Age-Structured Modeling of COVID-19 Epidemic in the USA, UAE and Algeria

Impact of immunity loss on the optimal vaccination strategy for an age-structured epidemiological model

The pursuit of effective vaccination strategies against COVID-19 remains a critical endeavour in global public health, particularly amidst challenges posed by immunity loss and evolving epidemiological dynamics. This study investigated optimal vaccination strategies by considering age structure, immunity dynamics, and varying maximal vaccination rates. To this end, we formulated an SEIR model stratified into $ n $ age classes, with the vaccination rate as an age-dependent control variable in an optimal control problem. We developed an objective function aimed at minimising critical infections while optimising vaccination efforts and then conducted rigorous mathematical analyses to ensure the existence and characterization of the optimal control. Using data from three countries with diverse age distributions, in expansive, constrictive, and stationary pyramids, we performed numerical simulations to evaluate the optimal age-dependent vaccination strategy, number of critical infections, and vaccination frequency. Our findings highlight the significant influence of maximal vaccination rates on shaping optimal vaccination strategies. Under constant maximal vaccination rates, prioritising age groups based on population demographics proves effective, with higher rates resulting in fewer critically infected individuals across all age distributions. Conversely, adopting age-dependent maximal vaccination rates, akin to the WHO strategy, may not always lead to the lowest critical infection peaks but offers a viable alternative in resource-constrained settings.

Understanding HIV/AIDS dynamics: insights from CD4+T cells, antiretroviral treatment, and country-specific analysis

In this article, we present a mathematical model for human immunodeficiency virus (HIV)/Acquired immune deficiency syndrome (AIDS), taking into account the number of CD4+T cells and antiretroviral treatment. This model is developed based on the susceptible, infected, treated, AIDS (SITA) framework, wherein the infected and treated compartments are divided based on the number of CD4+T cells. Additionally, we consider the possibility of treatment failure, which can exacerbate the condition of the treated individual. Initially, we analyze a simplified HIV/AIDS model without differentiation between the infected and treated classes. Our findings reveal that the global stability of the HIV/AIDS-free equilibrium point is contingent upon the basic reproduction number being less than one. Furthermore, a bifurcation analysis demonstrates that our simplified model consistently exhibits a transcritical bifurcation at a reproduction number equal to one. In the complete model, we elucidate how the control reproduction number determines the stability of the HIV/AIDS-free equilibrium point. To align our model with the empirical data, we estimate its parameters using prevalence data from the top four countries affected by HIV/AIDS, namely, Eswatini, Lesotho, Botswana, and South Africa. We employ numerical simulations and conduct elasticity and sensitivity analyses to examine how our model parameters influence the control reproduction number and the dynamics of each model compartment. Our findings reveal that each country displays distinct sensitivities to the model parameters, implying the need for tailored strategies depending on the target country. Autonomous simulations highlight the potential of case detection and condom use in reducing HIV/AIDS prevalence. Furthermore, we identify that the quality of condoms plays a crucial role: with higher quality condoms, a smaller proportion of infected individuals need to use them for the potential eradication of HIV/AIDS from the population. In our optimal control simulations, we assess population behavior when control interventions are treated as time-dependent variables. Our analysis demonstrates that a combination of condom use and case detection, as time-dependent variables, can significantly curtail the spread of HIV while maintaining an optimal cost of intervention. Moreover, our cost-effectiveness analysis indicates that the condom use intervention alone emerges as the most cost-effective strategy, followed by a combination of case detection and condom use, and finally, case detection as a standalone strategy.

Modeling preferential attraction to infected hosts in vector-borne diseases

Vector-borne infectious diseases cause more than 700,000 deaths a year and represent an increasing threat to public health worldwide. Strategies to mitigate the spread of vector-borne diseases can benefit from a thorough understanding of all mechanisms that contribute to viral propagation in human. A recent study showed that Aedes mosquitoes (the vectors for dengue and Zika virus, among others) are preferentially attracted to infected hosts. In order to determine the impact of this factor on viral spread, we built a dedicated agent-based model and parameterized it on dengue fever. We then performed a systematic study of how mosquitoes' preferential attraction for infected hosts affects viral load and persistence of the infection. Our results indicate that even small values of preferential attraction have a dramatic effect on the number of infected individuals and the persistence of the infection in the population. Taken together, our results suggests that interventions aimed at decreasing the preferential attraction of vectors for infected hosts can reduce viral transmission and thus can have public health implications.

Modeling the competitive transmission of the Omicron strain and Delta strain of COVID-19

Since November 2021, there have been cases of COVID-19's Omicron strain spreading in competition with Delta strains in many parts of the world. To explore how these two strains developed in this competitive spread, a new compartmentalized model was established. First, we analyzed the fundamental properties of the model, obtained the expression of the basic reproduction number, proved the local and global asymptotic stability of the disease-free equilibrium. Then by means of the cubic spline interpolation method, we obtained the data of new Omicron and Delta cases in the United States of new cases starting from December 8, 2021, to February 12, 2022. Using the weighted nonlinear least squares estimation method, we fitted six time series (cumulative confirmed cases, cumulative deaths, new cases, new deaths, new Omicron cases, and new Delta cases), got estimates of the unknown parameters, and obtained an approximation of the basic reproduction number in the United States during this time period as R 0 ≈ 1.5165 . Finally, each control strategy was evaluated by cost-effectiveness analysis to obtain the optimal control strategy under different perspectives. The results not only show the competitive transmission characteristics of the new strain and existing strain, but also provide scientific suggestions for effectively controlling the spread of these strains.

Study on the regional risk classification method for the prevention and control of emerging infectious diseases based on directed graph theory

Background

Emerging infectious diseases are a class of diseases that are spreading rapidly and are highly contagious. It seriously affects social stability and poses a significant threat to human health, requiring urgent measures to deal with them. Its outbreak will very easily lead to the large-scale spread of the virus, causing social problems such as work stoppages and traffic control, thereby causing social panic and psychological unrest, affecting human activities and social stability, and even endangering lives. It is essential to prevent and control the spread of infectious diseases effectively.

Purpose

We aim to propose an effective method to classify the risk level of a new epidemic region by using graph theory and risk classification methods to provide a theoretical reference for the comprehensive evaluation and determination of epidemic prevention and control, as well as risk level classification.

Methods

Using the graph theory method, we first define the network structure of social groups and construct the risk transmission network of the new epidemic region. Then, combined with the risk classification method, the classification of high, medium, and low risk levels of the new epidemic region is discussed from two cases with common and looped graph nodes, respectively. Finally, the reasonableness of the classification method is verified by simulation data.

Results

The directed weighted scale-free network can better describe the transmission law of an epidemic. Moreover, the proposed method of classifying the risk level of a region by using the correlation function between two regions and the risk value of the regional nodes can effectively evaluate the risk level of different regions in the new epidemic region. The experiments show that the number of medium and high risk nodes shows no increasing trend. The number of high-risk regions is relatively small compared to medium-risk regions, and the number of low-risk regions is the largest.

Conclusions

It is necessary to distinguish scientifically between the risk level of the epidemic area and the neighboring regions so that the constructed social network model of the epidemic region's spread risk can better describe the spread of the epidemic risk in the social network relations.

Analysis and prediction of improved SEIR transmission dynamics model: taking the second outbreak of COVID-19 in Italy as an example

This study aimed to predict the transmission trajectory of the 2019 Corona Virus Disease (COVID-19) and analyze the impact of preventive measures on the spread of the epidemic. Considering that tracking a long-term epidemic trajectory requires explanatory modeling with more complexities than short-term predictions, an improved Susceptible-Exposed-Infected-Removed (SEIR) transmission dynamic model is established. The model depends on defining various parameters that describe both the virus and the population under study. However, it is likely that several of these parameters will exhibit significant variations among different states. Therefore, regression algorithms and heuristic algorithms were developed to effectively adapt the population-dependent parameters and ensure accurate fitting of the SEIR model to data for any specific state. In this study, we consider the second outbreak of COVID-19 in Italy as a case study, which occurred in August 2020. We divide the epidemic data from February to September of the same year into two distinct stages for analysis. The numerical results demonstrate that the improved SEIR model effectively simulates and predicts the transmission trajectories of the Italian epidemic during both periods before and after the second outbreak. By analyzing the impact of anti-epidemic measures on the spread of the disease, our findings emphasize the significance of implementing anti-epidemic preventive measures in COVID-19 modeling.

Inverse problem for parameters identification in a modified SIRD epidemic model using ensemble neural networks

In this paper, we propose a parameter identification methodology of the SIRD model, an extension of the classical SIR model, that considers the deceased as a separate category. In addition, our model includes one parameter which is the ratio between the real total number of infected and the number of infected that were documented in the official statistics. Due to many factors, like governmental decisions, several variants circulating, opening and closing of schools, the typical assumption that the parameters of the model stay constant for long periods of time is not realistic. Thus our objective is to create a method which works for short periods of time. In this scope, we approach the estimation relying on the previous 7 days of data and then use the identified parameters to make predictions. To perform the estimation of the parameters we propose the average of an ensemble of neural networks. Each neural network is constructed based on a database built by solving the SIRD for 7 days, with random parameters. In this way, the networks learn the parameters from the solution of the SIRD model. Lastly we use the ensemble to get estimates of the parameters from the real data of Covid19 in Romania and then we illustrate the predictions for different periods of time, from 10 up to 45 days, for the number of deaths. The main goal was to apply this approach on the analysis of COVID-19 evolution in Romania, but this was also exemplified on other countries like Hungary, Czech Republic and Poland with similar results. The results are backed by a theorem which guarantees that we can recover the parameters of the model from the reported data. We believe this methodology can be used as a general tool for dealing with short term predictions of infectious diseases or in other compartmental models.

