更改

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I,  or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

Compartmental models simplify the mathematical modelling of infectious diseases. The population is assigned to compartments with labels - for example, S, I,  or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.

传染病模型简化了传染病的数学模型。人口被划分到带有标签的类别，例如，S，I，或 R，(易感者，感病者，或康复者)。不同类别中的人数会发生变化。标签的顺序通常显示类别之间的流动模式，例如 SEIS模型意味着易感染成为暴露者、然后成为感病者，然后再次回到易感者。

'''<font color="#ff8000">传染病模型Compartmental models</font>'''简化了传染病的数学模型。人口被划分到带有标签的类别，例如，S，I，或 R，(易感者，感病者，或康复者)。不同类别中的人数会发生变化。标签的顺序通常显示类别之间的流动模式，例如 SEIS模型意味着易感染成为暴露者、然后成为感病者，然后再次回到易感者。

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

==[[用户:Agnes|Agnes]]（[[用户讨论:Agnes|讨论]]）[翻译]译者知识水平限制，通过多方查询资料对传染病模型有了基本了解之后，在“SEIS means susceptible, exposed, infectious, then susceptible again”的翻译中，选择增添了一些成分，但此部分的翻译仍然存疑

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:  

The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments:  

SIR 模型是最简单的传染病模型之一，许多模型都是这种基本形式的衍生物。该模式由三个部分组成:

SIR 模型是最简单的'''<font color="#ff8000">传染病模型Compartmental models</font>'''之一，许多模型都是这种基本形式的衍生物。该模式由三个部分组成:

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

:'''S''': The number of '''s'''usceptible individuals. When a susceptible and an infectious individual come into "infectious contact", the susceptible individual contracts the disease and transitions to the infectious compartment.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e.

[[File:SIR Flow Diagram.svg|600px|center|SIR compartment model]]

[[File:SIR Flow Diagram.svg|600px|center|SIR compartment model]]

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/N², 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/N² 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/N², 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/N² 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)。

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'''<font color="#ff8000">质量作用定律the law of mass action</font>'''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，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)。

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'''<font color="#ff8000">爱尔郎分布Erlang distribution</font>''')。

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:

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'''<font color="#ff8000">传染病模型comparenmental models</font>'''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:

\end{align}

\end{align}

结束{ align }

\结束{ align }

</math>

</math>

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

This system is non-linear, however it is possible to derive its analytic solution in implicit form. Firstly note that from:

这个系统是非线性的，但是可以用隐式形式得到它的解析解。首先，请注意:

这个系统是非线性的，但是可以用隐式形式得到它的'''<font color="#ff8000">解析解analytic solution</font>'''。首先，请注意:

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>

所谓的'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''(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>

By dividing the first differential equation by the third, separating the variables and integrating we get

By dividing the first differential equation by the third, separating the variables and integrating we get

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:

'''<font color="#ff8000">基本再生数basic reprodution numer</font>'''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:

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.

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>) :

建立了从易感者到感病者的传染率模型，因此称之为'''<font color="#ff8000">感染力the force of infection</font>'''。然而，对于大类传染病来说，更现实的做法是考虑一种'''<font color="#ff8000">感染力the force of infection</font>'''，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>) :

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，它对应以下参数化时间

2014年，Harko 和合作者推导出了 SIR 模型的精确'''<font color="#ff8000">解析解analytical solution</font>'''。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

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 <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>。

其中<math>u_T</math>满足<math>\mathcal{I}(u_T)=0</math>。根据上面的'''<font color="#ff8000">超越方程transcendental equation</font>''' <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>。

An equivalent analytical solution found by Miller yields

An equivalent analytical solution found by Miller yields

\end{align}

\end{align}

结束{ align }

\结束{ align }

</math>

</math>

其无病平衡点为:

其'''<font color="#ff8000">无病平衡点the disease-free quilibrium</font>'''为:

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:

这种基本再生数具有临界性质。事实上，我们可以独立于具有生物学意义的初始值证明:

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'''<font color="#ff8000">解析解analytical solution</font>'''(通过对变量进行变换：<math>I = y^{-1}</math> 并将其代入'''<font color="#ff8000">平均场方程the mean-field euations</font>''') ，使基本再生率大于单位数。给出了解决方案如下:

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:

对于这种模式，'''<font color="#ff8000">基本再生数basic reproduction number</font>'''是: