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These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.
These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.
这些变量（S、I和R）表示特定时间每个类别内的人数。为了表示易感者、感病者和康复者数量可能随时间变化(总种群规模保持不变) ，我们将精确数字设为时间t的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发并控制它们。
这些变量（S、I和R）表示特定时间每个类别内的人数。为了表示易感者、感病者和康复者数量可能随时间变化(总种群规模保持不变) ，我们将精确数字设为时间''t''的函数: S(t)、 I(t)和 R(t)。对于特定人群中的特定疾病，这些函数可以用于预测潜在的传染病暴发并控制它们。
As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.
As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.
从''t''的变量函数中可以看出，该模型是动态的，因为每个分类中的数量可能随时间而波动。这一动态方面的重要性尤其体现在传染时间较短的地方性疾病中，如1968年引进疫苗之前英国的麻疹。由于易感者(S (t))随着时间发生变化，这类疾病往往会周期性爆发。在流行病爆发期间，易感者人数迅速下降，因为更多的人受到感染，从而进入感病者和康复者的类别。这种疾病只有易感者数量增加时才能再次爆发，例如：当后代出生在易感者区域中时。
Yellow=Susceptible, Maroon=Infectious, Teal=Recovered
Yellow=Susceptible, Maroon=Infectious, Teal=Recovered
【图2：Yellow=Susceptible, Maroon=Infectious, Teal=Recovered黄色=易感者，栗色=感病者，青色 =康复者】
SIR compartment model
SIR compartment model
【图3：SIR compartment modelSIR传染病模型】
===The SIR model without vital dynamics===
===The SIR model without vital dynamics===
缺少生命动力学的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:
流行病动力学，例如流感，往往比出生和死亡的动力学变化更快，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:
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>
所谓的基本再生数(亦称基本再生率)。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>
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:
基本再生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:
then
then
那么
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，基本生数和最初的易感性都极其重要。
An equivalent analytical solution found by Miller yields
An equivalent analytical solution found by Miller yields
等价的解析解由米勒发现
===The SIR model with vital dynamics and constant population===
===The SIR model with vital dynamics and constant population===
具有生命动力学和稳定人口的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:
==Variations on the basic SIR model==
==Variations on the basic SIR model==
基础SIR模型的变化
===The SIS model===
===The SIS model===
Yellow=Susceptible, Maroon=Infected
Yellow=Susceptible, Maroon=Infected
【图4：Yellow=Susceptible, Maroon=Infected黄色=易感，栗色=感染】
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.
SIS compartmental model
SIS compartmental model
【图5：SIS compartmental modelSIS传染病模型】
<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.
<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.
< math > frac { dS }{ dt } + frac { dI }{ dt } = 0 right tarrow s (t) + i (t) = n </math > .
<math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>.
& \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.
\end{align}
\end{align}
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>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:
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:
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>
【图6：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>)。
【图7：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>)。】
MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:
MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:
MSIR 分隔模型为了从数学上表示这一点，增加了一个额外的分隔，m (t)。这导致了下列微分方程:
【图8：MSIR compartmental modelTo indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:MSIR 分隔模型为了从数学上表示这一点，增加了一个额外的分隔，M(t)。这导致了下列微分方程:】
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。然后他们可能回归到感病者并出现症状（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。携带者被标记为C。
A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.
A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.
【图9：A simple modification of previous image by Viki Male to make the word "Carrier" plainly visible.对维基马累之前形象的一个简单修改使单词“携带者”清晰可见。】
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).
对于许多重要的感染，有一个重要的疾病潜伏期，在这期间个人已经被感染，但他们自己还没有感病。在此期间，这个人是属于类别E(为暴露)。
SEIR compartmental model
SEIR compartmental model
【图10：SEIR compartmental modelSEIR传染病模型】
Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:
Assuming that the incubation period is a random variable with exponential distribution with parameter <math>a</math> (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate <math>\Lambda</math> equal to death rate <math>\mu</math>, we have the model:
假设疾病潜伏期是一个带有参数 < math > a </math > 的随机变量，那么它就是一个带有指数分布的随机变量。平均疾病潜伏期是 < math > a ^ {-1} </math >) ，并且假设出生率 < math > Lambda </math > 等于死亡率 < math > mu </math > 存在生命动力学，我们有这样的模型:
假设疾病潜伏期是一个带有参数<math>''a''</math>，那么它就是一个服从指数分布的随机变量。平均疾病潜伏期是<math>a^{-1}</math>)，并且假设存在出生率<math>\Lambda</math>等于死亡率<math>\mu</math>的生命动力学，我们有这样的模型:
SEIS模型
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.
