Mathematical treatment

Main article: Logistic function
The diffusion of an innovation typically follows an S shaped curve which often resembles a logistic function. Mathematical programming models such as the S-D model apply the diffusion of innovations theory to real data problems.[31]

International Institute for Applied Systems Analysis (IIASA)[edit]Several papers on the relationship between technology and the economy have been written by researchers at the International Institute for Applied Systems Analysis (IIASA). The pertinent papers deal with energy substitution and the role of work in the economy as well as with the long economic cycle. Using the logistic function, these researchers were able to provide new insight into market penetration, saturation and forecasting the diffusion of various innovations, infrastructures and energy source substitutions.[32] Cesare Marchetti published on Kondretiev waves and on diffusion of innovations.[33] Grübler (1990) presents a mathematical discussion of diffusion and substition models.[34]