SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 3, Page 1209-1231, January 2021.
We propose a mathematical model to measure how multiple repetitions may influence the ultimate proportion of the population never hearing a rumor during a given outbreak. The model is a multidimensional continuous-time Markov chain that can be seen as a generalization of the Maki--Thompson model for the propagation of a rumor within a homogeneously mixing population. In the well-known basic model, the population is made up of “spreaders," “ignorants," and “stiflers," and any spreader attempts to transmit the rumor to the other individuals via directed contacts. In case the contacted individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. The process in a finite population will eventually reach an equilibrium situation, where individuals are either stiflers or ignorants. We generalize the model by assuming that each ignorant becomes a spreader only after hearing the rumor a predetermined number of times. We identify and analyze a suitable limiting dynamical system of the model, and we prove limit theorems that characterize the ultimate proportion of individuals in the different classes of the population.

中文翻译：

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多次重复对谣言大小的作用

SIAM Journal on Applied Dynamical Systems，第 20 卷，第 3 期，第 1209-1231 页，2021 年 1 月。
我们提出了一个数学模型来衡量多次重复如何影响在给定爆发期间从未听过谣言的人群的最终比例。该模型是一个多维连续时间马尔可夫链，可以看作是 Maki--Thompson 模型的推广，用于在均匀混合的群体中传播谣言。在众所周知的基本模型中，人群由“传播者”、“无知者”和“扼杀者”组成，任何传播者都试图通过直接接触将谣言传播给其他人。如果被联系的人是无知，它变成了一个传播者，而在其他两种情况下，发起传播者变成了一个扼杀者。有限种群中的过程最终会达到平衡状态，个人要么是扼杀者，要么是无知。我们通过假设每个无知者只有在听到谣言预定次数后才成为传播者来概括模型。我们确定并分析了模型的一个合适的极限动力系统，并证明了表征不同群体中个体的最终比例的极限定理。