The Maki–Thompson model with random awareness

Abstract We propose a rumor propagation model which generalizes the k -spreading Maki–Thompson model (Rada et al 2021 SIAM J. Appl. Dyn. Syst. 20 1209–31) by assuming k as being random. In this setting, individuals within a homogeneously mixed population can assume one of infinitely many possible states. To analyze this model, we extend the classical law of large numbers for density-dependent population models in R d (Ethier and Kurtz 2009 Markov Processes: Characterization and Convergence (John Wiley & Sons)) to an infinite–dimensional setting. Specifically, we prove a convergence result for stochastic processes in the space of p -summable real sequences, generalizing the finite–dimensional theory. We derive the asymptotic proportions of individuals in each state at the end of a rumor outbreak. Furthermore, we characterize the maximum proportion of individuals actively spreading the rumor and explore conditions under which the model exhibits waves of rumor propagation.