## 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 interconnected (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 in a circle and each consumer can be influenced only by the neighbor to the left (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 in a circle and each consumer can be influenced by the neighbors to the left and right (two-sided 1D network), the diffusion is strictly faster than on a one-sided 1D network. This is different from the case of nonrecovering adopters, where the diffusion on one-sided and 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.

Original language | English |
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Pages (from-to) | 1230-1247 |

Number of pages | 18 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 77 |

Issue number | 4 |

DOIs | |

State | Published - 2017 |

## Keywords

- Agent-based models
- Bass model
- Diffusion in social networks
- Marketing
- SIR model

## All Science Journal Classification (ASJC) codes

- Applied Mathematics