A Mathematical Model of Opinion Dynamics with Application to Vaccine Denial

TL;DR

The paper develops a macroscopic, influencer-driven model of opinion dynamics to study how official sources and prominent influencers shape public attitudes toward vaccination. It combines a deterministic ODE for individual opinion change with a Fokker-Planck-type PDE for the population density, enabling analysis of diffusion and advective transport in opinion space. The authors prove existence, uniqueness, and global asymptotic stability of the equilibrium and provide a practical method to compute the leading eigenvalue and eigenfunction that govern approach to equilibrium. Through vaccination-related scenarios, they illustrate how changes in influencer strength and alignment can polarize or re-balance the opinion landscape and discuss limitations and potential links to epidemiological dynamics.

Abstract

Public health outcomes can be heavily influenced by the landscape of public opinion; hence, it is important to understand how that landscape changes over time. For one, opinions on public health issues are responsive to official pronouncements, whether from the governmental or professional medical establishments. Additionally, in today's world of high speed communication, opinion can also be highly responsive to the broadcast opinions of "influencers" whose large numbers of followers assure them of a broad reach. To understand the opinion landscape in a general sense, we develop an ordinary differential equation model for opinion change that is based primarily on attraction to the opinions of prominent sources. The individual opinion change model is then used to develop a Fokker-Planck-type partial differential equation model for the overall opinion landscape. This model is shown to have a stable equilibrium solution, and the dependence of the equilibrium solution on key model parameters is illustrated with examples based on opinion regarding vaccination.

Figures (5)

• Figure 1: Left: The opinion dynamics function $g(x)=\dot{x}$ for two scenarios, both with moderate local attractors at $x=-1$; stable equilibria are marked with disks, while unstable equilibria are marked with squares. Right: Equilibrium solutions for the opinion density function. The solid curves are for the scenario described by \ref{['examp1']} with a strong local attractor at $x=1$, while the dashed curve represents the scenario described by \ref{['examp2']} with moderate local attractors at $x=1$ and $x=0$.
• Figure 2: Time series plots for the problem \ref{['prob1a']} using the equilibrium solution for the default scenario \ref{['examp1']} as the initial condition and the instance of $g$ from the speculative scenario \ref{['examp2']} for the partial differential equation and boundary conditions.
• Figure 3: The functions $\|Q\|(t;k)$ and $Q(x,20;0.09)$ from \ref{['Qeqn']}; the former shows that $\lambda_1 \approx 0.09$ and the latter shows a visual approximation of $\phi_1$.
• Figure 4: (left) Plot of $S(X)$ as a function of a parameter $\lambda$, where eigenvalues are the roots of this function. (right) The normalized eigenfunction $\phi_1$.
• Figure 5: (left) The distribution of opinions at equilibrium with varying $x_g$; (right) level curves of individuals with opinion at least 0 at equilibrium with varying $b_p$ and $\sigma_p$; in both cases the other parameters were set at the 2025 default levels.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof