Opinion-driven risk perception and reaction in SIS epidemics

TL;DR

The paper addresses how risk perception and adaptive behavior influence epidemic spread by coupling a nonlinear opinion dynamics system to the SIS framework. It introduces the NOD-SIS model, revealing two parameter regimes: a SIS-like regime at low infectiousness and a bistable regime with distinct endemic equilibria when infectiousness is high. It demonstrates that risk aversion can reduce infections and, under strong peer pressure, can eradicate disease, while risk seeking elevates infection levels. The authors extend the analysis to structured populations via two networks and show how cooperation or antagonism in opinion spread shapes regional infection outcomes, offering insights for policy design and further networked analyses.

Abstract

We present and analyze a mathematical model to study the feedback between behavior and epidemic spread in a population that is actively assessing and reacting to risk of infection. In our model, a population dynamically forms an opinion that reflects its willingness to engage in risky behavior (e.g., not wearing a mask in a crowded area) or reduce it (e.g., social distancing). We consider SIS epidemic dynamics in which the contact rate within a population adapts as a function of its opinion. For the new coupled model, we prove the existence of two distinct parameter regimes. One regime corresponds to a low baseline infectiousness, and the equilibria of the epidemic spread are identical to those of the standard SIS model. The other regime corresponds to a high baseline infectiousness, and there is a bistability between two new endemic equilibria that reflect an initial preference towards either risk seeking behavior or risk aversion. We prove that risk seeking behavior increases the steady-state infection level in the population compared to the baseline SIS model, whereas risk aversion decreases it. When a population is highly reactive to extreme opinions, we show how risk aversion enables the complete eradication of infection in the population. Extensions of the model to a network of populations or individuals are explored numerically.

Figures (4)

• Figure 1: Bifurcation diagrams for (A) $u_0=0.2$ and (B) $u_0=0.7$. For $u_0=0.7$, in the region where $\bar{\beta}<\delta$ (yellow), the only stable fixed point in the interpretable range is the IIFE. A transcritical bifurcation occurs when $\bar{\beta}=\delta$. For $\delta<\bar{\beta}<\bar{\beta}^*$ (red), the IIFE is unstable and the IEE is stable. In these two regions, the system behaves locally like the standard scalar SIS model. Let $\bar{\beta}_0$ be the value for which, given a set of parameters $\delta,k_p,k_x,u_0$, $f_2(x)$ has exactly one solution. For $\bar{\beta} \in (\bar{\beta}_0,\bar{\beta}^*)$ (blue), two new fixed points given implicitly by the roots of \ref{['eq:f2']} exist, the OEE$^+$ and the OEE$^-$, the first is stable, and the latter unstable. In this region, the IIFE is unstable, and the IEE is stable. Finally, at $\bar{\beta}=\bar{\beta}^*$ the IEE exchanges stability with OEE$^{-}$ in a transcritical bifurcation. For $\bar{\beta}>\bar{\beta}^*$ (green), the only stable equilibria are the OEE$^+$ and the OEE$^{-}$. Parameters: $k_x=0.3, k_p=0.7, \delta=0.3$.
• Figure 2: Trajectories for $12$ random initial conditions for $\bar{\beta} = 0.25$, $\bar{\beta}=0.36$, $\bar{\beta} =0.44$, and $\bar{\beta} =0.75$, that correspond to each of the regions (yellow, red, blue and green) of Fig. \ref{['fig:bifurcation-u0']}B when $u_0=0.7$. The black line in the infection plots is the endemic equilibrium of the standard SIS model. For $\bar{\beta}=0.25,0.36$, the system converges to the IIFE and IEE, respectively. For $\bar{\beta}=0.44$, agents who begin with an averter strategy converge to the endemic equilibrium of the SIS model, while agents who start with a risk seeking strategy converge to a higher infection level. For $\bar{\beta}=0.75$, the system's trajectories converge to one of the two opinionated equilibria determined by the initial opinions. Parameters: $\delta=0.3,k_p=0.7,k_x=0.3,\tau_x=1$.
• Figure 3: Initial opinion towards the risk seeking or aversion strategy is reinforced in a population with high peer pressure, and the sign of initial opinions are determinant of the infection levels at steady state. Initial averters reach an infection-free state, while initial risk seekers reach an endemic state with higher infection levels than the endemic equilibrium of the standard SIS infection level, represented by a thick black line. Parameters: $\delta = 0.3, \bar{\beta} = 0.75, u_0 = 0.9, k_p = 0.7, k_x = 0.7.$
• Figure 4: Column 1 shows contact graph $A$ and communication graphs $\hat{A}_{coop}$ and $\hat{A}_{ant}$ for $5$ subpopulations. Columns 2 and 3 show that under a cooperation regime ($\hat{A}=\hat{A}_{coop}$), subpopulations reach a state of agreement for either risk aversion (col. 2) or risk seeking (col. 3). The strategy chosen determines infection levels, and aversion reduces infection levels with respect to the standard network SIS model for the same initial conditions, while risk seeking increases infection levels. In column 4 we see that antagonism between subpopulations ($\hat{A}=\hat{A}_{ant}$) results in disagreement and some subpopulations choose risk aversion (red) and others choose a risk seeking strategy (green). For all graphs $a_{ii}=1$ and $\hat{a}_{ii}=1$ but not shown in graphs. Parameters: $\bar{\beta} = 0.5,\delta = 0.3, k_p = 0.5,k_x = 0.3, u_0 = 0.7$.

Theorems & Definitions (11)

• Theorem III.1: Positive Invariance
• proof
• Theorem IV.1
• proof
• Theorem IV.2: SIS Equivalence
• proof
• Theorem IV.3
• proof
• Corollary IV.1: Risk Seeking and Risk Aversion
• proof