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\to \infty) $, 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 $ \mathcal{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 $ \mathcal{R}_1 $ such that if $ \mathcal{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 $ \mathcal{R}_0 $. The discontinuity is a jump-type discontinuity and it occurs only at $ \mathcal{R}_0 = 1 $. If $ \mathcal{R}_1<1 $, then the phase transition is second order, i.e., the limit of the final size is continuous with respect to $ \mathcal{R}_0 $ and its derivative exists, except at $ \mathcal{R}_0 = 1 $, and the derivative is not continuous at $ \mathcal{R}_0 = 1 $. We also present numerical simulations to demonstrate our analytical results for the threshold phenomena and phase transition order criterion.