Modeling change in public sentiment with nonlocal reaction-diffusion equations

This is a brief ''proof of concept'' article that shows a nonlocal variant of a reaction-diffusion equation, which is already well suited to the study of pattern formation, is a plausible tool for the particular application of modeling change in public sentiment. Public sentiment is ubiquitous in modern society; from the marketing of consumer goods to the choice of political rhetoric, we are all affected by it.Modeling change in public sentiment has an important role when analyzing and predicting the course society is on.Of course, change is permanent. Perhaps the best-known feature in public sentiment is polarization.By this, we mean the development or establishment of a certain subset of a population whose views on a given subject are fixed to one end of the spectrum.It is when public sentiment undergoes a change do we witness the emergence of polarization.In large part we see polarization arise through the use of persuasion and rhetoric.Moreover, some marketing or arguments are more effective on certain individuals, thus leading these people into a polarized regime. Our model captures this emergence of the polarized regime. In addition, in some cases, we see the development of true polar opposites (which we call mixed polarity).Our method features: • We use a nonlocal Chafee–Infante reaction-diffusion equation to model the evolution of public sentiment in a population that interacts with other individuals. • We employ a pseudo-random convolution kernel as a symmetric matrix of lognor-mally distributed values. This kernel models the influence of individuals when interacting with others. • Change in sentiment emerges and may converge to a polarized state expressed by a double-well potential. Other more complicated states occur whereby a mixed polarization emerges.