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 \begin{document}$ (n\to \infty) $\end{document} , where \begin{document}$ 1/n $\end{document} is the initial population of spreaders. We present a rigorous proof for the existence of threshold on the final size of the rumor with respect to the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} . Moreover, we prove that a phase transition phenomenon occurs for the final size of the rumor (as an order parameter) with respect to the basic reproduction number and provide a criterion to determine whether the phase transition is of first or second order. Precisely, we prove that there is a critical number \begin{document}$ \mathcal{R}_1 $\end{document} such that if \begin{document}$ \mathcal{R}_1>1 $\end{document} , then the phase transition is of the first order, i.e., the limit of the final size is not a continuous function with respect to \begin{document}$ \mathcal{R}_0 $\end{document} . The discontinuity is a jump-type discontinuity and it occurs only at \begin{document}$ \mathcal{R}_0 = 1 $\end{document} . If \begin{document}$ \mathcal{R}_1<1 $\end{document} , then the phase transition is second order, i.e., the limit of the final size is continuous with respect to \begin{document}$ \mathcal{R}_0 $\end{document} and its derivative exists, except at \begin{document}$ \mathcal{R}_0 = 1 $\end{document} , and the derivative is not continuous at \begin{document}$ \mathcal{R}_0 = 1 $\end{document} . We also present numerical simulations to demonstrate our analytical results for the threshold phenomena and phase transition order criterion.