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第120行：
第120行： + + 第125行：
第127行： − 对于模型的完整说明，箭头应该标明舱室之间的转换率。在 s 和 i 之间，转移率假定为 d (S/N)/dt =-SI/N < sup > 2 </sup > ，其中 n 是总人口，是每人每时间的平均接触次数，乘以感染者和受感染者之间接触传播疾病的概率，SI/N < sup > 2 </sup > 是感染者和受感染者之间导致易感者受感染的接触易感个体的百分比。(这在数学上类似于化学中的质量作用定律，即分子之间的随机碰撞导致化学反应，分数率与两种反应物的浓度成正比)。+ 第133行：
第135行： − + 第141行：
第143行： − 对于无法从传染室中移除的特殊情况(= 0) ，SIR 模型退化为一个非常简单的 SI 模型，该模型具有逻辑解，其中每个人最终都会被感染。+ 第147行：
第149行： − + 第153行：
第155行： − 流行病的动态，例如流感，往往比出生和死亡的动态更快，因此，出生和死亡往往被简单的区域模型所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:+ 第161行：
第163行： − 《数学》+ 第197行：
第199行： − + 第205行：
第207行： − 其中，s </math > 是易感人群的存量，< math > i </math > 是感染人群的存量，< math > r </math > 是移除人群的存量(死亡或恢复) ，< math > n </math > 是这三者的总和。+ 第213行：
第215行： − 这个模型是由威廉 · 奥格威 · 科尔马克和安德森 · 格雷 · 麦克德里克首次提出的，作为我们现在所说的科尔马克-麦克德里克理论的一个特例，并继承了麦克德里克对罗纳德 · 罗斯所做的工作。+ − 第253行：
第254行： − 用数学术语来表达人口的稳定性。注意，上述关系意味着人们只需要研究三个变量中的两个的方程。+ 第261行：
第262行： − 其次，我们注意到传染类的动态取决于以下比例:+ 第277行：
第278行： − + 第301行：
第302行： − 其中，s (0) </math > 和 r (0) </math > 分别是易感人群和被移除人群的初始人数。+ 第307行：
第308行： − + 第313行：
第314行： − + 第319行：
第320行： − 敏感个体和被感染个体的比例分别为 r (infty)/n </math > + 第325行：
第326行： − 在这个极限里，一个人+ 第333行：
第334行： − + 第341行：
第342行： − + 第347行：
第348行： − 这个超越方程有一个关于 Lambert 函数的解决方案，即+ 第363行：
第364行： − 这表明，在流行病结束时，除非人群中的某些个体被移除，否则其中一些人仍然易受感染。这意味着传染病的结束是由于感染人数的减少，而不是由于绝对缺乏易感人群。+ 第371行：
第372行： − 基本传染数的作用和最初的易感性都极其重要。事实上，将传染性个体的等式重写如下:+ 第379行：
第380行： − + 第395行：
第396行： − + 第411行：
第412行： − 0，</math > frac { dI }{ dt }(0) > 0+ 第419行：
第420行： − 也就是说，随着传染病数量的增加(可以传染到相当一部分人口) ，将会有一场适当的流行病爆发。相反，如果+ 第427行：
第428行： − + 第443行：
第444行： − + 第451行：
第452行： − 也就是说，与易感人群的初始规模无关，这种疾病永远不会引起适当的流行病爆发。因此，很明显，基本传染数和最初的易感性都极其重要。+ 第457行：
第458行： − + 第471行：
第472行： − 数学 f = beta i,+ 第479行：
第480行： − 建立了从易感人群到感染人群的转变率模型，因此称之为感染力。然而，对于大类传染病来说，更现实的做法是考虑一种传染力，这种传染力并不取决于感染对象的绝对数量，而是取决于感染对象的比例(就总人口而言) :+ 第487行：
第488行： − + 第495行：
第496行： − 卡帕索和后来，其他作者提出了非线性的感染力量模型更现实的传染过程。+ + + 第505行：
第508行： − + 第513行：
第516行： − 数学 = s (0) u </math > + 第519行：
第522行： − 数学{ i }(u) = n-mathcal { r }(u)-mathcal { s }(u) </math > + 第525行：
第528行： − 数学{ r }(u) = r (0)-rho ln (u) </math > + 第541行：
第544行： − + 第549行：
第552行： − 有初始条件的+ 第557行：
第560行： − + 第565行：
第568行： − + 第581行：
第584行： − 《数学》+ 第587行：
第590行： − + 第623行：
第626行： − 数学+ + 第631行：
第635行： − 在这里，xi (t) </math > 可以解释为一个人在时间 < math > t </math > 之前收到的预期传输数量。这两个解是通过 < math > e ^ {-xi (t)} = u </math > 关联的。+ 第647行：
第651行： − + 第653行：
第657行： − + 第659行：
第663行： − 考虑一个人口拥有属性的死亡率和出生率，以及传染病正在传播的地方。具有质量作用传递的模型是:+ 第667行：
第671行： − 《数学》+ 第673行：
第677行： − + + + + − { dS }{ dt } & = Lambda-mu s-frac { beta i s }{ n }[8 pt ] 第685行：
第691行： − Frac { dI }{ dt } & = frac { beta i s }{ n }-gamma i-mu i [8 pt ]+ + + + + + + 第703行：
第715行： − 数学+ 第711行：
第723行： − 其无病平衡状态为:+ + 第719行：
第732行： − 左(s (t) ，i (t) ，r (t)右) = 左(frac { Lambda }{ mu } ，0,0右)+ + 第727行：
第741行： − 在这种情况下，我们可以得出一个基本传染数:+ + 第735行：
第750行： − + + 第743行：
第759行： − 具有临界性质。事实上，独立于具有生物学意义的初始值，我们可以证明:+ 第751行：
第767行： − + 第757行：
第773行： − + 第765行：
第781行： − EE 点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，人们可以表明 < math > r {0} </math > 可以理解为在一个完全易感人群中，由一个感染对象引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。+ 第774行：
第790行： + + 第785行：
第803行： − 有些传染病，例如普通感冒和流感，并不能提供任何持久的免疫力。这种感染在感染康复后不会产生免疫力，个人再次易受感染。+ 第793行：
第811行： − SIS 分室模型+ 第809行：
第827行： − 《数学》+ 第815行：
第833行： − + 第821行：
第839行： − Frac { dS }{ dt } & =-frac { beta s i }{ n } + gamma i [6 pt ]+ 第827行：
