SEIRD Model Research Articles

Optimal control in models of virus propagation

Based on the SEIRD model, we consider that when multiple viruses of different virulence coexist, the more virulent virus will reinfect nodes already infected by the less virulent virus, which we call here Superexposed. Based on the state transitions, the corresponding differential equations and cost functions are established, then building the corresponding optimal control problem, where the vaccine efficiency and drug efficiency are controlled variables. This nonlinear optimal control problem is solved by Pontryagin’s maximum principle to finding the structure of the optimal control strategies. Based on the definition of the basic regeneration number, yielding the R0 value for the model, then discussed the final epidemic size. In the numerical analysis section, we validate the accuracy of the structure, fitting the behavior of each state and the effect of different parameter values.

Vaccination Strategies based on a Mathematical Model of Epidemics Considering the Age Structure of the Population

During the COVID-19 pandemic, it is important to promote the skills needed for analyzing the disease course, including determining the relevance of vaccinations, especially among people who are unfamiliar with computer programming. This paper describes the basic epidemiological model (SIR), its extensions that allow vaccinations, and the emergence of renewed waves of disease growth. It also discusses a literature model, extended SEIRD, which includes a more detailed division of the population into susceptible, latent, symptomatic, and asymptomatic infected, recovered, and dead in eight age groups. Modifying the SEIRD model as shown on the basic SIR model, we analyzed five vaccination strategies, considering the limited vaccine supply, the number of vaccinations performed per day, and their effectiveness. The analysis was performed for a group of one million people, using the parameters of the model characteristic of the COVID-19 pandemic and Sweden's generational structure. We analyzed in terms of reducing both the number of deaths and the incidence of symptomatic infections, which represent the main burden of healthcare.

Predicting Tuberculosis Incidence and Its Trend in Tigray, Ethiopia: A Reality-Counterfactual Modeling Approach.

The Tigray region of Ethiopia, which has been affected by civil war from 2020 to 2022, is facing an increase in tuberculosis in the damaged health system. Our study employed mathematical modeling to predict the incidence of tuberculosis and its trends during the war and in the post-conflict setting of Tigray, Northern Ethiopia. We predicted the incidence of tuberculosis from 2020 to 2025 in Tigray using the SEIRD model in the context of the recent war and compared it with its counterfactual trend in the absence of war. The counterfactual trend was forecasted using an autoregressive integrated moving average (ARIMA) model for stationary time-series data. We performed rolling origin cross-validation for ARIMA and sensitivity analysis for the SEIRD model. The initial tuberculosis data and model parameters were obtained from the Institute for Health Metrics and Evaluation and the literature, respectively. Between 2000 and 2017, the incidence of tuberculosis in Tigray decreased at an annual rate of 3.0%. Shortly before the war, the incidence of tuberculosis in the region was 178 per 100,000 people. In a counterfactual scenario where there was no war, the incidence was projected to decrease to 144.3 in 2022 and 126.3 in 2025. However, owing to the war and siege, the SEIRD-projected incidence of tuberculosis would have increased to 965.5 (95% CI: 958.5-972.7) in 2022 and 372.4 (95% CI: 367.7-376.6) in 2025. Over 800 cases of tuberculosis per 100,000 people were attributed to the war in 2022. In the postwar period, the incidence is projected to decrease by 30% by 2023. The Tigray War reversed a two-decade decline in tuberculosis cases, causing a five-fold increase compared to the no-war scenario. Urgent interventions are needed to support tuberculosis prevention, testing, and treatment, particularly in key and vulnerable populations.

A mean field game model for COVID-19 with human capital accumulation

In this manuscript, we study a model of human capital accumulation during the spread of disease following an agent-based approach, where agents behave maximising their intertemporal utility. We assume that the agent interaction is of mean field type, yielding a mean field game description of the problem. We discuss how the analysis of a model including both the mechanism of change of species from one epidemiological state to the other and an optimisation problem for each agent leads to an aggregate behaviour that is not easy to describe, and that sometimes exhibits structural issues. Therefore we eventually propose and study numerically a SEIRD model in which the rate of infection depends on the distribution of the population, given exogenously as the solution to the mean field game system arising as the macroscopic description of the discrete multi-agent economic model for the accumulation of human capital. Such a model arises in fact as a simplified but tractable version of the initial one.

