new design and analysis of optimal control problems arising from COVID‐19 outbreak

In this manuscript, we append the hospitalization, diagnosed and isolation compartments to the classic SEIR model to design a new COVID‐19 epidemic model. We further subdivide the isolation compartment into asymptomatic infected and symptomatic infected compartments. For validity of the purposed model, we prove the existence of a unique solution and prove the positivity and boundedness of the solution. To study disease dynamics, we compute equilibrium points and the reproduction number . We also investigate the local and global stabilities at both of the equilibrium points. Sensitivity analysis will be performed to observe the effect of transmission parameters on . For optimal control analysis, we design two different optimal control problems by taking different optimal control approaches. Firstly, we add an isolation compartment in the newly designed model, and secondly, three parameters describing non‐pharmaceutical behaviors such as educating people to take precautionary measures, providing intensive medical care with medication, and utilizing resources by government are added in the model. We set up optimality conditions by using Pontryagin's maximum principle and develop computing algorithms to solve the conditions numerically. At the end, numerical solutions will be displayed graphically with discussion.

Mumps epidemic dynamics in the United States before vaccination (1923–1932)

Mumps is a vaccine-preventable, reemerging, and highly transmissible infectious disease. Widespread vaccination dramatically reduced cases; however, case counts have been increasing over the past 20 years. To provide a quantitative overview of historical mumps dynamics that can act as baseline information to help identify causes of mumps reemergence, we analyzed timeseries of cases reported from 1923 to 1932 in the United States. During that time, 239,230 mumps cases were reported in 70 cities. Larger cities reported annual epidemics and smaller cities reported intermittent, sporadic outbreaks. The critical community size above which transmission continuously occurred was likely between 365,583 and 781,188 individuals but could range as high as 3,376,438 individuals. Mumps cases increased as city size increased, suggesting density-dependent transmission. Using a density-dependent SEIR model, we calculated a mean effective reproductive number (Re) of 1.2. Re varied by city and over time, with periodic high values that could characterize short periods of very high transmission known as superspreading events. Case counts most often peaked in March, with higher-than-average transmission from December through April and showed a correlation with weekly births. While certain city pairs in Midwestern states had synchronous outbreaks, most outbreaks were less synchronous and not driven by distance between cities. This work demonstrates the importance of long-term infectious disease surveillance data and will inform future studies on mumps reemergence and control.

Big data in COVID-19 prevention and control: Modeling and analysis report

The COVID-19 epidemic has brought great external impact to China. China is facing complex internal and external environmental challenges. SIR epidemic model is a classical partition model, which is widely used to predict the progress of COVID-19. Although the SIR model may be useful in simulating multiple epidemics, it may not be sufficient to describe the spread of COVID-19. Therefore, some modifications were made and used to study the spread and control of COVID-19 epidemic on the SIR model of COVID-19 disease. Expand it by increasing the link between tracking and other interventions. By studying the SEIR model considering the interaction between human and infectious source. In this paper, we will use the classical SIR model to simulate and predict the spread of COVID-19. By distinguishing between confirmed and undiagnosed individuals, the development of COVID-19 is characterized by phased changes. Based on the preliminary data analysis of the epidemic on various industries, the actual impact of the epidemic on society was quantitatively analyzed.

UHIR: An effective information dissemination model of online social hypernetworks based on user and information attributes

With the expansion of the number of users in online social networks, the diversity of users and community characteristics become more prominent. Hypernetwork theory provides a path for characterizing complex relationships in networks. This paper used hypergraph’s hyperedges to represent the community relationship between users, and created an online social hypernetwork information dissemination model (UHIR model) based on user and information attributes by combining the hypernetwork model with the SEIR model. Through this model, this article simulated and analyzed the dynamic process and laws of information dissemination under different network structures, and studied the influence of user influence, confidence, interest value, and information timeliness of the process. The simulation results show that this model can accurately describe the information dissemination trend and process in the real online social network. This work extends a new research direction of information dissemination in hypernetworks and contributes to the in-depth study of more complex information dissemination mechanisms.

