Compartmental Epidemic Model Research Articles

Introduction to Group-Structured-Epidemic Model

The spread of an infectious disease in a population is a random process when considering a small group of individuals. However, to a great group of individuals, the use of deterministic behavior is better. Based on these facts, in the literature, there were proposed stochastic and deterministic epidemic models. This work proposes a mixed compartmental epidemic model that allows stratifying the population into groups, considers demographic and environmental variability, presents an approximation to stochastic effects, and contemplates the network effects. The proposed model has a compact form to assist in the synthesis of the control law and parameters estimation strategies. Its objective is to overcome the difficulties encountered when used purely deterministic or purely stochastic models. In the end, to detail and verify the functioning of the proposed model, we present a set of flowcharts and simulations.

Optimal control-based vaccination and testing strategies for COVID-19

Background and Objective:Assuming the availability of a limited amount of effective COVID-19 rapid tests, the effects of various vaccination strategies on SARS-CoV-2 virus transmission are compared for different vaccination scenarios characterized by distinct limitations associated with vaccine supply and administration. Methods:The vaccination strategies are defined by solving optimal control problems of a compartmental epidemic model in which the daily vaccination rate and the daily testing rate for the identification and isolation of asymptomatic subjects are the control variables. Different kinds of algebraic constraints are considered, representing different vaccination scenarios in which the total amount of vaccines available during the time period under consideration is limited or the number of daily available vaccines is limited. These optimal control problems are numerically solved by means of a direct transcription technique, which allows both equality and inequality constraints to be straightforwardly included in the formulation of the optimal control problems. Results:Several numerical experiments are conducted, in which the objective functional to be minimized is a combination of the number of symptomatic and asymptomatic infectious subjects with the cost of vaccination of susceptible subjects and testing of asymptomatic infectious subjects. The results confirm the hypothesis that the implementation of early control measures significantly reduces the number of symptomatic infected subjects, which is a key aspect for the resilience of the healthcare system. The sensitivity analysis of the solutions to the weighting parameters of the objective functional reveals that it is possible to obtain a vaccination strategy that allows vaccination supplies to be saved while keeping the same number of symptomatic infected subjects. Furthermore, it indicates that if the vaccination plan is not supported by a sufficient rate of testing, the number of symptomatic infected subjects could increase. Finally, the sensitivity analysis shows that a significant reduction in the efficacy of the vaccines could also lead to a relevant increase in the number of symptomatic infected subjects. Conclusions:The numerical experiments show that the proposed approach, which is based on optimal control of compartmental epidemic models, provides healthcare systems with a suitable method for scheduling vaccination plans and testing policies to control the spread of the SARS-CoV-2 virus.

Outcomes Associated With Social Distancing Policies in St Louis, Missouri, During the Early Phase of the COVID-19 Pandemic

In the absence of a national strategy in response to the COVID-19 pandemic, many public health decisions fell to local elected officials and agencies. Outcomes of such policies depend on a complex combination of local epidemic conditions and demographic features as well as the intensity and timing of such policies and are therefore unclear. To use a decision analytical model of the COVID-19 epidemic to investigate potential outcomes if actual policies enacted in March 2020 (during the first wave of the epidemic) in the St Louis region of Missouri had been delayed. A previously developed, publicly available, open-source modeling platform (Local Epidemic Modeling for Management & Action, version 2.1) designed to enable localized COVID-19 epidemic projections was used. The compartmental epidemic model is programmed in R and Stan, uses bayesian inference, and accepts user-supplied demographic, epidemiologic, and policy inputs. Hospital census data for 1.3 million people from St Louis City and County from March 14, 2020, through July 15, 2020, were used to calibrate the model. Hypothetical delays in actual social distancing policies (which began on March 13, 2020) by 1, 2, or 4 weeks. Sensitivity analyses were conducted that explored plausible spontaneous behavior change in the absence of social distancing policies. Hospitalizations and deaths. A model of 1.3 million residents of the greater St Louis, Missouri, area found an initial reproductive number (indicating transmissibility of an infectious agent) of 3.9 (95% credible interval [CrI], 3.1-4.5) in the St Louis region before March 15, 2020, which fell to 0.93 (95% CrI, 0.88-0.98) after social distancing policies were implemented between March 15 and March 21, 2020. By June 15, a 1-week delay in policies would have increased cumulative hospitalizations from an observed actual number of 2246 hospitalizations to 8005 hospitalizations (75% CrI: 3973-15 236 hospitalizations) and increased deaths from an observed actual number of 482 deaths to a projected 1304 deaths (75% CrI, 656-2428 deaths). By June 15, a 2-week delay would have yielded 3292 deaths (75% CrI, 2104-4905 deaths)-an additional 2810 deaths or a 583% increase beyond what was actually observed. Sensitivity analyses incorporating a range of spontaneous behavior changes did not avert severe epidemic projections. The results of this decision analytical model study suggest that, in the St Louis region, timely social distancing policies were associated with improved population health outcomes, and small delays may likely have led to a COVID-19 epidemic similar to the most heavily affected areas in the US. These findings indicate that an open-source modeling platform designed to accept user-supplied local and regional data may provide projections tailored to, and more relevant for, local settings.