Optimal control and stability analysis of an age-structured SEIRV model with imperfect vaccination

Vaccination programs are crucial for reducing the prevalence of infectious diseases and ultimately eradicating them. A new age-structured SEIRV (S-Susceptible, E-Exposed, I-Infected, R-Recovered, V-Vaccinated) model with imperfect vaccination is proposed. After formulating our model, we show the existence and uniqueness of the solution using semigroup of operators. For stability analysis, we obtain a threshold parameter $ R_0 $. Through rigorous analysis, we show that if $ R_0 < 1 $, then the disease-free equilibrium point is stable. The optimal control strategy is also discussed, with the vaccination rate as the control variable. We derive the optimality conditions, and the form of the optimal control is obtained using the adjoint system and sensitivity equations. We also prove the uniqueness of the optimal controller. To visually illustrate our theoretical results, we also solve the model numerically.

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.

Dynamics and calculation of the basic reproduction number for a nonlocal dispersal epidemic model with air pollution

In order to reflect the dispersal of pollutants in non-adjacent areas and the large-scale movement of individuals, this paper proposes an epidemic model of nonlocal dispersal with air pollution, where the transmission rate is related to the concentration of pollutants. This paper checks the uniqueness and existence of the global positive solution and defines the basic reproduction number, R 0 . We simultaneously explore the global dynamics: when R 0 < 1 , the disease-free stable point is global asymptotic stability; when R 0 > 1 , the disease is uniformly persistent. Additionally, in order to approximate R 0 , a numerical method has been introduced. Illustrative examples are used to verify the theoretical outcomes and show the effect of the dispersal rate on the basic reproduction number R 0 .

Modeling and analysis of novel COVID-19 outbreak under fractal-fractional derivative in Caputo sense with power-law: a case study of Pakistan

In this paper, a five-compartment model is used to explore the dynamics of the COVID-19 pandemic, taking the vaccination campaign into account. The present model consists of five components that lead to a system of five ordinary differential equations. In this paper, we examined the disease from the perspective of a fractal fractional derivative in the Caputo sense with a power law type kernal. The model is also fitted with real data for Pakistan between June 1, 2020, and March 8, 2021. The fundamental mathematical characteristics of the model have been investigated thoroughly. We have calculated the equilibrium points and the reproduction number for the model and obtained the feasible region for the system. The existence and stability criteria of the model have been validated using the Banach fixed point theory and the Picard successive approximation technique. Furthermore, we have conducted stability analysis for both the disease-free and endemic equilibrium states. On the basis of sensitivity analysis and the dynamics of the threshold parameter, we have estimated the effectiveness of vaccination and identified potential control strategies for the disease using the proposed model outbreaks. The stability of the concerned solution in Ulam-Hyers and Ulam-Hyers-Rassias sense is also investigated. For the proposed problem, some results regarding basic reproduction numbers and stability analysis for various parameters are represented graphically. Matlab software is used for numerical illustrations. Graphical representations are given for different fractional orders and for various parametric values.

Dynamical analysis of a stochastic non-autonomous SVIR model with multiple stages of vaccination

In this paper, we analyze the dynamics of a new proposed stochastic non-autonomous SVIR model, with an emphasis on multiple stages of vaccination, due to the vaccine ineffectiveness. The parameters of the model are allowed to depend on time, to incorporate the seasonal variation. Furthermore, the vaccinated population is divided into three subpopulations, each one representing a different stage. For the proposed model, we prove the mathematical and biological well-posedness. That is, the existence of a unique global almost surely positive solution. Moreover, we establish conditions under which the disease vanishes or persists. Furthermore, based on stochastic stability theory and by constructing a suitable new Lyapunov function, we provide a condition under which the model admits a non-trivial periodic solution. The established theoretical results along with the performed numerical simulations exhibit the effect of the different stages of vaccination along with the stochastic Gaussian noise on the dynamics of the studied population.

A review about COVID-19 in the MENA region: environmental concerns and machine learning applications

Coronavirus disease 2019 (COVID-19) has delayed global economic growth, which has affected the economic life globally. On the one hand, numerous elements in the environment impact the transmission of this new coronavirus. Every country in the Middle East and North Africa (MENA) area has a different population density, air quality and contaminants, and water- and land-related conditions, all of which influence coronavirus transmission. The World Health Organization (WHO) has advocated fast evaluations to guide policymakers with timely evidence to respond to the situation. This review makes four unique contributions. One, many data about the transmission of the new coronavirus in various sorts of settings to provide clear answers to the current dispute over the virus's transmission were reviewed. Two, highlight the most significant application of machine learning to forecast and diagnose severe acute respiratory syndrome coronavirus (SARS-CoV-2). Three, our insights provide timely and accurate information along with compelling suggestions and methodical directions for investigators. Four, the present study provides decision-makers and community leaders with information on the effectiveness of environmental controls for COVID-19 dissemination.

