Title: Diffusion of new products with recovering consumers,
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.