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Number of contacts, immunes and susceptibles in successive time periods of transmission
Time period of infectious transmission
(1) |
Total number of contacts
(2) |
Number immune
(3) |
Number susceptible and infected
(4) |
Number susceptible and indirectly protected (5) |
Fraction of indirectly protected among all susceptibles (5)/[(4)+(5)] |
t_{1} |
1 |
- |
1 |
- |
- |
t_{2} |
3 |
1 |
2 |
- |
- |
t_{3} |
9 |
3 |
4 |
2 |
0.33 |
t_{4} |
27 |
9 |
8 |
10 |
0.56 |
t_{5} |
81 |
27 |
16 |
38 |
0.70 |
t_{6} |
243 |
81 |
32 |
130 |
0.80 |
The table is based on the imaginary example of slides 6-7 (R_{0}=3, 1/3 immune) and gives a feel of the order of magnitude of the effect of indirect protection brought about by herd immunity.
Observe the exponential increase of numbers in columns (2), (3) and (4). Note the progression of the fraction in the last column.
Remember the unrealistic nature of this exercise and the implicit assumptions mentioned in slide 8. A further assumption is that the population is large enough so that the chances of an infected case to meet – in the first 6 time periods of transmission presented– someone infected in a previous time period (and already immune) are negligible.
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