|
Introduction
Influenza outbreaks are notoriously difficult to predict,
even when a seasonal outbreak is underway, both in their likely time
course and severity at an individual and a population level (1). Even
two subsequent annual outbreaks caused by an identical strain of the
virus can have very different impacts on both the timing and the levels
of resulting illness in the population.
The major real time indicator of influenza activity in
Scotland comes from the sentinel network of volunteer general practices
(2). This spotter scheme currently involves 90 practices in 12 health
board areas covering a total of 10% of the Scottish population. Participating
practices submit weekly totals for the approximate number of consultations
for ‘flu-like illness’ from which can be derived a consultation rate
per 100,000 based on population projections from the sample reporting.
Although essentially a voluntary set-up, in which not all health boards
are represented, the flu-spotter network has proven to be a consistently
reliable early indicator of the onset of seasonal influenza illness,
since the scheme’s inception in 1972.
As well as serving to illustrate the wide between season
variability of influenza outbreaks in both timing and magnitude, an
examination of cumulative plots for spotter data from past seasons reveals
classic sigmoid curves (see Figure 1). It was postulated that the rate
of increase at the midpoint of the outbreak, where the rise in reporting
for flu like illness is greatest, may be used to predict the likely
total number of cases for that season as estimated by the cumulative
flu spotter totals. From the cumulative plot, the best approximation
that can be measured for the midpoint of a seasonal outbreak would be
the maximum rate of increase between any two consecutive weeks.
Methods
Data on consultations for influenza like illness was available
from Scottish GP spotter practices for the years 1972 to 1999. Estimates
for the total numbers of cases seen in each week were derived during
the flu spotter season (weeks 40 to 20 of the following year) by multiplying
the overall Scottish rate per 100 000 by 51.2 (population 5.12 million).
The differences between the numbers of cases one week
and the next week were calculated for each week of the season and the
maximum increase for each year was noted.
The dataset was log-transformed and a linear regression
model was fitted to the total number of cases vs. the maximum increase
seen for each season between any two consecutive weeks. A 95% prediction
interval was calculated for the expected total numbers of cases dependent
on the maximum increase. The resulting model was then used to provide
weekly estimates of the total numbers of expected cases by week 20 during
the ongoing flu seasons of 1999–2000 and 2000–2001.
Results
Performing a simple linear regression with the total estimated
cumulative cases for each season versus the maximum increase (d) (corresponding
to the sharpest rise in the rate) (both log transformed) gives rise
to a significant positive correlation (p < 0.005, R2 = 72%), which
can be described as follows (with 95% prediction interval*) (Figure
2):
log (expected total) = 7.5134 + 0.4693 x log (max. increase)
Giving
Expected total = exp(7.5134) x max increase 0.4693
Upper / lower PI = exp (7.1534+/-1.96*0.1998) + max increase
0.4693
[*95% PI based on the residual standard deviation about
the fitted line].
Application of the model
The utility of the model was then investigated for the
winter flu season of 1999/2000. The sharpest increase in the GP spotter
rates occurred between week 52 and 53 and gave rise to an expected total
of 169 057 consultations with 95% prediction interval ([114 277–250
096]) for the whole season. At the end of the season, the actual estimate
based on cumulative figures from week 40 to week 20 was 175 787, less
than 5% difference from the predicted total. Since there is no way of
knowing in advance what the maximum change will be, the estimate of
total likely consultations was revised weekly throughout the season,
based on the extent of change over the previous week (see appendix 3).
The continuously revised estimate made it possible to expect by week
53 that 1999–2000 was likely to be more severe than the flu season of
1998–99 with a probability of 84% (based on the standard deviation of
the prediction interval), where the total estimated cases at the end
of the season was 137 336.
A revised model (incorporating the results of the 1999–2000
season – revised expression:
Log (expected total) = 7.526 + 0.468 x log (max. increase),
giving:
Expected total = exp(7.526) x max increase 0.468
was then applied during the 2000/01 season, a winter that
saw the lowest flu activity since 1972, and spotter rates that rarely
exceeded the baseline threshold level of 50 consultations per 100000
population (5). Even at this very low level of activity, the final cumulative
total for consultations (54 033) was still within the predicted range
(Predicted total = 46 556; 95% PI = 7089,305775).
