Supporting Information 2
Additional Methods and Statistical Data
Supplement to: Still protected against smallpox? Estimation of the duration of
vaccine-induced immunity against smallpox. Epidemiology
Details of data source
Our estimations are based on the reported numbers of smallpox cases during epidemics
in the United Kingdom; variola major outbreaks were reported in London (1870s),
Sheffield (1887-8), London (1893), Liverpool (1902-3), Dewsbury (1904), and a variola
minor outbreak was reported in London in 1929-30 S1-S6. A total of 10403 (London,
1870s), 6088 (Sheffield, 1887-8), 2101 of 2376 (London, 1893), 1163 of 2280 (Liverpool,
1902-3) and 625 of 635 (Dewsbury, 1904), and 3203 (London, 1929-30) cases,
respectively, could be stratified by age group and vaccination status, which was mainly
determined from vaccination scars. The cases described in Tables S3-S8 were either
vaccinated during infancy or unvaccinated, and were classified according to severity of
disease. Overall case fatality proportions (CFP) of variola major were 17.4% (London,
1870s), 9.7% (Sheffield, 1887-8), 7.6% (London, 1893), 7.6% (Liverpool, 1902-3) and
11.7 % (Dewsbury, 1904), respectively. Only 8 (0.2 %) out of a total of 3203 cases died
-1-
in variola minor outbreak in London (1929-30).
Additional Method 1: Test for significance
The age-specific numbers of cases by vaccination history are given in Tables S3-S8.
When assessing the goodness-of-fit of our models, using χ2 statistic, we simply followed
the age-classes given in the data source to which Cochran's rule was partly applied: No
expected frequency should be less than 1 and at least 80% of expected frequencies should
be greater than five S7. Using these rules, the χ2 or likelihood ratio test statistics could not
always be applied due to a large number of expected values below 5 even after
re-grouping adjacent age-classes. To these cases, we first assessed the goodness-of-fit
based on visual assessments and then applied an exact test (see below) which was
specifically developed to solve this problem.
As shown in Figure 1 of the main text, the observed number of infections corresponded
well to the expected number of infections. The χ2 goodness-of-fit tests revealed no
significant deviations of the model from the data in all six outbreaks for the estimation of
protection against smallpox: χ23 = 2.64, p = 0.45 in London (1870s); χ22 = 5.71, p = 0.06
in Sheffield (1877-8); χ26 = 9.05, p = 0.17 in London (1893); χ24 = 6.67, p = 0.15 in
Liverpool (1902-3); χ25 = 1.31, p = 0.93 in Dewsbury (1904); and χ212 = 6.11, p = 0.91 in
-2-
London (1929-30). For protection against severe manifestations, two outbreaks were
satisfactory (χ25 = 8.46, p = 0.13 in London in 1893 and χ25 = 2.80, p = 0.73 in Dewsbury
in 1904, respectively), while the outbreak in Liverpool (1902-3) revealed significant
deviations with a greatly exaggerated p-value due to low expected numbers (χ25 = 13.04,
p = 0.0230). Applying the exact test (see below), the p-value became 0.0423 that still
indicates a significant lack of fit. Lack of fit in estimating the partial protection in
Liverpool was obviously caused by outliers including small numbers of cases that did not
follow our assumed model (Table S6).
Additional Method 2: Exact goodness-of-fit test
To develop an exact test for the goodness of fit which can be used for very low
expected values (where the standard χ2 test cannot yield reliable results), we first
calculated the probabilities of all possible results of the Binomial experiments for each
age/vaccination class given the estimated parameter values. Then we calculated the
probability of every possible combination of outcomes across all age/vaccination classes.
As there were 7 age classes and 2 vaccination (or severity) classes for Liverpool, a single
combination formed a vector of 14 different values, originating from 14 different
Binomial distributions. By construction, the probabilities of all possible 14 dimensional
-3-
vectors sum up to 1. Then we sorted these vectors by their probability of occurrence and
determined whether the actual (historical) observation was within the bulk that accounted
for 95% of the probability or whether it came to rest within the 5% of highly unlikely
combinations. For the Liverpool data set, the observed data vector hit the 95.8 percentile,
forcing us to reject the goodness-of-fit with p=0.0423. As had to be expected, none of the
other comparisons which resulted in non-significant results using the χ2 test became
significant when using the exact test as low expected values could only lead to
under-estimates of the χ2 test p-value.
References
S1. Gayton W. The value of vaccination as shown by an analysis of 10,403 cases of
smallpox. London: Gillett & Henty; 1885.
S2. Barry FW. Report on an epidemic of small-pox at Sheffield, during 1887-88. London:
Local Government Board; 1889.
S3. Ricketts TF. A classification of cases of smallpox by the numerical severity of the
eruption. In: Metropolitan Asylums Board eds., Reports for the year 1893 of the
statistical committee and the medical superintendents of the infectious hospitals and
imbecile asylums, also of the ambulance and training ship "Exmouth" committees (8th
-4-
year of issue). London: McCorquodale; 1894.
S4. Hanna W. Studies in small-pox and vaccination. Bristol: John Wright; 1913.
S5. Low JS. Report upon an epidemic of small-pox in the Borough of Dewsbury in 1904.
Dewsbury: Joseph Ward; 1905.
S6. Chalke HD. Smallpox hospitals – condition as to vaccination of patients suffering
from smallpox admitted during 1929. In: Metropolitan Asylums Board eds., Annual
Report for the year 1929-30 (32nd year of issue). London: Harrison; 1930.
S7. Cochran WG. Some methods for strengthening the common chi-square test.
Biometrics 1954; 10: 417-451.
-5-
Fig. S2-1
b)
1
0.8
1
0.8
Protected fraction
Protected fraction
a)
0.6
Expected
0.4
Upper 95% CI
0.2
Lower 95% CI
0.6
Expected
0.4
0
0
10
20
30
40
50
60
70
0
Time since successful vaccination [years]
Protected fraction
Lower 95% CI
0.2
0
c)
Upper 95% CI
10
20
30
40
50
60
Time since successful vaccination [years]
1
0.8
0.6
Expected
0.4
Upper 95% CI
Lower 95% CI
0.2
0
0
10
20
30
40
50
60
70
Time since successful vaccination [years]
Fig. S2-1. The conditional probability of protection from severe clinical
manifestations of smallpox based on expected values (and 95 % confidence
intervals) of parameter  as shown in the Supporting Materials and Methods
sections: a) London (1893), b) Liverpool (1902-3) and c) Dewsbury (1904).
-6-
70
Table S1.
Parameter estimates of the duration of vaccine-induced immunity against
infection for 6 outbreaks in the United Kingdom†
Outbreak
London (1870s)
Estimate