Incorporating Interventions to an Extended SEIRD Model with Vaccination: Application to COVID-19 in Qatar

The COVID-19 outbreak of 2020 has required many governments to develop and adopt mathematical-statistical models of the pandemic for policy and planning purposes. To this end, this work provides a tutorial on building a compartmental model using Susceptible, Exposed, Infected, Recovered, Deaths and Vaccinated (SEIRDV) status through time. The proposed model uses interventions to quantify the impact of various government attempts made to slow the spread of the virus. Furthermore, a vaccination parameter is also incorporated in the model, which is inactive until the time the vaccine is deployed. A Bayesian framework is utilized to perform both parameter estimation and prediction. Predictions are made to determine when the peak Active Infections occur. We provide inferential frameworks for assessing the effects of government interventions on the dynamic progression of the pandemic, including the impact of vaccination. The proposed model also allows for quantification of number of excess deaths averted over the study period due to vaccination.

An extended SEIRDV compartmental model: case studies of the spread of COVID-19 and vaccination in Tunisia and South Africa.

In the wake of the unprecedented health crisis triggered by the global COVID-19 pandemic, countries are still grappling with the pandemic's immediate health and socioeconomic consequences. This paper presents an extended SEIRD model with vaccination to study the evolution of COVID-19 in South Africa and Tunisia since the commencement of the vaccination campaign in each country, respectively. Epidemiologists often quantify a risk reduction following the implementation of non-pharmaceutical containment measures and vaccines when attempting to stem the spread of pandemics. However, an important question they often ask is the effectiveness of the non-pharmaceutical containment measures (social distancing and lockdowns) and the efficacy of such measures, including vaccines. Africa's COVID-19 vaccine roll-out stands at 16% as of April 2022; however, the continent lags behind many developing countries even though it harbours about 16% of the world population. While proliferating literature quantifies the efficacy and effectiveness of COVID-19 vaccines, very little has been done using the SEIRDV model in African countries. This study compares the model-predicted results with the available data to estimate the dynamics of the infected population, using data from 20-03-2021 to 30-12-2021. A simulation of the SEIRDV model is performed and fitted to the data. Simulating the model involves solving a system of Ordinary Differential Equations numerically by taking the initial values for the key model parameters as inputs. After simulating the SEIRDV model, the model parameters are compared with real-world COVID-19 and vaccination data in order to estimate the values of the different parameters that best fit the observed data. The results of the study showed an inverted U-shaped trend for the infection rate after vaccination, indicating that increasing the vaccination rates reduces the transmission rates. Therefore African countries must continue to scale up the vaccination campaigns, and the world needs to endeavour to ensure an equitable vaccination roll-out to developing countries.

Fitting Time-varying Coefficients SEIRD Model to COVID-19 Cases in Malaysia

This paper proposes a compartmental Susceptible-Exposed-Infected-Recovered-Death (SEIRD) model for COVID-19 cases in Malaysia. This extended model is more relevant to describe the disease transmission than the SIRD model since the exposed (E) compartment represents individuals in the disease's incubation period. The mathematical model is a system of ordinary differential equations (ODEs) with time-varying coefficients as opposed to the conventional model with constant coefficients. This time dependency treatment is necessary as the epidemiological parameters such as infection rate β, recovery rate γ, and death rate μ usually change over time. However, this feature leads to an increasing number of unknowns needed to be solved to fit the model with the actual data. Several optimization algorithms under Python’s LMfit package, such as Levenberg-Marquardt, Nelder-Mead, Trust-Region Reflective and Sequential Linear Squares Programming; are employed to estimate the related parameters, in such that the numerical solution of the ODEs will fit the data with the slightest error. Nelder-Mead outperforms the other optimization algorithm with the least error. Qualitatively, the result shows that the proportion of the quarantine rule-abiding population should be maintained up to 90% to ensure Malaysia successfully reaches disease-free or endemic equilibrium.

