## A stochastic two-stage innovation diffusion model on a lattice.(English)Zbl 1356.60164

Summary: We propose a stochastic model describing a process of awareness, evaluation, and decision making by agents on the $$d$$-dimensional integer lattice. Each agent may be in any of the three states belonging to the set $$\{0,1,2\}$$. In this model $$0$$ stands for ignorants, $$1$$ for aware, and $$2$$ for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate $$\lambda$$. At rate $$\alpha$$ an agent in aware state becomes an adopter due to the influence of adopters’ neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant, at rate $$1$$. Our purpose is to analyze the influence of the parameters on the qualitative behavior of the process. We obtain sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J60 Diffusion processes 60J27 Continuous-time Markov processes on discrete state spaces 60J28 Applications of continuous-time Markov processes on discrete state spaces 60K10 Applications of renewal theory (reliability, demand theory, etc.)
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