K - College of Natural Resources - University of California, Berkeley

∞
∫
0
− u(
1−s
) () s = e f ( u)
g
where fv(u) is the pdf of the individual reproductive number ν for the outbreak in
question.
For a control strategy C : [0,∞) → [0,1] in which the probability of absolutely
controlling a case with individual reproductive number ν is C(ν), the pgf of the
branching process becomes
where
g
ν
du
C
∞
∫
0
ν
−u
( 1−s
) () s = c + e ( − C(
u)
) f ( u)
∞
∫
0
1 du
( u)
f ( u)
c = C ν
is the fraction of individuals controlled on average. For example, randomized absolute
(RA) control corresponds to choosing C(ν)=c for all ν. Maximally-targeted control, in
which the top c × 100% of infectious individuals are controlled absolutely, corresponds
to choosing
where νc satisfies
∞
∫
ν c
f
ν
( u)
du = c.
C
( ) ⎨
⎩ ⎧
ν =
0
1
du
if ν < ν
if ν ≥ν
Note that when ν is gamma-distributed with mean R0 and shape parameter k, the
pgf under maximally-targeted control is
g
max
⎛
⎜
⎝
R
k
0
() s = c + 1+
( 1−
s)
⎞
⎟
⎠
−k
143
⎛ Γ
⎜1−
⎝
c
c
( k,
ν c ( k R0
+ 1−
s)
)
Γ(
k)
⎞
⎟
⎠