外文翻译---带有垂直传染和接种疫苗SEIRS流行病模型的全局稳定性.docVIP

带有垂直传染和接种疫苗seirs流行病模型的全局稳定性global stability of an seirs epidemic model with vertical transmission and vaccination作者： 苟清明 刘春花起止页码：56-61出版日期(期刊号)：2010年11月(1673-9868)出版单位：西南大学学报( 自然科学版) 外文翻译译文：摘要： 本文建立一个考虑了疾病的水平传播和垂直传播以及接种疫苗等因素的传染病模型，通过排除周期解、同宿轨和异宿环的存在来研究模型的全局稳定性， 最后证明系统的全局动力学特性完全由基本再生数所确定：当时，无病平衡点是全局渐近稳定的；当时，地方病平衡点是全局渐进稳定的.关键词：传染病模型；垂直传播；疫苗；全局稳定性中图分类号：o17513 文献标识码：a在许多传染病模型中，总是假设人口传染病是通过直接接触感染源或通过诸如蚊子等媒介叮咬，或通过水平传播.但是许多传染病不仅有水平传播还有垂直传.垂直传播也可通过媒介的胎盘转移完成，如乙肝，风疹，疱疹的病原体.对昆虫或植物而言，往往是通过垂直传播如卵或种子.busenberg等人讨论了疾病的水平传播和垂直传播问题.在本文中，我们假设疾病既有水平传播又有垂直传播.我们假定人口具有指数出生，人口被均匀分为四个仓室：易感染者(s)，潜伏者(e)，染病者(i)和恢复者(r). 因此总人口为.我们认为这种疾病不是致命的，人均自然出生率和死亡率分别记为参数b和d.我们假设，潜伏者的新生儿进入易感者类，而染病者的新生儿有q比例是感染者.因此进入潜伏者类的新生儿为bqi，.对于染病者类，我们假设比例的染病者具有永久免疫力，进入r类，r比例的染病者没有免疫力，进入s类，模型假设易感者类的接种比例为. 根据上述假设，得到如下微分方程 (1)这里是常规接触率，参数是从e类到i类的转换率.参数b，d，为是正数，r为非负数.设x=s/n; y=e/n; z=i/n和=r/n分别表示s，e，i，r在总人口中的比例.易证x，y，z，满足下列微分方程： (2) 受的限制，由于变量不出现在方程组（2）的前三式中，这使我们减少方程（2）得到一个子式. (3)在可行的区域内，我们从生物角度研究(3)式 (4)在v中(3)式的动态学行为和疾病传播是由如下基本再生数决定的 (5)本文的目的是要证明(3)的动力学行为由决定.1 数学框架我们简要概述一个一般的数学框架，证明了一个常微分方程系统的全局稳定性，这是在文献3中提到的.令是一个函数，x属于开集.让我们考虑如下微分 (6)我们记是式(6)中使得的一个解.如果每个对于和充分大的t，则(6)式中集合k收敛于d.我们提出两个基本假设：存在一个紧的吸引集合kd.在d中(6)具有唯一的平衡点.若是局部稳定且在d中所有的轨迹收敛到，则唯一的平衡点是全局稳定的.对于可行区域是有界圆锥体的传染病模型，是等价于(6)的一致持久性.对于xd，设是一个矩阵值函数为的.假设当xk，k为紧集时，存在，且为连续的.一个数量定义为 (7) (8)矩阵是通过p沿f方向的导数来代替p的每个元素得到的，、和是第二加性复合矩阵f的雅可比矩阵和及(b)是b的lozinskii测度，其向量范数为中的范数.文献3中定理3.5给出了如下全局稳定性结果.定理1 设d是单连通的，而且假设，成立.如果0的，则(6)的唯一的平衡点在d是全局稳定的.文献3证明了在定理1的条件下，条件0排除了(6)中有不变闭曲线的可能性，如周期解，同宿轨和异宿轨，因而它蕴含了x的局部稳定性.使用定理1来分析(3)的全局稳定性，设v，定义分别为(4)，(5). 易证v是系统(3)的正不变集.2 模型(3)的定性分析易证，如果是模型(3)在v中的唯一平衡点；如果v的内部存在唯一的地方病平衡点.定理2 如果，系统(3)的无病平衡点在v中是全局渐近稳定；如果则它是不稳定的，从足够靠近0e出发的轨线远离，从x轴出发的轨线沿x轴趋向于.证明 令 则如果， 而且，如果；否则，如果，则在v中y=z=0.因此集合中的最大紧不变集是单点集.当时，的全局稳定性由lasalles不变集原理得到. 如果，则除了y=z=0的情形，当x足够接近时候，.因此，从足够接近出发的轨线远离，从x轴出发的轨线满足系统(3)的方程，从而当t，.当时，v中系统(3)的全局动力学完全由定理2决定.其流行病学含义是，受感染人口在总人口中的比例(即潜伏者和染病者比例之和)随着时间而趋于零.引理1 如果，此时系统(3)在v中是一致持久的，也即存在一个常数0 1使得从出发的任意解都满足 证明 我们运用参考文献4中的定理4.5来证明此引理.为了证明当，系统(3)满足定理4.5的所有条件，我们选择v=x，v.那么是x中的一个不变集，令，由定理2知，包含的同宿轨道不存在，而且m是一个弱排斥子.因此m是的非循环的，孤立的，覆盖.因此，文献4 中定理4.5 的所有条件系统(3)都满足，因此引理得证.为了证明地方病平衡点*e的全局渐近稳定性，我们需要另一个引理.引理2 假设是(3)的解且.如果，则存在，当时，解满足.证明 由(3)的第一个方程知如果bqr，显然成立.如果bqr，令. 当时，易得，因此， ，从而当时，.因此，当t从分大时，有.从而引理结论得以证明.定理 3 假定，那么在v中，唯一的地方病平衡点是全局渐近稳定的.证明 通过第一节的讨论以及引理1，我们看到系统(3)满足假设和.系统(3)的通解的雅可比矩阵j为： 其第二复合加法矩阵为： (9)关于复合矩阵及其性质的详细讨论，请读者参考文献9.设(8)中的函数p(x)为，则 ，(8)中的矩阵，可以被写成矩阵块的形式 当， ，在中，向量范数选作 .让(.)表示该对应范数的lozinskii度量.利用文献11中估计(.)的方法，得 (10)其中表示在中的范数对应的的lozinskii度量.由于为标量，对于中的任意范数的lozinskii度量都是等于.，是对应于范数的矩阵范数.因此，这里由引理2知，当时，有，.因此，当时， (11) (12)重写(3)，我们有 (13) (14)将(13)代入到(11)，(14)代入(12)，我们可得，当时， 由此得 根据(3)式的是紧促吸收集.对于，有 从(7)可知，这就完成了证明定理3.3 人口数量的动态学行为现在考虑和构成的动力学行为，它们有系统(1)来控制.由于(1)的前三个方程中没有出现r (t)，因此我们研究其等价系统： (15) 显然，人口总量n(t)可能增加、减少或为常数，其完全依赖于增长率r=bd.比例(x ，y ，z)可能趋向于或地方病平衡点，但是感染者比例的变化并不能给提供我们关于染病者(包括e类和i类)行为变化的信息.特别地，即使被感染的个体的总数呈指数增加，但以比人口总量增长率低，那么这两者所占的比重将趋向于零，然而，被感染个体的总数趋于无穷.我们也能够想象出相反的情况感染者的数量和人口总量的下降到零，但是二者比例一直保持(非零)常量不变.在这种情况下，只要人口总量非零，就一定存在感染者.为了描述的变动情况，我们需要另外两个阈值参数(文献12中有介绍).以下是相关的阈值参数： 我们得到以下两个定理.定理4 (a) 易感者的数量以指数渐近率bd增加(减小).(b) 假设，如果或，则.证明 (a)由定理2知，意味着由(1)的第一个方程，我们有.由定理3知，当时，. 方程(1)的第一个方程除以s并取极限得： 由(3)的第一个方程可知，平衡点满足方程 因此 (b)通过e(t)，i(t)的方程来研究其行为： (16)这是线性系统的一个扰动.(16)的线性主部的解正如定理中所描述的那样，因为当t时，扰动部分以指数衰减(见参考文献10，第3章定理2 3)，则(16)的解与其线性系统的解行为相同.定理5 假定.（a） 如果数量下降(增加).而且指数渐进增长(下降率)为 (17)（b） 如果数量下降(增加).