PHASE TRANSITIONS OF THE SIR RUMOR SPREADING MODEL WITH A VARIABLE TRUST RATE

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 R₀. 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 R₁ such that if R₁ > 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 R₀. The discontinuity is a jump-type discontinuity and it occurs only at R₀ = 1. If R₁ < 1, then the phase transition is second order, i.e., the limit of the final size is continuous with respect to R₀ and its derivative exists, except at R₀ = 1, and the derivative is not continuous at R₀ = 1. We also present numerical simulations to demonstrate our analytical results for the threshold phenomena and phase transition order criterion.