A population structure-sensitive mathematical model assessing the effects of vaccination during the third surge of COVID-19 in Italy

We provide a non-autonomous mathematical model to describe some of the most relevant parameters associated to the COVID-19 pandemic, such as daily and cumulative deaths, active cases, and cumulative incidence, among others. We will take into consideration the ways in which people from four different age ranges react to the virus. Using an appropriate transmission function, we estimate the impact of the third surge of COVID-19 in Italy. Also, we assess two different vaccination programmes. In one of them, a single shot is administered to all citizens over 16 years old before second shots are available. In the second model, first and second shots are administered to each citizen within, approximately, 20 days of time-gap.

Model-Based Analysis of SARS-CoV-2 Infections, Hospitalization and Outcome in Germany, the Federal States and Districts

The coronavirus disease 2019 (COVID-19) pandemic challenged many national health care systems, with hospitals reaching capacity limits of intensive care units (ICU). Thus, the estimation of acute local burden of ICUs is critical for appropriate management of health care resources. In this work, we applied non-linear mixed effects modeling to develop an epidemiological SARS-CoV-2 infection model for Germany, with its 16 federal states and 400 districts, that describes infections as well as COVID-19 inpatients, ICU patients with and without mechanical ventilation, recoveries, and fatalities during the first two waves of the pandemic until April 2021. Based on model analyses, covariates influencing the relation between infections and outcomes were explored. Non-pharmaceutical interventions imposed by governments were found to have a major impact on the spreading of SARS-CoV-2. Patient age and sex, the spread of variant B.1.1.7, and the testing strategy (number of tests performed weekly, rate of positive tests) affected the severity and outcome of recorded cases and could reduce the observed unexplained variability between the states. Modeling could reasonably link the discrepancies between fine-grained model simulations of the 400 German districts and the reported number of available ICU beds to coarse-grained COVID-19 patient distribution patterns within German regions.

Modeling and Analysis of COVID-19 Spread: The Impacts of Nonpharmaceutical Protocols

In this study, the extended SEIR dynamical model is formulated to investigate the spread of coronavirus disease (COVID-19) via a special focus on contact with asymptomatic and self-isolated infected individuals. Furthermore, a mathematical analysis of the model, including positivity, boundedness, and local and global stability of the disease-free and endemic equilibrium points in terms of the basic reproduction number, is presented. The sensitivity analysis indicates that reducing the disease contact rate and the transmissibility factor related to asymptomatic individuals, along with increasing the quarantine/self-isolation rate and the contact-tracing process, from the view of flattening the curve for novel coronavirus, are crucial to the reduction in disease-related deaths.

Modeling of corona virus and its application in confocal microscopy

Background

The proposal of spiky apertures showed resolution improvement compared with the circular apertures. Three models of corona virus are given. The 1st model consists of uniform circular aperture provided with 8 spikes while the 2nd model has 16 spikes for the same uniform circular aperture. The 3rd model has circular linear distribution with 8 spikes.

Results

The Normalized Point Spread Function (PSF) or the impulse response is computed for the three models using fast Fourier transform technique. In addition, the autocorrelation function corresponding to these apertures is calculated and compared with that corresponding to the ordinary circular and conic apertures. Coronavirus image is used as an object in the formation of images using confocal scanning laser microscope provided with suggested models. The fabricated MATLAB code is used to compute and plot all images and line plots.

Conclusions

The PSF plots are computed from Eqs. (8) and (12) using MATLAB code showing narrower cutoff in the PSF for spiky aperture compared with that corresponding to the uniform circular aperture and modulated linear and quadratic apertures. Hence, I reached resolution improvement in the case of spiky aperture.

Effect of the Universal Health Coverage Healthcare System on Stock Returns During COVID-19: Evidence From Global Stock Indices

The increased uncertainty caused by a sudden epidemic disease has had an impact on the global financial market. We aimed to assess the primary healthcare system of universal health coverage (UHC) during the coronavirus disease (COVID-19) pandemic and its relationship with the financial market. To this end, we employed the abnormal returns of 68 countries from January 2, 2019, to December 31, 2020, to test the impact of the COVID-19 outbreak on abnormal returns in the stock market and determine how a country's UHC changes the impact of a sudden pandemic on abnormal returns. Our findings show that the sudden onset of an epidemic disease results in unevenly distributed medical system resources, consequently diminishing the impact of UHC on abnormal returns.