tm
V
Sheffield (1887-8)

tm
V
London (1893)

tm
V
Liverpool (1902-3)

tm
V
Dewsbury (1904)

tm
V
London (1929-30)

tm
V
95% confidence interval
0.0216
11.7
7.8
0.0142
10.6
7.0
0.0286
12.6
8.6
0.0593
15.1
10.7
0.0408
13.7
10.2
0.0899
17.3
12.8
0.0046
17.2
11.5
0.0022
15.4
9.2
0.0090
19.7
15.1
0.0413
23.4
14.3
0.0195
18.6
9.7
0.0904
32.2
25.9
0.0034
28.4
4.3
0.0007
24.9
2.9
0.0096
33.7
7.2
0.00013
46.0
4.4
0.00006
42.7
3.0
0.00029
50.6
7.6
†
Refer to Tables S3-8 for data and to the Materials and Methods in the main text for
explanations.
-7-
Table S2.
Parameter estimates of the duration of vaccine-induced immunity against
severe manifestation for 3 outbreaks in the United Kingdom†
Outbreak
Estimate
95% confidence interval
London (1893)

0.00477
0.00343
0.00675
Liverpool (1902-3)

0.01211
0.00990
0.01490
Dewsbury (1904)

0.00680
0.00432
0.00928
†
Refer to Tables S5-7 for data and to the Materials and Methods in the main text for
explanations.
-8-
Table S3. Age and vaccination status of smallpox cases during an outbreak in London in
1870s
Vaccinated
nv
Unvaccinated
nu
0-10
664 (664.0)
1,187 (1,187.0)
11-20
3,440 (3,440.0)
521 (521.0)
21-25
1,399 (1,393.4)
174 (179.6)
26-30
834 (831.8)
105 (107.2)
31-40
810 (808.7)
103 (104.3)
41-50
300 (309.1)
49 (39.9)
Age a (years)
The observed total numbers of vaccinated and unvaccinated cases (nv and nu) were
obtained from the records of ref. S1
-9-
Table S4. Age and vaccination status of smallpox cases during an outbreak in Sheffield in
1887-8
Age a (years)
Vaccinated
nv
Unvaccinated
nu
0-9
340 (341.7)
426 (424.3)
10-19
2,034 (2,029.8)
380 (384.2)
20-29
1,579 (1,595.8)
169 (152.2)
30-39
677 (670.4)
56 (62.7)
40-49
250 (242.4)
15 (22.7)
The observed total numbers of vaccinated and unvaccinated cases (nv and nu) were
obtained from the records of ref. S2
- 10 -
Table S5. Age, vaccination status, and severity of disease during an outbreak in London
in 1893
Age a (years)
0-4
nv
Vaccinated cases
n vm
n vs
6 (8.2)
50 (50.7)
10-14
139 (132.3)
Unvaccinated cases
n um
n us
148 (145.8)
54 (55.7)
5-9
nu
2 (0.3)
207 (205.7)
65 (66.3)
83 (83.9)
30 (29.1)
124 (123.3)
66 (72.7)
396 (394.8)
6 (7.2)
15-19
263 (266.4)
47 (43.6)
20-24
335 (343.0)
319 (322.5)
16 (12.5)
39 (31.0)
27 (24.7)
12 (14.3)
25-30
242 (244.7)
230 (229.8)
12 (12.2)
24 (21.3)
14 (14.1)
10 (9.9)
31-34
211 (207.0)
200 (196.1)
11 (14.9)
14 (18.0)
5 (7.1)
9 (6.9)
35-39
