The SIS Competition Model for Conflicting Rumors

Abstract

We propose an SIS competition model describing the propagation of conflicting rumors, such as fake news and its corrections. This simple model captures the interaction between rumor propagation and opinion dynamics, where rumors drive opinion changes and, conversely, individuals' opinions determine the infection rates of rumors. We analytically derive all steady states and their stability. These results uncover a novel coexistence mechanism. This coexistence corresponds to a scenario where belief in one rumor (e.g., fake news) paradoxically aids the spread of the opposing rumor (e.g., corrective information). Due to this mechanism, a nontrivial but realistic phenomenon occurs where a lower infection rate actually enhances the spread of a rumor. Furthermore, although the model does not explicitly incorporate majority conformity, a phenomenon where the majority gains an advantage emerges spontaneously. Consequently, even if one rumor has a higher infection rate, it may be eliminated by the other if its initial share fails to exceed a critical threshold. We analytically derive this threshold using the singular perturbation method.

Figures (9)

• Figure 1: Schematic of transitions in the SIS competition model.
• Figure 2: Phase diagram of the SIS competition model for $\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}}, \mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}} < 1$. The horizontal and vertical axes represent $\mathcal{R}_{0}^{\mathrm{A}}$ and $\mathcal{R}_{0}^{\mathrm{B}}$, respectively. For example, $\mathrm{D} + \mathrm{A}$ indicates that the system is in either the $\mathrm{D}$ phase or the $\mathrm{A}$ phase. Parameters: $\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}}=0.1, \mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}}=0.8$. The curve is given by $(\mathcal{R}_{0}^{\mathrm{A}} - 1) (\mathcal{R}_{0}^{\mathrm{B}} - 1) = (\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}} - 1) (\mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}} - 1)$.
• Figure 3: Phase diagram of the SIS competition model for $\mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}} < 1 < \mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}}$. The horizontal and vertical axes represent $\mathcal{R}_{0}^{\mathrm{A}}$ and $\mathcal{R}_{0}^{\mathrm{B}}$, respectively. Parameters: $\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}}=1.7, \mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}}=0.8$. The curve is given by $(\mathcal{R}_{0}^{\mathrm{A}} - 1) (\mathcal{R}_{0}^{\mathrm{B}} - 1) = (\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}} - 1) (\mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}} - 1)$.
• Figure 4: Phase diagram of the SIS competition model for $1 < \mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}}, \mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}}$. The horizontal and vertical axes represent $\mathcal{R}_{0}^{\mathrm{A}}$ and $\mathcal{R}_{0}^{\mathrm{B}}$, respectively. Parameters: $\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}}=1.7, \mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}}=1.2$. The curve is given by $(\mathcal{R}_{0}^{\mathrm{A}} - 1) (\mathcal{R}_{0}^{\mathrm{B}} - 1) = (\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}} - 1) (\mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}} - 1)$.
• Figure 5: Steady state at $\mathcal{R}_{0}^{\mathrm{B}} = 1.5$ in Fig. \ref{['Fig_phase_diagram_2']}. The horizontal axis represents the basic reproduction number $\mathcal{R}_{0}^{\mathrm{B}}$, and the vertical axis represents the fraction of each state. The red and blue lines represent the fractions of individuals supporting opinions $\mathrm{A}$ and $\mathrm{B}$, respectively, while the black solid line represents $\rho_{\mathrm{A}}$. The steady states in the $\mathrm{D}$ phase are partially omitted. The boundary of the $\mathrm{C}$ phase is given by $\mathcal{R}_{0}^{\mathrm{A}} = 1 - (1 - \mathcal{R}_{0}^{\mathrm{A} \to \mathrm{B}} ) (\mathcal{R}_{0}^{\mathrm{B} \to \mathrm{A}} - 1)(\mathcal{R}_{0}^{\mathrm{B}} - 1)^{-1}$.