Diffusion of new products with recovering consumers

TL;DR

This paper models the diffusion of new products considering consumer recovery, analyzing how different social network structures influence adoption rates and proposing a nonlinear recovery model with negligible heterogeneity effects.

Contribution

It introduces a new discrete Bass-SIR model incorporating consumer recovery and compares diffusion dynamics across various network structures.

Findings

Complete networks lead to an aggregate model combining Bass and SIR dynamics.

One-sided 1D networks have explicit linear solutions for diffusion.

Two-sided 1D networks accelerate diffusion compared to one-sided networks.

Abstract

We consider the diffusion of new products in the discrete Bass-SIR model, in which consumers who adopt the product can later "recover" and stop influencing their peers to adopt the product. To gain insight into the effect of the social network structure on the diffusion, we focus on two extreme cases. In the "most-connected" configuration where all consumers are inter-connected (complete network), averaging over all consumers leads to an aggregate model, which combines the Bass model for diffusion of new products with the SIR model for epidemics. In the "least-connected" configuration where consumers are arranged on a circle and each consumer can only be influenced by his left neighbor (one-sided 1D network), averaging over all consumers leads to a different aggregate model which is linear, and can be solved explicitly. We conjecture that for any other network, the diffusion is bounded from below and from above by that on a one-sided 1D network and on a complete network, respectively. When consumers are arranged on a circle and each consumer can be influenced by his left and right neighbors (two-sided 1D network), the diffusion is strictly faster than on a one-sided 1D network. This is different from the case of non-recovering adopters, where the diffusion on one-sided and on two-sided 1D networks is identical. We also propose a nonlinear model for recoveries, and show that consumers' heterogeneity has a negligible effect on the aggregate diffusion.