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

decreasing function of x for s œ [0,1) and P(X2>x)>P(X1>x) for all x > 0, we have
P(Y1>x)>P(Y2>x) for all x>0. Hence, s)
c ( 1 c)
E(
Y ) c ( 1 c)
E(
Y ) g ( s),
=
+
−
gC C
>
( 1
1
2
2
and as argued above we have g 0)
> g ( 0)
for all generations n and therefore
C1
q >
q
C2
.
C n ( C , n
1,
2
To see the utility of this claim, let us consider two control strategies C1 and C2
that control two portions of the population in different ways. Suppose strategy Ci
*
controls the less-infectious portion of the population (i.e. ν < ν ) with probability ai and
*
controls the more-infectious portion of the population (i.e. ν ≥ ν ) with probability bi.
In other words,
C
i
( ) ⎨
⎩ ⎧
ν
a
*
if ν < ν
i
= *
b
i
if ν ≥ν
Moreover, let us assume that both strategies control the same fraction of individuals, i.e.
∞
∫C
i
0
( u)
f ( u)
du = c
ν
for i=1,2. Suppose that strategy 1 targets more-infectious
individuals to a greater degree than strategy 2, i.e. b1> b2 and thus a1 b
∞
∫
1
ν
∞
∫
2
ν
f
ν
ν
∞
( u)
du = C ( u)
f ( u)
145
du
∫
ν
2
ν
ν
*
∫
0
+
ν
du
−
=
( ) 80