## We show the global stability of the disease free equilibrium if a given threshold T0 is less or equal to 1 and we provide how to compute the basic reproduction number R0. 如果给定的阈值 T0 小于或等于 1，我们将展示无病平衡的全局稳定性，并且我们提供了如何计算基本再生数 R0。

Estimation and Optimal Control of the Multiscale Dynamics of Covid-19: A Case Study From Cameroon

## In many countries the COVID-19 pandemic seems to witness second and third waves with dire consequences on human lives and economies Given this situation the modeling of the transmission of the disease is still the subject of research with the ultimate goal of understanding the dynamics of the disease and assessing the efficacy of different mitigation strategies undertaken by the affected countries We propose a mathematical model for COVID-19 transmission The model is structured upon five classes: an individual can be susceptible, exposed, infectious, quarantined or removed The model is based on a nonlinear incidence rate, takes into account the influence of media on public behavior, and assumes the recovery rate to be dependent on the hospital-beds to population ratio A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, stability analysis of the disease-free equilibrium (symmetry) and sensitivity analysis We found that if the basic reproduction number is less than unity the system can exhibit Hopf and backward bifurcations for some range of parameters Numerical simulations using parameter values fitted to Saudi Arabia are carried out to support the theoretical proofs and to analyze the effects of hospital-beds to population ratio, quarantine, and media effects on the predicted nonlinear behavior. 在许多国家，COVID-19 大流行似乎出现了第二波和第三波，对人类生活和经济造成了可怕的后果。鉴于这种情况，疾病传播的建模仍然是研究的主题，最终目标是了解病毒的动态疾病和评估受影响国家采取的不同缓解策略的有效性 我们提出了 COVID-19 传播的数学模型 该模型分为五类：个体可能易感、暴露、传染、隔离或移除 该模型基于非线性发病率，考虑媒体对公众行为的影响，并假设康复率取决于病床与人口的比例 对所提出的模型进行详细分析，包括解的存在性和唯一性，无病平衡（对称）的稳定性分析和敏感性分析我们发现，如果ba sic 再生数小于 1 系统可以对某些参数范围表现出 Hopf 和后向分岔 使用适合沙特阿拉伯的参数值进行数值模拟以支持理论证明并分析医院床位与人口比率的影响，隔离和媒体对预测的非线性行为的影响。

Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds

## The basic reproduction number is determined using the next generation matrix approach. 使用下一代矩阵方法确定基本再生数。

Mathematical model of schistosomiasis with health education and molluscicide intervention

## By using next generation matrix method, we worked out for basic reproduction number R ∘ , which apprises us about the disease dissemination or control in the community. 通过使用下一代矩阵方法，我们得出了基本繁殖数R∘，它告诉我们社区疾病的传播或控制。

Dynamical aspects of pine wilt disease and control measures

## This study uses mobility statistics combined with business census data for the eight Japanese prefectures with the highest coronavirus disease-2019 (COVID-19) infection rates to study the effect of mobility reductions on the effective reproduction number (i. 本研究使用流动性统计数据与日本 2019 年冠状病毒病 (COVID-19) 感染率最高的 8 个县的商业普查数据相结合，研究流动性减少对有效繁殖数 (i.

Explaining the effective reproduction number of COVID-19 through mobility and enterprise statistics: Evidence from the first wave in Japan

## The first wave of the coronavirus disease 2019 (COVID-19) epidemic occurred between March 2020 and May 2020 in Japan The first epidemic wave in Chiba prefecture was analyzed based on the case series reported by the official health department of Chiba prefecture The mean interval from the onset of symptoms to a definitive diagnosis by PCR was 6 9 days The number of cases of infection, onset of symptoms, and of definitive diagnosis peaked on March 31, April 2 and April 10, respectively The effective reproduction number was below 1 0 in February, 2 0 to 3 0 between March 17 and March 27, and decreased to under 1 0 again after April 2, 2020 Some patients with an infectious duration of more than 10 days showed an effective reproduction number in the 3 0 to 4 0 range at the peak of the first epidemic. 第一波冠状病毒病 2019 (COVID-19) 流行于 2020 年 3 月至 2020 年 5 月期间在日本发生 千叶县的第一波流行病根据千叶县官方卫生部门报告的病例系列进行分析。通过 PCR 确诊的症状出现时间为 6 9 天 感染病例数、症状发作数和确诊病例数分别在 3 月 31 日、4 月 2 日和 4 月 10 日达到峰值 有效繁殖数低于 1 0 2 月 2 0 至 3 0 3 月 17 日至 3 月 27 日，2020 年 4 月 2 日后再次降至 1 0 以下 部分感染持续时间超过 10 天的患者显示有效繁殖数在 3 0 至 4 0 范围内在第一次流行病的高峰期。

Analysis of the First Epidemic Wave of Coronavirus Disease 2019 in Chiba Prefecture

## Furthermore, the estimations are given by our model for the real active infection, recovery rate against case fatality rate, reproduction number and the growth factor of the real accumulation infection have all shown significant similarities between the reported data and the estimations produced by our model. 此外，我们的模型对实际活动感染、治愈率与病死率、繁殖数和实际累积感染的生长因子的估计值都显示出报告数据与我们模型产生的估计值之间存在显着相似性。

Mathematical modeling for COVID-19 pandemic in Iraq

## However, in the absence of specific therapeutic intervention, some preventive strategies and supportive treatment minimize the viral transmission as studied by some factors such as basic reproduction number, case fatality rate, and incubation period in the epidemiology of viral diseases. 然而，在缺乏特定治疗干预的情况下，一些预防策略和支持性治疗可最大限度地减少病毒传播，正如病毒性疾病流行病学中的基本繁殖数、病死率和潜伏期等因素所研究的那样。

Polymeric Materials as Potential Inhibitors Against SARS-CoV-2

## Using Lyapunov theory, we prove the global asymptotic stability of the unique endemic equilibrium of the integer-order model, and the fractional models, whenever the basic reproduction number [Formula: see text] is greater than one. 使用 Lyapunov 理论，我们证明了整数阶模型和分数模型的唯一地方性平衡的全局渐近稳定性，只要基本再生数 [公式：见正文] 大于 1。

A malaria model with Caputo-Fabrizio and Atangana-Baleanu derivatives

## First, the basic reproduction number of the impulsive system is obtained, and the global asymptotic stability of the disease-free periodic solution is proved. 首先，得到了脉冲系统的基本再生数，证明了无病周期解的全局渐近稳定性。

Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China

## The model is shown to have a locally asymptotically stable disease-free equilibrium (DFE) when the associated effective reproduction number is less than unity. 当相关的有效繁殖数小于 1 时，该模型显示具有局部渐近稳定的无病平衡 (DFE)。

Modelling the dynamics of Zika in a population with two strains of the virus with optimal control and cost-effectiveness analysis

## It has a locally asymptotically stable disease-free equilibrium (DFE) point whenever the basic reproduction number (R0) is less than unity. 只要基本再生数 (R0) 小于 1，它就有一个局部渐近稳定的无病平衡 (DFE) 点。

The development of a deterministic dengue epidemic model with the influence of temperature: A case study in Malaysia

## The estimated SARS-CoV-2 attack rate in Manaus would be above the theoretical herd immunity threshold (67%), given a basic case reproduction number (R0) of 3. 假设基本病例繁殖数 (R0) 为 3，估计马瑙斯的 SARS-CoV-2 发病率将高于理论群体免疫阈值 (67%)。

Resurgence of COVID-19 in Manaus, Brazil, despite high seroprevalence

## From this modified SIR model, the basic reproduction number, effective reproduction number, herd immunity, and herd immunity threshold are redefined. 从这个修改后的 SIR 模型中，重新定义了基本繁殖数、有效繁殖数、群体免疫和群体免疫阈值。

Universality and herd immunity threshold: Revisiting the SIR model for COVID-19

## By using the next generation operator, we obtained the basic reproduction number of the model which shows whether the disease persists or dies out in time. 通过使用下一代算子，我们得到了模型的基本繁殖数，它显示了疾病是持续存在还是及时消失。

Mathematical modelling of HIV infection with the effect of horizontal and vertical transmissions

## The basic reproduction number R0 is defined as the spectral radius of a next generation operator K, which is calculated in an explicit form, and it serves as a vital value determining whether or not the disease persists. 基本再生数R0被定义为下一代算子K的光谱半径，其以显式形式计算，并且它用作确定疾病是否持续存在的生命值。

Temporal-spatial analysis of an age-space structured foot-and-mouth disease model with Dirichlet boundary condition.

## Through the analysis of the model, the basic reproduction number  were obtained, and the malware free equilibrium was proved to be locally asymptotical stable if  is less than unity and globally asymptotically stable if Ro is less than some threshold using a Lyapunov function. 通过对模型的分析，得到了基本的再生数，利用李雅普诺夫函数证明了无恶意软件平衡在小于1的情况下是局部渐近稳定的，而在Ro小于某个阈值时是全局渐近稳定的。

Epidemic Model and Mathematical Study of Impact of Vaccination for the Control of Malware in Computer Network

## Analytical expressions for the basic reproduction number $\mathcal{R}_{0}$ and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. 建立了基本再生数$\mathcal{R}_{0}$的解析表达式以及未感染和感染稳态全局渐近稳定的必要条件。

Stability of general pathogen dynamic models with two types of infectious transmission with immune impairment

## The basic reproduction number (R0) of the model is found by the next generation method and then disease-free (DF) and endemic equilibrium points of the system are found and their existence conditions are presented. 通过下一代方法求出模型的基本再生数（R0），然后求出系统的无病（DF）和地方病平衡点，并给出它们的存在条件。

A study of SIQR model with Holling type–II incidence rate

## The model uses the reproduction number $\mathscr{R}_0$ as a threshold, calculated using the Next-Generation Method. 该模型使用再生数 $\mathscr{R}_0$ 作为阈值，使用下一代方法计算。

Reproduction number and sensitivity analysis of cassava mosaic disease spread for policy design.

## Under appropriate assumptions on the infection rates, we show that if the basic reproduction number is less than or equal to one, then the disease will be eradicated in the long run and any solution to the Cauchy problem converges to the unique disease-free equilibrium of the model. 在对感染率的适当假设下，我们表明，如果基本繁殖数小于或等于 1，那么从长远来看，该疾病将被根除，任何柯西问题的解决方案都会收敛到唯一的无病平衡该模型。

Analysis of a time-delayed HIV/AIDS epidemic model with education campaigns

## It is shown that the model has a unique disease-free equilibrium if the basic reproduction number $${\mathcal {R}}_{0}\le 1$$ R 0 ≤ 1 and a unique endemic equilibrium if $${\mathcal {R}}_{0}> 1$$ R 0 > 1. 结果表明，如果基本繁殖数 $${\mathcal {R}}_{0}\le 1$$ R 0 ≤ 1，则该模型具有唯一的无病均衡，如果 $${\数学 {R}}_{0}> 1$$ R 0 > 1。

Complex dynamics in a discrete SIS epidemic model with Ricker-type recruitment and disease-induced death

## And we derive sufficient conditions guaranteeing extinction and persistence of epidemics, respectively, which are related to the basic reproduction number $R_0$ of the corresponding deterministic model. 并且我们分别推导出了保证流行病灭绝和持续存在的充分条件，这些条件与相应确定性模型的基本再生数$R_0$有关。

An SIS epidemic model with time delay and stochastic perturbation on heterogeneous networks.

## We deduce the basic reproduction number R0s for the stochastic model which is smaller than R0 of the corresponding deterministic model. 我们推导出随机模型的基本再生数 R0s，它小于相应确定性模型的 R0。

Stochastic Delay Differential Model for Coronavirus Infection COVID-19

## The strength of Lyapunov functional theory has been exploited to show that smoke-free equilibrium point is globally asymptotically stable whenever basic reproduction number $$R_\circ <1$$ R ∘ < 1. Lyapunov 泛函理论的优势已被利用来表明，只要基本再生数 $$R_\circ <1$$ R ∘ < 1，无烟平衡点是全局渐近稳定的。

A mathematical and parametric study of epidemiological smoking model: a deterministic stability and optimality for solutions

## Major qualitative analysis like the social media addiction free equilibrium point (E0), endemic equilibrium point (E∗), basic reproduction number (R0), were computed. 计算了主要的定性分析，如社交媒体无成瘾平衡点 (E0)、地方性平衡点 (E*)、基本再生数 (R0)。

Mathematical modeling with optimal control analysis of social media addiction

## Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. 当无病平衡点和地方病平衡点的繁殖数 R0 小于 1 时，就会出现稳定性。

Modeling the impact of early interventions on the transmission dynamics of coronavirus infection.

## Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. 当无病平衡点和地方病平衡点的繁殖数 R0 小于 1 时，就会出现稳定性。

Modeling the impact of early interventions on the transmission dynamics of coronavirus infection

## To explore the transmission advantage, we estimate that the N501Y substitution increases the infectivity by 52% (95%CI: 46, 58) in terms of the reproduction number. 为了探索传播优势，我们估计 N501Y 替代在繁殖数方面增加了 52% (95%CI: 46, 58) 的传染性。

Quantifying the transmission advantage associated with N501Y substitution of SARS-CoV-2 in the United Kingdom: An early data-driven analysis.

## By comparing the reproduction number before and after lockdown, we find a national transmissibility reduction of 85% (95% CI 78%-89%). 通过比较锁定前后的复制数量，我们发现全国的传播率降低了 85%（95% CI 78%-89%）。

Regional probabilistic situational awareness and forecasting of COVID-19

## Second the existence of all biological equilibrium points, basic reproduction number and stability analysis of all equilibrium points. 二是所有生物平衡点的存在性、基本繁殖数和所有平衡点的稳定性分析。

Dynamical Analysis of Transmission of Hepatitis B and C Viruses with External Source of disease by Mathematical Model

## Furthermore, we have calculated the equilibrium points, basic reproduction number and stability of the basic reproduction number. 此外，我们还计算了平衡点、基本再生数和基本再生数的稳定性。

Qualitative and Quantitative study of Zika virus epidemic model under Caputo’s fractional differential operator

## Towards the end of 2019, the world witnessed the outbreak of Severe Acute Respiratory Syndrome Coronavirus-2 (COVID-19), a new strain of coronavirus that was unidentified in humans previously In this paper, a new fractional-order Susceptible-Exposed-Infected-Hospitalized-Recovered (SEIHR) model is formulated for COVID-19, where the population is infected due to human transmission The fractional-order discrete version of the model is obtained by the process of discretization and the basic reproductive number is calculated with the next-generation matrix approach All equilibrium points related to the disease transmission model are then computed Further, sufficient conditions to investigate all possible equilibria of the model are established in terms of the basic reproduction number (local stability) and are supported with time series, phase portraits and bifurcation diagrams Finally, numerical simulations are provided to demonstrate the theoretical findings © 2021 World Scientific Publishing Company. 接近 2019 年底，世界见证了严重急性呼吸系统综合症冠状病毒 2（COVID-19）的爆发，这是一种新的冠状病毒株，此前在人类中未发现。 - 为 COVID-19 制定住院恢复 (SEIHR) 模型，其中人群因人类传播而被感染。通过离散化过程获得模型的分数阶离散版本，并计算基本再生数生成矩阵方法 然后计算与疾病传播模型相关的所有平衡点 此外，根据基本再生数（局部稳定性）建立研究模型所有可能平衡的充分条件，并得到时间序列、相图的支持和分岔图 最后，提供数值模拟来证明理论发现 © 2021 World Scientific Publi盛公司。

Modeling and stability analysis of the spread of novel coronavirus disease COVID-19

## The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) reproduction number has become an essential parameter for monitoring disease transmission across settings and guiding interventions. 严重急性呼吸系统综合症冠状病毒 2 (SARS-CoV-2) 的繁殖数量已成为监测跨环境疾病传播和指导干预措施的重要参数。

Estimates of regional infectivity of COVID-19 in the United Kingdom following imposition of social distancing measures

## It is demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity under some defined parameters. 证明如果在某些定义的参数下再生数小于一，则无病周期解是全局稳定的。

Computational modeling of human papillomavirus with impulsive vaccination

## We define the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_{0}$$\end{document}R0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0 < 1,$$\end{document}R0<1, then the disease-free periodic solution is globally asymptotically stable, while if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0 > 1,$$\end{document}R0>1, then the disease persists. 我们定义基本再生数 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \ setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_{0}$$\end{document}R0 作为线性积分算子的谱半径，并表明全局动力学由这个阈值参数决定： If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0 < 1,$$\end{document}R0<1，则无病周期解是全局渐近的稳定，如果 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{ \奇边距}{-69pt} \begin{document}$${\mathscr {R}}_0 > 1,$$\end{document}R0>1，则疾病持续存在。

Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

## We focused on the estimation of: (1) the delay between the appearance of the first infectious case in the population and the outbreak (“epidemic latency period”); (2) the duration of the exponential growth phase; (3) the basic and the time-varying reproduction numbers; and (4) the peaks (time and size) in confirmed positive cases, active cases and new infections. 我们专注于估计：（1）人口中首例感染病例出现与爆发之间的延迟（“流行潜伏期”）； (2) 指数增长阶段的持续时间； (3) 基本和时变再生数； （4）确诊阳性病例、活跃病例和新增感染病例的高峰（时间和规模）。

A Hybrid Modeling Technique of Epidemic Outbreaks with Application to COVID-19 Dynamics in West Africa

## We focused on the estimation of: (1) the delay between the appearance of the first infectious case in the population and the outbreak (“epidemic latency period"); (2) the duration of the exponential growth phase; (3) the basic and the time-varying reproduction numbers; and (4) the peaks (time and size) in confirmed positive cases, active cases and new infections. 我们专注于估计：（1）人口中首例感染病例出现与爆发之间的延迟（“流行潜伏期”）；（2）指数增长阶段的持续时间；（3）基本以及随时间变化的繁殖数量；以及 (4) 确诊阳性病例、活跃病例和新感染病例的峰值（时间和规模）。

A Hybrid Modelling Technique of Epidemic Outbreaks With Application to COVID-19 Dynamics in West Africa

## During decision-making, policymakers consider an estimate of the effective reproduction number Rt, which is the expected number of secondary infections spread by a single infected individual. 在决策过程中，政策制定者会考虑有效繁殖数 Rt 的估计值，即单个受感染个体传播的继发感染的预期数量。

Adaptive Susceptible-Infectious-Removed Model for Continuous Estimation of the COVID-19 Infection Rate and Reproduction Number in the United States: Modeling Study

## We established the basic reproduction number which is the average number of new secondary infection generated by a single infected individual during infectious period. 我们建立了基本繁殖数，即单个感染者在感染期间产生的新继发感染的平均数。

Mathematical Transmission of HIV/AID with Early Treatment

## Among control measures implemented, only national lockdown brought the reproduction number below 1 consistently; introduced one week earlier it could have reduced first wave deaths from 36,700 to 15,700 (95%CrI: 8,900-26,800). 在实施的控制措施中，只有全国封锁使再生产数持续低于1；一周前推出，它本可以将第一波死亡人数从 36,700 人减少到 15,700 人（95%CrI：8,900-26,800）。

