Basic Reproduction number
新冠传播指数计算方法
新冠传播指数计算方法
计算新冠传播指数的方法有很多种，其中比较常见和常用的
包括基本传播数（BasicReproductionNumber，简称R0）、有效传播数（EffectiveReproductionNumber，简称Reff）和疫情传播力量指数（EpidemicForceIndex，简称EFI）等。
1.基本传播数（R0）：R0是指在没有免疫力和防控措施的
情况下，一个感染者平均能够传染给多少个人。
计算R0的方法一般基于传染病传播过程的数学模型，结合流行病学调查数据
和传播特性等因素。
R0大于1表示疫情有扩散趋势，小于1表示疫情呈现下降态势。
2.有效传播数（Reff）：Reff是指在实施了控制措施后，目前感染者平均能够传染给多少个人。
Reff的计算方法会考虑到
人口免疫率、感染者的防控措施遵守情况等因素。
Reff大于1，表示疫情仍然在扩散，小于1表示疫情得到了有效控制。
3.疫情传播力量指数（EFI）：EFI是一个综合的指标，它综
合考虑了感染率、传播速度、传播路径、传播距离等多个因素，用来评估疫情的传播力量和风险。
EFI的计算方法一般结合了流行病学调查数据、统计学方法和地理信息系统（GIS）等技术。
这些指标的计算方法涉及到很多复杂的数学模型和统计方法，需要综合考虑病毒特性、疾病传播途径、人群行为等多个因素。
在实际应用中，通常需要借助专业的疫情数据分析和统计软件
来计算和预测传播指数，以便更好地制定疫情防控措施和应对策略。
传染病问题数学公式
传染病问题数学公式
传染病问题通常涉及到传染率、接触率、疾病治愈率、死亡率等参数。
在数学上，这些参数可以通过一系列模型和公式来描述和计算。
以下是一些常见的传染病问题的数学公式：
1. 基本复制数（Basic Reproduction Number, R0）：
R0 = β× D
其中，β表示传染率，D表示传染时间。
基本复制数表示一个感染者在人群中能够传染的人数。
当R0大于1时，疾病将继续传播；当R0小于1时，疾病将逐渐消失。
2. SEIR模型
SEIR模型用于描述一次传染病流行的过程。
该模型包括四类人群：易感人群（Susceptible）、潜伏期人群（Exposed）、感染者（Infectious）和康复者（Recovered）。
该模型的微分方程如下：
dS/dt = -βSI
dE/dt = βSI - αE
dI/dt = αE - γI
dR/dt = γI
其中，S、E、I、R分别表示四类人群的人数，β表示接触率，α表示潜伏期转化为感染期的概率，γ表示感染者恢复的概率。
3. SEIRD模型
SEIRD模型加入了死亡人群（Dead），用于描述一次传染病流行中的死亡情况。
该模型的微分方程如下：
dS/dt = -βSI
dE/dt = βSI - αE
dI/dt = αE - (γ+μ)I
dR/dt = γI
dD/dt = μI
其中，S、E、I、R、D分别表示五类人群的人数，β、α、γ、μ的含义与SEIR模型中相同，μ表示感染者死亡的概率。
什么是R0？
什么是R0？
什么是R0？
（2020.02.09）
R0 是 Basic reproductionnumber of the infection 的缩写，
意思是基本传染数。
元⽉四⽇⾄七⽇，
发往英国科学期刊的论⽂所载，
基本传染数 R0=2.2。
此数与1918年西班⽛流感的 R0 相当，
何以向国内公众发布说：
没有⼈传⼈？
【注1】
衡量病毒传播能⼒的最重要指标，叫基本传染数 R0 (Basic reproductionnumber of the infection)。
它指的是，在⾃然情况下（没有外⼒介⼊，同时所有⼈都没有免疫⼒），⼀个感染到某种传染病的⼈，会把疾病传染给多少⼈的平均数。
如果 R0 < 1，表⽰传染病得到控制，将会逐渐消失；
R0 = 1，传染病则会变成⼈⼝中的地⽅性流⾏病，就像击⿎传花般⼀个传⼀个；
R0> 1，传染病则会以指数⽅式暴发；
R0 的数字愈⼤，疾病将越难以控制。
1918年西班⽛流感的 R0 值在2-3之间，最终造成全世界10亿⼈感染，2500-4000万⼈死亡，当时全世界⼈⼝才17亿。
（引⾃王建东）
【注1】“R0”中的“0”是“零”，不是“o”。
流行病学专业英语词汇
流行病学专业英语词汇A安慰剂Placebo安慰剂效应placebo effectsB把握度study power半试验semi-trial保护率protective rate ，PR报告偏倚reporting bias暴露exposure暴露标志exposure marker暴露怀疑偏倚exposure suspicion bias爆发调查outbreak survey被动监测passive surveillance比例研究proportional study比值比Odds Ratio ，OR必需病因necessary factor of cause标点地图spot map标化比例死亡比standardized proportional mortality ratio，SPMR 标化比值比standardized odds ratio，SOR标化死亡比standardized mortality ratio，SMR标准化standardization标准化率比standardized rate ratio，SRR"冰山"现象ice-berg phenomenon丙型肝炎hepatitis C，HC并联试验parallel test病毒性肝炎viral hepatitis病例报告case report病例-病例研究case-case study病例调查case survey病例对照研究case-control study病例交叉设计case-crossover design病例交叉研究case-crossover study病例-系列研究case-series study病死率fatality rate病因causation of disease病因的异质性etiological heterogeneity病因分值etiologic fraction，EF病因链chains of causation病因网web of causation不合格ineligibility不依从noncomplianceC测量偏倚measurement bias长期变异secular change，trend variation超额死亡率excess mortality超额危险度excess risk巢式病例对照研究nested case-control study成本效果分析cost -effectiveness analysis成本效益分析cost-benefit analysis成本效用分析cost-utility成组匹配category matching抽样调查sampling survey抽样框架sampling frame抽样误差sampling error出生队列birth cohort出生队列分析法birth cohort analysis初级卫生保健primary health care ，PHC传播概率transmission probability传播机制mechanism of transmission传播途径path of transmission传播途径transmission route传播因素transmitting factor传染病communicable diseases传染过程infectious process传染性infectiousness串联试验serial test粗死亡率crude death rate改成crude mortality rate 较好粗一致性crude agreement促成病因contributory factor of cause错分偏倚misclassification bias错误分类misclassificationD大骨节病Kashin-Beck disease大流行pandemic代表性representativeness单倍型haplotype单纯随机抽样simple random sampling单盲single blind单项筛检single screening等位基因allele等位基因频率gene frequency地病endemic diseases地病学Endemiology地性endemicity地性氟中毒endemic fluorosis地性砷中毒endemic arsenism点突变point mutation点源流行point source epidemic碘缺乏病iodine deficiency disease调查随访研究survey follow-up study调整一致性adjusted agreement丁型肝炎hepatitis D，HD动态人群dynamic population毒力virulence短期波动irregular variation，rapid fluctuation队列cohort队列研究cohort study对象报告偏倚report bias多归因程序的法Multiple imputation