Assessing COVID-19 Prevention Strategies to Permit the Safe Opening of Residential Colleges in Fall 2021.

Effective vaccines, improved testing technologies, and decreases in COVID-19 incidence prompt an examination of the choices available to residential college administrators seeking to safely resume in-person campus activities in fall 2021. To help college administrators design and evaluate customized COVID-19 safety plans. Decision analysis using a compartmental epidemic model to optimize vaccination, testing, and other nonpharmaceutical interventions depending on decision makers' preferences, choices, and assumptions about epidemic severity and vaccine effectiveness against infection, transmission, and disease progression. U.S. residential colleges. Hypothetical cohort of 5000 persons (students, faculty, and staff) living and working in close proximity on campus. Cumulative infections over a 120-day semester. Under base-case assumptions, if 90% coverage can be attained with a vaccine that is 85% protective against infection and 25% protective against asymptomatic transmission, the model finds that campus activities can be resumed while holding cumulative cases below 5% of the population without the need for routine, asymptomatic testing. With 50% population coverage using such a vaccine, a similar cap on cumulative cases would require either daily asymptomatic testing of unvaccinated persons or a combination of less frequent testing and resumption of aggressive distancing and other nonpharmaceutical prevention policies. Colleges returning to pre-COVID-19 campus activities without either broad vaccination coverage or high-frequency testing put their campus population at risk for widespread viral transmission. Uncertainty in data, particularly vaccine effectiveness (preventive and transmission); no distinguishing between students and employees; and assumes limited community intermixing. Vaccination coverage is the most powerful tool available to residential college administrators seeking to achieve a safe return to prepandemic operations this fall. Given the breadth of potential outcomes in the face of uncontrollable and uncertain factors, even colleges with high vaccination rates should be prepared to reinstitute or expand testing and distancing policies on short notice. National Institute on Drug Abuse.

A numerical framework for estimating the effective reproduction number of infectious diseases from compartmental epidemic models

We present a numerical framework for estimating the effective reproduction number of infectious diseases from compartmental epidemic models. The idea is to augment the reproduction number as an extended state variable in the model. We assume the reproduction number is a piece-wise constant function with jumps every new data are recorded. The numerical framework is written in discrete-time and is combined with estimation algorithms. Simulation results using epidemiological data from Denmark and Germany show our approach is comparable with existing method such as methods based on Bayesian statistical inference.

Stochastic modelling of the dynamics of infections caused by the SARS-CoV-2 and COVID-19 under various conditions of lockdown, quarantine, and testing