The 2020 SARS-CoV-2 epidemic in England: key epidemiological drivers and impact of interventions

## Of the control measures implemented, only national lockdown brought the reproduction number (Rteff) below 1 consistently; if introduced 1 week earlier, it could have reduced deaths in the first wave from an estimated 48,600 to 25,600 [95% credible interval (CrI): 15,900 to 38,400]. 在实施的控制措施中，只有国家封锁使复制数（Rteff）始终低于 1；如果提前 1 周引入，它可能会将第一波的死亡人数从估计的 48,600 减少到 25,600 [95% 可信区间 (CrI)：15,900 到 38,400]。

Key epidemiological drivers and impact of interventions in the 2020 SARS-CoV-2 epidemic in England

## BACKGROUND The basic reproduction number (R0) is an important concept in infectious disease epidemiology and the most important parameter to determine the transmissibility of a pathogen. 背景 基本繁殖数（R0）是传染病流行病学中的一个重要概念，也是确定病原体传播能力的最重要参数。

Nine-month Trend of Time-Varying Reproduction Numbers of COVID-19 in West of Iran.

## The most important quantity in infectious disease epidemiology is the basic reproduction number (R 0). 传染病流行病学中最重要的量是基本繁殖数（R 0）。

Determination of Basic Reproduction Numbers using Transition Intensities Multi-state SIRD Model for COVID-19 in Indonesia

## The basic reproduction number is seen to be slightly above the critical value of one suggesting that stricter measures such as the use of face-masks, social distancing, contact tracing, and even longer stay-at-home orders need to be enforced in order to mitigate the spread of the virus. 基本复制数被认为略高于临界值，这表明需要执行更严格的措施，例如使用口罩、保持社交距离、追踪接触者，甚至更长时间的居家令，以便减轻病毒的传播。

A fractional-order compartmental model for the spread of the COVID-19 pandemic

## This paper addresses the problem of describing the spread of COVID-19 by a mathematical model introducing all the possible control actions as prevention (informative campaign, use of masks, social distancing, vaccination) and medication The model adopted is similar to SEIQR, with the infected patients split into groups of asymptomatic subjects and isolated ones This distinction is particularly important in the current pandemic, due to the fundamental the role of asymptomatic subjects in the virus diffusion The influence of the control actions is considered in analysing the model, from the calculus of the equilibrium points to the determination of the reproduction number This choice is motivated by the fact that the available organised data have been collected since from the end of February 2020, and almost simultaneously containment measures, increasing in typology and effectiveness, have been applied The characteristics of COVID-19, not fully understood yet, suggest an asymmetric diffusion among countries and among categories of subjects Referring to the Italian situation, the containment measures, as applied by the population, have been identified, showing their relation with the government’s decisions;this allows the study of possible scenarios, comparing the impact of different possible choices. 本文通过一个数学模型解决了描述 COVID-19 传播的问题，该模型引入了所有可能的控制措施，如预防（信息宣传活动、使用口罩、社交距离、疫苗接种）和药物。采用的模型类似于 SEIQR，具有受感染的患者分为无症状受试者和孤立受试者组 这种区别在当前的大流行中尤为重要，因为无症状受试者在病毒扩散中的基本作用 在分析模型时考虑了控制措施的影响，从微积分的平衡点的确定再生数这一选择的动机是，自 2020 年 2 月以来已经收集了可用的有组织数据，并且几乎同时采取了增加类型和有效性的遏制措施。尚未完全了解的 COVID-19 的特征表明存在不对称差异国家之间和主题类别之间的使用参考意大利的情况，已经确定了民众采用的遏制措施，显示了它们与政府决定的关系；这允许研究可能的情景，比较不同可能的影响选择。

A Control Based Mathematical Model for the Evaluation of Intervention Lines in COVID-19 Epidemic Spread: The Italian Case Study

## In this study, we estimated the piecewise instantaneous reproduction number (R t ) and the reporting delay-adjusted case-fatality ratio (dCFR) of COVID-19 in seven WPR jurisdictions: Hong Kong Special Administrative Region, Japan, Malaysia, Shanghai, Singapore, South Korea, and Taiwan. 在这项研究中，我们估计了分段瞬时再现数（R 吨 ) 以及七个 WPR 司法管辖区的 COVID-19 报告延迟调整病死率 (dCFR)：香港特别行政区、日本、马来西亚、上海、新加坡、韩国和台湾。

Assessing the impact of non-pharmaceutical interventions on the transmissibility and severity of COVID-19 during the first five months in the Western Pacific Region

## Results Altogether, data on the mean time-varying reproduction number (mean R_t) were available for 153 countries, but standardised averages for the age of cases and deaths and for the case-fatality ratio (CFR) could only be computed for 61, 39 and 31 countries respectively. 结果 总共有 153 个国家的平均时变繁殖数（平均 R_t）数据可用，但只能计算 61、39 的病例年龄和死亡年龄以及病死率（CFR）的标准化平均值和 31 个国家。

Country differences in transmissibility, age distribution and case-fatality of SARS-CoV-2: a global ecological analysis

## We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number $$\mathfrak {R}_0>1$$ R 0 > 1 , there exists a critical wave speed $$c^*>0$$ c ∗ > 0 , such that for each $$c \ge c^*$$ c ≥ c ∗ the system admits a nontrivial TWS and for $$c<c^*$$ c < c ∗ there exists no nontrivial TWS for the system. 我们发现TWS的存在是由所谓的基本再生数和临界波速决定的：当基本再生数$$\mathfrak {R}_0>1$$ R 0 > 1 时，存在一个临界波速度 $$c^*>0$$ c ∗ > 0 ，使得对于每个 $$c \ge c^*$$ c ≥ c ∗ 系统承认一个非平凡的 TWS 并且对于 $$c<c^*$$ c < c * 系统不存在非平凡交易平台。

Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model

## When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. 当无扩散模型的基本再生数大于 1 时，我们找到一个临界波速，低于该临界波速将不存在正行波解。

Traveling waves in cooperative predation: relaxation of sublinearity

## We define the basic reproduction number R 0 and show its threshold role: if R 0 1 , the disease-free steady state is globally asymptotically stable; if R 0 > 1 , the model system is uniformly persistent. 我们定义了基本再生数R 0 并显示了它的阈值作用：如果R 0 1 ，则无病稳态是全局渐近稳定的；如果 R 0 > 1 ，模型系统是一致持久的。

Influence of environmental pollution to a waterborne pathogen model: Global dynamics and asymptotic profiles

