These models are used to forecast new product (eg internet) adoption and the emphasis is on predicting the ultimate level of penetration (saturation) and the rate of approach to saturation. The theory of diffusion of innovations considers how a new idea, or adoption of new behaviour, spreads throughout the market over time. There are strong parallels with epedemiology - the study of how a contagious disease spreads.
A purely innovative process has no contagion - members of the population spontaneously adopt at a certain rate. The process is:
dY/dt = p*(S-Y)
where Y is the number who have adopted, S is the saturation adoption (some members of the population may be "immune", so S is usually less than the total population); p is the rate of adoption. At any instant, (S-Y) is the population at risk and p is the hazard rate.
This innovative process yields the modified exponential:
Y = S*(1 - exp(-p*t))
A purely imitative process has contagion only. The process is:
dY/dt = q*(Y/S)*(S-Y)
the population at risk is (S-Y) as before, but the rate of contagion increases as people become infected: q*(Y/S).
This imitative process yields a logistic (S-shaped) growth curve:
Y = S/(1 + C*exp(-q*t))
The Bass model contains both innovation and imitation:
dY/dt = (p + q*(Y/S))*(S-Y).
When few people have adopted, innovation drives adoption. As more consumers adopt, awareness will grow, acceptability and social pressure will grow and contagion starts to drive growth.
Diffusion models are hard to calibrate prior to launch and the models tend to be unstable until the point of inflection occurs. It is important to derive a good estimate of the saturation level, S, via choice models or other means. Surveys can also provide guidance as to the p and q parameters.
A number of generalisations of the Bass model are possible, especially the introduction of marketing mix variables. The saturation level may be affected by price (via choice models) for example and the rates of adoption could be influenced by advertising.
See "Marketing Engineering" by Lilien and Rangaswamy for more information.