procedure多国心血管病趋势和决定因素监测Multinational Monitoring of Trends and Determinants inardiovascular Diseases，MONICA多级抽样multistage sampling多群组比较研究multi-group comparison study多态性polymorphism多项筛检multiple screeningE恶性肿瘤流行病学cancer Epidemiology二代侵袭率，续发率secondary attack rate，SAR二级预防secondary preventionF发病率incidence rate，morbidity发病密度incidence density发病密度incidence density发生率研究incidence study飞沫droplet飞沫核droplet nuclei非即时性non-concurrent非均衡性错分differential misclassification非连续性状Discrete traits肺炎与流感死亡率pneumonia and influenza deaths，即P&I deaths分层stratification分层抽样stratified sampling分层随机分组stratified randomization分类变量categorical variable分离分析segregation analysis分子流行病学Molecular Epidemiology封闭性抗体blocking antibodies氟斑牙dental fluorosis氟骨症skeletal fluorosis负混杂negative confounding复等位基因位点multiple alles复杂性状complex traitG概念框架conceptual framework干预随访研究intervention follow-up study感染储主reservoir of infection感染来源source of infection感染力infectivity感染谱spectrum of infection感染梯度gradient of infection感染性腹泻infectious diarrhea感染性腹泻病infectious diarrheal disease感染症infections or infectious diseases高危人群策略high risk strategy个案case个案调查case study个案调查individual survey个体匹配individual matching工程干预engineering intervention共变法method of concomitant variation共显性codominance构成比proportion固定队列fixed cohort故意伤害intentional injuries关联的合理性plausibility of association关联的可重复性consistency of association关联的强度strength of association关联的时间性temporality of association关联的特异性specificity of association关联的一致性coherence of association观察偏倚observational bias观察性研究observational study观察一致率observation agreement广义遗传度broad heritability归因危险度attributable risk归因危险度百分比attributable risk percent，ARP，或AR%国际病毒分类委员会International Committee on Taxonomy of Viruses，ICTV 国际冠心病预防工作组The International Task Force for Prevention of Coronary Heart Disease国际疾病分类第十版International Classification of Diseases，ICD-10结核病防治项目National Tuberculosis Program，NTP卫生统计中心the National Center for Health Statistics，NCHSH合作率cooperation rate核心家系Nuclear families横断面研究cross sectional analysis横断面研究cross-sectional study患病率prevalence rate回顾性研究retrospective studies回忆偏倚recall bias混合型流行mixed epidemic混合研究mixed study混杂confounding混杂偏倚confounding bias混杂因素confounder，confounding factor活跃病区active endemic area获得性免疫缺陷综合症acquired immunodeficiency syndrome，AIDS 霍桑效应Howthorne effectJ机遇chance基本繁殖率basic reproduction rate基本繁殖数basic reproductive number，R0基本人群base population or population at risk基线信息baseline information基因gene基因毒性genotoxic基因型Genotyping基因型频率genotype frequency及时性timeliness疾病爆发disease outbreak疾病标志Markers of disease疾病的分布distribution of disease疾病的自然史natural history of disease疾病监测surveillance of disease疾病链the disease chain剂量反应关系dose-response relationship季节性seasonal variation，seasonality继发关联secondary association家庭簇研究family cluster study家庭聚集性Familial aggregation家庭聚集性研究familial aggregation study家庭相似性Family resemblance家系研究pedigree study甲型肝炎hepatitis A，HA假阳性率false positive rate假阴性率false negative rate间接传播indirect transmission间接关联indirect association间接接触indirect contact检出症侯偏倚detection signal bias简单随机分组simple randomization简明损伤定级法abbreviated injury scale，AIS健康促进health promotion健康工作者效应healthy worker effect健康监测系统health information system，HIS健康生命损失年years of life lived with disability，YLLD 教育干预educational intervention结局outcome结局变量outcome variable截断点cutoff金标准gold standard经济干预economic intervention经节肢动物媒介vector-borne经空气air-borne经食物food-borne经饮水drinking water-borne经载体vehicle-borne精确性precision聚合酶链式反应polymerase chain reaction，PCR均衡性错分non-differential misclassification均数回归趋势regression to the meanK抗原漂移antigenic drift抗原转变antigenic shift克山病Keshan disease空间、时间集研究space-time cluster study扩大免疫计划expanded program on immunization，EPI L来源同一identity by descent，IBD类试验，准试验quasi-trial，quasi-experiment类推法method of analogy累积发病率cumulative incidence罹患率attack rate礼貌偏倚politeness bias理论流行病学theoretical epidemiology历史性historical连锁linkage连锁不平衡linkage disequilibrium连锁分析Linkage analysis连锁平衡linkage equilibrium联合无应答偏倚combined nonrespondent bias联系率contact rate临床试验clinical