Modified SEIRD model for describing the COVID-19 epidemic

This article is devoted to mathematical models in epidemiology, in particular SIR, SEIR, and SEIRD models. It explores the importance of these models in predicting the spread of infectious diseases and evaluating the effectiveness of control measures. These models allow for assessing important epidemic parameters such as the speed of infection transmission, the number of people infected, and the number of deaths. This data can help in making decisions regarding the imposition and lifting of quarantine restrictions, opening and closing of schools and other institutions, as well as in developing vaccination strategies and other control measures. In summary, mathematical models such as SIR, SEIR, and SEIRD are important tools in the fight against epidemics. They enable epidemiologists and medical professionals to predict and control the spread of diseases, thus preserving the health and lives of people.

Final size and partial distance estimate for a two-group SEIRD model.

In this paper we consider a SEIRD epidemic model for a population composed by two groups of individuals with asymmetric interaction. Given an approximate solution for the two-group model, we estimate the error of this approximation to the unknown solution to the second group based on the known error that the approximation has with respect to the solution to the first group. We also study the final size of the epidemic for each group. We illustrate our results with the spread of the coronavirus disease 2019 (COVID-19) pandemic in the New York County (USA) for the initial stage of the contamination, and in the cities of Petrolina and Juazeiro (Brazil).

The application of next generation matrix in the calculation of basic reproduction number for COVID-19.

Infectious diseases are disorders caused by microorganisms such as bacteria, viruses, fungi, or parasites. Many organisms live in and on our bodies. They are normally harmless or even helpful. However under certain conditions, some organisms may cause disease. Infectious diseases are also called contagious diseases due to the fact that they can be passed from person to person. Some are transmitted by insects or other animals. COVID-19 is an infectious disease that has "pervaded" the whole world during the last three years. The World Health Organization (WHO) has declared COVID-19 a Public Health Emergency of International Concern. In this paper, we will study the outbreak of this pandemic in Albania based on some mathematical models, such as SIR, SIRD, and SEIRD. We will present a detailed analysis of these models and also demonstrate how they can be used to predict the spread of infectious diseases. More precisely, we will see the spread of COVID-19 in our country, Albania. Software such as MATLAB and RStudio will be used to do this. The data that we will use when working with these programs is taken from the Institute of Public Health, Tirana, Albania. We've developed an application utilizing actual data to estimate SEIRD model parameters. It's able to compute the basic reproduction number and, more significantly, provides forecasts on the disease's progression. Our aim is to calculate the Basic Reproduction Number, using the Next Generation Matrix, and use it to see the future of the disease. This is the average number of new infections generated by an infected individual. A large value indicates that the infection is transmitted very quickly. We will try to calculate what the values of Basic Number Reproduction have been over different time periods.

Model Development and Prediction of Covid-19 Pandemic in Bangladesh with Nonlinear Incident.

We introduce a SEIRD compartmental model to analyze the dynamics of the pandemic in Bangladesh. The multi-wave patterns of the new infective in Bangladesh from the day of the official confirmation to August 15, 2021, are simulated in the proposed SEIRD model. To solve the model equations numerically, we use the RK-45 method. Primarily, we establish some theorems including local and global stability for the proposed model. The analysis shows that the death curve simulated by the model provides a very good agreement with the officially confirmed death data for the Covid-19 pandemic in Bangladesh. Furthermore, the proposed model estimates the duration and peaks of Covid-19 in Bangladesh which are compared with the real data.

Modelling influenza and SARS-CoV-2 interaction: Analysis for Catalonia region

The aim is to analyse that, during the current pandemic situation, the reduction in the number of cases of influenza suggests that the non-pharmaceutical interventions (NPIs) applied to contain the expansion of SARS-CoV-2 also affect the influenza expansion. We analyse the interaction of influenza and SARS-CoV-2 spread based on an extended SEIRD model for the Catalonia region in Spain. We show that the dynamic evolution of the spread of SARS-CoV-2 and influenza generates a small interference. This interference causes a small reduction in the number of cases of seasonal influenza, reducing its expansion over the population. Other elements like the face mask mandates, social distancing and hand cleaning boost the reduction in both expansions. Influenza expansion will present a small reduction in the number of cases due to the interaction with SARS-CoV-2 expansion but mainly because the NPIs applied to the population.

An Improved Epidemiological Model for the Underprivileged People in the Contemporary Pandemics.