Malware propagation model of fractional order, optimal control strategy and simulations

In this paper, an improved SEIR model of fractional order is investigated to describe the behavior of malware propagation in the wireless sensor network. Firstly, the malware propagation model of fractional order is established based on the classical SEIR epidemic theory, the basic reproductive number is obtained by the next-generation method and the stability condition of the model is also analyzed. Then, the inverse approach for the uncertainty propagation based on the discrete element method and least square algorithm is presented to determine the unknown parameters of the propagation process. Finally, the optimal control strategy is also discussed based on the adaptive model. Simulation results show the proposed model works better than the propagation model of integer order. The error of proposed model is smaller than integer order models.

ANALISIS KESTABILAN DAN SOLUSI NUMERIK PADA MODEL SEIR UNTUK PENYAKIT TUBERKULOSIS

One of the infectious diseases that can be modelled into the SEIR model is Tuberculosis (TB), this is because TB has a bacterial incubation period, so it is at this time that a person enters the exposed subpopulation. TB is divided into two types, namely latent TB and active TB. This study aims to explain the formation of the SEIR Model for the Spread of Tuberculosis, determine the equilibrium point and Basic Reproductive Numbers on the SEIR Model for the Spread of Tuberculosis, analyze the stability of the SEIR Model for the spread of Tuberculosis at the equilibrium point, and make numerical simulations. The result of this research is the formation of a mathematical model on the spread of Tuberculosis, and from the model obtained two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. Then the basic reproduction number ( ) was found through the Next Generation Matrix. Furthermore, the stability analysis was carried out at the disease-free equilibrium point and it was found that the local asymptotic stable model with , while at the endemic equilibrium point it was found that the local asymptotic stable model with . Numerical simulations are presented to show numerical solutions and strengthen the explanation of the stability analysis of the model using the fourth-order Runge-Kutta method with parameters that meet the stability requirements.

The lockdown and vaccination distribution in Thailand's COVID-19 epidemic: A model study.

Several countries used varied degrees of social isolation measures in response to the COVID-19 outbreak. In 2021, the lockdown in Thailand began on July 20 and lasted for the following six weeks. The lockdown has extremely detrimental effects on the economy and society, even though it may reduce the number of COVID-19 instances. Our goals are to assess the impact of the lockdown policy, the commencement time of lockdown, and the vaccination rate on the number of COVID-19 cases in Thailand in 2021. We modeled the dynamics of COVID-19 in Thailand throughout 2021 using the SEIR model. The Google Mobility Index, vaccine distribution rate, and lockdown were added to the model. The Google Mobility Index represents the movement of individuals during a pandemic and shows how people react to lockdown. The model also examines the effect of vaccination rate on the incidence of COVID-19. The modeling approach demonstrates that a 6-week lockdown decreases the incidence number of COVID-19 by approximately 15.49-18.17%, depending on the timing of the lockdown compared to a non-lockdown scenario. An increasing vaccination rate potentially reduce the incidence number of COVID-19 by 5.12-18.35% without launching a lockdown. Lockdowns can be an effective method to slow down the spread of COVID-19 when the vaccination program is not fully functional. When the vaccines are easily accessible on a large scale, the lockdown may terminated.

Understanding the impact of disease and vaccine mechanisms on the importance of optimal vaccine allocation.

Vaccination is an important epidemic intervention strategy. However, it is generally unclear how the outcomes of different vaccine strategies change depending on population characteristics, vaccine mechanisms and allocation objective. In this paper we develop a conceptual mathematical model to simulate strategies for pre-epidemic vaccination. We extend the SEIR model to incorporate a range of vaccine mechanisms and disease characteristics. We then compare the outcomes of optimal and suboptimal vaccination strategies for three public health objectives (total infections, total symptomatic infections and total deaths) using numerical optimisation. Our comparison shows that the difference in outcomes between vaccinating optimally and suboptimally depends on vaccine mechanisms, disease characteristics, and objective considered. Our modelling finds vaccines that impact transmission produce better outcomes as transmission is reduced for all strategies. For vaccines that impact the likelihood of symptomatic disease or dying due to infection, the improvement in outcome as we decrease these variables is dependent on the strategy implemented. Through a principled model-based process, this work highlights the importance of designing effective vaccine allocation strategies. We conclude that efficient allocation of resources can be just as crucial to the success of a vaccination strategy as the vaccine effectiveness and/or amount of vaccines available.

