On a differential equation model of booms

A boom is a social phenomenon in which some commodity, fashion or the like is suddenly prevailed among people and is forgotten by most of them shortly after that. In this study, we introduce a mathematical model of booms and try to analyze such phenomena. This model is based on two assumptions. The first is that each of the consumers is in one of the four stages at a time: the stage in which he has not consumed the commodity yet, the stage in which he has begun to consume it after the start of the boom, the stage in which he stopped consuming and the stage of the regular consumer. Second, the increasing speed of the number of the consumers of each stage is assumed to depend only on populations of the former stages. A system of linear differential equations is formulated to describe the change in the numbers of the consumers in these four stages. The validity of the model is verified by fitting the solutions of the equations for some real data of booms such as “instant noodles boom”, “clear liquor boom”, “football boom”, and so on. The model can explain some aspects of the mechanism of the boom. We develop some quantitative arguments about each boom, and the characteristics of regional consumers in Japan are also described by the estimated parameters in the model. Our model has so simple structure that we may be able to describe some more complicated phenomena by adding some elements to this model.