## The analysis shows that the free steady state is locally stable when the basic reproduction number  R 0 $R_{0}$ is less than unity and unstable when R 0 > 1 $R_{0} > 1$. 分析表明，当基本再生数 R 0 $R_{0}$ 小于 1 时，自由稳态是局部稳定的；当 R 0 > 1 $R_{0} > 1$ 时，自由稳态是不稳定的。

Mathematical model of SIR epidemic system (COVID-19) with fractional derivative: stability and numerical analysis

## In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient \begin{document}$d$\end{document} of susceptible population is zero and the basic reproduction number is greater than 1. 在本文中，我们证明了在易感人群的扩散系数 \begin{document}$d$\end{document} 为零且基本再生数大于1。

Traveling wave solution for a diffusive simple epidemic model with a free boundary

## We find that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the critical wave speed. nan

Traveling waves of a nonlocal dispersal SEIR model with standard incidence

## The transmission dynamics of A(H1N1)pdm09 virus in Kenya were characterized by (i) multiple virus introductions into Kenya over the study period, although only a few of those introductions instigated local seasonal epidemics that then established local transmission clusters, (ii) persistence of transmission clusters over several epidemic seasons across the country, (iii) seasonal fluctuations in effective reproduction number (Re) associated with lower number of infections and seasonal fluctuations in relative genetic diversity after an initial rapid increase during the early pandemic phase, which broadly corresponded to epidemic peaks in the northern and southern hemispheres, (iv) high virus genetic diversity with greater frequency of seasonal fluctuations in 2009–2011 and 2018 and low virus genetic diversity with relatively weaker seasonal fluctuations in 2012–2017, and (v) virus spread across Kenya. A(H1N1)pdm09 病毒在肯尼亚的传播动态的特点是（i）在研究期间将多种病毒引入肯尼亚，尽管其中只有少数引入引发了当地季节性流行病，然后建立了当地传播集群，（ii）持续存在全国几个流行季节的传播集群，(iii) 与感染数量减少相关的有效繁殖数 (Re) 的季节性波动和在大流行初期初期快速增加后相对遗传多样性的季节性波动，这大致对应到北半球和南半球的流行高峰，(iv) 病毒遗传多样性高，2009-2011 年和 2018 年季节性波动频率更高，病毒遗传多样性低，2012-2017 年季节性波动相对较弱，以及 (v) 病毒传播遍及肯尼亚。

Characterizing the Countrywide Epidemic Spread of Influenza A(H1N1)pdm09 Virus in Kenya between 2009 and 2018

## Results: The transmission dynamics of influenza A(H1N1)pdm09 virus in Kenya was characterized by: (i) multiple virus introductions into Kenya over the study period, although these were remarkably few, with only a few of those introductions instigating seasonal epidemics that then established local transmission clusters; (ii) persistence of transmission clusters over several epidemic seasons across the country; (iii) seasonal fluctuations in effective reproduction number (Re) associated with lower number of infections and seasonal fluctuations in relative genetic diversity after an initial rapid increase during the early pandemic phase, which broadly corresponded to epidemic peaks in the northern and southern hemispheres; (iv) high virus genetic diversity with greater frequency of seasonal fluctuations in 2009-11 and 2018 and low virus genetic diversity with relatively weaker seasonal fluctuations in 2012-17; and (v) virus migration from multiple geographical regions to multiple geographical destinations in Kenya. 结果：A(H1N1)pdm09 流感病毒在肯尼亚的传播动态特点是：(i) 在研究期间将多种病毒引入肯尼亚，尽管这些病毒非常少，只有少数引入会引发季节性流行病，然后建立本地传输集群； (ii) 在全国几个流行季节持续存在传播集群； (iii) 有效繁殖数 (Re) 的季节性波动与感染数量的减少和相对遗传多样性的季节性波动相关，在大流行初期初期迅速增加后，这大致对应于北半球和南半球的流行高峰； (iv) 病毒遗传多样性高，2009-11 年和 2018 年季节性波动频率更高，病毒遗传多样性低，2012-17 年季节性波动相对较弱； (v) 病毒从多个地理区域迁移到肯尼亚的多个地理目的地。

Characterizing the countrywide epidemic spread of influenza A(H1N1)pdm09 virus in Kenya between 2009 and 2018

## We show that the existence of traveling waves only depends on the basic reproduction number of the corresponding spatial-homogeneous system of delay differential equations, which is determined by the recovery rate, the local properties of f and g and a minimal wave speed c ∗ that is affected by the distributed delay. 我们证明了行波的存在只取决于相应的时滞微分方程空间齐次系统的基本再现数，它由恢复率、f 和 g 的局部性质以及最小波速 c ∗ 决定受分布式延迟的影响。

Traveling waves of a diffusive SIR epidemic model with general nonlinear incidence and infinitely distributed latency but without demography

## By using the basic reproduction number $$R_0$$ of the corresponding periodic ordinary differential system and the minimal wave speed $$c^*$$ , the spreading properties of the corresponding solution of the model are established. 通过使用基本再生数 $$R_0$$ 对应的周期常微分系统和最小波速$$c^*$$ ，建立模型对应解的扩展性质。

Propagation dynamics for a time-periodic reaction–diffusion SI epidemic model with periodic recruitment

## While the relative contribution of within-class and within-grade transmissions to the reproduction number varied with the number of classes per grade, tThe overall within-school reproduction number, which determines the initial growth of cases and the risk of sustained transmission, was only minimally associated with class sizes and the number of classes per grade. 虽然班内和年级内传播对繁殖数量的相对贡献随每个年级的班级数量而变化，但决定病例初始增长和持续传播风险的总体校内繁殖数量仅与班级规模和每个年级的班级数量最小相关。

Within and between classroom transmission patterns of seasonal influenza among primary school students in Matsumoto city, Japan

## The overall within-school reproduction number, which determines the initial growth of cases and the risk of sustained transmission, was only minimally associated with class sizes and the number of classes per grade. 决定病例最初增长和持续传播风险的校内总繁殖数量与班级规模和每个年级的班级数量的相关性很小。

Within and between classroom transmission patterns of seasonal influenza and implications for pandemic management strategies at schools

## Using Lyapunov functionals, we prove that if the basic reproduction number $$R_0 \le 1$$ R 0 ≤ 1 , then the infection-free equilibrium is globally asymptotically stable, and when $$R_0 > 1$$ R 0 > 1 , the infection will persist by the global asymptotic stability of infection equilibrium. 利用李雅普诺夫泛函，我们证明如果基本再生数$$R_0 \le 1$$ R 0 ≤ 1 ，则无感染均衡是全局渐近稳定的，当$$R_0 > 1$$ R 0 > 1 时，感染将通过感染平衡的全局渐近稳定性持续存在。