第845行： − 和 = frac { beta s i }{ n }-gamma i+ 第833行：
第851行： − + 第839行：
第857行： − 数学+ 第847行：
第865行： − 注意，用 n 表示它所拥有的总人口:+ 第871行：
第889行： − + − 第879行：
第896行： − 也就是。传染病的动态性是由 Logistic函数控制的，所以对于所有的 i (0) > 0 </math > :+ 第887行：
第904行： − 《数学》+ 第893行：
第910行： − + 第899行：
第916行： − + 第905行：
第922行： − + − − + 第917行：
第933行： − 数学+ 第925行：
第941行： − + 第933行：
第949行： − [ math > i (t) = frac { i _ infty }{1 + v e ^ {-chi t }} </math > .+ − 第941行：
第956行： − + 第949行：
第964行： − + 第957行：
第972行： − 《数学》+ 第963行：
第978行： − Frac { dI }{ dt } propto i cdot (N-I).+ 第969行：
第984行： − 数学+ 第982行：
第997行： + + 第987行：
第1,004行： − 初始值 < math > s (0) = 997，i (0) = 3，r (0) = 0 </math > ，感染率 < math > beta = 0.4 </math > ，恢复率 < math > gamma = 0.035 </math > ，死亡率 < math > mu = 0.005 </math > + 第993行：
第1,010行： − 动画的 SIRD 模型的初始值 < math display ="inline"> s (0) = 997，i (0) = 3，r (0) = 0 </math > ，初始感染率 < math display ="inline"> beta = 0.5 </math > 和恒定恢复率 < math display =""> = 0.035 </math > 和死亡率 </math display ="inline"> mu = 0.005 </math > 。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 < math display ="inline"> beta = 0.5 </math > 到 < math display ="inline"> beta = 0.12 </math >)。+ 第1,001行：
第1,018行： − + 第1,007行：
第1,024行： − 《数学》+ 第1,013行：
第1,030行： − + 第1,019行：
第1,036行： − + 第1,043行：
第1,060行： − + 第1,049行：
第1,066行： − 数学+ 第1,055行：
第1,072行： − + + + 第1,065行：
第1,084行： − + 第1,081行：
第1,100行： − 《数学》+ 第1,087行：
第1,106行： − + 第1,117行：
第1,136行： − + 第1,123行：
第1,142行： − 数学+ 第1,129行：
第1,148行： − + 第1,135行：
第1,154行： − + 第1,149行：
第1,168行： − + 第1,179行：
第1,198行： − 《数学》+ 第1,185行：
第1,204行： − + 第1,221行：
第1,240行： − 数学+ 第1,229行：
第1,248行： − + 第1,237行：
第1,256行： − 对于这种模式，基本传染数是:+ 第1,245行：
第1,264行： − + − 第1,253行：
第1,271行： − + 第1,261行：
第1,279行： − + − 第1,283行：
第1,300行： − 左(s (t) ，e (t) ，i (t) ，r (t)右) = EE。数学+ 第1,299行：
第1,316行： − 《数学》+ 第1,329行：
第1,346行： − 数学+ 第1,337行：
第1,354行： − + 第1,343行：
第1,360行： + 第1,355行：
第1,373行： − [数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学]+ − 第1,371行：
第1,388行： − 《数学》+ + 第1,407行：
第1,425行： − 数学+ + + 第1,425行：
第1,445行： − 数学，数学，数学，数学，数学，数学，数学，数学+ 第1,431行：
第1,451行： − 《数学》+ 第1,479行：
第1,499行： − 数学 + + 第1,497行：
第1,518行： − 数学是数学，数学是数学，数学是数学，数学是数学+ + + 第1,507行：
第1,530行： − 众所周知，患病的概率在时间上并不是一成不变的。随着大流行的进展，对大流行的反应可能会改变在较简单模型中假定为恒定的接触率。面罩、社会疏远和封锁等应对措施将改变接触率，从而降低大流行的速度。+ 第1,523行：
第1,546行： − + 第1,531行：
第1,554行： − 周期 t 等于一年。+ 第1,577行：
第1,600行： − 数学+ 第1,616行：
第1,639行： + + 第1,621行：
第1,646行： − + 第1,627行：
第1,652行： − + + + 第1,645行：
第1,672行： − 《数学》+ 第1,669行：
第1,696行： − 文章首先介绍了一种新的计算机辅助计算机辅助计算机辅助计算机辅助计算机辅助计算机辅助计算机辅助计算机辅助计算的方法+ 第1,681行：
第1,708行： − 数学+ 第1,705行：
第1,732行： − 因此，我们将处理《数学》和《数学》的长期行为，它认为:+ 第1,713行：
第1,740行： − 左(s (t) ，i (t)右) = DFE = 左(n 左(1-P 右) ，0右)+ 第1,746行：
第1,773行： + + 第1,780行：
第1,809行： + + 第1,793行：
第1,824行： − 《数学》+ 第1,823行：
第1,854行： − 数学+ 第1,844行：
第1,875行： + + 第1,857行：
第1,890行： − 《数学》+ 第1,887行：
第1,920行： − 数学+ 第1,993行：
第2,026行： − 在哪里:+ 第2,127行：
第2,160行： + + − === Vertical transmission ===+ 第2,140行：
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第2,234行： + 第2,208行：
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第2,257行： − + 第2,224行：
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无编辑摘要
===Transition rates===
===Transition rates===
传染率
For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N<sup>2</sup>'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N<sup>2</sup>'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).
For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between ''S'' and ''I'', the transition rate is assumed to be ''d(S/N)/dt = -βSI/N<sup>2</sup>'', where ''N'' is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and ''SI/N<sup>2</sup>'' is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the [[law of mass action]] in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).
For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N<sup>2</sup>, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N<sup>2</sup> is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).
For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N<sup>2</sup>, where N is the total population, β is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N<sup>2</sup> is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).
为了完整说明这个模型，箭头应该标明类别之间的传染率。在S和I之间，传染率假定为d (S/N)/dt =-SI/N < sup > 2 </sup > ，其中 ''N'' 是总人口，β是平均每人每次接触的人数，乘以易感者和感病者之间接触传播疾病的概率，SI/N < sup > 2 </sup > 是易感个体和感病个体之间接触之后导致易感个体感染的百分比。(这与化学中的质量作用定律在数学计算上类似，即分子之间的随机碰撞导致化学反应，反应速率与两种反应物的浓度成正比)。
Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).
Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g the Erlang distribution).
在 i 和 r 之间，转化率假定与感染个体的数目成正比，即 i。这相当于假设一个感染性个体在任何时间间隔内恢复的概率仅为 dt。如果一个人在一个平均时间段内具有传染性，那么 = 1/D。这也等价于假设一个人在感染状态下的时间长度是一个随机变量和一个指数分布。“经典的” SIR 模型可以通过使用更加复杂和现实的分布来修正 I-R 转变速率(例如爱尔朗分布)。
在 I 和 R 之间，假设传染率与感病者的数目成正比，即γI。这相当于假设一个感病者在任何时间间隔内恢复的概率仅为γdt。如果平均每个人在时间段D内具有传染性，那么γ= 1/D。这也相当于假设一个人在感染状态下的时间长度是一个服从指数分布的随机变量。“经典的” SIR 模型可以通过更加复杂和现实的分布来修正I-R 传染率(例如爱尔朗分布)。
For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.
For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.
对于感病者全都没有康复的特殊情况（γ=0） ，SIR 模型就简化为一个非常简单的 SI 模型，该模型具有一个逻辑解，即其中每个人最终都会被感染。
===The SIR model without vital dynamics===
===The SIR model without vital dynamics===
缺少关键动态的SIR模型
The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>
The dynamics of an epidemic, for example, the [[Influenza|flu]], are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary [[differential equations]]:<ref name="Hethcote2000">{{cite journal |author=Hethcote H |title=The Mathematics of Infectious Diseases |journal=SIAM Review |volume=42 |issue= 4|pages=599–653 |year=2000 |doi=10.1137/s0036144500371907|bibcode=2000SIAMR..42..599H }}</ref><ref name="Beckley"/>
The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:
The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dynamics (birth and death, sometimes called demography) described above can be expressed by the following set of ordinary differential equations:
流行病的动态变化，例如流感的动态变化，往往比出生和死亡的动态变化更快，因此，出生和死亡往往被简单的传染病模型所忽略。没有上述所谓的生命动力学(出生和死亡，有时称为人口统计学)的 SIR 系统可以用下列一组常微分方程表示:
<math>
<math>
<数学>
\begin{align}
\begin{align}
</math>
</math>
数学
</数学>
where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.
where <math>S</math> is the stock of susceptible population, <math>I</math> is the stock of infected, <math>R</math> is the stock of removed population (either by death or recovery), and <math>N</math> is the sum of these three.
其中，S</math> 是易感人群的存量，<math>I</math> 是感染人群的存量，<math>R</math> 是康复人群的存量(死亡或康复) ，<math>N</math> 是这三者的总和。
This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.
This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.
这个模型是一个特例，是在继承了麦克德里克对罗纳德·罗斯所做研究的基础上，由威廉·奥格威·科尔马克和安德森·格雷·麦克德里克首次提出，现在称作科尔马克-麦克德里克理论。
expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.
expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.
用数学术语来表达人口 <math> N </math>的稳定性。注意，上述关系意味着人们只需要研究三个变量中的两个的方程。
Secondly, we note that the dynamics of the infectious class depends on the following ratio:
Secondly, we note that the dynamics of the infectious class depends on the following ratio:
其次，我们注意到传染种类的动态取决于以下比例:
the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>
the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is <math>T_{c} = \beta^{-1}</math>, and the typical time until removal is <math>T_{r} = \gamma^{-1}</math>. From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: <math>T_{r}/T_{c}.</math>
所谓的基本传染数(亦称基本再生产率)。这个比率是根据所有受试者都易受感染的人群中单一感染的预期新感染人数(这些新感染有时称为二次感染)计算出来的。如果我们说两次联系之间的典型时间是 < math > t _ { c } = beta ^ {-1} </math > ，而移除之前的典型时间是 < math > t _ { r } = gamma ^ {-1} </math > ，那么这个想法可能更容易被看出来。由此可以得出，平均而言，在感染者被移除之前，感染者与其他人的接触次数为: < math > t _ { r }/t _ { c }。数学
所谓的基本传染数(亦称基本再感染率)。这个比率是根据一群易感者中单次感染后的预期的新感染人数(这些新感染有时称为二次感染)计算出来的。如果我们说两次感染之间的典型时间是 <math>T_{c}=\beta^{-1}</math>，而康复之前的典型时间是 <math>T_{r}=|\gamma^{-1}</math> ，那么这个想法可能更容易被看出来。由此可以得出，平均而言，在感染者康复之前，感染者与其他人的接触次数为: <math>T_{r}/T_{c}。</数学>
where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.
where <math>S(0)</math> and <math>R(0)</math> are the initial numbers of, respectively, susceptible and removed subjects.