trial灵敏度sensitivity领先时间偏倚lead time bias流感病毒influenza virus流行epidemic流行病模型epidemic model流行病学epidemiology流行病学监测epidemiological surveillance流行病学实验epidemiological experiment流行过程epidemic process流行曲线epidemic curve流行性出血热epidemic hemorrhagic fever，EHF流行性感冒Influenza流行性肾病nephropathia epidemica，NE率rate率比rate ratio率差rate difference，RD轮状模式wheel modeM蔓延流行propagated or progressive epidemic盲法blindness美国胆固醇教育规划National Cholesterol Education Program，NCEP 描述流行病学descriptive epidemiology描述性研究descriptive study目标人群target populationN暴露剂量Internal Dose部有效性internal validity剂量internal dose剂量的测定internal dose meters纳入/排除偏倚inclusion/exclusion bias奈曼偏倚Neyman biasP排除exclusions排除法method of exclusion排除偏倚exclusive bias配对pair matching"皮鞋"流行病学shoeleather epidemiology匹配matching匹配变量matching variable匹配过头overmatching匹配因素matching factors偏倚bias频数匹配frequency matching普查censusQ前瞻性prospective 8前瞻性研究prospective study潜伏期incubation period潜隐期latent period潜在工作损失年数Working Years of Potential Life Lost，WYPLL潜在价值损失年数Valued Years of Potential Life Lost，VYPLL潜在寿命损失年potential years of life lost，PYLL强制干预enforcement intervention求同法method of agreement求异法method of difference全人群策略population strategy缺课天数days lost from school缺勤天数days lost from work缺失数据missing data确定性检验explicit testing确定性模型deterministic modelR人传人person to person spread人类传染病anthroponosis人类基因组计划human genome project人类免疫缺陷病毒human immunodeficiency virus，HIV人群population人群病因分值population etiologic fraction，PEF人群的分层population stratification人群归因危险度population attributable risk，PAR人群归因危险度百分比population attributable risk percent，PARP 或PAR% 人群免疫性herd immunity人群易感性herd susceptibility人兽共患病，动物传染病zoonosis人畜共患病anthropozoonosis日期型date type入院率偏倚（伯克森偏倚）admission rate bias （Berkson's bias）。
一个传染病模型中的后向分支问题
一个传染病模型中的后向分支问题白婵;万辉【摘要】为了研究在考虑免疫接种情况下有限的医疗资源对疾病传播的影响,我们建立了一个带有特殊恢复率的SIVS传染病模型,研究了模型的基本动力学性质并对后向分支进行了详细的证明.结果表明,有限的医疗资源会导致重要的动力学性质,比如双稳现象等.后向分支意味着,即使基本再生数小于1模型依然可能会有稳定的地方病平衡点,基本再生数不能完全反映疾病流行与否.此时,人们应该注意疾病爆发时的初始状态.研究结果同时表明,充足的医疗资源和服务对于疾病的消除与控制非常重要.另外,文章也分析了免疫接种的影响.%In this paper,we formulate a SIVS epidemic model with special recovery rate to study the impact of limited medical resource on the transmission dynamics of diseases with vaccination.The basic investigation of the model has been finished. The backward bifurcation has been proved precisely. It is shown that limited medical resource leads to vital dynamics,such as bistability. Backward bifurcation implies that even if the basic reproduction number is smaller than unity,there may be a stable endemic equilibrium and the basic reproductive number itself is not enough to describe whether a disease will prevail or not and we should pay more attention to the initial conditions. It is also shown that suffi-cient medical services and medicines are very important for the disease control and eradication. Besides,the impact of vaccination has been explored too.【期刊名称】《南京师大学报（自然科学版）》【年(卷),期】2017(040)003【总页数】9页(P5-12,20)【关键词】免疫接种;传染病模型;医疗资源;平衡点;稳定性;后向分支【作者】白婵;万辉【作者单位】江苏省大规模复杂系统数值模拟重点实验室,南京师范大学数学科学学院,江苏南京210023;江苏省大规模复杂系统数值模拟重点实验室,南京师范大学数学科学学院,江苏南京210023【正文语种】中文Recently,attention has been given to vaccination and treatment policies in terms of the different vaccine classes,efficacy,treatment resource and associated costs([1-12],etc.).In[2],Kribs-Zaleta et al. introduced a vaccination compartment with temporarily immune state and set up a SIV model with general incidence rate. Their analysis indicated that when the vaccine for all population is not totally effective,the basic reproduction number R0 is no longer a threshold for the spread of diseases and the model will exhibit multiple endemic states. In[9],Shan and Zhu took the per capita recovery rate as a function of the number of hospital beds. Their analysis indicated that the system could undergo backward bifurcation,saddle-node bifurcation,Hopf bifurcation and cusp type of Bogdanov-Takens bifurcation within different conditions. In[11],Xiao and Tang analyzed a SIV epidemic model with nonlinear incidence rate. The main result shows that the system undergoesforward bifurcation with hysteresis except for the backward bifurcation. Besides,Erika et al.