而且指数渐近增长(下降率)为 (18) 证明 由定理3知：如果，则且 (19)因此 (20)由(1)的第二个方程，方程(19)及方程(20)，可得(17).由(1)的第三个方程及方程(20)，也可得到(18).致谢：审稿人提出了非常有价值的意见和建议， 对此我们表示感谢.参考文献：1 苟清明.一类具有阶段结构和标准发生率的sis 模型j.西南大学学报：自然科学版，2007，29(9)：6-13.2 苟清明，王稳地.一类有迁移的传染病模型的稳定性j.西南师范大学学报：自然科学版，2006，31(1)：18-23.3 li m y， muldowney j s. a geometric approach to global stability problems j. siam j math anal， 1996，27(4)：1070-1083.4 thieme h r. persistence under relaxed point dissipativity (with an application to an endemic model)j.siam j math anal，1993，24(2)：407-435.5 driessche p van den，watmough j. reproduction numbers and sub threshold endemic equilibria for compartmental models of disease transmission j.j math biosci， 2002(180)：29-48.6 busenberg s， cooke k. vertical transmission diseaes，models and dynamics m. berlin： springer verlag，1993.7 li m y， smith h l， wang l. global stability of an seir model with vertical transmission j. siam j appl math， 2001，62(1)：58-69.8 li m y， wang l. a criterion for stability of matricesj.j math anal appl， 1998(225)： 249-264.9 muldowney j s. compound matrice and ordinary differential equationsj.rocky moun j math，1990，20(4)： 857-872.10hale j k. ordinary differential equationsm.new york：wiley-interscience， 1969：296 -297.11 martin r h j r. logarithmic norms and projections applied to linear differential system j. j math anal appl，1974(45)：432-454.12 thieme h r. epidemic and demographic interaction in the spread of potentially fataldiseases in a growing population. j. math biosci， 1992， 111(1)： 99-130.该英文原文由指导教师提供，选自：gou q m， liu c h. global stability of an seirs epidemic model with vertical transmis-sion and vaccination j. j southwest univ， 2010， 32(11)： 55-61.原文：global stability of an seirs epidemic model with vertical transmission and vaccinationabstract： in this paper， by ruling out the presence of periodic solutions， homoclinic orbits and heteroclinic cycles， we study the global stability of an seirs epidemic model which incorporates exponential growth， horizontal transmission， vertical transmission， standard incidence and vaccination. it is shown that the global dynamics are completely determined by the basic reproduction number . if ， the disease free equilibrium is globally asymptotically stable; whereas if ， the unique endemic equilibrium is globally asymptotically stable.key words： epidemic model; vertical transmission; vaccination; global stabilityclc number： o175 13 document code： ain many epidemic models， one assumes that infectious diseases transmit in a population through direct contact with infectious host s， or through disease vectors such as mosquitos or other biting sects， viz. horizontal transmission . but many infectious diseases spread through not only horizontal transmission but also vertical transmission. vertical transmission can be accomplished through transplacental transfer of disease aents， such as hepat it is b， rubella， herpes simplex . among insect s or plant s， vertical transmission is often through eggs or seeds. busenbergetal and lietal discussed the problem of diseases horizontal transmission and vertical transmission. in the present paper， the disease that transmits both horizontally and vertically are considered. we assume that the population which has a exponential birth can be divided into four homogeneneous compartment s： susceptible (s) ，exposed (e) ， infectious (i) and immune (r) . so the