Associations between the COVID-19 Pandemic and Hospital Infrastructure Adaptation and Planning—A Scoping Review

The SARS-CoV-2 pandemic has put unprecedented pressure on the hospital sector around the world. It has shown the importance of preparing and planning in the future for an outbreak that overwhelms every aspect of a hospital on a rapidly expanding scale. We conducted a scoping review to identify, map, and systemize existing knowledge about the relationships between COVID-19 and hospital infrastructure adaptation and capacity planning worldwide. We searched the Web of Science, Scopus, and PubMed and hand-searched gray papers published in English between December 2019 and December 2021. A total of 106 papers were included: 102 empirical studies and four technical reports. Empirical studies entailed five reviews, 40 studies focusing on hospital infrastructure adaptation and planning during the pandemics, and 57 studies on modeling the hospital capacity needed, measured mostly by the number of beds. The majority of studies were conducted in high-income countries and published within the first year of the pandemic. The strategies adopted by hospitals can be classified into short-term (repurposing medical and non-medical buildings, remote adjustments, and establishment of de novo structures) and long-term (architectural and engineering modifications, hospital networks, and digital approaches). More research is needed, focusing on specific strategies and the quality assessment of the evidence.

A robust study of a piecewise fractional order COVID-19 mathematical model

In the current manuscript, we deal with the dynamics of a piecewise covid-19 mathematical model with quarantine class and vaccination using SEIQR epidemic model. For this, we discussed the deterministic, stochastic, and fractional forms of the proposed model for different steps. It has a great impact on the infectious disease models and especially for covid-19 because in start the deterministic model played its role but with time due to uncertainty the stochastic model takes place and with long term expansion the use of fractional derivatives are required. The stability of the model is discussed regarding the reproductive number. Using the non-standard finite difference scheme for the numerical solution of the deterministic model and illustrate the obtained results graphically. Further, environmental noises are added to the model for the description of the stochastic model. Then take out the existence and uniqueness of positive solution with extinction for infection. Finally, we utilize a new technique of piecewise differential and integral operators for approximating Caputo-Fabrizio fractional derivative operator for the purpose of constructing of the fractional-order model. Then study the dynamics of the models such as positivity and boundedness of the solutions and local stability analysis. Solved numerically fractional-order model used Newton Polynomial scheme and present the results graphically.

Acute threshold dynamics of an epidemic system with quarantine strategy driven by correlated white noises and Lévy jumps associated with infinite measure

Several studies have previously been conducted on the dynamics of probabilistic epidemic models driven by Lévy disorder. All of these works have used the Poisson counting process with finite Lévy measures. However, this scope disregards a considerable category of correlated Lévy jump processes governed by an infinite Lévy measure. In this research, we take into consideration this general framework applied to an epidemic model with a quarantine strategy. Under an appropriate hypothetical setting, we infer the exact threshold value between the ergodicity and the disease disappearance. Our analysis completes the work presented by Privault and Wang (J Nonlinear Sci 31(1):1-28, 2021) and puts forward a novel analytical aspect to deal with other stochastic models in several areas. As a numerical application, we implement the algorithm of Rosinski (Stoch Process Appl 117:677-707, 2007) for tempered stable Lévy processes with an infinite Lévy measure.

Spatio-temporal solutions of a diffusive directed dynamics model with harvesting

The study considers a directed dynamics reaction-diffusion competition model to study the density of evolution for a single species population with harvesting effect in a heterogeneous environment, where all functions are spatially distributed in time series. The dispersal dynamics describe the growth of the species, which is distributed according to the resource function with no-flux boundary conditions. The analysis investigates the existence, positivity, persistence, and stability of solutions for both time-periodic and spatial functions. The carrying capacity and the distribution function are either arbitrary or proportional. It is observed that if harvesting exceeds the growth rate, then eventually, the population drops down to extinction. Several numerical examples are considered to support the theoretical results.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s12190-022-01742-x.

The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case

This study concentrates on the analysis of a stochastic SIC epidemic system with an enhanced and general perturbation. Given the intricacy of some impulses caused by external disturbances, we integrate the quadratic Lévy noise into our model. We assort the long-run behavior of a perturbed SIC epidemic model presented in the form of a system of stochastic differential equations driven by second-order jumps. By ameliorating the hypotheses and using some new analytical techniques, we find the exact threshold value between extinction and ergodicity (persistence) of our system. The idea and analysis used in this paper generalize the work of N. T. Dieu et al. (2020), and offer an innovative approach to dealing with other random population models. Comparing our results with those of previous studies reveals that quadratic jump-diffusion has no impact on the threshold value, but it remarkably influences the dynamics of the infection and may worsen the pandemic situation. In order to illustrate this comparison and confirm our analysis, we perform numerical simulations with some real data of COVID-19 in Morocco. Furthermore, we arrive at the following results: (i) the time average of the different classes depends on the intensity of the noise (ii) the quadratic noise has a negative effect on disease duration (iii) the stationary density function of the population abruptly changes its shape at some values of the noise intensity. Mathematics Subject Classification 2020: 34A26; 34A12; 92D30; 37C10; 60H30; 60H10.

A multi-strain epidemic model for COVID-19 with infected and asymptomatic cases: application to French data

Many SARS-CoV-2 variants have appeared over the last months, and many more will continue to appear. Understanding the competition between these different variants could help make future predictions on the evolution of epidemics. In this study we use a mathematical model to investigate the impact of three different SARS-CoV-2 variants on the spread of COVID-19 across France, between January-May 2021 (before vaccination was extended to the full population). To this end, we use the data from Geodes (produced by Public Health France) and a particle swarm optimisation algorithm, to estimate the model parameters and further calculate a value for the basic reproduction number R₀. Sensitivity and uncertainty analysis is then used to better understand the impact of estimated parameter values on the number of infections leading to both symptomatic and asymptomatic individuals. The results confirmed that, as expected, the alpha, beta and gamma variants are more transmissible than the original viral strain. In addition, the sensitivity results showed that the beta/gamma variants could have lead to a larger number of infections in France (of both symptomatic and asymptomatic people).

Mathematical Modeling to Study Optimal Allocation of Vaccines against COVID-19 Using an Age-Structured Population

Vaccination against the coronavirus disease 2019 (COVID-19) started in early December of 2020 in the USA. The efficacy of the vaccines vary depending on the SARS-CoV-2 variant. Some countries have been able to deploy strong vaccination programs, and large proportions of their populations have been fully vaccinated. In other countries, low proportions of their populations have been vaccinated, due to different factors. For instance, countries such as Afghanistan, Cameroon, Ghana, Haiti and Syria have less than 10% of their populations fully vaccinated at this time. Implementing an optimal vaccination program is a very complex process due to a variety of variables that affect the programs. Besides, science, policy and ethics are all involved in the determination of the main objectives of the vaccination program. We present two nonlinear mathematical models that allow us to gain insight into the optimal vaccination strategy under different situations, taking into account the case fatality rate and age-structure of the population. We study scenarios with different availabilities and efficacies of the vaccines. The results of this study show that for most scenarios, the optimal allocation of vaccines is to first give the doses to people in the 55+ age group. However, in some situations the optimal strategy is to first allocate vaccines to the 15–54 age group. This situation occurs whenever the SARS-CoV-2 transmission rate is relatively high and the people in the 55+ age group have a transmission rate 50% or less that of those in the 15–54 age group. This study and similar ones can provide scientific recommendations for countries where the proportion of vaccinated individuals is relatively small or for future pandemics.

Application of Mathematical Modeling in Prediction of COVID-19 Transmission Dynamics

The entire world has been affected by the outbreak of COVID-19 since early 2020. Human carriers are largely the spreaders of this new disease, and it spreads much faster compared to previously identified coronaviruses and other flu viruses. Although vaccines have been invented and released, it will still be a challenge to overcome this disease. To save lives, it is important to better understand how the virus is transmitted from one host to another and how future areas of infection can be predicted. Recently, the second wave of infection has hit multiple countries, and governments have implemented necessary measures to tackle the spread of the virus. We investigated the three phases of COVID-19 research through a selected list of mathematical modeling articles. To take the necessary measures, it is important to understand the transmission dynamics of the disease, and mathematical modeling has been considered a proven technique in predicting such dynamics. To this end, this paper summarizes all the available mathematical models that have been used in predicting the transmission of COVID-19. A total of nine mathematical models have been thoroughly reviewed and characterized in this work, so as to understand the intrinsic properties of each model in predicting disease transmission dynamics. The application of these nine models in predicting COVID-19 transmission dynamics is presented with a case study, along with detailed comparisons of these models. Toward the end of the paper, key behavioral properties of each model, relevant challenges and future directions are discussed.

Global proprieties of a delayed epidemic model with partial susceptible protection

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

The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps

This paper reports novel theoretical and analytical results for a perturbed version of a SIR model with Gamma-distributed delay. Notably, our epidemic model is represented by Itô–Lévy stochastic differential equations in order to simulate sudden and unexpected external phenomena. By using some new and ameliorated mathematical approaches, we study the long-run characteristics of the perturbed delayed model. Within this scope, we give sufficient conditions for two interesting asymptotic properties: extinction and persistence of the epidemic. One of the most interesting results is that the dynamics of the stochastic model are closely related to the intensities of white noises and Lévy jumps, which can give us a good insight into the evolution of the epidemic in some unexpected situations. Our work complements the results of some previous investigations and provides a new approach to predict and analyze the dynamic behavior of epidemics with distributed delay. For illustrative purposes, numerical examples are presented for checking the theoretical study.