([7])studied the dynamics of a SIR epidemic model with nonlinear incidence rate,vertical transmission vaccination for the newborns and the capacity of treatment,that takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions,they proved that the existence of backward bifurcation,the stability and the direction of Hopf bifurcation. They also explored how the mechanism of backward bifurcation affects the control of the infectious disease.In order to consider the impact of limited medical resources and vaccination on the transmission dynamics of infection diseases more precisely,we formulate a SIVS epidemic model with human population demography and vaccinated individuals.The organizations of this paper are as follow. Firstly,we will introduce our SIVS model in section 2. In section 3,we will analyze the existence of equilibria. Stability of equilibria and backward bifurcation analysis will be given in Section 4. Some discussion will be given in Section 5.We classify the population in a given region/area into three categories:susceptible,infective and vaccinated. Let S(t),I(t)and V(t)denote the number of susceptible,infective,vaccinated individuals at timet,respectively. Based on standard SIS model with the incidence of mass action,we can construct a modelwith initial data S(0)≥0,I(0)≥0,V(0)≥0,S(0)+I(0)+V(0)≤Λ/d,where all parameters listed in Table 1 are positive.In classical epidemic models,the per capita recovery rate is assumed to be a constant. Nevertheless,in general,the recovery rate depends on the resources of the health system available to the public,particularly the capacity of the hospital settings and efficiency of the treatment. There are many factors determining the recovery rate. The significant factor is the number of the hospital beds and medicines are another significant factors which are essential for safe and effective prevention,diagnosis and treatment of illness. We take the per capita recovery rate as μ=μ0+(μ1-μ0) which is used in[9]. Therefore,the recovery rate in a unit time μI is a that can describe the impact of limited medical resource.It is not difficult to prove the following theorem:Theorem 1 With an initial value condition in(1),there is a unique solution,and the solution remains positive and bounded for any finite time t≥0.Therefore,Model(1)is mathematically well-defined and biologically reasonable.Summing up the system(1),then =Λ-dN. So N(t)tends to Λ/d as t increases to infinity. Therefore,we can reduce the size of the model by letting S=Λ/d-I-V. Now the model becomesLet the right-hand side of(2)to be zero. One can verify that themodel(2)has one disease free equilibrium at E0=. The local stability of E0 can be obtained through a straightforward calculation for the eigenvalues. It follows from([13])that for the compartmental models,the local stability of the disease-free equilibrium is governed by the reproduction number ofthe model. If we use the notation in([13]),then we haveThe infected compartment is I,hence a straightforward calculation gives andHence the reproduction number is given by ρ(FV-1),andRemark 1 According to a straightforward calculation,R0 is a monotone decreasing function with respect to the vaccination rate φ.Let the right hand side of(2)be zero,then the endemic equilibriumE(I,V)satisfiesand I must satisfy the following equation:whereC=-β2Λσb+aβ+d(μ0+d)(φ+θ+d),a=-Λ(d+θ+σφ)+db(θ+d+σφ+μ0σ+dσ)+σbd(μ1-μ0),D=bd(d+μ1)(d+θ+φ)(1-R0).Because it is complicated to discuss the root of function F(I),we will study the number of the root from the geometry.Let the right-hand side of(2)be zero,thenwhereOne can easily verify thatandTherefore,f(I)is a monotonous decreasing concave function whereas g(I)is a convex function on the interval[0,Λ/d]. Besides,which implies thatNow we can discuss the number of the equilibrium points in three cases.