Investigation 5: Modelling the Spread of a Rumor Using Logistic Growth | 探究五：用逻辑斯谛增长模型模拟谣言传播

📚 Investigation 5: Modelling the Spread of a Rumor Using Logistic Growth | 探究五：用逻辑斯谛增长模型模拟谣言传播

Mathematical modelling offers a powerful lens through which we can understand the patterns of information dissemination in social networks. One classic scenario is the spread of a rumor within a closed community, such as a school. This investigation uses the logistic growth model, originally developed for population dynamics, to describe how the number of people who have heard a rumor increases over time until nearly everyone knows it. We will formulate the differential equation, find its analytical solution, estimate parameters from observed data, and critically evaluate the model's validity.

数学建模为我们理解社交网络中信息传播的模式提供了强大的视角。一个经典的场景是在一个封闭社区（例如一所学校）中谣言的传播。本探究使用最初为人口动态学开发的逻辑斯谛增长模型，来描述听到谣言的人数如何随时间增长，直到几乎所有人都知道。我们将建立微分方程，求出解析解，从观测数据中估计参数，并批判性地评估模型的有效性。

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1. The Logistic Growth Framework | 逻辑斯谛增长框架

In a finite population, the rate at which a rumor spreads is not constant. At first, few people know the rumor, so the number of new 'knowers' grows exponentially. As more people become aware, the pool of ignorant individuals shrinks, causing the growth rate to decline. This is exactly the principle behind the logistic model: the per capita growth rate decreases linearly with the current population size, eventually reaching zero when the carrying capacity is attained.

在有限的人口中，谣言传播的速率并非恒定。起初，只有少数人知道谣言，因此新的“知情者”数量呈指数增长。随着知情者增多，不知情的人群缩小，导致增长率下降。这正是逻辑斯谛模型背后的原理：人均增长率随当前种群规模线性下降，当达到承载能力时最终降为零。

The model assumes that mixing between those who know and those who do not is homogeneous, and that every encounter has a constant probability of transmitting the rumor. These assumptions, while simplified, give rise to a rich mathematical structure that can be compared with real observations.

该模型假设知情者与不知情者之间的混合是均匀的，且每次接触都有恒定的概率传递谣言。这些假设虽然简化了，却产生了可以与实际观察对比的丰富数学结构。

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2. Differential Equation Formulation | 微分方程公式

Let P(t) be the number of people who have heard the rumor at time t (in days). Let N be the total population, which acts as the carrying capacity K. The logistic differential equation is given by:

设 P(t) 为 t 时刻（以天为单位）听说谣言的人数。设总人数为 N，它也充当承载能力 K。逻辑斯谛微分方程如下：

dP/dt = r P (1 − P/K)

Here, r > 0 is the intrinsic growth rate—the maximum per capita rate of spread when P is very small. The term (1 − P/K) represents the fraction of the population still unaware. When P is close to K, the factor approaches zero, halting further growth.

这里，r > 0 是内在增长率——当 P 非常小时，每人传播的最大速率。因子 (1 − P/K) 表示仍不知情的人口比例。当 P 接近 K 时，该因子趋近于零，从而阻止进一步增长。

This nonlinear first-order ordinary differential equation is separable and can be solved analytically, providing a continuous function that fits the discrete nature of rumor spreading as an approximation.

这个非线性一阶常微分方程是可分离的，可以解析求解，从而给出一个连续函数，作为对离散传播过程的近似。

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3. Analytical Solution of the Logistic Equation | 逻辑斯谛方程的解析解

P/K), we separate variables and integrate. The result is the well-known logistic function:

为了解 dP/dt = r P (1 − P/K)，我们分离变量并积分。结果就是著名的逻辑斯谛函数：

P(t) = K / (1 + A e⁻ʳᵗ)

initial number of knowers. At t = 0, the formula yields P(0) = K/(1+A) = P₀. As t → ∞, e⁻ʳ int, where the rate of spread is maximal, occurs at P = K/2, corresponding to t = ln(A)/r.

其中 A = (K − P₀) / P₀，而 P₀ = P(0) 是初始知情人数。在 t = 0 时，该公式给出 P(0) = K/(1+A) = P₀。当 t → ∞ 时，e⁻ʳᵗ → 0，因此 P(t) 渐近地趋近于 K。拐点发生在 P = K/2 处，此时传播速率最大，对应的时间为 t = ln(A)/r。

This closed-form solution allows us to make predictions for any time t and facilitates parameter estimation by transforming the data to a linear relationship.

这个封闭形式的解使我们能够预测任何时刻 t 的值，并可以通过将数据转换为线性关系来方便地进行参数估计。

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4. Data Collection: A School Rumor Case Study | 数据收集：一个学校谣言的案例研究

000 students. At lunchtime on Day 0, one student starts a rumor. The table below shows the cumulative number of students who have heard the rumor at the end of each subsequent day, obtained through a confidential survey.

假设一所中学有 N = 1000 名学生。在第 0 天午餐时，一名学生开始传播一个谣言。下表显示了通过保密问卷调查得到的随后每天结束时累计已听说谣言的学生人数。

Day (t)	P(t) (observed)
0	1
1	5
2	12
3	33
4	78
5	180
6	360
7	610
8	850
9	950
10	980

The data exhibit the classic S-shaped pattern: slow initial growth, a phase of rapid expansion, and finally a leveling off as the rumor saturates the population.

该数据呈现出经典的 S 形模式：初期增长缓慢，接着是一个快速扩张阶段，最后当谣言饱和整个人口时趋于平稳。

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5. Estimating Parameters from Data | 从数据估计参数

e r. Starting from the logistic solution, we algebraically rearrange to obtain a linear form:

将 K 固定为 1000，我们需要估计 r。从逻辑斯谛解出发，我们进行代数变形以获得线性形式：

ln((K − P)/P) = ln(A) − r t

ln(999). Using the observed data, we compute y for each day and perform linear regression.

令 y = ln((1000 − P)/P)。如果逻辑斯谛模型合适，y 对 t 作图应得到一条斜率为 −r、截距为 ln(A) = ln(999) 的直线。使用观测数据，我们计算每天的 y 值并进行线性回归。

t	P(t)	y = ln((1000-P)/P)
0	1	6.907
1	5	5.293
2	12	4.410
3	33	3.378
4	78	2.470
5	180	1.516
6	360	0.575
7	610	-0.448
8	850	-1.735
9	950	-2.944
10	980	-3.892

intercept is calculated as 6.90, closely matching ln(999). Therefore, the fitted model is:

最小二乘法线性拟合给出的斜率约为 −0.62，从而得出 r ≈ 0.62 天⁻¹。截距计算结果为 6.90，与 ln(999) 吻合。因此，拟合模型为：

P(t) = 1000 / (1 + 999 e⁻⁰·⁶²ᵗ)

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6. Model Implementation and Visual Fit | 模型实现与视觉拟合

fitted equation yields the predicted values. The table below compares the observed and predicted numbers, showing a close agreement.

将 t = 0, 1, ..., 10 代入拟合方程得到预测值。下表比较了观测值与预测值，显示出高度一致性。

Day	Observed P(t)	Predicted P(t)	Residual
0	1	1.0	0.0
1	5	5.3	-0.3
2	12	12.1	-0.1
3	33	30.5	2.5
4	78	74.2	3.8
5	180	171.5	8.5
6	360	352.8	7.2
7	610	607.9	2.1
8	850	853.0	-3.0
9	950	952.4	-2.4
10	980	982.5	-2.5

The maximum residual magnitude is about 8.

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