其中，<math>S(0)</math>和<math>R(0)</math>分别是易感者和康复者的初始人数。
Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and
Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and
Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and
Writing <math>s_0 = S(0) / N</math> for the initial proportion of susceptible individuals, and
为易感人群的初始比例编写 s _ 0 = s (0)/n </math > ，以及
为易感人群的初始比例编写<math>S_0=S(0)/N</math>，以及
<math>s_\infty = S(\infty) / N</math> and
<math>s_\infty = S(\infty) / N</math> and
<math>s_\infty = S(\infty) / N</math> and
<math>s_\infty = S(\infty) / N</math> and
[数学] s infty = s (infty)/n
[math] s infty = S(\infty)/N</math>并且
<math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively
<math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively
<math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively
<math>r_\infty = R(\infty) / N</math> for the proportion of susceptible and removed individuals respectively
敏感个体和被感染个体的比例分别为R(\infty)/N</math>
in the limit <math>t \to \infty,</math> one has
in the limit <math>t \to \infty,</math> one has
in the limit <math>t \to \infty,</math> one has
in the limit <math>t \to \infty,</math> one has
在这个极限里，<math>t \to \infty,</math> 一个人又
<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>
<math>s_\infty = 1 - r_\infty = s_0 e^{-R_0(r_\infty - r_0)}</math>
< math > s _ infty = 1-r _ infty = s _ 0 e ^ {-r _ 0(r _ infty-r _ 0)} </math >
< math > s_\infty = 1-r _\infty = s_0 e ^{-R_0(r _\infty -r _0)}</math >
(note that the infectious compartment empties in this limit).
(note that the infectious compartment empties in this limit).
(请注意，在这个限度内，传染性腔室是空的)。
(请注意，在这个限度内，没有感病者)。
This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely
This [[transcendental equation]] has a solution in terms of the [[Lambert W function|Lambert {{mvar|W}} function]],<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref> namely
This transcendental equation has a solution in terms of the Lambert function, namely
This transcendental equation has a solution in terms of the Lambert function, namely
这个超越方程有一个关于Lambert函数的解决方案，即
This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.
This shows that at the end of an epidemic, unless <math>s_0=0</math>, not all individuals of the population have been removed, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infectious individuals rather than an absolute lack of susceptible subjects.
这表明，在流行病结束时，除非<math>s_0=0</math>，否则只要有人没有康复，一些人仍然易受感染。这意味着传染病的结束是由于感病者的减少，而不是由于完全没有易感者。
The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:
The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows:
基本再感染数的作用和最初的易感性都极其重要。事实上，将传染性个体的等式重写如下:
<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>
<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} - 1\right) \gamma I,</math>
< math > frac { dI }{ dt } = left (r _ 0 frac { s }{ n }-1 right) gamma i，</math >
<math> \frac{dI}{dt} = \left(R_0 \frac{S}{N} -1 \right) \gamma I,</math>
<math> R_{0} \cdot S(0) > N,</math>
<math> R_{0} \cdot S(0) > N,</math>
< math > r _ {0} cdot s (0) > n，</math >
<math> R_{0} \cdot S(0) > N,</math>
<math> \frac{dI}{dt}(0) >0 ,</math>
<math> \frac{dI}{dt}(0) >0 ,</math>
</math> \frac{dI}{dt}(0) > ,</math>
i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if
i.e., there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if
也就是说，随着感病者数量的增加(达到人口相当大的一个比例) ，将会有一场适当的流行病爆发。相反，如果
<math> R_{0} \cdot S(0) < N,</math>
<math> R_{0} \cdot S(0) < N,</math>
< math > r _ {0} cdot s (0) < n，</math >
<math> R_{0} \cdot S(0) < N,</math>
<math> \frac{dI}{dt}(0) <0 ,</math>
<math> \frac{dI}{dt}(0) <0 ,</math>
(0) < 0，</math >
<math> \frac{dI}{dt}(0) <0 ,</math>
i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.
i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.
也就是说，如果与易感者的初始规模无关，这种疾病将永远不会引起适当的流行病爆发。因此，很明显，基本再传染数和最初的易感性都极其重要。
====The force of infection====
====The force of infection====
感染力
Note that in the above model the function:
Note that in the above model the function:
<math> F = \beta I,</math>
<math> F = \beta I,</math>
<math> F = \beta I,</math>
models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):
models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):
建立了从易感者到感病者的传染率模型，因此称之为感染力。然而，对于大类传染病来说，更现实的做法是考虑一种感染力，这种传染力并不取决于感染对象的绝对数量，而是取决于感染对象的比例(就总人口而言<math>N</math>) :
<math> F = \beta \frac{I}{N} .</math>
<math> F = \beta \frac{I}{N} .</math>
{ n }
<math> F = \beta \frac{I}{N} .</math>
Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.
Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.