host total population . we assume that the disease is not fatal and that per capita nature birth rate and per capita death rate are denoted by parameters b and d respectively. we assume that new born infants from the exposed class enter the susceptible class， while a fraction q of new born infants from the infectious class is infected. consequently， the birth flux into the exposed class is given by bqi with . for the infectious class， we assume that a pro portion of infectious host s acquire permanent immunity and enter r class， while a proportion r o f infectious host s have no immunity and enter s class. our model includes vaccination at a rate for susceptible individuals. based on the above assumptions， the following differential equations are derived： (1) here， is the adequate contact rate， the parameteris the transfer rate from the e class to i class. the parameters b， d， ，are positive， ，r， rare nonnegative.let x = s / n， y = e / n ， z = i / n and w = r / n denote the fraction of the classes s ， e ， i ， r in the population， respectively. it is easy to verify that x， y， z， w satisfy the following differential equations： (2) subject to the restriction x + y + z + w = 1. because the variable w does not appear in the first three equations of ( 2) . this allow s us to reduce ( 2) to a subsystem： (3) from biological considerations， we study ( 3) in the feasible closed region (4)the dynamical behavior of ( 3) in v and the fate of the disease is determined by the basic reproduction number (5)the objective of this paper is to show that the dynamical behavior of ( 3) is characterized by .1 mathematical frameworkwe briefly outline a general mathematical framework to prove the global stability o fa system of ordinary differential equations， which is proposed in reference 3 .let be a function for x in an open set . let us considerthe system of differential equations ( 6)we denote by the solution to ( 6) such that . a set k is said to beabsorbing in d for (6)， if x ( t ， k 1 ) k for each compact k 1 d and sufficiently large t. we make two basic assumptions：( h1 ) there exists a compact absorbing set kd.( h2 ) ( 6) has a unique equilibrium in d. the unique equilibrium x is said to be globally stable in d if it is locally stable and all trajectories in d converge to x. for epidemic models w here the feasible region is abounded cone， ( h 1 ) is equivalent to the uniform persistence of ( 6).let be an matrix-valued function that is for xd assume that exists and is continuous for x|k， the compact set. a quantity is defined as (7) (8)the matrix pf is obtained by replacing each entry pij of p by its derivative in the direct ion of f， ( pij ) f ， andis the second additive compound matrix of the jacobian matrixof f ， and (b)is the lozinski measure of b with respect to a vector norm in . the following global stability result is proved in theorem 3.5 of reference 3 .theorem 1 assume that d is simply connected and that assumptions (h 1 ) ， (h 2 ) hold. then the unique equilibrium of ( 6) is globally stable in d if 0. it is show in reference 3 that under the assumptions of theorem 1， the condition 1， a unique endemic equilibrium v( the interior of v ) exists.theorem 2 the disease-free equilibrium of ( 3) is globally asymptotically stable in v if ; it is unstable if and the trajectories star ting sufficiently close