Innovation diffusion models

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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

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

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

Generalised logistic function

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

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

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

$$ \fd A t = - α \log \left( \frac A K \right) A $$

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

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

$$ \fd A t = c \log \left( \frac {X(b,c)} A \right) A $$

Simple logistic function

Take the GLF and set $ν = 0$:

$$ \fd A t = A · (1 - A) $$

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