Stability analysis of SEIR model related to efficiency of vaccines for COVID-19 situation

Modeling outbreaks of COVID-19 in China: The impact of vaccination and other control measures on curbing the epidemic

This study aims to examine the development trend of COVID-19 in China and propose a model to assess the impacts of various prevention and control measures in combating the COVID-19 pandemic. Using COVID-19 cases reported by the National Health Commission of China from January 2, 2020, to January 2, 2022, we established a Susceptible-Exposed-Infected-Asymptomatic-Quarantined-Vaccinated-Hospitalized-Removed (SEIAQVHR) model to calculate the COVID-19 transmission rate and Rt effective reproduction number, and assess prevention and control measures. Additionally, we built a stochastic model to explore the development of the COVID-19 epidemic. We modeled the incidence trends in five outbreaks between 2020 and 2022. Some important features of the COVID-19 epidemic are mirrored in the estimates based on our SEIAQVHR model. Our model indicates that an infected index case entering the community has a 50%-60% chance to cause a COVID-19 outbreak. Wearing masks and getting vaccinated were the most effective measures among all the prevention and control measures. Specifically targeting asymptomatic individuals had no significant impact on the spread of COVID-19. By adjusting prevention and control parameters, we suggest that increasing the rates of effective vaccination and mask-wearing can significantly reduce COVID-19 cases in China. Our stochastic model analysis provides a useful tool for understanding the COVID-19 epidemic in China.

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

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

Sensitivity of endemic behaviour of COVID-19 under a multi-dose vaccination regime, to various biological parameters and control variables

For an infectious disease such as COVID-19, we present a new four-stage vaccination model (unvaccinated, dose 1 + 2, booster, repeated boosters), which examines the impact of vaccination coverage, vaccination rate, generation interval, control reproduction number, vaccine efficacies and rates of waning immunity upon the dynamics of infection. We derive a single equation that allows computation of equilibrium prevalence and incidence of infection, given knowledge about these parameters and variable values. Based upon a 20-compartment model, we develop a numerical simulation of the associated differential equations. The model is not a forecasting or even predictive one, given the uncertainty about several biological parameter values. Rather, it is intended to aid a qualitative understanding of how equilibrium levels of infection may be impacted upon, by the parameters of the system. We examine one-at-a-time sensitivity analysis around a base case scenario. The key finding which should be of interest to policymakers is that while factors such as improved vaccine efficacy, increased vaccination rates, lower waning rates and more stringent non-pharmaceutical interventions might be thought to improve equilibrium levels of infection, this might only be done to good effect if vaccination coverage on a recurrent basis is sufficiently high.

Mathematical Modeling of SARS-CoV-2 Omicron Wave under Vaccination Effects

Over the course of the COVID-19 pandemic millions of deaths and hospitalizations have been reported. Different SARS-CoV-2 variants of concern have been recognized during this pandemic and some of these variants of concern have caused uncertainty and changes in the dynamics. The Omicron variant has caused a large amount of infected cases in the US and worldwide. The average number of deaths during the Omicron wave toll increased in comparison with previous SARS-CoV-2 waves. We studied the Omicron wave by using a highly nonlinear mathematical model for the COVID-19 pandemic. The novel model includes individuals who are vaccinated and asymptomatic, which influences the dynamics of SARS-CoV-2. Moreover, the model considers the waning of the immunity and efficacy of the vaccine against the Omicron strain. This study uses the facts that the Omicron strain has a higher transmissibility than the previous circulating SARS-CoV-2 strain but is less deadly. Preliminary studies have found that Omicron has a lower case fatality rate compared to previous circulating SARS-CoV-2 strains. The simulation results show that even if the Omicron strain is less deadly it might cause more deaths, hospitalizations and infections. We provide a variety of scenarios that help to obtain insight about the Omicron wave and its consequences. The proposed mathematical model, in conjunction with the simulations, provides an explanation for a large Omicron wave under various conditions related to vaccines and transmissibility. These results provide an awareness that new SARS-CoV-2 variants can cause more deaths even if their fatality rate is lower.

