Cellular automata and Riccati equation models for diffusion of innovations

Innovation diffusion represents a central topic both for researchers and for managers and policy makers. Traditionally, it has been examined using the successful Bass models (BM, GBM), based on an aggregate differential approach, which assures flexibility and reliable forecasts. More recently, the rising interest towards adoptions at the individual level has suggested the use of agent based models, like Cellular Automata models (CA), that are generally implemented through computer simulations. In this paper we present a link between a particular kind of CA and a separable non autonomous Riccati equation, whose general structure includes the Bass models as a special case. Through this link we propose an alternative to direct computer simulations, based on real data, and a new aggregate model, which simultaneously considers birth and death processes within the diffusion. The main results, referred to the closed form solution, the identification and the statistical analysis of our new model, may be both of theoretical and empirical interest. In particular, we examine two applied case studies, illustrating some forecasting improvements obtained.