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

• * Corresponding author: Hyowon Seo
• 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.

Mathematics Subject Classification: Primary: 34A34, 34D05; Secondary: 82B26.

 Citation: • • Figure 1.  Time evolutions of $(I(t), S(t), R(t))$ with initial data (4.1) when $\mathcal{R}_0 = 0.5$

Figure 2.  Time evolutions of $(I(t), S(t), R(t))$ with initial data (4.1) when $\mathcal{R}_0 = 2$

Figure 3.  Numerical simulations for $\phi^\infty_n$ when $T = 10^4$

Figure 4.  Numerical simulations for $\phi^\infty_n$ when $T = 10^6$

Figure 5.  Phase transition diagrams for $\phi^e$ when $n = 10^{10}$ and final time $T = 10^6$

Figure 6.  Phase portrait for $\sigma_R>\sigma_S$ or $\lambda_S^0>0$

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