Analysis of a Virus Model with Cure Rate, General Incidence Function and Time Delay

## Under the assumption that the tranmission-transfer network is strongly connected, we establish that the basic reproduction number R 0 is a sharp threshold parameter: if R 0 ≤ 1 , the disease-free equilibrium is globally asymptotically stable and the disease always dies out; if R 0 > 1 , the disease-free equilibrium is unstable, the system is uniformly persistent and initial outbreaks lead to persistent disease infection. 在传输-传输网络强连通的假设下，我们建立了基本再生数 R 0 是一个尖锐的阈值参数：如果 R 0 ≤ 1 ，无病平衡是全局渐近稳定的，疾病总是消失；如果 R 0 > 1 ，无病平衡是不稳定的，系统是均匀持续的，最初的爆发导致持续的疾病感染。

Epidemic models with discrete state structures☆

## We estimate that the new variant has a 43 to 90% higher reproduction number (range of 95% credible intervals, 38 to 130%) than preexisting variants. 我们估计新变体的再生数比先前存在的变体高 43% 到 90%（95% 可信区间的范围为 38% 到 130%）。

Estimated transmissibility and impact of SARS-CoV-2 lineage B.1.1.7 in England

## The outbreak of novel coronavirus disease (COVID-19) has spread around the world since it was detected in December 2019 The Chinese government executed a series of interventions to curb the pandemic The "battle" against COVID-19 in Shenzhen, China is valuable because populated industrial cities are the epic centres of COVID-19 in many regions We made use of synthetic control methods to create a reference population matching specific characteristics of Shenzhen With both the synthetic and observed data, we introduced an epidemic compartmental model to compare the spread of COVID-19 between Shenzhen and its counterpart regions in the United States that didn't implement interventions for policy evaluation Once the effects of policy interventions adopted in Shenzhen were estimated, the delay effects of those interventions were referred to provide the further control degree of interventions Thus, the hypothetical epidemic situations in Shenzhen were inferred by using time-varying reproduction numbers in the proposed SIHR (Susceptible, Infectious, Hospitalized, Removed) model and considering if the interventions were delayed by 0 day to 5 days The expected cumulative confirmed cases would be 1546, which is 5 75 times of the observed cumulative confirmed cases of 269 in Shenzhen on February 3, 2020, based on the data from the counterpart counties (mainly from Broward, New York, Santa Clara, Pinellas, and Westchester) in the United States If the interventions were delayed by 5 days from the day when the interventions started, the expected cumulative confirmed cases of COVID-19 in Shenzhen on February 3, 2020 would be 676 with 95% credible interval (303,1959) Early implementation of mild interventions can subdue the epidemic of COVID-19 The later the interventions were implemented, the more severe the epidemic was in the hard-hit areas Mild interventions are less damaging to the society but can be effective when implemented early. 新型冠状病毒病 (COVID-19) 自 2019 年 12 月被发现以来已在全球蔓延 中国政府为遏制大流行采取了一系列干预措施 中国深圳与 COVID-19 的“战斗”很有价值，因为人口稠密的工业城市是许多地区 COVID-19 的史诗中心 我们利用综合控制方法创建了与深圳特定特征相匹配的参考人口 我们结合综合数据和观察数据，引入了流行病区划模型来比较COVID-19 深圳与未实施政策评估干预措施的美国对口地区之间的 COVID-19 一旦估算了深圳采取的政策干预措施的效果，就可以参考这些干预措施的延迟效应，以提供干预措施的进一步控制程度因此，利用时变再现性推断深圳的假设疫情。根据提议的 SIHR（易感、传染性、住院、已移除）模型中的数字，并考虑干预措施是否延迟 0 天至 5 天 预计累计确诊病例为 1546 例，是观察到的累计确诊病例的 5 75 倍2020 年 2 月 3 日深圳的 269 例，根据美国对口县（主要来自布劳沃德、纽约、圣克拉拉、皮内拉斯和威彻斯特）的数据，如果干预措施从实施之日起延迟 5 天干预开始，预计 2020 年 2 月 3 日深圳 COVID-19 累计确诊病例为 676 例，可信区间为 95% (303,1959) 早期实施轻度干预可以抑制 COVID-19 的流行实施，疫情在重灾区越严重 温和的干预对社会的危害较小，但早期实施可以有效。

Rejoinder of “The timing and effectiveness of implementing mild interventions of COVID-19 in large industrial regions via a synthetic control method”

## The basic reproduction number (BRN) is found via the new generation matrix method. 基本再生数（BRN）是通过新一代矩阵方法找到的。

A fractional‐order model of coronavirus disease 2019 (COVID‐19) with governmental action and individual reaction

## The model analysis uses the generation matrix method to obtain the basic reproduction number and the stability of the model’s equilibrium points. 模型分析采用生成矩阵法获得模型平衡点的基本再生数和稳定性。

An SIR epidemic model for COVID-19 spread with fuzzy parameter: the case of Indonesia

## Their local asymptotic stability properties and the attainability of the endemic equilibrium point are investigated based on the next generation matrix properties, the value of the basic reproduction number, and nonnegativity properties of the solution and its equilibrium states. 基于下一代矩阵性质、基本再生数的取值、解及其平衡态的非负性质，研究了它们的局部渐近稳定性性质和地方性平衡点的可达性。

On an SE(Is)(Ih)AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic

## For the deterministic model, we calculate the basic reproduction number, simultaneously, and investigate the local asymptotic stability of the disease-free equilibrium and the endemic equilibrium. 对于确定性模型，我们同时计算基本繁殖数，并研究无病平衡和地方性平衡的局部渐近稳定性。

Dynamic analysis of a SIQR epidemic model considering the interaction of environmental differences

## Depending on the basic reproduction number, we prove that the total immunity induces local stability-instability of equilibria and the epidemic may disappear after a first epidemic wave and more epidemic waves may appear in the case of non-total immunity. 根据基本繁殖数，我们证明了全免疫导致局部平衡的不稳定-不稳定性，流行可能在第一波流行后消失，在非全免疫的情况下可能出现更多流行波。

SIARD model and effect of lockdown on the dynamics of COVID-19 disease with non total immunity

