A Mathematical Model of Opinion Dynamics with Application to Vaccine Denial

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 prominent sources whose opinions are independent of the opinions of others. 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. Eigenvalues and eigenfunctions for the associated Sturm–Liouville problem are determined numerically by two methods, one by using numerical solutions of the partial differential equation and the other by applying a shooting method directly to the eigenvalue problem.