卡帕索和后来的其他作者提出了非线性的感染力以建立更现实的传染过程。
====Exact analytical solutions to the SIR model====
====Exact analytical solutions to the SIR model====
SIR 模型的精确解析解
In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization
In 2014, Harko and coauthors derived an exact analytical solution to the SIR model.<ref name="Harko" /> In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization
In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization
In 2014, Harko and coauthors derived an exact analytical solution to the SIR model. In the case without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization
2014年，Harko 和合作者推导出了 SIR 模型的精确解析解。在没有重要动力学设置的情况下，对于 < math > mathcal { s }(u) = s (t) </math > 等，它对应于以下时间参数化
2014年，Harko 和合作者推导出了 SIR 模型的精确解析解。在没有重要动力学设置的情况下，对于 <math>\mathcal{S}(u) =S(t)</math> 等，它对应以下参数化时间
<math>\mathcal{S}(u)= S(0)u </math>
<math>\mathcal{S}(u)= S(0)u </math>
<math>\mathcal{S}(u)= S(0)u </math>
:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>
:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>
<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>
<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>
<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>
:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>
:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>
<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>
<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>
<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>
<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>
<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>
< math > t = frac { n }{ beta } int _ u ^ 1 frac { du ^ * }{ i }(u ^ *)} ，quad rho = frac { gamma n }{ beta } ，</math >
<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>
with initial conditions
with initial conditions
和初始条件
<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>
<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>
< math > (数学{ s }(1) ，数学{ i }(1) ，数学{ r }(1)) = (s (0) ，n-r (0)-s (0) ，r (0)) ，四边形 u _ t < u < 1，</math >
<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad u_T<u<1,</math>
where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.
where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.
其中 < math > u _ t </math > 满足 < math > mathcal { i }(u _ t) = 0 </math > 。根据上面的超越方程，如果 < math > s (0) neq </math > </math > </math > </math > > u _ t = e ^ {-(r _ { infty }-r (0))/rho }(= s _ { infty }/s (0) </math > 如果 < math > s (0) neq 0) </math > 和 < math > i _ infty = 0 </math > 。
其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的超越方程 <math>R_{\infty}</math> ，如果<math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>，那么遵守<math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>。
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
</math>
</math>
</math>
Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.
Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>. The two solutions are related by <math>e^{-\xi(t)} = u</math>.
在这里，<math>\xi(t)</math>可以解释为随着时间<math>t</math>变化一个人预期收到的传染数量。这两个解是通过<math>e^{-\xi(t)} = u</math>关联的。
These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.
These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.
注意到原微分方程右边的所有项都与“数学”成正比，这些解就很容易理解了。这样，方程组就可以通过 < math > i </math > 来分解，时间重新调整，使得左边的微分算子变成 < math > d/d tau </math > ，其中 < math > d tau = i dt </math > ，即。< math > tau = int i dt.这些微分方程现在都是线性的，而第三个方程，即形式为 < math > dR/d tau = </math > 的常量，表明 < math > tau </math > 和 < math > r </math > (和 < math > xi </math > > > > > > > > > > >)仅仅是线性关系。
注意到原微分方程右边的所有项都与“数学”成正比，这些解就很容易理解了。这样，方程组就可以通过<math>I</math>来分解，时间重新调整，使得左边的微分算子变成<math>d/d\tau</math>，其中<math>d\tau=I dt</math>，即 <math>\tau=\int I dt</math>。这些微分方程现在都是线性的，而第三个方程，即形式为<math>dR/d\tau =</math> 的常量，表明 <math>\tau</math>和<math>R</math>(和上方的<math>\xi</math>)仅仅是线性关系。
===The SIR model with vital dynamics and constant population===
===The SIR model with vital dynamics and constant population===
具有关键动态和稳定人口的SIR模型
Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:
Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading<ref name="Beckley"/>. The model with mass-action transmission is:
Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:
Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\Lambda</math>, and where a communicable disease is spreading. The model with mass-action transmission is:
考虑人口有死亡率<math>\mu</math>和出生率<math>\Lambda</math>的特点，以及正在传播的传染病。具有质量作用传递的模型是:
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dI}{dt} & = \frac{\beta I S}{N} - \gamma I -\mu I \\[8pt]
\frac{dR}{dt} & = \gamma I - \mu R
\frac{dR}{dt} & = \gamma I - \mu R
\frac{dR}{dt} & = \gamma I - \mu R
\frac{dR}{dt} & = \gamma I - \mu R
\frac{dR}{dt} & = \gamma I - \mu R
\frac{dR}{dt} & = \gamma I - \mu R
\end{align}
\end{align}
\end{align}
\end{align}
结束{ align }
结束{ align }
</math>
</math>
</math>
</math>
</math>
for which the disease-free equilibrium (DFE) is:
for which the disease-free equilibrium (DFE) is:
其无病平衡点为:
<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>
<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>
<math>\left(S(t),I(t),R(t)\right) =\left(\frac{\Lambda}{\mu},0,0\right).</math>
In this case, we can derive a basic reproduction number:
In this case, we can derive a basic reproduction number:
在这种情况下，我们可以得出一个基本再生数:
<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>
<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>
< math > r _ 0 = frac { beta Lambda }{ mu (mu + gamma)} ，</math >
<math> R_0 = \frac{ \beta\Lambda }{\mu(\mu+\gamma)}, </math>
which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:
which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:
这种基本在生数具有临界性质。事实上，我们可以独立于具有生物学意义的初始值证明:
<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>
<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>
(s (t) ，i (t) ，r (t)) = textrm { DFE } = 左(frac { Lambda }{ mu } ，0,0右) </math >
<math> R_0 \le 1 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{DFE} = \left(\frac{\Lambda}{\mu},0,0\right) </math>
:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>
:<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>
<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>
<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>
< math > r _ 0 > 1，i (0) > 0 right tarrow lim _ { t to infty }(s (t) ，i (t) ，r (t)) = textrm { EE } = 左(frac { gamma + mu }{ beta } ，frac { mu }{ beta }左(r _ 0-1右) ，frac { beta }左(r _ 0-1右))。数学
<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to \infty} (S(t),I(t),R(t)) = \textrm{EE} = \left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_0-1\right), \frac{\gamma}{\beta} \left(R_0-1\right)\right). </math>
The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.