(1)If R0>1,then f(0)<g(0),and g(Λ/d)<0 always exists. Thus there is only onepositive intersection which gives one endemic equilibrium(See Fig.1(a)). (2)If R0=1,then f(0)=g(0),and g(Λ/d)<0 always exists. Thus f(I)and g(I)will intersect one point if and only if f0(0)<g0(0)(See Fig.1(b)).(3)If R0<1,then f(0)>g(0),and g(Λ/d)0 always exists. Thus f(I)and g(I)will intersect with each other at two points which gives two endemic equilibrium if and only if f′(0)<g′(0)and there exists such that <g();The two points coalesce with each other if f()=and f′()=g′()(See Fig.1(c)(d)).As a result,we have the following theorem:Theorem 2 For the system(2).(1)The disease-free equilibrium E0 always exists.(2)If R0>1,there exists only one endemic equilibrium E1(I2,V2).(3)If R0=1,there exists one endemic equilibrium if and only if g0(0)>f0(0). Otherwise,there is no endemic equilibrium.(4)If R0<1,there exist two endemic equilibrium E1(I1,V1)and E2(I2,V2)if and only if f′(0)<g′(0)and there exists such that f()<g(),the two equilibria will coalesce if f()=g()and f′()=g′().Fix parametersΛ=1,d=0.1,β=0.2,θ=0.1,σ=0.4,b=0.1,μ0=0.2,φ=0.5,R0=2,1,0.2,0.047 respectively,we will get the Fig.1,respectively.Firstly,we will discuss stability of the disease-free equilibrium.The Jacobian matrix of system(2)So we will getwe have the following theorem:Theorem 3 When R0<1,E0 is locally asymptotically stable;when R0>1,E0 isunstable;when R0=1,E0 is a saddle-node.Proof One can verify that system at E0 has an eigenvalue,-(φ+θ+d)<0. The other eigenvalue is(μ1+d)(R0-1)<0. The result from the fact that(μ1+d)(R0-1)<0 is equivalent to R0<1 and(μ1+d)(R0-1)>0 is equivalent to R0>1. Hence,E0 is locally asymptotically stable if R0<1 and E0 is unstable if R0>1. Besides,obviously E0 is a Lyapunov singularity when R0=1. A straightforward calculation gives that one eigenvalue of Jacobian matrix of system at E0 is zero when R0=1,the other is -(φ+θ+d). The transformation I′=I,V′=V+ brings E0 to the origin. Then the system in a neighborhood of the origin becomesSimplifying the system,we will getLinearizing the above syst em and still substituting I=I′,V=V′. By straightforward calculating,we getwhereLet the right-hand side of the second equation be zero. According to implicit function,one can obtain a function for V in terms of I. SupposeV(I)=a1I+a2I2+O(I3)with V(0)=0. S ubstituting V(I)into ψ(I,V),Comparing the coefficients of the same powers gives that a1=-. Thus,andClearly,m=2 which implies that E0 is half saddle node. LetThen,E0 is left saddle and right node if C<0 while E0 is right saddle and left node if C>0.In the following,we will discuss the stability of the endemic equilibrium E-(I-,V-).The Jacobian matrix of the model(2)at E(I-,V-)givesand the corresponding characteristic equation is given bywhereObviously,F′(I)and H(I)determine the eigenvalues of the ma trix J(E-). It is easy to know F′(I1)>0,F′(I2)<0. So E2(I,V)is a hyperbolic saddle and is always unstable,and E1(I,V)is a anti-saddle. Furthermore,if H(I)>0,E1(I,V)is locally asymptotically stable;If H(I)=0,E2(I,V)is a weak focus or center;IfH(I)<0,E1(I,V)is a unstable node or focus. Thus we have the following theorem.Theorem 4 When R0>1,the system exists unique endemic equilibrium and it’s an anti-saddle;when R0<1,and the two endemic equilibriaexist,E1(I,V)is always unstable,E2(I,V)is locally asymptotically stable if H(I)>0 and unstable if H(I)<0.Lemma 1 (Theorem 3 in[14])AssumeA1:A=Dxf(0,0)=is the linearization matrix of system around the equilibrium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts;A2:Matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.Let fk be the kth component of f andThe local dynamics of system around 0 are totally determined by a and b.(1)a>0,b>0. When φ<0with |φ|≪1. 0 is locally asymptotically stable. and there exists a positive unstable equilibrium;when 0<φ≪1,0 is unstable and there exists a negative and locally asymptotically stable equilibrium;(2)a<0,b<0. When φ<0 with |φ|≪1. 