SME
Grade 9 Science · Problem type
Herd immunity
Use this for “Each case of a disease infects 5 people, and the vaccine works in 95% of people. What share must be vaccinated to stop it spreading?”
Diagram
84.2% of 19,000 is 16,000.
The dashed line and the band along the bar’s bottom mark H = 80%.
People to vaccinate: 16,000
This is an example. Type your own values to start a new problem.
H (%)
=
100
×
(1
−
1
÷
R₀
)
e
Vaccine effectiveness
example · %
C
Share to vaccinate
worked out (not typed) · %
P
People in the community
example
V
People to vaccinate
worked out (not typed)
Formulas
H
=
100
×
(1
−
1
R₀
)
80
=
100
×
(1
−
1
5
)
C
=
100
×
H
e
84.2
=
100
×
80
95
V
=
⌈
P
×
H
e
⌉
16,000
=
⌈19,000
×
80
95
⌉
Assumptions
•
R₀ is how many people one case infects when no one is immune; it differs by disease, about 12–18 for measles.
•
Spread stops once each case infects fewer than one more: a share H = 1 − 1 ÷ R₀ must be immune.
•
A vaccine that works in e% of people means more must be vaccinated; everyone is assumed to mix evenly.
Step-by-step
WE KNOW
R₀ = 5, e = 95%, P = 19,000
FIND
share immune to stop spread (H), share to vaccinate (C), people to vaccinate (V)
Step 1 · Find share immune to stop spread (H)
Use H = 100 × (1 − 1 ÷ R₀)
Each
case
must
infect
fewer
than
one
person,
so
all
but
1
in
R₀
must
be
immune.
H
=
100
×
(1
−
1
5
)
H
=
100
×
(1
−
0.2)
H
=
100
×
0.8
H
=
80%
Step 2 · Find share to vaccinate (C)
Use C = 100 × H ÷ e
Only
e
%
of
those
vaccinated
become
immune,
so
divide
the
share
needed
by
e
%.
C
=
100
×
80
95
C
=
8,000
95
C
=
84.2%
Step 3 · Find people to vaccinate (V)
Use V = ⌈P × H ÷ e⌉
C
%
of
P
is
P
×
H
e
,
because
C
=
100
×
H
e
;
round
up
to
whole
people,
so
enough
are
immune.
V
=
⌈19,000
×
80
95
⌉
V
=
16,000
Check
Put the numbers back in. Both sides should match.
80
=
100
×
(1
−
1
5
)
✓
84.2
≈
100
×
80
95
✓
16,000
=
⌈19,000
×
80
95
⌉
✓
Topic check
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