An application of discrete-time SEIR model to the COVID-19 spread

An application of discrete-time SEIR model to the COVID-19 spread

Study Models of COVID-19 in Discrete-Time and Fractional-Order

The novel coronavirus disease (SARS-CoV-2) has caused many infections and deaths throughout the world; the spread of the coronavirus pandemic is still ongoing and continues to affect healthcare systems and economies of countries worldwide. Mathematical models are used in many applications for infectious diseases, including forecasting outbreaks and designing containment strategies. In this paper, we study two types of SIR and SEIR models for the coronavirus. This study focuses on the discrete-time and fractional-order of these models; we study the stability of the fixed points and orbits using the Jacobian matrix and the eigenvalues and eigenvectors of each case; moreover, we estimate the parameters of the two systems in fractional order. We present a statistical study of the coronavirus model in two countries: Saudi Arabia, which has successfully recovered from the SARS-CoV-2 pandemic, and China, where the number of infections remains significantly high.

Mathematical Modeling of Dog Rabies Transmission Dynamics Using Optimal Control Analysis

An ideal control method for the dynamics of dog rabies transmission is provided in this study. Given the nature of the disease and the fact that contact behavior varies, we divided the infected compartment into prodromal and furious compartments in the current updated model, which is an extension of the previous SEIR model. Vaccination and culling are two disease-controlling strategies used in the current model, and their effects are examined. It is possible to compute the basic reproduction number using the next-generation matrix. We study the stability, sensitivity analysis, endemic equilibrium, disease-free equilibrium, and stability of the optimal control model. According to the numerical simulation, which utilizes approximations for parameter values, the most efficient strategy to prevent the spread of rabies is a combination of vaccination and the culling of infected dogs. Using ode45 from MATLAB, this numerical simulation investigation was carried out. According to our research, the annual dog birth rate is a factor that influences the incidence of rabies. The state equations, adjoint equations, and the optimal condition that sets the controls by Pontryagin's Maximum/Minimum principle can all be used to construct the optimal control system. The body of the article contains the findings and discussions in an ordered manner.

The effectiveness of control measures during the 2022 COVID-19 outbreak in Shanghai, China.

In March 2022, the Omicron variant of SARS-CoV-2 spread rapidly in Shanghai, China. The city adopted strict non-pharmacological intervention (NPI) measures, including lockdown (implemented on March 28 in Pudong and April 1 in Puxi) and blanket PCR testing (April 4). This study aims to understand the effect of these measures. We tabulated daily case counts from official reports and fitted a two-patch stochastic SEIR model to the data for the period of March 19 to April 21. This model considered two regions in Shanghai, namely Pudong and Puxi, as the implementation of control measures in Shanghai was carried out on different dates in these regions. We verified our fitting results using the data from April 22 to June 26. Finally, we applied the point estimate of parameter values to simulate our model while varying the dates of control measure implementation, and studied the effectiveness of the control measures. Our point estimate for the parameter values yields expected case counts that agree well the data for both the periods from March 19 to April 21 and from April 22 to June 26. Lockdown did not significantly reduce the intra-region transmission rates. Only about 21% cases were reported. The underlying basic reproduction number R0 was 1.7, and the control reproduction number with both lockdown and blanket PCR testing was 1.3. If both measures were implemented on March 19, only about 5.9% infections would be prevented. Through our analysis, we found that NPI measures implemented in Shanghai were not sufficient to reduce the reproduction number to below unity. Thus, earlier intervention only has limited effect on reducing cases. The outbreak dies out because of only 27% of the population were active in disease transmission, possibly due to a combination of vaccination and lockdown.

An approximate diffusion process for environmental stochasticity in infectious disease transmission modelling.

Modelling the transmission dynamics of an infectious disease is a complex task. Not only it is difficult to accurately model the inherent non-stationarity and heterogeneity of transmission, but it is nearly impossible to describe, mechanistically, changes in extrinsic environmental factors including public behaviour and seasonal fluctuations. An elegant approach to capturing environmental stochasticity is to model the force of infection as a stochastic process. However, inference in this context requires solving a computationally expensive "missing data" problem, using data-augmentation techniques. We propose to model the time-varying transmission-potential as an approximate diffusion process using a path-wise series expansion of Brownian motion. This approximation replaces the "missing data" imputation step with the inference of the expansion coefficients: a simpler and computationally cheaper task. We illustrate the merit of this approach through three examples: modelling influenza using a canonical SIR model, capturing seasonality using a SIRS model, and the modelling of COVID-19 pandemic using a multi-type SEIR model.