Phase transitions of the SIR Rumor spreading model with a variable trust rate

Abstract

We study a threshold phenomenon of rumor outbreak on the SIR rumor spreading model with a variable trust rate depending on the populations of ignorants and spreaders. Rumor outbreak in the SIR rumor spreading model is defined as a persistence of the final rumor size in the large population limit or thermodynamics limit (n → ∞), where 1/n is the initial population of spreaders. We present a rigorous proof for the existence of threshold on the final size of the rumor with respect to the basic reproduction number R 0 . Moreover, we prove that a phase transition phenomenon occurs for the final size of the rumor (as an order parameter) with respect to the basic reproduction number and provide a criterion to determine whether the phase transition is of first or second order. Precisely, we prove that there is a critical number R 1 such that if R 1 > 1, then the phase transition is of the first order, i.e., the limit of the final size is not a continuous function with respect to R 0 . The discontinuity is a jump-type discontinuity and it occurs only at R 0 = 1. If R 1 < 1, then the phase transition is second order, i.e., the limit of the final size is continuous with respect to R 0 and its derivative exists, except at R 0 = 1, and the derivative is not continuous at R 0 = 1. We also present numerical simulations to demonstrate our analytical results for the threshold phenomena and phase transition order criterion.

A critical review on applications of the Avrami equation beyond materials science

“…Finally, we note that the SIR model has itself been adapted to unconventional contexts, to model transformations as diverse as the spread of song popularity [ 115 ], political influence [ 116 ], rumors [ 117 ] and gun ownership [ 118 ]. While we have focused on life sciences applications of the Avrami equation, the spirit of interdisciplinary science invites us to also consider whether the SIR model might have relevance in describing materials phase transformations.…”