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→SIR 模型
===SIR 模型===
===SIR 模型===
In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments, susceptible: <math>S(t)</math>, infected, <math>I(t)</math>, and recovered, <math>R(t)</math>. The compartments used for this model consist of three classes:
* <math>S(t)</math> is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease
* <math>I(t)</math> denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category
* <math>R(t)</math> is the compartment used for those individuals who have been infected and then recovered from the disease. Those in this category are not able to be infected again or to transmit the infection to others.
The flow of this model may be considered as follows:
: <math>\mathcal{S} \rightarrow \mathcal{I} \rightarrow \mathcal{R} </math>
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.
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：
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：