In this work, we introduce an improved form of the basic SEIRD model based on Python simulation for the troublesome people who are oblivious about the contemporary pandemics due to diverse social impediments, especially those economically underprivileged. In the extant epidemiological models, some unorthodox issues are yet to be considered, such as poverty, illiteracy, and carelessness towards health issues, significantly influencing the data modeling. Our focus is to overcome these issues by adding two more branches, for instance, uncovered and apathetic people, which significantly influence the practical purposes. For the data simulation, we have used the Python-based algorithm that trains the desired system based on a set of real-time data with the proposed model and provides predicted data with a certain level of accuracy. Comparative discussions, statistical error analysis, and correlation-regression analysis have been introduced to validate the proposed epidemiological model. To show the numerical evidence, the investigation comprised the figurative and tabular modes for both real-time and predicted data. Finally, we discussed some concluding remarks based on our findings.

Modeling the COVID-19 epidemic in Croatia: a comparison of three analytic approaches

AimTo facilitate the development of a COVID-19 predictive model in Croatia by analyzing three different methodological approaches.MethodWe used the historical data to explore the fit of the extended SEIRD compartmental model, the Heidler function, an exponential approximation in analyzing electromagnetic phenomena related to lightning strikes, and the Holt-Winters smoothing (HWS) for short-term epidemic predictions. We also compared various methods for the estimation of R0.ResultsThe R0 estimates for Croatia varied from 2.09 (95% CI 1.77-2.40) obtained by using an empirical post-hoc method to 2.28 (95% CI 2.27-2.28) when we assumed an exponential outbreak at the very beginning of the COVID-19 epidemic in Croatia. Although the SEIRD model provided a good fit for the early epidemic stages, it was outperformed by the Heidler function fit. HWS achieved accurate short-term predictions and depended the least on model entry parameters. Neither model performed well across the entire observed period, which was characterized by multiple wave-form events, influenced by the re-opening for the tourist season during the summer, mandatory masks use in closed spaces, and numerous measures introduced in retail stores and public places. However, an extension of the Heidler function achieved the best overall fit.ConclusionsPredicting future epidemic events remains difficult because modeling relies on the accuracy of the information on population structure and micro-environmental exposures, constant changes of the input parameters, varying societal adherence to anti-epidemic measures, and changes in the biological interactions of the virus and hosts.

The mathematical and machine learning models to forecast the COVID-19 outbreaks in Bangladesh

The COVID-19 virus mutates in many different variants after its outbreak. Although several vaccines have been developed by many countries and implemented worldwide, it is difficult to prevent the outbreaks due to the pops out of different variants from its regular mutations. This study is an attempt to develop models which could precisely forecast the COVID-19 outbreaks in Bangladesh. In this study, we have developed a SEIRD based machine learning model to forecast the next possible one year outbreaks scenario in this country. We have tested the accuracy of this model by fitting the results with the considered historical data from March 08, 2020 to October 14, 2021. Also, we have validated this model by predicting the future inside the existing dataset, which is almost similar to the real dataset. It is observed that the final future forecasting results are very realistic compared to the current outbreak situation. Additionally, we have shown that the classical SEIRD model cannot predict the COVID-19 future outbreaks even it does not fit with the real datasets of outbreaks. Moreover, another machine learning time series forecasting model, FBProphet, has been implemented to forecast the future outbreaks of Bangladesh. Finally, we have analyzed and compared the forecasting results and hence identify the limitations of the proposed models which can improve future research in this field.

A comparison and calibration of integer and fractional-order models of COVID-19 with stratified public response.

The spread of SARS-CoV-2 in the Canadian province of Ontario has resulted in millions of infections and tens of thousands of deaths to date. Correspondingly, the implementation of modeling to inform public health policies has proven to be exceptionally important. In this work, we expand a previous model of the spread of SARS-CoV-2 in Ontario, "Modeling the impact of a public response on the COVID-19 pandemic in Ontario, " to include the discretized, Caputo fractional derivative in the susceptible compartment. We perform identifiability and sensitivity analysis on both the integer-order and fractional-order SEIRD model and contrast the quality of the fits. We note that both methods produce fits of similar qualitative strength, though the inclusion of the fractional derivative operator quantitatively improves the fits by almost 27% corroborating the appropriateness of fractional operators for the purposes of phenomenological disease forecasting. In contrasting the fit procedures, we note potential simplifications for future study. Finally, we use all four models to provide an estimate of the time-dependent basic reproduction number for the spread of SARS-CoV-2 in Ontario between January 2020 and February 2021.