We develop a mathematical model for the transmission and spread of infections caused by the severe acute respiratory syndrome coronavirus 2 (SAR-CoV-2) which causes the coronavirus disease 2019 (COVID-19), a disease that has since been classified, by the World Health Organization, as a global pandemic. We focus attention on virus transmissions in a closed population and hence use a compartmental epidemic model to study the inherent dynamics of infections between the various subgroups of the population. We assume random interactions between members from different subgroups and hence we employ stochastic modelling techniques. In the absence of a vaccine for this novel coronavirus, governments worldwide have put in place various intervention strategies, including travel bans, lockdowns, screening, testing, quarantine, etc. in order to reduce and hopefully eliminate the transmission and spread of the COVID-19 virus. These interventions are built into our model and we investigate their effects and effectiveness. In particular, we observe that the two subgroups containing infectious individuals, namely; the subgroup comprising of unidentified asymptomatic individuals as well as; the subgroup comprising of un-quarantined symptomatic individuals; pose the greatest risk of the transmission and spread of infections. We therefore also observe from our results that; rapid (realtime) mass testing as well as effective (or mandatory) quarantine of all infected individuals are the fundamentally critical and necessary steps in reducing the internal transmissions and spread of the COVID-19 virus. In particular, our results indicate that lockdowns are only important to the extent that, if implemented effectively, they help in reducing the rate of transmissions (i.e. help to ‘flatten’ the transmission curves) and hence allow policy makers and healthcare practitioners to put in place the important processes of realtime mass testing and mandatory quarantine, including hospitalization and treatment. Our results are presented over a single wave (or part of a single wave) of COVID-19 infections in a given population. If conditions are repeated, the results would correspondingly extend to multiple waves of infection.

Understanding and predicting the spatio-temporal spread of COVID-19 via integrating diffusive graph embedding and compartmental models.

In order to find useful intervention strategies for the novel coronavirus (COVID‐19), it is vital to understand how the disease spreads. In this study, we address the modeling of COVID‐19 spread across space and time, which facilitates understanding of the pandemic. We propose a hybrid data‐driven learning approach to capture the mobility‐related spreading mechanism of infectious diseases, utilizing multi‐sourced mobility and attributed data. This study develops a visual analytic approach that identifies and depicts the strength of the transmission pathways of COVID‐19 between areal units by integrating data‐driven deep learning and compartmental epidemic models, thereby engaging stakeholders (e.g., public health officials, managers from transportation agencies) to make informed intervention decisions and enable public messaging. A case study in the state of Colorado, USA was performed to demonstrate the applicability of the proposed transmission modeling approach in understanding the spatio‐temporal spread of COVID‐19 at the neighborhood level. Transmission path maps are presented and analyzed, demonstrating their utility in evaluating the effects of mitigation measures. In addition, integrated embeddings also support daily prediction of infected cases and role analysis of each area unit during the transmission of the virus.

A comprehensive estimation of country-level basic reproduction numbers R0 for COVID-19: Regime regression can automatically estimate the end of the exponential phase in epidemic data.

In a compartmental epidemic model, the initial exponential phase reflects a fixed interaction between an infectious agent and a susceptible population in steady state, so it determines the basic reproduction number R0 on its own. After the exponential phase, dynamic complexities like societal responses muddy the practical interpretation of many estimated parameters. The computer program ARRP, already available from sequence alignment applications, automatically estimated the end of the exponential phase in COVID-19 and extracted the exponential growth rate r for 160 countries. By positing a gamma-distributed generation time, the exponential growth method then yielded R0 estimates for COVID-19 in 160 countries. The use of ARRP ensured that the R0 estimates were largely freed from any dependency outside the exponential phase. The Prem matrices quantify rates of effective contact for infectious disease. Without using any age-stratified COVID-19 data, but under strong assumptions about the homogeneity of susceptibility, infectiousness, etc., across different age-groups, the Prem contact matrices also yielded theoretical R0 estimates for COVID-19 in 152 countries, generally in quantitative conflict with the R0 estimates derived from the exponential growth method. An exploratory analysis manipulating only the Prem contact matrices reduced the conflict, suggesting that age-groups under 20 years did not promote the initial exponential growth of COVID-19 as much as other age-groups. The analysis therefore supports tentatively and tardily, but independently of age-stratified COVID-19 data, the low priority given to vaccinating younger age groups. It also supports the judicious reopening of schools. The exploratory analysis also supports the possibility of suspecting differences in epidemic spread among different age-groups, even before substantial amounts of age-stratified data become available.

Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy

In this study, using a system of delay nonlinear ordinary differential equations, we introduce a new compartmental epidemic model considered the effect of filiation (contamination) control strategy to the spread of Covid-19. Firstly, the formulation of this new $SI_{u}I_{a}QR$ epidemic model with delay process and the parameters arised from isolation and filiation is formed. Then the disease-free and endemic equilibrium points of the model is obtained. Also, the basic reproduction number $\mathcal{R}_{0}$ is found by using the next-generation matrix method, and the results on stabilities of the disease-free and endemic equilibrium points are investigated. Finally some examples are presented to show the effect of filiation control strategy.

Control with uncertain data of socially structured compartmental epidemic models

The adoption of containment measures to reduce the amplitude of the epidemic peak is a key aspect in tackling the rapid spread of an epidemic. Classical compartmental models must be modified and studied to correctly describe the effects of forced external actions to reduce the impact of the disease. The importance of social structure, such as the age dependence that proved essential in the recent COVID-19 pandemic, must be considered, and in addition, the available data are often incomplete and heterogeneous, so a high degree of uncertainty must be incorporated into the model from the beginning. In this work we address these aspects, through an optimal control formulation of a socially structured epidemic model in presence of uncertain data. After the introduction of the optimal control problem, we formulate an instantaneous approximation of the control that allows us to derive new feedback controlled compartmental models capable of describing the epidemic peak reduction. The need for long-term interventions shows that alternative actions based on the social structure of the system can be as effective as the more expensive global strategy. The timing and intensity of interventions, however, is particularly relevant in the case of uncertain parameters on the actual number of infected people. Simulations related to data from the first wave of the recent COVID-19 outbreak in Italy are presented and discussed.

A Compartmental Epidemiological Model Applied to the Covid-19 Epidemic

The objective of this work is to provide a sophisticated but accessible compartmental epidemic model. Our algorithm is highly inspired from the compartmental model developed by Sofonea and al. in 2020. This model has been used as a reference for several working groups in France during the Covid-19 crisis. Each individual is allocated to a compartment according to her age, her current state with respect to the disease, as well as the length of time she has been in that state. The model then reproduces the mechanisms of transition from one state to another: mathematically, this translates into a system of recurrence relations. It captures how much individuals interact with one another through a parameter that estimates compliance with hygiene measures and lifestyle habits. The present work aims to make the model implementation fully transparent as well as the corresponding code available and give control to users so that they are able to test the model in total transparency. Focus has been put on reproducibility and explanation of the various parameters. The hard-coded parameters correspond to the data for the Covid-19 epidemic in France.

Dynamics of SEIR model with delay effects - latent period and recovery period

A compartmental SEIR epidemic model with constant latent and recovery periods as delay effects is proposed in this paper. Incidence rate applied in the model is non-monotonic. Basic reproduction number for the model is calculated and the dynamical behavior is analyzed by studying the local & global asymptotic stability of disease free and endemic equilibriums of this model. Numerical simulation of this model is given to illustrate the results.

Mathematical model, forecast and analysis on the spread of COVID-19.

Pandemic COVID-19 which has infected more than 35,027,546 people and death more than 1,034,837 people in 235 countries as on October 05, 2020 has created a chaos across the globe. In this paper, we develop a compartmental epidemic model to understand the spreading behaviour of the disease in human population with a special case of Bhilwara, a desert town in India where successful control measures TTT (tracking, testing and treatment) was adopted to curb the disease in the very early phase of the spread of the disease in India. Local and global asymptotic stability is established for endemic equilibrium. Extensive numerical simulations with real parametric values are performed to validate the analytical results. Trend analysis of fatality rate, infection rate, and impact of lockdown is performed for USA, European countries, Russia, Iran, China, Japan, S. Korea with a comparative assessment by India. Kruskal - Wallis test is performed to test the null hypothesis for infected cases during the four lockdown phases in India. It has been observed that there is a significant difference at both 95% and 99% confidence interval in the infected cases, recovered cases and the case fatality rate during all the four phases of the lockdown.

Short-term predictions and prevention strategies for COVID-19: A model-based study

An outbreak of respiratory disease caused by a novel coronavirus is ongoing from December 2019. As of December 14, 2020, it has caused an epidemic outbreak with more than 73 million confirmed infections and above 1.5 million reported deaths worldwide. During this period of an epidemic when human-to-human transmission is established and reported cases of coronavirus disease 2019 (COVID-19) are rising worldwide, investigation of control strategies and forecasting are necessary for health care planning. In this study, we propose and analyze a compartmental epidemic model of COVID-19 to predict and control the outbreak. The basic reproduction number and the control reproduction number are calculated analytically. A detailed stability analysis of the model is performed to observe the dynamics of the system. We calibrated the proposed model to fit daily data from the United Kingdom (UK) where the situation is still alarming. Our findings suggest that independent self-sustaining human-to-human spread (R0>1,Rc>1) is already present. Short-term predictions show that the decreasing trend of new COVID-19 cases is well captured by the model. Further, we found that effective management of quarantined individuals is more effective than management of isolated individuals to reduce the disease burden. Thus, if limited resources are available, then investing on the quarantined individuals will be more fruitful in terms of reduction of cases.

