传染病的数学模型

Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programmes. The modelling can help decide which intervention/s to avoid and which to trial, or can predict future growth patterns, etc.

历史 History

The modeling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic.[1]

The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt's analysis of causes of death is considered the beginning of the "theory of competing risks" which according to Daley and Gani [1] is "a theory that is now well established among modern epidemiologists".

The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox.[2] The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months.[3] Daniel Bernoulli's work preceded the modern understanding of germ theory.

In the early 20th century, William Hamer[4] and Ronald Ross[5] applied the law of mass action to explain epidemic behaviour.

The 1920s saw the emergence of compartmental models. The Kermack–McKendrick epidemic model (1927) and the Reed–Frost epidemic model (1928) both describe the relationship between susceptible, infected and immune individuals in a population. The Kermack–McKendrick epidemic model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics.[6]

1920年代出现了仓室模型。 Kermack-McKendrick流行病模型（1927年）和Reed-Frost流行病模型（1928年）均描述了人群中易感，感染和免疫个体之间的关系。 Kermack–McKendrick流行病模型成功地预测了爆发行为，与许多已记录的流行病中观察到的行为非常相似。

Recently, agent-based models (ABMs) have been used in exchange for simpler compartmental models, e.g.,[7]. For example, epidemiological ABMs have been used to inform public health (nonpharmaceutical) interventions against the spread of SARS-CoV-2[8]. Epidemiological ABMs, in spite of their complexity and requiring high computational power, have been criticized for simplifying and unrealistic assumptions[9][10]. Still, they can be useful in informing decisions regarding mitigation and suppression measures in cases when ABMs are accurately calibrated[11].

假设条件 Assumptions

Models are only as good as the assumptions on which they are based. If a model makes predictions that are out of line with observed results and the mathematics is correct, the initial assumptions must change to make the model useful.

• Rectangular and stationary age distribution, i.e., everybody in the population lives to age L and then dies, and for each age (up to L) there is the same number of people in the population. This is often well-justified for developed countries where there is a low infant mortality and much of the population lives to the life expectancy.
• 矩形和稳态时固定年龄分布，即人群中的每个人都活到L岁，然后死亡，并且对于每个年龄（至L），人群中的人数都是相同的。 对于婴儿死亡率较低且人群中大部分人都达到预期寿命的发达国家来说，这个假设通常是合理的。
• Homogeneous mixing of the population, i.e., individuals of the population under scrutiny assort and make contact at random and do not mix mostly in a smaller subgroup. This assumption is rarely justified because social structure is widespread. For example, most people in London only make contact with other Londoners. Further, within London then there are smaller subgroups, such as the Turkish community or teenagers (just to give two examples), who mix with each other more than people outside their group. However, homogeneous mixing is a standard assumption to make the mathematics tractable.
• 群体的均匀混合，即受到仔细检查且每个个体随机分配并进行接触的群体，并且其大多不在较小的亚组中混合。 由于社会结构多种多样，因此这种假设很少成立。 例如，伦敦的大多数人只与其他是伦敦人的人接触。 此外在伦敦，则有较小的子群体，例如土耳其社区或青少年（仅举两个例子），他们的交往比其他人更多。 但是不管怎样，均匀混合假设是更易于数学上的处理分析的标准假设。

流行病模型的类型 Types of epidemic models

随机性 Stochastic

"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics.

“随机”是指是或具有一个随机变量。 随机模型是用来估计潜在结果的概率分布的一种工具，它允许一个或多个随时间变化的随机变量的输入。 随机模型依赖于暴露风险，疾病和其他疾病动态的机会变化。

确定性 Deterministic

When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. In a deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic.

The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model.[6]

流行病模型 Epidemic models

Generally, people are divided into different compartments in the epidemic models, individuals in the same compartment are assigned with the same state. The most common states used are: (i) S: the susceptible state; (ii) I: the infected state; (iii) R: the recovered state. Different combinations of these states can result in different models, such as SI, SIS and SIR model.