Dynamic behavior analysis of an SVIR epidemic model with two time delays associated with the COVID-19 booster vaccination time

Since the outbreak of COVID-19, there has been widespread concern in the community, especially on the recent heated debate about when to get the booster vaccination. In order to explore the optimal time for receiving booster shots, here we construct an SVIR model with two time delays based on temporary immunity. Second, we theoretically analyze the existence and stability of equilibrium and further study the dynamic properties of Hopf bifurcation. Then, the statistical analysis is conducted to obtain two groups of parameters based on the official data, and numerical simulations are carried out to verify the theoretical analysis. As a result, we find that the equilibrium is locally asymptotically stable when the booster vaccination time is within the critical value. Moreover, the results of the simulations also exhibit globally stable properties, which might be more beneficial for controlling the outbreak. Finally, we propose the optimal time of booster vaccination and predict when the outbreak can be effectively controlled.

An SEQAIHR model to study COVID-19 transmission and optimal control strategies in Hong Kong, 2022

During the COVID-19 pandemic, one of the major concerns was a medical emergency in human society. Therefore it was necessary to control or restrict the disease spreading among populations in any fruitful way at that time. To frame out a proper policy for controlling COVID-19 spreading with limited medical facilities, here we propose an SEQAIHR model having saturated treatment. We check biological feasibility of model solutions and compute the basic reproduction number ( R 0 ). Moreover, the model exhibits transcritical, backward bifurcation and forward bifurcation with hysteresis with respect to different parameters under some restrictions. Further to validate the model, we fit it with real COVID-19 infected data of Hong Kong from 19th December, 2021 to 3rd April, 2022 and estimate model parameters. Applying sensitivity analysis, we find out the most sensitive parameters that have an effect on R 0 . We estimate R 0 using actual initial growth data of COVID-19 and calculate effective reproduction number for same period. Finally, an optimal control problem has been proposed considering effective vaccination and saturated treatment for hospitalized class to decrease density of the infected class and to minimize implemented cost.

Stability Analysis of an Extended SEIR COVID-19 Fractional Model with Vaccination Efficiency

This work is aimed at presenting a new numerical scheme for COVID-19 epidemic model based on Atangana-Baleanu fractional order derivative in Caputo sense (ABC) to investigate the vaccine efficiency. Our construction of the model is based on the classical SEIR, four compartmental models with an additional compartment V of vaccinated people extending it SEIRV model, for the transmission as well as an effort to cure this infectious disease. The point of disease-free equilibrium is calculated, and the stability analysis of the equilibrium point using the reproduction number is performed. The endemic equilibrium's existence and uniqueness are investigated. For the solution of the nonlinear system presented in the model at different fractional orders, a new numerical scheme based on modified Simpson's 1/3 method is developed. Convergence and stability of the numerical scheme are thoroughly analyzed. We attempted to develop an epidemiological model presenting the COVID-19 dynamics in Italy. The proposed model's dynamics are graphically interpreted to observe the effect of vaccination by altering the vaccination rate.

A kinetic model considering the decline of antibody level and simulation about vaccination effect of COVID-19

We build a model that consider the falling antibody levels and vaccination to assess the impact of falling antibody levels and vaccination on the spread of the COVID-19 outbreak, and simulate the influence of vaccination rates and failure rates on the number of daily new cases in England. We get that the lower the vaccine failure rate, the fewer new cases. Over time, vaccines with low failure rates are more effective in reducing the number of cases than vaccines with high failure rates and the higher the vaccine efficiency and vaccination rate, the lower the epidemic peak. The peak arrival time is related to a boundary value. When the failure rate is less than this boundary value, the peak time will advance with the decrease of failure rate; when the failure rate is greater than this boundary value, the peak time is delayed with the decrease of failure rate. On the basis of improving the effectiveness of vaccines, increasing the vaccination rate has practical significance for controlling the spread of the epidemic.