Uncertainty quantification of a mathematical model of COVID-19 transmission dynamics with mass vaccination strategy.

In this paper, the uncertainty quantification and sensitivity analysis of a mathematical model of the SARS-CoV-2 virus transmission dynamics with mass vaccination strategy has been carried out. More specifically, a compartmental epidemic model has been considered, in which vaccination, social distance measures, and testing of susceptible individuals have been included. Since the application of these mitigation measures entails a degree of uncertainty, the effects of the uncertainty about the application of social distance actions and testing of susceptible individuals on the disease transmission have been quantified, under the assumption of a mass vaccination program deployment. A spectral approach has been employed, which allows the uncertainty propagation through the epidemic model to be represented by means of the polynomial chaos expansion of the output random variables. In particular, a statistical moment-based polynomial chaos expansion has been implemented, which provides a surrogate model for the compartments of the epidemic model, and allows the statistics, the probability distributions of the interesting output variables of the model at a given time instant to be estimated and the sensitivity analysis to be conducted. The purpose of the sensitivity analysis is to understand which uncertain parameters have most influence on a given output random variable of the model at a given time instant. Several numerical experiments have been conducted whose results show that the proposed spectral approach to uncertainty quantification and sensitivity analysis of epidemic models provides a useful tool to control and mitigate the effects of the COVID-19 pandemic, when it comes to healthcare resource planning.

Sensitivity assessment and optimal economic evaluation of a new COVID-19 compartmental epidemic model with control interventions.

Optimal economic evaluation is pivotal in prioritising the implementation of non-pharmaceutical and pharmaceutical interventions in the control of diseases. Governments, decision-makers and policy-makers broadly need information about the effectiveness of a control intervention concerning its cost-benefit to evaluate whether a control intervention offers the best value for money. The outbreak of COVID-19 in December 2019, and the eventual spread to other parts of the world, have pushed governments and health authorities to take drastic socioeconomic, sociocultural and sociopolitical measures to curb the spread of the virus, SARS-CoV-2. To help policy-makers, health authorities and governments, we propose a Susceptible, Exposed, Asymptomatic, Quarantined asymptomatic, Severely infected, Hospitalized, Recovered, Recovered asymptomatic, Deceased, and Protective susceptible (individuals who observe health protocols) compartmental structure to describe the dynamics of COVID-19. We fit the model to real data from Ghana and Egypt to estimate model parameters using standard incidence rate. Projections for disease control and sensitivity analysis are presented using MATLAB. We noticed that multiple peaks (waves) of COVID-19 for Ghana and Egypt can be prevented if stringent health protocols are implemented for a long time and/or the reluctant behaviour on the use of protective equipment by individuals are minimized. The sensitivity analysis suggests that: the rate of diagnoses and testing, the rate of quarantine through doubling enhanced contact tracing, adhering to physical distancing, adhering to wearing of nose masks, sanitizing-washing hands, media education remains the most effective measures in reducing the control reproduction number to less than unity in the absence of vaccines and therapeutic drugs in Ghana and Egypt. Optimal control and cost-effectiveness analysis are rigorously studied. The main finding is that having two controls (transmission reduction and case isolation) is better than having one control, but is economically expensive. In case only one control is affordable, then transmission reduction is better than case isolation. Hopefully, the results of this research should help policy-makers when dealing with multiple waves of COVID-19.