Extinction and stationary distribution of a stochastic SIQR epidemic model with demographics and non-monotone incidence rate on scale-free networks

By taking full consideration of contact heterogeneity of individuals, quarantine measures, demographics, information transmission and random environments, we present a stochastic SIQR epidemic model with demographics and non-monotone incidence rate on scale-free networks, which introduces stochastic perturbations to death rate. The formula of the basic reproduction number of the deterministic model is obtained by utilizing the existence of the endemic equilibrium. Next, we define a stopping time, then the existence of a unique global positive solution for the stochastic model is proved by constructing appropriate Lyapunov function to demonstrate the stopping time is infinite. In addition, we also manifest sufficient conditions for diseases extinction and the existence of ergodic stationary distribution by constructing appropriate stochastic Lyapunov functions. At last, numerical simulations illustrate the analytical results.

The effect of self-limiting on the prevention and control of the diffuse COVID-19 epidemic with delayed and temporal-spatial heterogeneous

BACKGROUND

The global spread of the novel coronavirus pneumonia is still continuing, and a new round of more serious outbreaks has even begun in some countries. In this context, this paper studies the dynamics of a type of delayed reaction-diffusion novel coronavirus pneumonia model with relapse and self-limiting treatment in a temporal-spatial heterogeneous environment.

METHODS

First, focus on the self-limiting characteristics of COVID-19, incorporate the relapse and self-limiting treatment factors into the diffusion model, and study the influence of self-limiting treatment on the diffusion of the epidemic. Second, because the traditional Lyapunov stability method is difficult to determine the spread of the epidemic with relapse and self-limiting treatment, we introduce a completely different method, relying on the existence conditions of the exponential attractor of our newly established in the infinite-dimensional dynamic system to determine the diffusion of novel coronavirus pneumonia. Third, relapse and self-limiting treatment have led to a change in the structure of the delayed diffusion COVID-19 model, and the traditional basic reproduction number [Formula: see text] no longer has threshold characteristics. With the help of the Krein-Rutman theorem and the eigenvalue method, we studied the threshold characteristics of the principal eigenvalue and found that it can be used as a new threshold to describe the diffusion of the epidemic.

RESULTS

Our results prove that the principal eigenvalue [Formula: see text] of the delayed reaction-diffusion COVID-19 system with relapse and self-limiting treatment can replace the basic reproduction number [Formula: see text] to describe the threshold effect of disease transmission. Combine with the latest official data and the prevention and control strategies, some numerical simulations on the stability and global exponential attractiveness of the diffusion of the COVID-19 epidemic in China and the USA are given.

CONCLUSIONS

Through the comparison of numerical simulations, we find that self-limiting treatment can significantly promote the prevention and control of the epidemic. And if the free activities of asymptomatic infected persons are not restricted, it will seriously hinder the progress of epidemic prevention and control.

Dynamics and control of delayed rumor propagation through social networks

Investigation of rumor spread dynamics and its control in social networking sites (SNS) has become important as it may cause some serious negative effects on our society. Here we aim to study the rumor spread mechanism and the influential factors using epidemic like model. We have divided the total population into three groups, namely, ignorant, spreader and aware. We have used delay differential equations to describe the dynamics of rumor spread process and studied the stability of the steady-state solutions using the threshold value of influence which is analogous to the basic reproduction number in disease model. Global stability of rumor prevailing state has been proved by using Lyapunov function. An optimal control system is set up using media awareness campaign to minimize the spreader population and the corresponding cost. Hopf bifurcation analyses with respect to time delay and the transmission rate of rumors are discussed here both analytically and numerically. Moreover, we have derived the stability region of the system corresponding to change of transmission rate and delay values.

Two-Age-Structured COVID-19 Epidemic Model: Estimation of Virulence Parameters to Interpret Effects of National and Regional Feedback Interventions and Vaccination

The COVID-19 epidemic has recently led in Italy to the implementation of different external strategies in order to limit the spread of the disease in response to its transmission rate: strict national lockdown rules, followed first by a weakening of the social distancing and contact reduction feedback interventions and finally the implementation of coordinated intermittent regional actions, up to the application, in this last context, of an age-stratified vaccine prioritization strategy. This paper originally aims at identifying, starting from the available age-structured real data at the national level during the specific aforementioned scenarios, external-scenario-dependent sets of virulence parameters for a two-age-structured COVID-19 epidemic compartmental model, in order to provide an interpretation of how each external scenario modifies the age-dependent patterns of social contacts and the spread of COVID-19.