0 is unstable;When 0<φ≪1,0 is locally asymptotically stable,and there exists a positive unstable equilibrium. (3)a>0,b<0. When φ<0 with |φ|≪1. 0 is unstable;When 0<φ≪1,0 is stable,and there exists a locally asymptotically stable negative equilibrium,and a positive unstable equilibrium appears.(4)a<0,b>0. When φ<0 changes from negative to positive,0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable. Theorem 5 When R0=1,b<,the system will undergo backward bifurcation. Proof Let =-d. Then R0=1 if and only if μ1=.To apply Lemma 1,we essentially have to compute two quantities,labeled A and B,which depend on the higher order terms in the Taylor expansion of system,and require,for their computation,a change of coordinates involving the right and left eigenvectors of the Jacobian J(E0)associated with the eigenvalue λ=0. we will express A and B,in terms of parameters. The right and left eigenvectors of the Jacobian J(E0)arerespectively. In order to follow the notations introduced in Theorem 3of[18],we let x1=I,x2=V and φ=-μ1. Then the Taylor expansion system are represented by the fi(x,φ),(i=1,2),and we haveAll other derivatives equal to zero. Consequently,we can readily compute the following q uantity A:=∑ukwiwj=+2(-β+σβ)=2(-β+σβ)(f′(0)-g′(0)). Note that =1-R0+ and all other derivative equal to zero. so we can calculate B by substituting the vector v and w and the respective partial derivatives into the expression β= ∑ukwi=1-R0+.So we conclude that when R0=1 and f 0(0)<g0(0),A>0 and B>0,which is the defining condition for a backward bifurcation[14].In this paper,we formulate a SIVS epidemic model with special recovery rate to study the impact of limited medical resource on the transmission dynamics of diseases with vaccination.In Model(2),the disease-free equilibrium always exists which is locally asymptotically stable if R0<1 and unstable if R0>1. There is a unique endemic equilibrium if R0>1. Furthermore,on the one hand,if the medical resource in a given region is not sufficient enough,according to Theorem 5,backward bifurcation will occur and there are at most two endemic equilibria even if R0<1. One(E1)is a hyperbolic saddle and the other(E2)is a anti-saddle. Furthermore,E2 is a stable anti-saddle if H(I)>0. Backward bifurcation implies that the basic reproductive number itself is not enough to describe whether the disease will prevail or not and we should pay more attention to the initial value. On the other hand,if the medical resource is not less than the threshold value of b described in Theorem 5,R0 can act as a critical value and one just need to control R0 less than unity in order to control a disease. It is also shown that sufficient medical services and medicines are very important for the disease control and eradication. According to Remark 1,the basic reproduction number is monotone decreasing function with respect to the vaccination rate φ,which implies that vaccination may help to control a disease,especially if the medical resource is not sufficient. By vaccination,one can decrease the basic reproduction number to small enough value to guarantee there is noendemic equilibrium.[1] CUI J,MU X,WAN H. Saturation recovery leads to multiple endemic equilibria and backward bifurcation[J]. Journal of theoreticalbiology,2008,254:275-283.[2] KRIBS Z C,VELASCO H J. A simple vaccination model with multiple endemic states[J]. Math Biosci,2000,164:183-201.[3] LIU X,TAKEUCHI Y,IWAMI S. SVIR epidemic models with vaccination strategies[J]. Journal of theoretical biology,2008,253:1-11.[4] LI J,ZHAO Y,ZHU H. Bifurcation of an SIS model with nonlinear contact rate[J]. J Math Anal Appl,2015,432:1 119-1 138.[5] LI G,LI G F. Bifurcation analysis of an SIR epidemic model with the contact transmissions function[J]. Abstract and appliedanalysis,2014,Article ID 930541.