The maximum proportion of spreaders in stochastic rumor models

We examine a general stochastic rumor model characterized by specific parameters that govern the interaction rates among individuals. Our model includes the $(\alpha, p)$-probability variants of the well-known Daley--Kendall and Maki--Thompson models. In these variants, a spreader involved in an interaction attempts to transmit the rumor with probability $p$; if successful, any spreader encountering an individual already informed of the rumor has probability $\alpha$ of becoming a stifler. We prove that the maximum proportion of spreaders throughout the process converges almost surely, as the population size approaches~$\infty$. For both the classical Daley--Kendall and Maki--Thompson models, the asymptotic proportion of the rumor peak is $1 - \log 2 \approx 0.3069$.