148 (141.7)
134 (134.6)
14 (13.4)
6 (12.3)
3 (2.7)
3 (3.3)
40-49
148 (148.1)
136 (135.5)
12 (12.5)
13 (12.9)
7 (7.3)
6 (5.7)
The observed numbers of total (nv and nu), mild (nvm and num) and severe cases (nvs and
nus) were obtained from ref.S3
- 11 -
Table S6. Age, vaccination status, and severity of disease during an outbreak in Liverpool
in 1902
Age a (years)
nv
Vaccinated cases
n vm
n vs
nu
Unvaccinated cases
n vm
n vs
0-4
7 (10.8)
7 (6.8)
0 (0.2)
55 (51.2)
6 (12.3)
49 (42.7)
5-9
34 (27.8)
31 (31.9)
3 (2.1)
26 (32.2)
6 (7.1)
20 (18.9)
10-14
62 (62.0)
54 (53.7)
8 (8.3)
31 (31.0)
9 (1.3)
22 (29.7)
15-19
103 (105.5)
89 (87.0)
14 (16.0)
30 (27.5)
1 (5.6)
29 (24.4)
20-29
333 (334.1)
249 (237.9)
84 (95.1)
42 (40.9)
11 (0.0)
31 (45.9)
30-39
248 (244.0)
163 (171.1)
85 (76.9)
14 (18.0)
0 (1.4)
14 (12.6)
40-49
101 (103.7)
63 (68.1)
38 (32.9)
10 (7.3)
1 (2.2)
9 (7.8)
The observed numbers of total (nv and nu), mild (nvm and num) and severe cases (nvs and
nus) were obtained from ref. S4
- 12 -
Table S7. Age, vaccination status, and severity of disease during an outbreak in Dewsbury
in 1904
Age a (years)
nv
Vaccinated cases
n vm
n vs
nu
Unvaccinated cases
n vm
n vs
0-4
1 (0.4)
1 (1.0)
0 (0.0)
44 (44.6)
26 (26.0)
18 (18.0)
5-9
2 (2.2)
2 (2.0)
0 (0.0)
48 (47.8)
32 (32.0)
16 (16.0)
10-14
9 (8.9)
26 (27.5)
3 (1.5)
61 (60.2)
64 (64.8)
65 (65.1)
15-19
20 (21.2)
60 (58.8)
20-24
42 (41.1)
37 (37.8)
5 (4.2)
46 (46.9)
14 (13.7)
32 (32.3)
25-29
45 (45.9)
40 (40.0)
5 (4.9)
25 (24.1)
9 (9.0)
16 (16.0)
30-39
93 (90.8)
79 (79.6)
14 (13.4)
21 (23.2)
7 (6.8)
14 (14.2)
40-49
57 (58.5)
46 (43.3)
11 (13.7)
15 (13.6)
1 (1.3)
14 (13.7)
The observed numbers of total (nv and nu), mild (nvm and num) and severe cases (nvs and
nus) were obtained from ref. S5
- 13 -
Table S8. Age and vaccination status of smallpox cases during an outbreak in London in
1929.
Age a (years)
Vaccinated
nv
Unvaccinated
nu
0-4
0 (0.2)
379 (378.8)
5-9
0 (1.3)
668 (666.7)
10-14
3 (3.5)
587 (586.5)
15-19
9 (7.0)
440 (442.0)
20-24
12 (9.1)
218 (221.0)
25-29
12 (15.8)
156 (152.2)
30-34
27 (24.7)
93 (95.3)
35-39
37 (38.2)
62 (60.8)
40-44
49 (50.5)
37 (35.5)
45-49
69 (66.8)
22 (24.2)
50-54
46 (47.3)
13 (11.7)
55-59
45 (44.0)
9 (10.1)
60-64
13 (13.0)
3 (3.0)
65-69
6 (5.7)
1 (1.3)
70-74
4 (4.9)
2 (1.1)
The observed total numbers of vaccinated and unvaccinated cases (nv and nu) were
obtained from the records of ref. S6
- 14 -