SME
Microbiology
Immunology and pathogens
Use this for “Measles has R₀ = 12 and the vaccine is 95% effective. What share must be vaccinated?”
Diagram
92 of 100 people are immune (91.7%).
The case meets R₀ = 12 people: 11 arrows stop at immune people and 1 passes it on.
Each case now infects R₀(1 − p) = 12 × (1 − 0.9167) = 1: at or below 1, the outbreak dies out.
This is an example. Type your own values to start a new problem.
R₀
Basic reproduction number
example
pc
Herd immunity threshold
calculated · %
E
Vaccine effectiveness
example · %
Vc
Coverage needed
calculated · %
Formulas
pc
=
100
×
(1
−
1
R₀
)
91.6667
=
100
×
(1
−
1
12
)
Vc
=
100
×
pc
E
96.4912
=
100
×
91.6667
95
Assumptions
•
People mix at random, and immunity from a vaccine or past infection blocks infection fully.
•
R₀ counts the people one case infects when nobody is immune.
•
Spread dies out once each case infects fewer than one other: R₀(1 − p) < 1.
•
A vaccine E% effective makes E% of the people given it immune.
Step-by-step
WE KNOW
R₀ = 12, E = 95%
FIND
herd immunity threshold (pc), coverage needed (Vc)
Step 1 · Find herd immunity threshold (pc)
Use pc = 100 × (1 − 1 ÷ R₀)
With this share immune, each case meets on average just one person it can infect.
pc
=
100
×
(1
−
1
12
)
pc
=
100
×
11
12
pc = 91.66667%
Step 2 · Find coverage needed (Vc)
Use Vc = 100 × pc ÷ E
Only
E
%
of
those
vaccinated
become
immune,
so
more
people
than
the
threshold
must
be
vaccinated.
Vc
=
100
×
91.66667
95
Vc
=
9,166.667
95
Vc = 96.4912%
Check
Put the numbers back in. Both sides should match.
91.66667
=
100
×
(1
−
1
12
)
✓
96.4912
=
100
×
91.66667
95
✓
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Antibody titer from a two-fold dilution row
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