[6] MAGPANTAY F M G,RIOLO M A,DOMENECH de CELLS M,et al. Epidemiological consequences of imperfect vaccines for immunizing Iinfection[J]. SIAM Journal on applied mathematics,2014,74:1 810-1 830.[7] ERIKA R,ERIC A,GERARDO G. Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment[J]. Mathematics and computers in simulation,2016,121:109-132.[8] SHAN C,ZHU H. Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds[J]. J differential equations,2014,257:1 662-1 688.[9] SHAN C,ZHU H. Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources[J]. J differentialequations,2016,260:4 339-4 365.[10] WAN H,CUI J. Rich Dynamics of an epidemic model with saturation recovery[J]. Journal of applied mathematics,2013,Article ID 314958,9 pages.[11] XIAO Y,TANG S. Dynamics of infection with nonlinear incidence in a simple vaccination model[J]. Nonlinear analysis:real world applications,2010,11:4 154-4 163.[12] ZHOU T,ZHANG W,LU Q. Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function[J]. J applied mathematics and computation,2014,226:288-305.[13] VAN DEN DRIESSCHE P,WATMOUGH J. Reproduction numbers and sub-threshold endemic equilibria for compartmental model of disease transmission[J]. J Math Biosci,2002,180:29-48.[14] CASTILLO C C,SONG B. Dynamical models of tuberculosis and their applications[J]. Mathematical biosciences and enginieering,2004(1):361-404.【相关文献】CLC number:175.12 Document codeA Article ID1001-4616(2017)03-0005-08Received data:2017-04-13.Foundation item:Supported by Jiangsu Overseas Research and Training Program for University Prominent Young & Middle-aged Tachers and Presidents;the NSF of the Jiangsu Higher Education Committee of China(15KJD110004);Project Founded by PAPD of Jiangsu Higher Education Institutions.Corresponding author:Wan Hui,associate professor,majored in mathematical biology. E-mail:****************doi:10.3969/j.issn.1001-4616.2017.03.002。
社团结构对一个SIR模型的基本再生数的影响
社团结构对一个SIR模型的基本再生数的影响基本再生数（Basic Reproduction Number，R0）是流行病学中一个经常用来衡量传染病传播能力的指标。
对于SIR模型来说，R0等于感染率（β）除以恢复率（γ）。
在传统的SIR模型中，人与人之间的接触是随机的，即任意一个易感者与患病者接触的概率是相等的。
但在现实生活中，人们的接触网络往往具有特定的结构，如社团结构。
社团结构指的是人们在某种联系下能够组成的小团体。
社团结构对SIR模型的传播动态和基本再生数有着深远的影响。
社团结构可以增加传播的速度。
在传统的SIR模型中，人与人之间的接触是随机的，患者与易感者的接触机会相等。
在社团结构中，患者与易感者之间的接触概率会增加。
因为社团结构中的人们更倾向于与同社团的人接触，从而增加了患者与易感者的接触机会，使得传播更为迅速。
这种增加传播速度的效应使得SIR模型中的基本再生数增大，从而增加了疫情的爆发风险。
社团结构可以增加疫情的规模。
在传统的SIR模型中，整个人群由一个大的网络所连接。
而在社团结构中，人们被分成了多个小的社团，社团内的人之间接触更频繁，而社团之间的接触相对较少。
这样的结构导致了疫情在社团内传播的机会更多，从而增加了疫情的规模。
SIR模型中的基本再生数在社团结构下更大，疫情的数量也更多。
社团结构还会影响疫情的传播路径。
在传统的SIR模型中，传播路径是随机的，任意一个易感者与患病者接触的概率是相等的。
但在社团结构中，易感者更容易接触到同社团的患者，从而顺着社团的内部连接路径传播。
这种传播路径的特点会导致疫情在社团内的传播更为迅速，同时也使得社团之间的传播较为缓慢。
在社团结构下，疫情的传播路径与传统的随机传播路径有所不同。
社团结构对一个SIR模型的基本再生数有着重要的影响。
社团结构不仅增加了传播速度和疫情规模，还对传播路径产生了一定的影响。
研究社团结构对传染病传播的影响，有助于制定更有效的防控措施，并提高对疫情的预测和应对能力。
R0是12说明
R0是12说明
R0是指基本再生数（basic reproduction number），它表示在自然传播下，一个患者平均会把疾病传染给多少人，以此来衡量该病毒的传播力。
如果R0<1，表示传染病得到控制，将会逐渐消失；R0=1，传染病则会变成人群中的地方性流行病，就像击鼓传花般一个传一个；
R0>1，传染病则会以指数方式暴。
R0值愈大，意味着疾病越难控制，例如麻疹的R0值为12-18，SARS为2-3。
对于新型冠状病毒的R0值，国内外不同科研机构给出了不同结论。
虽然结论很多，但主要通过两种方式得出：一种是数学模型估算，另一种是现场流行病学调查。
前者主要是根据官方公布的疫情数据进行推算，后者则是在疫区进行实地案例调查。
不同场景，得出的R0值有很大差距。
比如家庭聚集病例就会比社会传播的病例更高。
R0值的估算有一个重要前提，那就是没有外力介入。
这就意味着，一旦采取相应的防控措施，R0值就会逐步下降，病毒的传播力也会减弱。
新冠病毒的R0值仍在不断变化。
周期传染病模型的基本再生数
周期传染病模型的基本再生数∗白振国【摘要】本文是对周期传染病模型中基本再生数如何定义的一个简单综述。
基于常微分方程(ODE)、时滞微分方程(DDE)、偏微分方程(PDE)、差分方程和脉冲微分方程系统所描述的模型，我们利用积分算子谱半径的方法给出了这些不同类型模型的基本再生数的具体定义。
其结果有助于流行病学家们有效地预测具有季节性波动传染病的发展趋势。
