An expression for the rise time of a logistic function is derived, where rise time is defined to be the
time taken for a logistic to rise from 10% to 90% of the final value.
An expression for the average slope between the 10% and 90% of final value marks is also derived.
The Logistic Equation can be written
P(t ) =
β
... Eqn (1)
δ + ( β − δ ) ⋅ e − β ⋅ (t − t 0 )
where
t 0 = time of first case
P(t 0 ) = 1
and
P(t = ∞) =
β
δ
since second (exponential) term on denominator → 0)
define rise time = time for P(t) to rise from 10% to 90% of P(∞)
Let
t a = time at which P(t ) = a ⋅ P(∞)
Q(t ) = ( β − δ ) ⋅ e − β ⋅(t − t 0 )
then at t a ,
P(t a ) =
β
δ + Q(t a )
= a⋅
β
δ
solving...
∴
K Duffy
Q(t a ) =
δ ⋅ (1 − a)
a
Page 1
11/05/2003
∴ ( β − δ ) ⋅ e − β ⋅(t a − t 0 ) =
δ ⋅ (1 − a)
a
Solving for t a :
∴ e − β ⋅ (t a − t 0 ) =
δ ⋅ (1 − a)
(β − δ ) ⋅ a
 δ (1 − a) 
∴ − β ⋅ (t a − t 0 ) = ln 

 (β − δ ) ⋅ a 


(1 − a) 

1
∴ ta = t0 +
β ⋅ ln  β

(
− 1) ⋅ a 
 δ

(ln is logarithm base e)
... Eqn 3
 



1
9
 



1
∴ Rise Time = t 0.9 − t 0.1 =
⋅ ln
− ln
β

 β


  9( β − 1) 

(
−
1
)
δ

 δ
 
 
but
ln(a ) = ln(a) − ln(b),
b
so
1
ln(81)
Rise Time = 1 ln( ) =
β 81
β
It can also be shown that the slope between 10% and 90% levels is :
Rise Slope =
0.8 ⋅ β
δ
ln(81)
=
0.8 β 2
β2
⋅
= 0.182 ⋅
ln(81) δ
δ
(cases per day)
β
For studying epidemics, infection spread occurs over several days. A curve which rose over
just one day might be sampling a single source or too small a number of sources to be
reliable as an indication of how the epidemic will progress
To eliminate this possibility, the search for a fitting epidemic curve should be constrained.
A constraint like
β < ln(81)
will ensure this (ln (81) = 2.48).
Author: K Duffy, May 12th, 2003
K Duffy
Page 2
11/05/2003