to e0 leave e0 except those starting on the x axis which approach e0 along this axis. proof set then， if， furthermore，if ；whereas，if ，then y=z=0 in v. therefore， the largest compact invariant set in is the singleton.the global stability o f w hen时，follow s from the lasalles invariance principle 10 .if，then for x sufficiently close toexcept when y=z=0，therefore， trajectorie starting sufficiently close to leave a neighborhood of except those starting on the xaxis，on which model(3)reduces to，and thus， as t.theorem 2 completely determines the global dynamics of ( 3) in v for the case . its epidemiological implication is that the infected fraction (the sum of the latent and the infectious fraction) of the population vanishes over time.lemma 1 if system ( 3) is uniformly persistent in v in the sense that there exists a constant 0 0， such that the solution satisfies for .proof from the first equation of ( 3) ， we have if bqr ， there is nothing to prove. if bq t 0 ， therefore for .along each solution is the compact absorbing set， we have， for which in turn implies that from ( 7) ， completing the proof of theorem 3.3 the dynamics of the population sizewe now turn to the dynamics of and，which are governed by systems ( 1) . the fact that r( t ) does not appear in the first three equations in ( 1) allows us to study the equivalent system： (15) it is obvious that the total population size n (t ) may be increasing ， decreasing or constant ， depending on the growth rate r = b- d .the proportions (x，y，z) might tend to or the endemic equilibrium ， but the behavior of the proportions does not g iv e us much insight on the behavior of the total number of infected individuals ( comprise e class and i class) . in particular， even if the total number of infected individuals increases exponentially but at a lower rate than the total population size n ( t ) ， then the proportion of the two tends to zero， however， the total number of infected individuals approach infinity. we can also imagine the reversed situation both the number of infected individuals and the total population decline to zero but the proportion remains almost constant ( and non zero) at all times. in this case the disease remains in the population as long as the population exist s. to describe the behavior of s( t) ， e( t ) ， i ( t ) ， one needs two additional threshold parameters which are introduced in reference 12 . the following are the pertinent threshold parameters： we derived the following theorem.theorem 4 (a) the number of susceptible individuals s(t) increase (decrease) with exponential asymptotic rate b- d.(b) assume that then ifor .proof (a) from theorem 2， implies that ， from the first equation in (1) ， we have . in the case ， from theorem 3， ， we divide by s in the first equation in (1) and take the limit ： from the first equation in (3) it follows that the equilibrium satisfies thus (b) to see the behavior of e(t) and i (t) ， consider the equations for e and i (16)which is a perturbation of a linear system. the solutions to the principal part of (16) behave as claimed in the theorem， as do those for the perturbed system (16) since the perturbation decays exponentially as t( see reference 10 ，chapter 3，theorem 2.3) .ttheorem 5 assume that ， (a) the number of individuals e(t) decreases if r 2 1. moreover， the exponential asymptotic rate of