Innovation diffusion models

Remco Bloemen

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)=M0+MA(tt0) A'(t) = M_0 + M A(t - t_0)

Super diffusion model

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

Bass Diffusion Model

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

A(t)=1e(p+q)t1+qpe(p+q)t A(t) = \frac{1 - \mathrm e^{-(p + q)t}}{1 + \frac qp \mathrm e^{-(p+q)t} }

Generalised logistic function

dAdt=αν(1(AK)1ν)A \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)=1(Q+eB(tM))1ν A(t) = \frac{1}{(Q + \mathrm e^{-B(t-M)} )^{\frac 1 \nu }}

Gompertz model

Take the generalised logistic function

dAdt=αν(1(AK)1ν)A \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 ν\nu \rightarrow \infty :

limvν(1x1ν)=logx \lim_{v \rightarrow \infty } \nu \left( 1 - x^{\frac 1 \nu } \right) = -\log x

then

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

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

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

dAdt=clog(X(b,c)A)A \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 ν=0\nu = 0:

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

A(t)=M1+ex A(t) = \frac{M}{1 + \mathrm e^{-x}}