Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

An SIS-type marketing model on random networks

Pages: 1723 – 1740



Reinhard Illner (Department of Mathematics and Statistics, University of Victoria, British Columbia, Canada)

Junling Ma (Department of Mathematics and Statistics, University of Victoria, British Columbia, Canada)


Marketing on random networks displays similarities to epidemiological models in the sense that “word-of-mouth” information passes between individuals and may “infect” susceptible buyers such that they end up buying the product. The difference to epidemics is that there are usually many competing products (rather than just one disease), and in addition to word-of-mouth transmission, products are also advertised by the producers, which can be thought of as external nodes connected to the network. In this paper we develop a model in which these various transmission pathways compete, and, in addition, where product fatigue and product switching are possible. This is a genuine and realistic extension of the model developed in [M. Li, R. Edwards, R. Illner, and J. Ma, Commun. Math. Sci., 13, 497–509, 2015], where a customer would never abandon a product after purchase. The model presented here is similar to and was inspired by SIS epidemiological models. We discuss the homogeneous limit for a fully connected graph, present some analytical properties of the models and conduct a number of numerical experiments, including an investigation of a modelling assumption we call “edge chaos”. The validity of this assumption turns out to depend on the type of the underlying random network.


random network, SIS marketing model, product fatigue, product switch

2010 Mathematics Subject Classification


Published 12 August 2016