Analysis of a discrete mathematical COVID-19 model

To describe the main propagation of the COVID-19 and has to find the control for the rapid spread of this viral disease in real life, in current manuscript a discrete form of the SEIR model is discussed. The main aim of this is to describe the viral disease in simplest way and the basic properties that are related with the nature of curves for susceptible and infected individuals are discussed here. The elementary numerical examples are given by using the real data of India and Algeria.

Threshold dynamics of difference equations for SEIR model with nonlinear incidence function and infinite delay

In this research, we explore the global conduct of age-structured SEIR system with nonlinear incidence functional (NIF), where a threshold behavior is obtained. More precisely, we will analyze the investigated model differently, where we will rewrite it as a difference equations with infinite delay by the help of the characteristic method. Using standard conditions on the nonlinear incidence functional that can fit with a vast class of a well-known incidence functionals, we investigated the global asymptotic stability (GAS) of the disease-free equilibrium (DFE) using a Lyapunov functional (LF) for R 0 ≤ 1 . The total trajectory method is utilized for avoiding proving the local behavior of equilibria. Further, in the case R 0 > 1 we achieved the persistence of the infection and the GAS of the endemic equilibrium state (EE) using the weakly ρ -persistence theory, where a proper LF is obtained. The achieved results are checked numerically using graphical representations.

Fractional optimal control problem for an age-structured model of COVID-19 transmission

The aim of this study is to model the transmission of COVID-19 and investigate the impact of some control strategies on its spread. We propose an extension of the classical SEIR model, which takes into account the age structure and uses fractional-order derivatives to have a more realistic model. For each age group j the population is divided into seven classes namely susceptible S j , exposed E j , infected with high risk I h j , infected with low risk I l j , hospitalized H j , recovered with and without psychological complications R 1 j and R 2 j , respectively. In our model, we incorporate three control variables which represent: awareness campaigns, diagnosis and psychological follow-up. The purpose of our control strategies is protecting susceptible individuals from being infected, minimizing the number of infected individuals with high and low risk within a given age group j , as well as reducing the number of recovered individuals with psychological complications. Pontryagin's maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. Numerical simulations performed using Matlab, are provided to show the effectiveness of three control strategies and the effect of the order of fractional derivative on the efficiency of these control strategies. Using a cost-effectiveness analysis method, our results show that combining awareness with diagnosis is the most effective strategy. To the best of our knowledge, this work is the first that propose a framework on the control of COVID-19 transmission based on a multi-age model with Caputo time-fractional derivative.

Modeling the Impact of Unreported Cases of the COVID-19 in the North African Countries

In this paper, we study a mathematical model investigating the impact of unreported cases of the COVID-19 in three North African countries: Algeria, Egypt, and Morocco. To understand how the population respects the restriction of population mobility implemented in each country, we use Google and Apple's mobility reports. These mobility reports help to quantify the effect of the population movement restrictions on the evolution of the active infection cases. We also approximate the number of the population infected unreported, the proportion of those that need hospitalization, and estimate the end of the epidemic wave. Moreover, we use our model to estimate the second wave of the COVID-19 Algeria and Morocco and to project the end of the second wave. Finally, we suggest some additional measures that can be considered to reduce the burden of the COVID-19 and would lead to a second wave of the spread of the virus in these countries.

Assessment of Social Distancing for Controlling COVID-19 in Korea: An Age-Structured Modeling Approach

The outbreak of the novel coronavirus disease 2019 (COVID-19) occurred all over the world between 2019 and 2020. The first case of COVID-19 was reported in December 2019 in Wuhan, China. Since then, there have been more than 21 million incidences and 761 thousand casualties worldwide as of 16 August 2020. One of the epidemiological characteristics of COVID-19 is that its symptoms and fatality rates vary with the ages of the infected individuals. This study aims at assessing the impact of social distancing on the reduction of COVID-19 infected cases by constructing a mathematical model and using epidemiological data of incidences in Korea. We developed an age-structured mathematical model for describing the age-dependent dynamics of the spread of COVID-19 in Korea. We estimated the model parameters and computed the reproduction number using the actual epidemiological data reported from 1 February to 15 June 2020. We then divided the data into seven distinct periods depending on the intensity of social distancing implemented by the Korean government. By using a contact matrix to describe the contact patterns between ages, we investigated the potential effect of social distancing under various scenarios. We discovered that when the intensity of social distancing is reduced, the number of COVID-19 cases increases; the number of incidences among the age groups of people 60 and above increases significantly more than that of the age groups below the age of 60. This significant increase among the elderly groups poses a severe threat to public health because the incidence of severe cases and fatality rates of the elderly group are much higher than those of the younger groups. Therefore, it is necessary to maintain strict social distancing rules to reduce infected cases.