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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