## We also estimated the time-varying reproduction number (Rt) during the first epidemic wave. 我们还估计了第一波流行期间的时变繁殖数（Rt）。

Reconstruction of the transmission dynamics of the first COVID-19 epidemic wave in Thailand

## Analysis from the model show that there is a locally and globally asymptotic stable disease-free equilibrium whenever a certain epidemiological threshold, the control reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_v$$\end{document}Rv, is less than unity. 模型分析表明，只要达到一定的流行病学阈值，控制繁殖数 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts，就存在局部和全局渐近稳定的无病平衡} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_v$$\end{document }Rv，小于统一。

Analysis of time delayed Rabies model in human and dog populations with controls

## Analysis of the gonorrhea-only sub-model shows the existence of a stable disease free equilibrium (DFE) and a stable endemic equilibrium (EE) when the associated reproduction number is less than one. 仅淋病子模型的分析表明，当相关繁殖数小于 1 时，存在稳定的无病平衡 (DFE) 和稳定的地方病平衡 (EE)。

Mathematical analysis of a model for Chlamydia and Gonorrhea codynamics with optimal control

## Firstly, when the basic reproduction number \begin{document}$\mathcal{R}_{0}>1$\end{document} and speed \begin{document}$c>c^{\ast}$\end{document} , we prove that the system admits a nontrivial traveling wave solution, where \begin{document}$c^{\ast}$\end{document} is the minimal wave speed. 首先，当基本再现数 \begin{document}$\mathcal{R}_{0}>1$\end{document} 和速度 \begin{document}$c>c^{\ast}$\end{ document} ，我们证明系统承认一个非平凡的行波解，其中 \begin{document}$c^{\ast}$\end{document} 是最小波速。

Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

## We prove that when the basic reproduction number R 0 > 1 , there exists a critical number c 1 ⁎ > 0 such that for each c > c 1 ⁎ , the system has a nontrivial traveling wave solution with speed c, while for 0 c c 1 ⁎ the system admits no nontrivial traveling wave solution. 我们证明了当基本再生数 R 0 > 1 时，存在一个临界数 c 1 ⁎ > 0 使得对于每个 c > c 1 ⁎ ，系统有一个速度为 c 的非平凡行波解，而对于 0 c c 1 ⁎ 该系统不承认非平凡行波解。

Existence of traveling waves for a nonlocal dispersal SIR epidemic model with treatment

## The basic reproduction number [Formula: see text] is given, and the unique endemic equilibrium exists when [Formula: see text], while the disease-free equilibrium always exists. 给定基本繁殖数【公式：见正文】，当【公式：见正文】时存在唯一的地方病平衡，而无病平衡始终存在。

Dynamics of an SIRS model with age structure and two delays

## We provide estimates of the viral dynamic parameters in ferrets, such as the infection rate, the virus production rate, the infectious virus proportion, the infected cell death rate, the virus clearance rate, as well as other related characteristics, including the basic reproduction number, pre-peak infectious viral growth rate, post-peak infectious viral decay rate, pre-peak infectious viral doubling time, post-peak infectious virus half-life, and the target cell loss in the respiratory tract. 我们提供了雪貂中病毒动态参数的估计，例如感染率、病毒产生率、感染性病毒比例、感染细胞死亡率、病毒清除率，以及其他相关特征，包括基本繁殖数、峰值前传染性病毒生长率、峰值后传染性病毒衰变率、峰值前传染性病毒倍增时间、峰值后传染性病毒半衰期和呼吸道靶细胞损失。

Modeling Within-Host Dynamics of SARS-CoV-2 Infection: A Case Study in Ferrets

## In this paper, we developed a deterministic mathematical model of the pandemic COVID-19 transmission in Ethiopia, which allows transmission by exposed humans We proposed an SEIR model using system of ordinary differential equations First the major qualitative analysis, like the disease free equilibruim point(DFEP), basic reproduction number, stability analysis of equilibrium points and sensitivity analysis was rigorously analysed Second, we introduced time dependent controls to the basic model and extended to an optimal control model of the disease We then analysed using Pontryagin’s Maximum Principle to derive necessary conditions for the optimal control of the pandemic The numerical simulation indicated that, an integrated strategy effective in controling the epidemic and the gvernment must apply all control strategies in combating COVID-19 at short period of time [ABSTRACT FROM AUTHOR] Copyright of Journal of Interdisciplinary Mathematics is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use This abstract may be abridged No warranty is given about the accuracy of the copy Users should refer to the original published version of the material for the full abstract (Copyright applies to all s ). 在本文中，我们开发了埃塞俄比亚大流行 COVID-19 传播的确定性数学模型，该模型允许通过暴露的人类传播我们提出了使用常微分方程系统的 SEIR 模型首先进行主要的定性分析，如无病平衡点（ DFEP)、基本再生数、平衡点稳定性分析和敏感性分析进行了严格分析其次，我们在基本模型中引入时间依赖控制，并扩展到疾病的最优控制模型，然后使用庞特里亚金最大原理进行分析，得出必要条件为优化控制大流行 数值模拟表明，有效控制流行和政府的综合策略必须在短时间内将所有控制策略应用于抗击 COVID-19 [ABSTRACT FROM AUTHOR] 交叉学科数学杂志版权所有是 Taylor & Francis Ltd 的财产，并且未经版权所有者的明确书面许可，不得将其内容复制或通过电子邮件发送到多个站点或发布到列表服务器。但是，用户可以打印、下载或通过电子邮件发送文章供个人使用 本摘要可能会被删节 不对以下内容的准确性提供任何保证副本 用户应参考材料的原始出版版本以获得完整的摘要（版权适用于所有 s ）。

Mathematical modeling and optimal control analysis of COVID-19 in Ethiopia

## We introduce a theoretical framework to explain and predict changes in the reproduction number of SARS-CoV-2 (Sudden Acute Respiratory Syndrome Coronavirus 2) in terms of individual mobility and interpersonal proximity (alongside other epidemiological and environmental variables) during and after the lockdown period. 我们引入了一个理论框架来解释和预测 SARS-CoV-2（突发急性呼吸系统综合症冠状病毒 2）在锁定期间和之后的个体流动性和人际接近度（以及其他流行病学和环境变量）的繁殖数量的变化.

