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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ (n\to \infty) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ 1/n $\end{document}</tex-math></inline-formula> is the initial population of spreaders. We present a rigorous proof for the existence of threshold on the final size of the rumor with respect to the basic reproduction number <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula>. Moreover, we prove that a phase transition phenomenon occurs for the final size of the rumor (as an order parameter) with respect to the basic reproduction number and provide a criterion to determine whether the phase transition is of first or second order. Precisely, we prove that there is a critical number <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_1 $\end{document}</tex-math></inline-formula> such that if <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{R}_1&gt;1 $\end{document}</tex-math></inline-formula>, then the phase transition is of the first order, i.e., the limit of the final size is not a continuous function with respect to <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula>. The discontinuity is a jump-type discontinuity and it occurs only at <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{R}_0 = 1 $\end{document}</tex-math></inline-formula>. If <inline-formula><tex-math id="M8">\begin{document}$ \mathcal{R}_1&lt;1 $\end{document}</tex-math></inline-formula>, then the phase transition is second order, i.e., the limit of the final size is continuous with respect to <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> and its derivative exists, except at <inline-formula><tex-math id="M10">\begin{document}$ \mathcal{R}_0 = 1 $\end{document}</tex-math></inline-formula>, and the derivative is not continuous at <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{R}_0 = 1 $\end{document}</tex-math></inline-formula>. We also present numerical simulations to demonstrate our analytical results for the threshold phenomena and phase transition order criterion.</p>