A large deviations principle for the Maki-Thompson rumour model

We consider the stochastic model for the propagation of a rumour within a population which was formulated by Maki and Thompson. Sudbury established that, as the population size tends to infinity, the proportion of the population never hearing the rumour converges in probability to $0.2032$. Watson later derived the asymptotic normality of a suitably scaled version of this proportion. We prove a corresponding large deviations principle, with an explicit formula for the rate function.Comment: 18 pages, 2 figure

Laws of large numbers for the frog model on the complete graph

The frog model is a stochastic model for the spreading of an epidemic on a graph, in which a dormant particle starts to perform a simple random walk on the graph and to awake other particles, once it becomes active. We study two versions of the frog model on the complete graph with $N + 1$ vertices. In the first version we consider, active particles have geometrically distributed lifetimes. In the second version, the displacement of each awakened particle lasts until it hits a vertex already visited by the process. For each model, we prove that as $N \to \infty$, the trajectory of the process is well approximated by a three-dimensional discrete-time dynamical system. We also study the long-term behavior of the corresponding deterministic systems

A new upper bound for the critical probability of the frog model on homogeneous trees

We consider the interacting particle system on the homogeneous tree of degree $(d + 1)$, known as frog model. In this model, active particles perform independent random walks, awakening all sleeping particles they encounter, and dying after a random number of jumps, with geometric distribution. We prove an upper bound for the critical parameter of survival of the model, which improves the previously known results. This upper bound was conjectured in a paper by Lebensztayn et al. ($J. Stat. Phys.$, 119(1-2), 331-345, 2005). We also give a closed formula for the upper bound

Limit theorems for a general stochastic rumour model

We study a general stochastic rumour model in which an ignorant individual has a certain probability of becoming a stifler immediately upon hearing the rumour. We refer to this special kind of stifler as an uninterested individual. Our model also includes distinct rates for meetings between two spreaders in which both become stiflers or only one does, so that particular cases are the classical Daley-Kendall and Maki-Thompson models. We prove a Law of Large Numbers and a Central Limit Theorem for the proportions of those who ultimately remain ignorant and those who have heard the rumour but become uninterested in it.Comment: 13 pages, to appear in SIAM Journal on Applied Mathematic

Epidemic Models with Immunization and Mutations on a Finite Population

We propose two variants of a stochastic epidemic model in which the disease is spread by mobile particles performing random walks on the complete graph. For the first model, we study the effect on the epidemic size of an immunization mechanism that depends on the activity of the disease. In the second model, the transmission agents can gain lives at random during their existences. We prove limit theorems for the final outcome of these processes. The epidemic model with mutations exhibits phase transition, meaning that if the mutation parameter is sufficiently large, then asymptotically all the individuals in the population are infected

A process of rumor scotching on finite populations

Rumor spreading is a ubiquitous phenomenon in social and technological networks. Traditional models consider that the rumor is propagated by pairwise interactions between spreaders and ignorants. Spreaders can become stiflers only after contacting spreaders or stiflers. Here we propose a model that considers the traditional assumptions, but stiflers are active and try to scotch the rumor to the spreaders. An analytical treatment based on the theory of convergence of density dependent Markov chains is developed to analyze how the final proportion of ignorants behaves asymptotically in a finite homogeneously mixing population. We perform Monte Carlo simulations in random graphs and scale-free networks and verify that the results obtained for homogeneously mixing populations can be approximated for random graphs, but are not suitable for scale-free networks. Furthermore, regarding the process on a heterogeneous mixing population, we obtain a set of differential equations that describes the time evolution of the probability that an individual is in each state. Our model can be applied to study systems in which informed agents try to stop the rumor propagation. In addition, our results can be considered to develop optimal information dissemination strategies and approaches to control rumor propagation.Comment: 13 pages, 11 figure

On the asymptotic enumeration of accessible automata

We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.CNPq[311909/2009-4