320
个编辑
更改
跳到导航
跳到搜索
第782行：
第782行： + + + + + + + + + + + + + + + + + + + + + + + + +
→The SIR model
Using a fixed population, <math>N = S(t) + I(t) + R(t)</math>, Kermack and McKendrick derived the following equations:
Using a fixed population, <math>N = S(t) + I(t) + R(t)</math>, Kermack and McKendrick derived the following equations:
: <math>
\begin{align}
\frac{dS}{dt} & = - \beta S I \\[8pt]
\frac{dI}{dt} & = \beta S I - \gamma I \\[8pt]
\frac{dR}{dt} & = \gamma I
\end{align}
</math>
Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of <math>\beta</math>, which is considered the contact or infection rate of the disease. Therefore, an infected individual makes contact and is able to transmit the disease with <math>\beta N</math> others per unit time and the fraction of contacts by an infected with a susceptible is <math>S/N</math>. The number of new infections in unit time per infective then is <math>\beta N (S/N)</math>, giving the rate of new infections (or those leaving the susceptible category) as <math>\beta N (S/N)I = \beta SI</math> (Brauer & Castillo-Chavez, 2001). For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, infectives are leaving this class per unit time to enter the recovered/removed class at a rate <math>\gamma</math> per unit time (where <math>\gamma</math> represents the mean recovery rate, or <math>1/\gamma</math> the mean infective period). These processes which occur simultaneously are referred to as the [[Law of mass action|Law of Mass Action]], a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned (Daley & Gani, 2005). Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.
More can be read on this model on the [[Epidemic model]] page.
===SIR 模型===
1927年， W. O. Kermack 和 A. G. McKendrick 建立了一个固定人口中仅包含三种人群的模型，易感者： <math>S(t)</math>，感染者， <math>I(t)</math>和康复者 <math>R(t)</math>。该模型中用的划分分为三类：
* <math>S(t)</math> 用来表示t时刻，没有被疾病感染或者容易感染某疾病的个体数
* <math>I(t)</math> 表示已经被感染的个体数，并且他们可以把疾病传播给易感人群d
* <math>R(t)</math> 是那部分感染过疾病但是已经康复的人。这部分个体不会再次被感染，并且也不会传染给易感染人群。
这个模型中的流可以这样表示：
: <math>\mathcal{S} \rightarrow \mathcal{I} \rightarrow \mathcal{R} </math>
对于数量固定的群体，有<math>N = S(t) + I(t) + R(t)</math>， 从而Kermack 和 McKendrick 得到以下方程：
: <math>
: <math>