The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.
EE点被称为地方病平衡点(这种疾病还没有完全根除，仍然存在于人群中)。通过启发式的论证，人们可以表明<math>R_{0}</math>可以理解为在一个完全易感人群中，由一个感染对象引起的平均感染人数，上述关系在生物学上意味着，如果这个数字小于或等于1，这种疾病就会灭绝，而如果这个数字大于1，这种疾病就会在人群中永久地流行下去。
===The SIS model===
===The SIS model===
SIS模型
[[File:SIS System Graph.svg|thumb|Yellow=Susceptible, Maroon=Infected]]
[[File:SIS System Graph.svg|thumb|Yellow=Susceptible, Maroon=Infected]]
Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.
Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.
有些传染病，例如普通感冒和流感，并不能提供任何持久的免疫力。这种感染在感染康复后不会产生免疫力，个体易再次感染。
SIS compartmental model
SIS compartmental model
SIS传染病模型
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dS}{dt} & = - \frac{\beta S I}{N} + \gamma I \\[6pt]
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\frac{dI}{dt} & = \frac{\beta S I}{N} - \gamma I
\end{align}
\end{align}
\end{align}
\end{align}
结束{ align }
\结束{ align }
</math>
</math>
</math>
</math>
</math>
Note that denoting with N the total population it holds that:
Note that denoting with N the total population it holds that:
注意，用N表示它所拥有的总人口:
<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,
<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,
(beta-gamma) i-frac { beta }{ n } i ^ 2 </math > ,
<math> \frac{dI}{dt} = (\beta - \gamma) I - \frac{\beta}{N} I^2 </math>,
i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:
i.e. the dynamics of infectious is ruled by a logistic function, so that <math>\forall I(0) > 0</math>:
也就是。传染病的动态性是由Logistic函数控制的，所以对于所有的<math>\forall I(0) > 0</math>:
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& frac { beta }{ gamma } le 1 right tarrow lim _ { t to + infty } i (t) = 0，[6 pt ]
& \frac{\beta}{\gamma} \le 1 \Rightarrow \lim_{t \to +\infty}I(t)=0, \\[6pt]
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
1 right tarrow lim _ { t to + infty } i (t) = left (1-frac { gamma }{ beta } right).
& \frac{\beta}{\gamma} > 1 \Rightarrow \lim_{t \to +\infty}I(t) = \left(1 - \frac{\gamma}{\beta} \right) N.
\end{align}
\end{align}
\end{align}
\end{align}
结束{ align }
\结束{ align }
</math>
</math>
</math>
</math>
</math>
It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as
It is possible to find an analytical solution to this model (by making a transformation of variables: <math>I = y^{-1}</math> and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as
这个模型可以找到一个解析解(通过对变量进行变换: < math > i = y ^ {-1} </math > 并将其代入平均场方程) ，使基本再生产率大于单位数。给出了解决方案如下:
这个模型可以找到一个解析解(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入平均场方程) ，使基本再生率大于单位数。给出了解决方案如下:
<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.
<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.
<math>I(t) = \frac{I_\infty}{1+V e^{-\chi t}}</math>.
where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.
where <math>I_\infty = (1 -\gamma/\beta)N</math> is the endemic infectious population, <math>\chi = \beta-\gamma</math>, and <math>V = I_\infty/I_0 - 1</math>. As the system is assumed to be closed, the susceptible population is then <math>S(t) = N - I(t)</math>.
其中 i _ infty = (1-gamma/beta) n </math > 是地方性传染病人口，< math > chi = beta-gamma </math > ，和 < math > v = i _ infty/I _ 0-1 </math > 。假设系统是封闭的，那么易感人群是 < math > s (t) = n-i (t) </math > 。
其中 <math>I_\infty = (1 -\gamma/\beta)N</math>是地方性传染病人口，<math>\chi = \beta-\gamma</math>，和 <math>V = I_\infty/I_0 - 1</math>。假设系统是封闭的，那么易感者是<math>S(t) = N - I(t)</math>。
As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:
As a special case, one obtains the usual logistic function by assuming <math>\gamma=0</math>. This can be also considered in the SIR model with <math>R=0</math>, i.e. no removal will take place. That is the SI model. The differential equation system using <math>S=N-I</math> thus reduces to:
作为一个特例，人们通过假设 < math > gamma = 0 </math > 得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型具有 < math > r = 0 </math > ，即。不会进行移除。这就是 SI 模型。微分方程系统使用 < math > s = N-I </math > 因此可以简化为:
作为一个特例，人们通过假设<math>R=0</math>得到通常的 Logistic函数。这也可以在 SIR 模型中考虑，该模型具有 < math > r = 0 </math > ，即没有康复者。这就是 SI 模型。微分方程系统使用<math>S=N-I</math>因此可以简化为:
<math>
<math>
<math>
\frac{dI}{dt} \propto I\cdot (N-I).
\frac{dI}{dt} \propto I\cdot (N-I).
\frac{dI}{dt} \propto I\cdot (N-I).
\frac{dI}{dt} \propto I\cdot (N-I).
\frac{dI}{dt} \propto I\cdot (N-I).
</math>
</math>
</math>
</math>
</math>
===The SIRD model===
===The SIRD model===
SIRD模型
[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]
[[File:SIRD.svg|thumb|Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>]]
Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>
Diagram of the SIRD model with initial values <math>S(0)=997, I(0)=3, R(0)=0</math> and the rates for infection <math>\beta=0.4</math>, recovery <math>\gamma=0.035</math> and mortality <math>\mu=0.005</math>
SIRD模型示意图，初始值<math>S(0)=997，I(0)=3, R(0)=0</math>，感染率<math>\beta=0.4</math>，康复率<math>\gamma=0.035</math>，死亡率 <math>\mu=0.005</math>
[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]
[[File:SIRD model anim.gif|thumb|Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).]]
Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).
Animation of the SIRD model with initial values <math display="inline">S(0)=997, I(0)=3, R(0)=0</math>, initial rate for infection <math display="inline">\beta=0.5</math> and constant rates for recovery <math display="inline">\gamma=0.035</math> and mortality <math display="inline">\mu=0.005</math>. If there is neither medicine nor vaccination available, it is only possible to reduce the infection rate (often referred to as „flattening the curve“) by appropriate measures (e. g. „social distancing“). This animation shows the impact of reducing the infection rate by 76 % (from <math display="inline">\beta=0.5</math> down to <math display="inline">\beta=0.12</math>).
SIRD模型动画，初始值<math display="inline">S(0)=997I(0)=3, R(0)=0</math>，初始感染率 <math display="inline">\beta=0.5</math>和恒定康复率 <math display="inline">\gamma=0.035</math> 和死亡率<math display="inline">\mu=0.005</math>。如果既没有药物也没有疫苗可用，只有通过适当的措施(例如“社会距离”)才有可能降低感染率(通常称为“平缓曲线”)。这个动画展示了降低感染率76% 的效果(从 <math display="inline">\beta=0.5</math>下降到<math display="inline">\beta=0.12</math>)。
The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:
The Susceptible-Infectious-Recovered-Deceased-Model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations:
易感-传染-恢复-死亡-模型区分恢复(特别是指个人已经生存下来的疾病和现在的免疫)和死亡。该模型使用了下列微分方程组:
易感-感病-康复-死亡-模型区分恢复(特别是指从疾病中存活下来并且现在免疫)和死亡。该模型使用了下列微分方程组:
:<math>
:<math>
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& frac { dS }{ dt } =-frac { beta i s }{ n } ，[6 pt ]
& \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
& \frac{dI}{dt} = \frac{\beta I S}{N} - \gamma I - \mu I, \\[6pt]
\end{align}
\end{align}
结束{ align }
\结束{ align }
</math>
</math>
</math>
</math>
</math>
where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>
where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.<ref>The first and second differential equations are transformed and brought to the same form as for the ''SIR model'' above.</ref>
where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.
where <math>\beta, \gamma, \mu</math> are the rates of infection, recovery, and mortality, respectively.
这里 < math > beta，gamma，mu </math > 分别是感染率，恢复率和死亡率。
这里<math>\beta, \gamma, \mu</math>分别是感染率，康复率和死亡率。
===The MSIR model===
===The MSIR model===
MSIR模型
For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.
For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This is called [[passive immunity]]. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.
For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.
For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.
对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，但由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开头加入一个 m 类(用于母系免疫)来显示。
对于包括麻疹在内的许多传染病，婴儿出生时并不易感染，但由于母体抗体的保护(通过胎盘和初乳传播) ，婴儿在出生后的头几个月对该疾病免疫。这叫做被动免疫。这个额外的细节可以通过在模型的开头加入一个M类(用于母系免疫)来显示。
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\frac{dM}{dT} & = \Lambda - \delta M - \mu M\\[8pt]
\end{align}
\end{align}
结束{ align }
\结束{ align }
</math>
</math>
</math>
</math>
</math>
===Carrier state===
===Carrier state===
载体状态
Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.
Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.
Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.
Some people who have had an infectious disease such as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.
一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。然后他们可能回到传染室并出现症状(如肺结核) ，或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是玛丽 · 马伦，她将伤寒传染给了22个人。载体舱被标记为 c。
一些患有肺结核等传染病的人永远不会完全康复，而是继续携带这种传染病，同时他们自己也不会患上这种疾病。然后他们可能回到传染室并出现症状(如肺结核) ，或者他们可能继续以携带者的状态传染给其他人，而不出现症状。最著名的例子可能是玛丽·马伦，她将伤寒传染给了22个人。载体舱被标记为C。
===The SEIR model===
===The SEIR model===
SEIR模型
For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).
For many important infections, there is a significant incubation period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed).
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
\frac{dS}{dt} & = \Lambda - \mu S - \frac{\beta I S}{N} \\[8pt]
</math>
</math>
</math>
We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.
We have <math>S+E+I+R=N,</math> but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general <math>N</math> is a variable.
我们有 s + e + i + r = n，</math > 但是这只是常数，因为(退化的)假设出生率和死亡率是相等的; 一般来说 < math > n </math > 是一个变量。
我们有<math>S+E+I+R=N,</math>但是这只是常数，因为(退化的)假设出生率和死亡率是相等的; 一般来说<math>N</math>是一个变量。
For this model, the basic reproduction number is:
For this model, the basic reproduction number is:
对于这种模式，基本再生数是:
<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>
<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>
< math > r0 = frac { a }{ mu + a } frac { beta }{ mu + gamma } . </math >
<math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.</math>
Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions
Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions
类似于 SIR 模型，在这种情况下，我们有一个无病平衡(n，0,0,0)和一个地方病平衡 EE，我们可以证明，独立于生物学上有意义的初始条件
类似于 SIR 模型，在这种情况下，我们有一个无病平衡(N,0,0,0)和一个地方病平衡 EE，我们可以证明，独立于生物学上有意义的初始条件
<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>
<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>
左(s (0) ，e (0) ，i (0) ，r (0)右)在[0，n ] ^ 4: sge0，ege0，i ge0，rge0，s + e + i + r = n 右} </math >
<math> \左(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \右\} </math>
<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>
<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>
<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \to +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>
<math>
<math>
<math>
\begin{align}
\begin{align}
</math>
</math>
</math>
is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).
is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).
