# Innovation diffusion models

2015-08-26

To simplify matters I will take out any time offset and market size factor, they can be reintroduced by setting:

$A'(t) = M_0 + M A(t - t_0)$

## Super diffusion model

$\frac{\operatorname*{{d}}\nolimits_{{A}}^{{}}\;}{\operatorname*{{d}}\nolimits_{{t}}^{{}}\;} = \left( p + q A \right) (1 - A^\nu )$

## Bass Diffusion Model

$\frac{\operatorname*{{d}}\nolimits_{{A}}^{{}}\;}{\operatorname*{{d}}\nolimits_{{t}}^{{}}\;} = \left( p + q A \right) (1 - A)$

$A(t) = \frac{1 - \mathrm e^{-(p + q)t}}{1 + \frac qp \mathrm e^{-(p+q)t} }$

## Generalised logistic function

$\frac{\operatorname*{{d}}\nolimits_{{A}}^{{}}\;}{\operatorname*{{d}}\nolimits_{{t}}^{{}}\;} = \alpha \nu \left( 1 - \left( \frac A K \right)^{\frac 1 \nu } \right) A$

$A(t) = \frac{1}{(Q + \mathrm e^{-B(t-M)} )^{\frac 1 \nu }}$

## Gompertz model

Take the generalised logistic function

$\frac{\operatorname*{{d}}\nolimits_{{A}}^{{}}\;}{\operatorname*{{d}}\nolimits_{{t}}^{{}}\;} = \alpha \nu \left( 1 - \left( \frac A K \right)^{\frac 1 \nu } \right) A$

and apply the limit $\nu \rightarrow \infty$:

$\lim_{v \rightarrow \infty } \nu \left( 1 - x^{\frac 1 \nu } \right) = -\log x$

then

$\frac{\operatorname*{{d}}\nolimits_{{A}}^{{}}\;}{\operatorname*{{d}}\nolimits_{{t}}^{{}}\;} = - \alpha \log \left( \frac A K \right) A$

$A(t) = \mathrm e^{-b \mathrm e^{-c t}}$

$b = -log(A(0))$

$\frac{\operatorname*{{d}}\nolimits_{{A}}^{{}}\;}{\operatorname*{{d}}\nolimits_{{t}}^{{}}\;} = c \log \left( \frac {X(b,c)} A \right) A$

## Simple logistic function

Take the GLF and set $\nu = 0$:

$\frac{\operatorname*{{d}}\nolimits_{{A}}^{{}}\;}{\operatorname*{{d}}\nolimits_{{t}}^{{}}\;} = A \cdot (1 - A)$

$A(t) = \frac{M}{1 + \mathrm e^{-x}}$