\end{align}
\end{align}
结束{ align }
\结束{ align }
</math>
</math>
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。随着大流行的发展，相关应对措施可能会改变接触率，而较简单模型中将接触率假定为恒定。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。
In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate
In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate
此外，有些疾病是季节性的，例如普通感冒病毒，这些病毒在冬季更为普遍。儿童疾病，如麻疹、腮腺炎和风疹，与上学日期有很强的相关性，因此在学校假期内患这种疾病的可能性大大降低。因此，对于许多种类的疾病，人们应该考虑周期性/“季节性”)变化接触率的感染力
<math>
<math>
<math>
\begin{align}
\begin{align}
\begin{align}
\begin{align}
开始{ align }
\开始{ align }
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\frac{dS}{dt} & = \mu N - \mu S - \beta(t) \frac{I}{N} S \\[8pt]
\end{align}
\end{align}
结束{ align }
\结束{ align }
</math>
</math>
(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:
(the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:
(恢复的动力学很容易遵循 < math > r = N-S-I </math >)。具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会经历非常有趣和复杂的非线性参数共振现象。显而易见，如果:
(康的动力学很容易遵循<math>R=N-S-I</math>)，也就是具有周期变化参数的非线性微分方程组。众所周知，这类动力系统可能会经历非常有趣和复杂的非线性参数共振现象。显然，如果:
This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.
This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.
这就可以解释某些传染病的多年(典型的两年)流行病爆发，即接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。
这就可以解释某些传染病的多年(常见为两年)流行病爆发，即接触率振荡周期和地方病平衡点附近阻尼振荡的伪周期之间的相互作用。值得注意的是，在某些情况下，这种行为也可能是准周期的，甚至是混沌的。
</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类别，SIR 模型， <math>V</math> ，为免疫个体。下面是一些例子。
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>:
面对传染病，主要任务之一是通过预防措施消除传染病，如果可能的话，通过建立大规模疫苗接种计划。假设一种新生儿接种疫苗(通过疫苗给予终身免疫力)的疾病，其发病率为 <math>P \in (0,1)</math>:
\end{align}
\end{align}
结束{ align }
\结束{ align }
</math>
</math>
where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:
where <math>V</math> is the class of vaccinated subjects. It is immediate to show that:
其中<math>V</math> 是指接种疫苗的人群。很快就可以看出:
In other words, if
In other words, if
也就是说，如果
:<math> P < P^{*}= 1-\frac{1}{R_0} </math>
:<math> P < P^{*}= 1-\frac{1}{R_0} </math>
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:
现代社会正面临着“理性”豁免的挑战，这个家庭决定不给孩子接种疫苗，是因为对感染的可感知风险和接种疫苗造成的损害进行了“合理”的比较。为了评估这种行为是否真的是理性的，即：如果它同样可以导致根除该疾病，人们可以简单地假定疫苗接种率是传染病人数的增加函数:
:<math> P=P(I), \quad P'(I)>0.</math>
:<math> P=P(I), \quad P'(I)>0.</math>
<math> P(0) \ge P^{*},</math>
<math> P(0) \ge P^{*},</math>
< math > p (0) ge p ^ {} ，</math >
<math> P(0) \ge P^{*},</math>
i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.
i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.
i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.
i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.