COVID-19 forecasting using new viral variants and vaccination effectiveness models

Recently, a high number of daily positive COVID-19 cases have been reported in regions with relatively high vaccination rates; hence, booster vaccination has become necessary. In addition, infections caused by the different variants and correlated factors have not been discussed in depth. With large variabilities and different co-factors, it is difficult to use conventional mathematical models to forecast the incidence of COVID-19. Machine learning based on long short-term memory was applied to forecasting the time series of new daily positive cases (DPC), serious cases, hospitalized cases, and deaths. Data acquired from regions with high rates of vaccination, such as Israel, were blended with the current data of other regions in Japan such that the effect of vaccination was considered in efficient manner. The protection provided by symptomatic infection was also considered in terms of the population effectiveness of vaccination as well as the vaccination protection waning effect and ratio and infectivity of different viral variants. To represent changes in public behavior, public mobility and interactions through social media were also included in the analysis. Comparing the observed and estimated new DPC in Tel Aviv, Israel, the parameters characterizing vaccination effectiveness and the waning protection from infection were well estimated; the vaccination effectiveness of the second dose after 5 months and the third dose after two weeks from infection by the delta variant were 0.24 and 0.95, respectively. Using the extracted parameters regarding vaccination effectiveness, DPC in three major prefectures of Japan were replicated. The key factor influencing the prevention of COVID-19 transmission is the vaccination effectiveness at the population level, which considers the waning protection from vaccination rather than the percentage of fully vaccinated people. The threshold of the efficiency at the population level was estimated as 0.3 in Tel Aviv and 0.4 in Tokyo, Osaka, and Aichi. Moreover, a weighting scheme associated with infectivity results in more accurate forecasting by the infectivity model of viral variants. Results indicate that vaccination effectiveness and infectivity of viral variants are important factors in future forecasting of DPC. Moreover, this study demonstrate a feasible way to project the effect of vaccination using data obtained from other country.

A Chess and Card Room-Induced COVID-19 Outbreak and Its Agent-Based Simulation in Yangzhou, China

Objective

To evaluate epidemiological characteristics of the COVID-19 outbreak that resurged in Yangzhou and to simulate the impact of different control measures at different regional scales.

Methods

We collected personal information from 570 laboratory-confirmed cases in Yangzhou from 28 July to 26 August 2021, and built a modified susceptible-exposed-infected-removed (SEIR) model and an agent-based model.

Results

The SEIR model showed that for passengers from medium-high risk areas, pre-travel nucleic acid testing within 3 days could limit the total number of infected people in Yangzhou to 50; among elderly persons, a 60% increase in vaccination rates could reduce the estimated infections by 253. The agent-based model showed that when the population density of the chess and card room dropped by 40%, the number of infected people would decrease by 54 within 7 days. A ventilation increase in the chess and card room from 25 to 50% could reduce the total number of infections by 33 within 7 days; increasing the ventilation from 25 to 75% could reduce the total number of infections by 63 within 7 days.

Conclusions

The SEIR model and agent-based model were used to simulate the impact of different control measures at different regional scales successfully. It is possible to provide references for epidemic prevention and control work.

COVID-19 waves: variant dynamics and control

The waves of COVID-19 infections driven by its variants continue to nullify the success we achieved through efficacious vaccines, social restrictions, testing and quarantine policies. This paper models the two major variants-driven waves by two sets of susceptible-infected-quarantined-recovered-vaccinated-deceased coupled dynamics that are modulated by the three main interventions: vaccination, quarantine and restrictions. This \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SI^2Q^2R^2VD$$\end{document}SI2Q2R2VD system is used to demonstrate that the second major novel coronavirus wave in the US is caused by the delta variant and the corresponding rapid surge in infectious cases is driven by the unvaccinated pool of the populace. Next, a feedback control based planned vaccination strategy is derived and is shown to be able to suppress the surge in infections effectively.

Mathematical COVID-19 model with vaccination: a case study in Saudi Arabia

The discovery of a new form of corona-viruses in December 2019, SARS-CoV-2, commonly named COVID-19, has reshaped the world. With health and economic issues at stake, scientists have been focusing on understanding the dynamics of the disease, in order to provide the governments with the best policies and strategies allowing them to reduce the span of the virus. The world has been waiting for the vaccine for more than one year. The World Health Organization (WHO) is advertising the vaccine as a safe and effective measure to fight off the virus. Saudi Arabia was the fourth country in the world to start to vaccinate its population. Even with the new simplified COVID-19 rules, the third dose is still mandatory. COVID-19 vaccines have raised many questions regarding in its efficiency and its role to reduce the number of infections. In this work, we try to answer these question and propose a new mathematical model with five compartments, including susceptible, vaccinated, infectious, asymptotic and recovered individuals. We provide theoretical results regarding the effective reproduction number, the stability of endemic equilibrium and disease free equilibrium. We provide numerical analysis of the model based on the Saudi case. Our developed model shows that the vaccine reduces the transmission rate and provides an explanation to the rise in the number of new infections immediately after the start of the vaccination campaign in Saudi Arabia.