是稳定的(即。它在复平面的单位圆内有它的 Floquet 特征值)。
是稳定的(它在复平面的单位圆内有它的 Floquet 特征值)。
=== The SEIS model ===
=== The SEIS model ===
SEIS模型
The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.
The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.
<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>
<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>
<math>{\color{blue}{\mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{S}}}</math>
<math>
<math>
<math>
\begin{align}
\begin{align}
</math>
</math>
</math>
=== The MSEIR model ===
=== The MSEIR model ===
MSEIR模型
For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.
For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.
<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
<math> \color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R}} </math>
::<math>
::<math>
<math>
<math>
<math>
\begin{align}
\begin{align}
</math>
</math>
=== The MSEIRS model ===
=== The MSEIRS model ===
MSEIRS模型
An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.
An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.
<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>
<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>
<math>{\color{blue}{\mathcal{M} \to \mathcal{S} \to \mathcal{E} \to \mathcal{I} \to \mathcal{R} \to \mathcal{S}}}</math>
===Variable contact rates===
===Variable contact rates===
多样化接触率
It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.
It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.
It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.
It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.
众所周知，患病的概率在时间上并不是一成不变的。随着大流行的进展，对大流行的反应可能会改变在较简单模型中假定为恒定的接触率。口罩、社交距离和封锁等应对措施将改变接触率，从而降低大流行的速度。
<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>
<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>
{ n } ，quad beta (t + t) = beta (t) </math >
<math> F = \beta(t) \frac{I}{N} , \quad \beta(t+T)=\beta(t)</math>
with period T equal to one year.
with period T equal to one year.
周期T等于一年。
</math>
</math>
</math>
==Modelling vaccination==
==Modelling vaccination==
疫苗接种模型
The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}
The SIR model can be modified to model vaccination<ref>{{cite journal |last1=Gao |first1= Shujing|last2=Teng |first2= Zhidong|last3=Nieto |first3=Juan J. |last4=Torres |first4=Angela |date=2007 |title=Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay |url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2217597/ |journal=Journal of Biomedicine and Biotechnology |volume=2007 |issue= |pages= |doi= |access-date=July 19, 2020}}
The SIR model can be modified to model vaccination<ref>
The SIR model can be modified to model vaccination<ref>
SIR 模型可以修改为疫苗接种模型 < ref >
SIR 模型可以修改为疫苗接种模型<ref>
</ref>. Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.
</ref>. Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.
</ref>. Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.
</ref>. Typically these introduce an additional compartment to the SIR model, <math>V</math>, for vaccinated individuals. Below are some examples.
</ref > .典型地，这些引入了一个额外的隔间，SIR 模型，< math > v </math > ，为免疫个体。下面是一些例子。
</ref >.典型地，这些引入了一个额外的隔间，SIR 模型，< math > v </math > ，为免疫个体。下面是一些例子。
===Vaccinating newborns===
===Vaccinating newborns===
新生儿疫苗接种
In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:
In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:
<math>
<math>
<math>
\begin{align}
\begin{align}
\frac{dV}{dt} & = \nu N P - \mu V
\frac{dV}{dt} & = \nu N P - \mu V
\frac{dV}{dt} & = \nu N P - \mu V
\end{align}
\end{align}
</math>
</math>
</math>
thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:
thus we shall deal with the long term behavior of <math>S</math> and <math>I</math>, for which it holds that:
因此，我们将处理<math>S</math>和<math>I</math>的长期行为，它认为:
<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>
<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>
<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \to +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>
===Vaccination and information===
===Vaccination and information===
疫苗接种与信息
Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:
Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:
===Vaccination of non-newborns===
===Vaccination of non-newborns===
非新生儿疫苗接种
In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:
In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:
<math>
<math>
<math>
\begin{align}
\begin{align}
</math>
</math>
</math>
===Pulse vaccination strategy===
===Pulse vaccination strategy===
脉冲接种疫苗策略
This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:
This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:
<math>
<math>
<math>
\begin{align}
\begin{align}
</math>
</math>
</math>
where:
where:
其中:
== Other considerations within compartmental epidemic models ==
== Other considerations within compartmental epidemic models ==
其他传染病模型的考虑因素
=== Vertical transmission ===
横向传染
In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>
In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.<ref name=":2">{{cite book |last1=Brauer |first1=F. |last2=Castillo-Chávez |first2=C. |year=2001 |title=Mathematical Models in Population Biology and Epidemiology |location=NY |publisher=Springer |isbn=0-387-98902-1 }}</ref>
=== Vector transmission ===
=== Vector transmission ===
纵向传染
Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>
Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.<ref name=":2" /><ref>For more information on this type of model see {{cite book |editor-last=Anderson |editor-first=R. M. |editor-link=Roy M. Anderson |year=1982 |title=Population Dynamics of Infectious Diseases: Theory and Applications |publisher=Chapman and Hall |location=London-New York |isbn=0-412-21610-8 }}</ref>
=== Others ===
=== Others ===
其他
Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />
Other occurrences which may need to be considered when modeling an epidemic include things such as the following:<ref name=":2" />
==See also==
==See also==
参见
*[[Mathematical modelling in epidemiology]]
*[[Mathematical modelling in epidemiology]]
==References==
==References==
参考文献
{{reflist}}
{{reflist}}
==Further reading==
==Further reading==
深入阅读
*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}
*{{cite book |last1=May |first1=Robert M. |last2=Anderson |first2=Roy M.|authorlink=Robert_May,_Baron_May_of_Oxford|title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford |year=1991 |pages= |isbn=0-19-854040-X }}
==External links==
==External links==
外部链接
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]