也就是说基线疫苗接种率应高于“强制接种”的门槛，而在豁免情况下，“强制接种”不能持续。因此，“合理的”豁免可能是短视的，因为它只是基于目前由于疫苗覆盖率高而发病率低的情况，而不是考虑到由于覆盖率下降而导致的未来感染死灰复燃的情况。
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
= mu n p + rho s-mu v
\frac{dV}{dt} & = \mu N P + \rho S - \mu V
\end{align}
\end{align}
<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>
<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>
1-left (1 + frac { rho }{ mu } right) frac {1}{ r0} </math >
<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>
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:
随着时间的推移，这种策略反复对特定年龄组群(如幼儿或老年人)的易感人群进行疫苗接种。使用这种策略，易感人群的区块立即被移除，从而有可能从整个人群中消灭传染病病毒(如麻疹)。每 t 时间单位一个常数分数 p 的易感受试者接种疫苗在一个相对较短的时间内(相对于疾病的动态)。这就导致了以下易受感染和接种疫苗的受试者的脉冲微分方程:
随着时间的推移，这种策略反复对特定年龄组群(如幼儿或老年人)的易感人群进行疫苗接种。使用这种策略，易感者类别立即康复，从而有可能从整个人群中消灭传染病病毒(如麻疹)。每T时间单位一个常数分数p的易感受试者接种疫苗在一个相对较短的时间内(相对于疾病的动态)。这就导致了以下易受感染和接种疫苗的受试者的脉冲微分方程:
It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:
It is easy to see that by setting one obtains that the dynamics of the susceptible subjects is given by:
很容易看出，通过设置，人们得到的易感体动力学是由以下因素决定的:
Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:
Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:
年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑到流行病的年龄 s (t，a) ，i (t，a) ，r (t，a) </math > (将我们限制在易感染的-传染的-移除的计划) ，这样:
年龄对疾病在人群中的传播速度有很大的影响，尤其是接触率。这个比率概括了易感人群和感染人群之间接触的有效性。考虑到流行病的年龄<math>s(t,a),i(t,a),r(t,a)</math>(将我们限制在易感者-感病者-康复的计划) ，这样:
<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>
<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>
<math>S(t)=\int_0^{a_M} s(t,a)\,da </math>
<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>
<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>
I (t) = int _ 0 ^ { a _ m } i (t，a) ，da </math >
<math>I(t)=\int_0^{a_M} i(t,a)\,da</math>
(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:
(where <math>a_M \le +\infty</math> is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:
(其中“ a _ m le + infty </math > 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:
(其中<math>a_M \le +\infty</math> 是最大可接受的年龄) ，它们的动力学并不像人们想象的那样用“简单的”偏微分方程描述，而是用积分-微分方程描述:
is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.
is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.
当然，虽然联系核心<math> k(a,a_1;t) </math>在于年龄之间的相互作用，但这将取决于感染力。
Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:
Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:
新生婴儿的初始条件(即。对于 a = 0) ，可以直接用于传染和移除:
新生婴儿的初始条件(即对于 a = 0) ，可以直接用于传染和康复:
<math>i(t,0)=r(t,0)=0</math>
<math>i(t,0)=r(t,0)=0</math>
<math>i(t,0)=r(t,0)=0</math>
where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.
where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.
其中，<math>\varphi_j(a), j=s,i,r</math>是成人的受精能力。
Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:
Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:
此外，现在定义总人口的密度 n (t，a) = s (t，a) + i (t，a) + r (t，a) </math > 1得到:
此外，现在定义总人口的密度<math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> 1得到:
In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:
In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:
在三个流行等级生育率相等的最简单情况下，我们得到了人口平衡与生育率相关的下列充要条件出生率<math>\varphi(.)和死亡率</math><math>\mu(a)</math>灵活性间的联系，必须坚持:
<math>DFS(a)= (n^*(a),0,0).</math>
<math>DFS(a)= (n^*(a),0,0).</math>
< math > DFS (a) = (n ^ * (a) ，0,0) </math >
<math>DFS(a)= (n^*(a),0,0).</math>
A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.
A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.
基本再生数可以用一个适当的函数算符的谱半径来计算。
=== Vertical transmission ===
=== Vertical transmission ===
垂直传染
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>
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.
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.
在艾滋病和乙型肝炎等一些疾病的情况下，受感染父母的后代有可能在出生时就受到感染。这种从母体向下传染的疾病叫做垂直传染。在模型中，可以通过将一部分新生儿成员包括在受感染的隔间中来考虑额外成员流入受感染类别的情况。
=== Vector transmission ===
=== Vector transmission ===
水平传染
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>
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.
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.
由人间接传播的疾病，即。疟疾通过蚊子传播，通过媒介传播。在这种情况下，从人到昆虫的感染转移和一个流行病模型必须包括这两个物种，通常需要更多的区域比模型直接传染。
To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.
To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.
更准确地说，这些模型只适用于中东热力学极限，那里的人口实际上是无限的。在随机模型中，由上述推导出的长期地方病平衡并不成立，因为在一个系统中，感染个体的数量下降到1以下的概率是有限的。在一个真正的系统中，病原体可能不会繁殖，因为没有宿主会被感染。但是，在确定性平均场模型中，被感染的病原体数量可以呈现实数值，即被感染宿主的非整数值，模型中的宿主数量可以小于1，但大于零，从而使模型中的病原体得以繁殖。传染病模型的可靠性仅限应用于传染病。
参见
参见
*[[Mathematical modelling in epidemiology]]
*[[Mathematical modelling in epidemiology]]
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==编辑推荐==
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