Herein we first consider these epidemic models in a well-mixed population.

SI模型

The simplest case is the SI model, in which two states are considered, S and I. Denote s(t) and i(t) as the proportion of susceptible and infected individuals at time t, thus we have $s(t)+i(t)=1$. Suppose the probability of infected individuals infecting the susceptible individuals is $\beta$. Therefore, the SI model can be illustrated by ordinary differential equations as follows:

SIS模型

The SIS model is used to characterize epidemics that have transient immunity, such as influenza. When it turns to the information process, it means that the individuals know about the information (in I state) will ignore the information and become susceptible again (in S state). We denote s(t) and i(t) as the fraction of susceptible and infected individuals in the population respectively. The transmission probability between I state individuals and S state individuals is $\beta$, and the recovered probability of the I state individuals is $\gamma$. Accordingly, the SIS model can also be expressed by the ordinary differential equations:

We define $\lambda=\beta/\gamma$ as the effective infection rate of this model. From the above equation, we can find that when $\lambda>1$, the steady-state value of I is $i=(\beta-\gamma)/\beta=1-1/\lambda$, an endemic disease state is obtained in this case (indicates that there will be a number of infected individuals in the population at the final state). However, when $\lambda<1$, we have$i\to0(t\to\infty)$, which means that there will be no I state individuals in the population finally and is called a healthy state. Therefore, $\lambda=1$ is a threshold value of the SIS model, which is also known as the basic reproduction number.[12]

SIR模型

Diagram of the SIR model with initial values $\displaystyle{ S(0)=997, I(0)=3, R(0)=0 }$, and rates for infection $\displaystyle{ \beta=0.4 }$ and for recovery $\displaystyle{ \gamma=0.04 }$
Animation of the SIR model with initial values $\displaystyle{ S(0)=997, I(0)=3, R(0)=0 }$, and rate of recovery $\displaystyle{ \gamma=0.04 }$. The animation shows the effect of reducing the rate of infection from $\displaystyle{ \beta=0.5 }$ to $\displaystyle{ \beta=0.12 }$. If there is no medicine or vaccination available, it is only possible to reduce the infection rate (often referred to as "flattening the curve") by appropriate measures such as social distancing.

The SIR model is introduced to explain the epidemics with permanent immunity in the population. Different from the SI and SIS model, a recovered state (R) is given in this model. Individuals in S state would be infected by the I state individuals with probability $\beta$, whereas the I state individuals would recover to R state with a recovery probability $\gamma$. Thus, the ordinary differential equations of SIR model are

SIR模型是用来解释具有永久免疫力的人群中的流行病。 与SI和SIS模型不同，此模型中还存在恢复状态（R）。 处于S状态的个体被I状态个体以速率$\beta$感染，而处于I状态的个体以恢复速率$\gamma$恢复到R状态，节点一旦成为恢复的R状态则成为永久免疫人群，不可以再被感染。 因此，SIR模型的常微分方程为 \label{eq:sir} \left\{ \begin{aligned} &\frac{ds(t)}{dt}=-\beta s(t)i(t),\\ &\frac{di(t)}{dt}=\beta s(t)i(t)-\gamma i(t),\\ &\frac{dr(t)}{dt}=\gamma i(t). \end{aligned} \right.

$\lambda=\beta/\gamma$ can also be defined as the basic reproduction number of the SIR model. We can get similar conclusion as the SIS model, i.e., when $\lambda<1$时, $r=0$, indicating that information cannot spread in the population. When $\lambda>1$, we have $r>0$, which means the information can spread out as the increase of $\lambda$. As a consequence, $\lambda=1$ is a threshold value of SIR model.

参考文献 References

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（1）维基百科：Mathematical modelling of infectious disease

（2）Zhang Z K, Liu C, Zhan X X, et al. Dynamics of information diffusion and its applications on complex networks[J]. Physics Reports, 2016, 651: 1-34.