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.

On the Supervision of a Saturated SIR Epidemic Model with Four Joint Control Actions for a Drastic Reduction in the Infection and the Susceptibility through Time

This paper presents and studies a new epidemic SIR (Susceptible–Infectious–Recovered) model with susceptible recruitment and eventual joint vaccination efforts for both newborn and susceptible individuals. Furthermore, saturation effects in the infection incidence terms are eventually assumed for both the infectious and the susceptible subpopulations. The vaccination action on newborn individuals is assumed to be applied to a fraction of them while that on the susceptible general population is of linear feedback type reinforced with impulsive vaccination actions (in practice, very strong and massive vaccination controls) at certain time points, based on information on the current levels of the susceptible subpopulation. Apart from the above vaccination controls, it is also assumed that the average of contagion contacts can be controlled via intervention measures, such as confinements or isolation measures, social distance rules, use of masks, mobility constraints, etc. The main objectives of the paper are the achievement of a strictly decreasing infection for all time periods and that of the susceptible individuals over the initial period if they exceed the disease-free equilibrium value. The monitoring mechanism is the combined activation of intervention measures to reduce the contagion contacts together with the impulsive vaccination to reduce susceptibility. The susceptibility and recovery levels of the disease-free equilibrium point are suitably prefixed by the design of the regular feedback vaccination on the susceptible subpopulation.

Evaluating the Impact of SARS-CoV-2 Variants on the COVID-19 Epidemic and Social Restoration in the United States: A Mathematical Modelling Study

Background: Multiple SARS-CoV-2 variants are still rampant across the United States (US). We aimed to evaluate the impact of vaccination scale-up and potential reduction in the vaccination effectiveness on the COVID-19 epidemic and social restoration in the US. Methods: We extended a published compartmental model and calibrated the model to the latest US COVID-19 data. We estimated the vaccine effectiveness against the variant and evaluated the impact of a potential reduction in vaccine effectiveness on the epidemics. We explored the epidemic trends under different levels of social restoration. Results: We estimated the overall existing vaccine effectiveness against the variant as 88.5% (95% CI: 87.4-89.5%) with the vaccination coverage of 70% by the end of August, 2021. With this vaccine effectiveness and coverage, there would be 498,972 (109,998-885,947) cumulative infections and 15,443 (3,828-27,057) deaths nationwide over the next 12 months, of which 95.0% infections and 93.3% deaths were caused by the variant. Complete social restoration at 60, 65, 70% vaccination coverage would increase cumulative infections to 1.6 (0.2-2.9) million 0.7 (0.1-1.2) million, and 511,159 (110,578-911,740), respectively. At same time it would increase cumulative deaths to 39,040 (5,509-72,570), 19,562 (3,873-35,250), 15,739 (3,841-27,638), respectively. However, if the vaccine effectiveness were reduced to 75%, 50% or 25% due to new SARS-CoV-2 variants, there would be 667,075 (130,682-1,203,468), 1.7 (0.2-3.2) million, 19.0 (5.3-32.7) million new infections and 19,249 (4,281-34,217), 42,265 (5,081-79,448), 426,860 (117,229-736,490) cumulative deaths to occur over the next 12 months. Further, social restoration at a lower vaccination coverage would lead to even greater secondary outbreaks. Conclusion: Current COVID-19 vaccines remain effective against the SARS-CoV-2 variant, and 70% vaccination coverage would be sufficient to restore social activities to a pre-pandemic level. Further reduction in vaccine effectiveness against SARS-CoV-2 variants would result in a potential surge of the epidemic. Multiple measures, including public health interventions, vaccination scale-up and development of a new vaccine booster, should be integrated to counter the new challenges of new SARS-CoV-2 variants.