Modeling Spatial Technology Diffusion with Reaction-Diffusion Equations

Abstract

Modeling of diffusion of technology, that is the process by which a new technology spreads across populations, was heavily influenced by E. Rogers' foundational book, "Diffusion of Innovations", published in 1962, followed by F. Bass' famous model published in 1969. The Bass model classified adopters into innovators, making adoption decisions independently, and imitators, whose decisions are guided by their interactions with existing adopters. Many extensions of and alternatives to this model have been proposed since - however, they have generally modeled aggregate numbers of adoptions over time. Few attempts have been made to describe the propagation of the adoption process across space, yet it has long been known that the spatial dynamics heavily influence the aggregate numbers.

Inspired by the approaches common in mathematical biology, we develop from first principles a general system of partial differential equations. In addition to featuring innovator and imitator densities, it also models two auxillary fields: information and communication. The model is highly flexible, featuring five unspecified functions.

In the theoretical part of the work we rigorously establish well-posedness of the model, show that the classic Bass model is its special case, and develop a procedure for exact reproduction of any aggregate adoption model with a known hazard function - naturally equipping all such models with the spatial diffusion component. We also explore structural properties of the simplest version of the model. Then we develop a Finite Element Method (FEM) approach to approximate solution of the system and prove its convergence to the exact solution. Finally, we develop a stochastic version of the model applicable to cases with low adopter densities and prove its convergence to the deterministic model.

In the main numerical part of the dissertation we provide a detailed description of the computer implementation of the model, from the mesh generation and mathematical underpinnings of the FEM solver to the multi-stage hyperparameter optimization process. We also develop benchmark tests to compare our model performance to the alternatives, for which we develop extensions to two existing competing models, as well as a standalone machine learning model.

Then we show that the model, when applied to the US market of residential photo-voltaic (PV) solar panels, reproduces the observed spatial adoption dynamics across a variety of geographical scales. In benchmark tests, its performance surpasses that of the two existing competing models, both in in-sample fitting and in forecasting, and the performance can be further significantly improved by refining the model function choices.

In the remaining parts of the work we numerically implement the stochastic version of the model and use it to reproduce results from other researchers who developed a network-based stochastic version of the Bass model. We also apply it to the US market of large scale PV (LSPV) facilities and find structural similarities between results of simulations and the LSPV data.