Clinical and Economic Effects of Widespread Rapid Testing to Decrease SARS-CoV-2 Transmission

The value of frequent, rapid testing to reduce community transmission of SARS-CoV-2 is poorly understood. To define performance standards and predict the clinical, epidemiologic, and economic outcomes of nationwide, home-based antigen testing. A simple compartmental epidemic model that estimated viral transmission, portrayed disease progression, and forecast resource use, with and without testing. Parameter values and ranges as informed by Centers for Disease Control and Prevention guidance and published literature. U.S. population. 60 days. Societal; costs included testing, inpatient care, and lost workdays. Home-based SARS-CoV-2 antigen testing. Cumulative infections and deaths, number of persons isolated and hospitalized, and total costs. Without a testing intervention, the model anticipates 11.6 million infections, 119000 deaths, and $10.1 billion in costs ($6.5 billion in inpatient care and $3.5 billion in lost productivity) over a 60-day horizon. Weekly availability of testing would avert 2.8 million infections and 15700 deaths, increasing costs by $22.3 billion. Lower inpatient outlays ($5.9 billion) would partially offset additional testing expenditures ($12.5 billion) and workdays lost ($14.0 billion), yielding incremental cost-effectiveness ratios of $7890 per infection averted and $1430000 per death averted. Outcome estimates vary widely under different behavioral assumptions and testing frequencies. However, key findings persist across all scenarios, with large reductions in infections, mortality, and hospitalizations. Costs per death averted are roughly an order of magnitude lower than commonly accepted willingness-to-pay values per statistical life saved ($5 to $17 million). Analysis was restricted to at-home testing. There are uncertainties concerning test performance. High-frequency home testing for SARS-CoV-2 with an inexpensive, imperfect test could contribute to pandemic control at justifiable cost and warrants consideration as part of a national containment strategy. National Institutes of Health.

Mathematical modelling and analysis of COVID-19 epidemic and predicting its future situation in Ethiopia

The epidemic of the coronavirus disease 2019 (COVID-19) has been rising rapidly and life-threatening worldwide since its inception. The lack of an established vaccine for this disease has caused millions of illnesses and hundreds of thousands of deaths globally. Mathematical models have become crucial tools in determining the potential and seriousness of the disease and in helping the types of strategic intervention measures to be taken to prevent and control the intensity of the spread of the disease. In this study, a compartmental epidemic model of COVID-19 is proposed and analyzed to predict the transmission dynamics of the disease in Ethiopia. Analytically, the basic reproduction number is determined. To observe the dynamics of the system, a detailed stability analysis of the disease-free equilibrium (DFE) of the proposed model is carried out. Our result shows that the DFE is stable if the basic reproduction number is less than unity and unstable otherwise. Also, the parameters of the assumed model are estimated using the actual data of COVID-19 from Ethiopia reported for three months between March and June 2020. Furthermore, we performed a sensitivity analysis of the basic reproductive number and found that reducing the rate of transmission is the most important factor in achieving disease control. Numerical simulations demonstrate the suitability of the proposed model for the actual COVID-19 data in Ethiopia. In particular, the numerical simulation shows an increase in the rate of transmission leads to a significant increase in the infected individuals. Thus, results of the numerical simulations are in agreement with the sensitivity results of the system. The possible implication of this is that declining the rate of transmission to the desired level could enable us to combat the disease. Numerical simulations are also performed to forecast the disease prevalence in the community.

A compartmental epidemic model incorporating probable cases to model COVID-19 outbreak in regions with limited testing capacity

We propose a new compartmental epidemic model taking into account people who have symptoms with no confirmatory laboratory testing (probable cases). We prove well-posedness of the model and provide an explicit expression for the basic reproduction number (R0). We use the model together with an extended Kalman filter (EKF) to estimate the time-varying effective reproduction number (Rt) of COVID-19 in West Java province in Indonesia and the state of Michigan in the USA, where laboratory testing capacities are limited. Based on our estimation, the value of Rt is higher when the probable cases are taken into account. This correction can be used by decision and policy makers when considering re-opening policy and evaluation of public measures.

Analysis of a COVID-19 compartmental model: a mathematical and computational approach.

In this note, we consider a compartmental epidemic mathematical model given by a system of differential equations. We provide a complete toolkit for performing both a symbolic and numerical analysis of the spreading of COVID-19. By using the free and open-source programming language Python and the mathematical software SageMath, we contribute for the reproducibility of the mathematical analysis of the stability of the equilibrium points of epidemic models and their fitting to real data. The mathematical tools and codes can be adapted to a wide range of mathematical epidemic models.