Human Mobility and Epidemic Evolution

## Also, we have calculated the basic reproduction number which is an important parameter in the infection models. 此外，我们还计算了基本繁殖数，这是感染模型中的一个重要参数。

Mathematical analysis of a within-host model of SARS-CoV-2

## The basic reproduction number is determined using the next generation matrix approach. 使用下一代矩阵方法确定基本再生数。

Mathematical model of schistosomiasis with health education and molluscicide intervention

## Methods Our approach to forecasting future COVID-19 cases involves 1) modeling the observed incidence cases using a Poisson distribution for the daily incidence number, and a gamma distribution for the series interval; 2) estimating the effective reproduction number assuming its value stays constant during a short time interval; and 3) drawing future incidence cases from their posterior distributions, assuming that the current transmission rate will stay the same, or change by a certain degree. 方法 我们预测未来 COVID-19 病例的方法包括 1) 对观察到的发病病例进行建模，对每日发病数使用泊松分布，对系列区间使用伽马分布； 2) 估计有效再生数，假设其值在短时间内保持不变； 3）从后验分布中提取未来的发病案例，假设当前的传播率将保持不变，或有一定程度的变化。

COVID-19: Short term prediction model using daily incidence data

## Without such direct epidemiological measurement, other approaches are required to infer whether the number of new cases is likely to be increasing or decreasing: for example, estimating the pathogen‐effective reproduction number, R, using data gathered from the clinical response to the disease. 如果没有这种直接的流行病学测量，则需要其他方法来推断新病例的数量是可能增加还是减少：例如，使用从对该疾病的临床反应中收集的数据来估计病原体的有效繁殖数 R。

Inferring UK COVID‐19 fatal infection trajectories from daily mortality data: Were infections already in decline before the UK lockdowns?

## The time-varying reproduction number (Rt) that was estimated through fitting the mathematical model was adopted to quantify the transmissibility. 采用通过拟合数学模型估计的时变再现数（Rt）来量化传播率。

Containing the Transmission of COVID-19: A Modeling Study in 160 Countries

## Methods We estimated the instantaneous transmissibility of COVID-19 by using the time-varying reproduction number ( R t ). 方法 我们通过使用时变复制数 (Rt) 来估计 COVID-19 的瞬时传播率。

Modelling the association between COVID-19 transmissibility and D614G substitution in SARS-CoV-2 spike protein: using the surveillance data in California as an example

## Our simulation shows that approximately 51% of the entire population needs to be fully vaccinated to bring the control reproduction number to a value less than one (threshold condition needed for disease elimination) as compared to the 14. 我们的模拟表明，与 14 相比，大约 51% 的人口需要完全接种疫苗，以使对照繁殖数低于 1（消除疾病所需的阈值条件）。

Assessment of the COVID-19 Vaccine Program: Impact of the No Mask Mandate Executive Order in the State of Texas

## Again, the model is shown via numerical simulations to possess the backward bifurcation, where a stable DFE co-exists with one or more stable endemic equilibria when the control reproduction number, R 0 is less than unity and the rate of denial of COVID-19 is above its upper bound. 同样，该模型通过数值模拟显示具有后向分岔，其中当控制再生数时，稳定的 DFE 与一个或多个稳定的地方性平衡共存， R 0 小于 1，并且拒绝 COVID-19 的比率高于其上限。

Mathematical assessment of the role of denial on COVID-19 transmission with non-linear incidence and treatment functions

## Using numbers of SARS-CoV-2 variants detected in Japan as at 13 June 2021, relative instantaneous reproduction numbers (RRI) of the R. 使用截至 2021 年 6 月 13 日在日本检测到的 SARS-CoV-2 变体数量，R.

Predicted dominance of variant Delta of SARS-CoV-2 before Tokyo Olympic Games, Japan, July 2021

## Then, using the instantaneous reproduction number to characterize the status of the epidemic (Rt {approx} 1, Rt > 1 or Rt < 1), this information is used to propose different scenarios for the number of cases and deaths for 2021. 然后，使用瞬时再现数来表征流行病的状态（Rt {约} 1，Rt > 1 或 Rt < 1），该信息用于针对 2021 年的病例数和死亡数提出不同的情景。

COVID-19 epidemic scenarios into 2021 based on observed key superdispersion events

## Time dependent reproduction number (Rt) is one of the most popular parameters to track the impact of COVID-19 pandemic. 与时间相关的再生数 (Rt) 是追踪 COVID-19 大流行影响的最常用参数之一。

Hospitalization as reliable indicator of second wave COVID-19 pandemic in eight European countries

## Materials and Methods We have developed and applied an age-structured susceptible, exposed, infectious, recovered, or dead compartmental model for both civilian and military populations, driven by estimates of the time-dependent reproduction number, R(t), which can be both fit to available data and also forecast future cases, intensive care unit (ICU) patients, and deaths. 材料和方法 我们已经开发并应用了一个年龄结构的易感、暴露、感染、恢复或死亡区室模型，适用于平民和军人，由对时间依赖性繁殖数 R(t) 的估计驱动，R(t) 可以是两者都适合现有数据，还可以预测未来的病例、重症监护病房 (ICU) 患者和死亡人数。

COVID-19: On the Disparity in Outcomes Between Military and Civilian Populations

## Here I measured longitudinal time-series correlations between outdoor temperature, humidity and covid-19 reproduction number (Rt) in the 50 U. 在这里，我测量了 50 U 中室外温度、湿度和 covid-19 繁殖数 (Rt) 之间的纵向时间序列相关性。

COVID-19 spread and Weather in U.S. states: a cross-correlative study on summer-autumn 2020.

## This paper estimates time-varying COVID-19 reproduction numbers worldwide solely based on the number of reported infected cases, allowing for under-reporting. 本文仅根据报告的感染病例数估计全球随时间变化的 COVID-19 复制数量，允许少报。

COVID-19 Time-varying Reproduction Numbers Worldwide: An Empirical Analysis of Mandatory and Voluntary Social Distancing

## Specifically, we quantify the associations of daily mean temperature, specific humidity, and ultraviolet radiation with daily estimates of the SARS-CoV-2 reproduction number (Rt) and calculate the fraction of Rt attributable to these meteorological conditions. 具体来说，我们量化了每日平均温度、比湿度和紫外线辐射与每日估计的 SARS-CoV-2 繁殖数 (Rt) 的关联，并计算了可归因于这些气象条件的 Rt 比例。

Role of meteorological factors in the transmission of SARS-CoV-2 in the United States

## Fourth, we assess the association of VOC frequency with independent estimates of the overall SARS-CoV-2 reproduction number through time. 第四，我们评估了 VOC 频率与随着时间的推移对 SARS-CoV-2 总繁殖数的独立估计之间的关联。

Transmission of SARS-CoV-2 Lineage B.1.1.7 in England: Insights from linking epidemiological and genetic data