Switched forced SEIRDV compartmental models to monitor COVID-19 spread and immunization in Italy.

This paper presents a new hybrid compartmental model for studying the COVID-19 epidemic evolution in Italy since the beginning of the vaccination campaign started on 2020/12/27 and shows forecasts of the epidemic evolution in Italy in the first six months. The proposed compartmental model subdivides the population into six compartments and extends the SEIRD model proposed in [E.L.Piccolomini and F.Zama, PLOS ONE, 15(8):1–17, 08 2020] by adding the vaccinated population and framing the global model as a hybrid-switched dynamical system. Aiming to represent the quantities that characterize the epidemic behaviour from an accurate fit to the observed data, we partition the observation time interval into sub-intervals. The model parameters change according to a switching rule depending on the data behaviour and the infection rate continuity condition. In particular, we study the representation of the infection rate both as linear and exponential piecewise continuous functions. We choose the length of sub-intervals balancing the data fit with the model complexity through the Bayesian Information Criterion. We tested the model on italian data and on local data from Emilia-Romagna region. The calibration of the model shows an excellent representation of the epidemic behaviour in both cases. Thirty days forecasts have proven to well reproduce the infection spread, better for regional than for national data. Both models produce accurate predictions of infected, but the exponential-based one perform better in most of the cases. Finally, we discuss different possible forecast scenarios obtained by simulating an increased vaccination rate.

National Holidays and Social Mobility Behaviors: Alternatives for Forecasting COVID-19 Deaths in Brazil.

In this paper, we investigate the influence of holidays and community mobility on the transmission rate and death count of COVID-19 in Brazil. We identify national holidays and hallmark holidays to assess their effect on disease reports of confirmed cases and deaths. First, we use a one-variate model with the number of infected people as input data to forecast the number of deaths. This simple model is compared with a more robust deep learning multi-variate model that uses mobility and transmission rates (, ) from a SEIRD model as input data. A principal components model of community mobility, generated by the principal component analysis (PCA) method, is added to improve the input features for the multi-variate model. The deep learning model architecture is an LSTM stacked layer combined with a dense layer to regress daily deaths caused by COVID-19. The multi-variate model incremented with engineered input features can enhance the forecast performance by up to 18.99% compared to the standard one-variate data-driven model.

Extended Epidemiological Models for Weak Economic Region: Case Studies of the Spreading of COVID-19 in the South Asian Subcontinental Countries.

The ascendancy of coronavirus has become widespread all around the world. For the prevention of viral transmission, the pattern of disease is explored. Epidemiological modeling is a vital component of the research. These models assist in studying various aspects of infectious diseases, such as death, recovery, and infection rates. Coronavirus trends across several countries may analyze sufficiently using SIR, SEIR, and SIQR models. Across this study, we propose two modified versions of the SEIRD method for evaluating the transmission of this infectious disease in the South Asian countries, more precisely, in the south Asian subcontinent. The SEIRD model is updated further by fusing some new factors, namely, isolation for the suspected people and recovery and death of the people who are not under the coverage of healthcare schemes or reluctant to receive treatment for various catastrophes. We will investigate the influences of those ingredients on public health-related issues. Finally, we will predict and display the infection scenario and relevant elements with the concluding remarks through the statistical analysis.

Monitoring SEIRD model parameters using MEWMA for the COVID-19 pandemic with application to the state of Qatar

During the current COVID-19 pandemic, decision-makers are tasked with implementing and evaluating strategies for both treatment and disease prevention. In order to make effective decisions, they need to simultaneously monitor various attributes of the pandemic such as transmission rate and infection rate for disease prevention, recovery rate which indicates treatment effectiveness as well as the mortality rate and others. This work presents a technique for monitoring the pandemic by employing an Susceptible, Exposed, Infected, Recovered, Death model regularly estimated by an augmented particle Markov chain Monte Carlo scheme in which the posterior distribution samples are monitored via Multivariate Exponentially Weighted Average process monitoring. This is illustrated on the COVID-19 data for the State of Qatar.