Innovation diffusion models

\gdef\e{\mathrm e} \gdef\fd#1#2{\frac{\mathrm{d} #1}{\mathrm{d} #2}}

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

\fd A t = \left( p + q A \right) (1 - A^ν)

Bass Diffusion Model

\begin{aligned} \fd A t &= \left( p + q A \right) (1 - A) \\ A(t) &= \frac{1 - \e^{-(p + q)t}}{1 + \frac qp \e^{-(p+q)t} } \end{aligned}

Generalised logistic function

\begin{aligned} \fd A t &= α ν \left( 1 - \left( \frac A K \right)^{\frac 1 ν} \right) A \\ A(t) &= \frac{1}{(Q + \e^{-B(t-M)} )^{\frac 1 ν}} \end{aligned}

Gompertz model

Take the generalised logistic function

\fd A t = α ν \left( 1 - \left( \frac A K \right)^{\frac 1 ν} \right) A

and apply the limit ν → ∞:

\lim_{v → ∞} ν \left( 1 - x^{\frac 1 ν} \right) = -\log x

then

\begin{aligned} \fd A t &= - α \log \left( \frac A K \right) A \\ A(t) &= \e^{-b \e^{-c t}} \\ b &= -log(A(0)) \\ \fd A t &= c \log \left( \frac {X(b,c)} A \right) A \end{aligned}

Simple logistic function

Take the GLF and set ν = 0:

\begin{aligned} \fd A t &= A · (1 - A) \\ A(t) &= \frac{M}{1 + \e^{-x}} \end{aligned}

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