%This paper is a brief survey on how to define the basic reproduction number in seasonally forced epidemiological models. Applying the method of the spectral radius of the integral operator, we present the definitions of the basic reproduction number in different types of models, which involve a variety of mathematical areas, such as ODE (ordinary differential equations), DDE (delay differential equations), PDE (partial differential equations), system of difference equations, system of impulsive differential equations. The results can help epi-demiologists effectively to predict the development trend of infectious diseases with seasonal fluctuation.【期刊名称】《工程数学学报》【年(卷),期】2013(000)002【总页数】9页(P175-183)【关键词】传染病模型;基本再生数;谱半径;季节性【作者】白振国【作者单位】西安电子科技大学理学院，西安 710071【正文语种】中文【中图分类】O175.1;O175.2;Q-3321 引言在传染病动力学的研究中，基本再生数(R0)是一个非常重要的概念．它表示在发病初期，当所有人均为易感者时，一个病人在其平均患病期内所传染的人数[1]．通常，R0=1可作为决定疾病是否消亡的一个阈值，即：当R0<1时，疾病逐步消亡；而当R0>1时，疾病将始终存在而形成地方病．例如，对于如下一个经典的Kermack-McKendrick模型[2]其基本再生数的推导过程可借助于图1形象地描述．这表示：在一个所有人均为易感者的环境中，在t=0时刻引入的一个染病者到t时刻仍为病人的概率为e−γt，则在t时刻新感染的人数为βS0e−γt．因此就给出了一个染病者在其平均的患病期内所感染的人数，这与R0在医学中的解释也相符．然而，对于复杂的模型这样的推导并不总是成立．幸运的是，对于自治的传染病模型，Diekmann等，Driessche和Watmough引入了下一代方法(next generation method)来定义和计算R0[3，4]．而对于非自治的人口和传染病模型，R0的定义和计算将更加复杂．图1: 感染的演变过程，其中(S0，0，0)为模型(1)的无病平衡点研究发现，对于一些具有周期参数的SIR或SIRS模型，决定疾病消亡与否的阈值与其参数平均化后的自治系统的基本再生数一致[5，6]．然而，对于周期的SEIR或SEIRS模型，情况却并非如此．为了阐明这个缘由，我们考虑一个具有周期传染率的SEIR模型其中β(t)=β0[1+εcos(2πωt)]，1/ω=365天．当β(t)=β0时，(2)的基本再生数为对于周期模型(2)，这个平均的R0可解释为：在一个所有人均为易感者的环境中，如果疾病流行开始的时刻t满足cos(2πt/365)=0，那么在此时刻引入的一个病人在其患病期内所感染的平均人数．然而对于具有季节性波动的传染病模型，疾病的最终规模(the f i nal size)和传染病曲线的形状(如在一年中疾病的流行会出现多少次波峰等)强烈地依赖于一种病菌在人群中传染开始的时间t0，而自治的模型却不依赖于t0[7，8]．因此，采用平均的R0并不能准确地判断出一种疾病是否会侵袭人类．事实上，对于周期的SEIR或SEIRS模型，一些研究已经表明平均意义下的R0有估高疾病发生的危险，见文献[6，9，10]．那么，对于一般周期模型的R0，我们应该如何定义和计算它呢？许多年来，这是一个在人口和传染病动力学中非常难的问题．一些学者甚至认为：在周期环境中，基本再生数的概念可能已不再适用[11，12]或者即使能给出其定义但没有一般的数值方法来计算它[13]．值得庆幸的是，这个问题终于在2006年得到解决．Baca¨er和Guernaoui在文献[14，15]中表明周期环境中的R0可定义为：在连续的T周期函数空间中一个线性积分算子的谱半径．也即，有一个唯一的实数R0，使得存在一个连续正的T周期向量值函数满足其中A(t，τ)为一n×n的非负连续矩阵函数且满足：对所有的τ，A(t，τ)=A(t+T，τ)及对所有的t，∫∞0A(t，τ)dτ有限．其中A(t，τ)中的每个元素Ai，j(t，τ)表示由一个在t−τ时刻类型为j的感染者在单位时刻t所感染的类型为i的人的数目．这个一般的定义适用于周期的ODE模型、DDE模型、PDE模型、脉冲微分方程模型等．特别地，对于ODE模型，王稳地和赵晓强[16]给出了一个更为具体的定义R0的方法．利用Floquet理论，他们也确立了R0的计算公式．这些先期的研究工作为我们研究具有季节性传染病的动力学行为提供了理论依据．2 “积分算子谱半径”方法的应用下面我们将针对几类不同的微分方程模型，给出这个方法具体应用的一些例子．2.1 ODE模型在文献[16]的例1中，王稳地和赵晓强考虑了一个具有两个菌株的周期的SIS模型其中S为易感者的数目，I1为菌株1的数目，I2为菌株2的数目．νI1I2表示菌株2可被菌株1再次感染，传染率β1(t)和β2(t)是两个连续非负的T周期函数．并假定模型(3)中的其它参数均为正．容易看出，模型(3)有一个无病稳定态x0(0，0，1)．依据文献[16]中给出的定义R0的方法，我们有则模型(3)的基本再生数可定义为对于这个方法在其它ODE模型中的应用，我们可参见文献[16–19]．2.2 差分方程模型考虑如下一个具有周期传染率的SIS模型其中β(t)=β(t+m)(m>2)为周期的接触率，µ>0为人口的出生率和死亡率，γ>0为感染者的恢复率．并假设0<µ+γ<1，0<µ+β(i)<1，i=0，1，···，m−1．系统(4)在无病平衡点E0(1，0)处线性化后染病者所对应的方程为其中F(t)=β(t)，T(t)=1−(µ+γ)．则依据文献[20]，模型(4)的基本再生数可定义为其中r(A)表示矩阵A的谱半径．2.3 脉冲微分方程模型考虑一个具有脉冲接种和周期传染率的SIR模型模型(5)是文献[21]中所考虑模型的一种特殊形式．其中是饱和的发生率，µ为人口的出生率和死亡率，γ>0是感染者的康复率．T>0为对易感者进行两次脉冲接种的间隔时间，θk(06θk61)是在时刻t=kT对易感者的接种率．α(t)和β(t)为连续正的ω周期函数．并假定q=ω/T为一整数及θk=θk+q，k∈N．利用文献[21]中的引理2，系统(5)有一无病周期解(S∗(t)，0，0)，其中具有由文献[16]可得因此，模型(5)的基本再生数可定义为2.4 PDE和DDE模型Baca¨er在文献[22]中考虑了一个具有周期接触率和固定潜伏期的PDE模型其中E(t，x)是在t时刻具有染病年龄为x的潜伏者的数目，L表示固定的潜伏期，a为感染者的恢复率，c(t)=c0(1+εcos(wt))为周期的接触率．并假定总人口数为常数，则(6)可写为这是一个周期的时滞微分方程．在无病稳定态(N，0)线性化系统(7)可得染病者的方程令i∗(t)=c(t)I∗(t)为在t时刻处新感染的人数，则有其中则模型(7)的基本再生数可定义为积分算子在连续的2π/ω周期函数空间上的谱半径[14，15]．尽管这样定义的R0没有显式表达式，但估计是可能的．Bacar基于文献[23]中的数值方法表明R0是下列具有连续分数方程的最大实根其中通过数值模拟，Baca¨er发现：1) 当ε=0时，R0不依赖于L；2) 当L=0或时，R0不依赖于ε；3) R0是关于L的2π/ω周期函数；4) 对于固定的L，R0是关于ε的增函数或减函数．此外，R0的估计表达式(9)也适用于方程(8)的积分核具有形式的一大类的ODE、DDE和PDE的传染病模型．上述例子表明，具有时滞的周期传染病模型的R0与季节性因素ε、周期2π/ω和潜伏期的长短L都有关系．这些参数的引入给数值估计R0带来了很大的困难．据作者所知，目前对于此类模型R0数值研究的文献非常少见，仅见文献[15]．然而，时滞因素在许多具有周期波动发生率的媒介传播疾病中普遍存在，如疟疾、登革热、西尼罗河病毒等．因此，提出一些有效的数值估计此类模型R0的方法具有很大的应用价值，这将是我们今后进行研究的一个重要课题．3 在周期模型中R0的生物意义尽管周期模型中的R0可通过一个积分算子的谱半径来定义，但是它的生物意义并不明确．特别值得一提的是，这个R0的生物意义不能像自治的SIR或SEIR模型中一样解释为“在一个所有人均为易感者的环境中，由一个病人在其患病期内平均所感染的人数”．事实上，对于周期的传病模型，这是不合理的．例如考虑如下一个染病者所对应的方程其中a(t+T)=a(t)，T>0，则一个在t0时刻引入的患者二次感染所产生的新的染病者的平均数目为称R0=/b为平均意义上的再生数实际上是指C(t0)在一个周期内的平均，即然而当b(t)也是一T周期函数时，利用文献[14，16]给出的方法可得R0=/．显然，在此情形下关于R0平均意义下的解释完全失效．Baca¨er在文献[24]中表明在周期模型中所定义的R0可解释为“渐近的每代增长率”．其它关于周期R0的研究工作，我们可参见文献[20，25–31]．4 结论在本文，针对不同类型具有周期参数的微分方程模型，我们对它们所对应的基本再生数的定义方法给出了一个概括性的综述．这可为众多研究具有季节性波动传染病阈值动力学行为的学者提供一个重要的理论参考．参考文献：[1]Anderson R M，May R M.Infectious Diseases ofHumans[M].Oxford:Oxford University Press，1991[2]Kermack W O，McKendrick A G.A contribution to the mathematical theory of epidemics[J].Proceedings of the Royal Society 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