Discussion
Seasonal outbreaks of influenza are difficult to predict
for a number of reasons. The continual antigenic changes between seasons,
the introduction of new viral strains, the high proportions of sub-clinical
infections and continuing controversy over factors which affect transmission
all combine to frustrate attempts to model or define a ‘typical’ influenza
outbreak. Since even modest influenza outbreaks can exert additional
pressures on health services however, the benefits for planning and
healthcare purposes of a model that is simple to apply and has some
capacity to predict the course of an ongoing outbreak are self-evident.
The main drawbacks to the above model as a predictive
tool are firstly the very wide prediction intervals which accompany
the estimated eventual size of the outbreak and secondly, like all linear
regression models, it becomes less reliable at the extreme ends of the
range of the source data on which it is based (3). Since in prediction
intervals, the scatter of the individual data about the fitted line
becomes more directly relevant, they are invariably much wider than
the equivalent confidence interval for the fitted values (6). In theory
it should be possible also to refine the model with each additional
season, although the nature of prediction intervals means again that
their likely reduction will be small. The increasing availability of
rapid virological testing also makes it possible to identify quickly
the underlying virus types that are contributing to an increase in illness
presentation (eg: A alone, B alone or A + B). The well established differences
in severity and population health impact between A and B strains (7)
may mean that introducing interaction terms to the regression, according
to the epidemic type as suggested by Dab et al, could improve the predictive
capability of the model (8). A model which took account of virus type
may also be able to begin to address the likely time course of an ongoing
outbreak, often as important a consideration with regard to health service
planning as overall population attack rate.
Although the limitations of the model prevent its adoption
as a definitive predictive tool, its usefulness relates more to the
capacity to provide a dynamic weekly revisable estimate of the likely
severity of an ongoing flu outbreak. While the current model does not
specifically address the timing of any peak, large increases in consulting
rates are likely to be followed with higher workloads in secondary health
services. Additionally, although consulting patterns are not by any
means the only indicator of influenza activity, they are certainly the
timeliest and sentinel practice networks like that in Scotland are used
widely throughout Europe 9. Variations of the presented model may also
therefore be of interest to other countries that have a significant
historical dataset.
Conclusion
Tillet and Spencer have previously highlighted the potential
of cumulative totals of GP consultations, among other indicators, for
describing the extent of influenza outbreaks in England and Wales (4).
The model presented here demonstrates that it is possible to describe
the relationship between cumulative total numbers of consultations and
the maximum weekly increase for seasonal outbreaks of influenza using
simple linear regression, allowing predictions for the eventual size
of an outbreak to be revised as the winter season progresses. The wide
ranging prediction interval seen during the exceptionally mild influenza
season of 2000–01, although in keeping with the diminishing applicability
of regression models at the extremes of their range, is probably not
a serious practical limitation in that the main use of the model would
be to flag up potentially large epidemics as early as possible. The
increased availability of rapid virological testing may make it possible
to further refine models such as that presented here, on the basis of
the type(s) of influenza in circulation in any one season.
Annex 1. Winter season
|
Semaine N° /
Week no.
|
94-95
|
95-96
|
96-97
|
97-98
|
98-99
|
*99-00
|
*00-01
|
|
40
|
1863.68
|
1709.568
|
1336.931
|
500
|
1204.164
|
823.36
|
1389.42
|
|
41
|
4290.56
|
3850.752
|
2791.19
|
1139.648
|
2037.816
|
1698.18
|
2315.7
|
|
42
|
6656
|
6475.264
|
4548.035
|
3631.856
|
4132.238
|
2727.38
|
4219.72
|
|
43
|
9057.28
|
10698.24
|
6777.797
|
6036.067
|
5603.994
|
4065.34
|
5660.6
|
|
44
|
11822.08
|
14717.44
|
8470.831
|
8044.551
|
7369.072
|
5454.76
|
7307.32
|
|
45
|
14510.08
|
20331.01
|
10296.12
|
10598.51
|
10214.81
|
7152.94
|
9005.5
|
|
46
|
17146.88
|
31774.21
|
12739.44
|
14304.15
|
12499.63
|
8645.28
|
10343.46
|
|
47
|
20413.44
|
46856.19
|
15036.1
|
17013
|
15257.89
|
10600.76
|
11629.96
|
|
48
|
23470.08
|
66041.34
|
18284.25
|
20868.9
|
18726.29
|
13791.28
|
12659.16
|
|
49
|
26101.76
|
86018.56
|
23800.25
|
24574.53
|
21674.95
|
16981.8
|
14048.58
|
|
50
|
29178.88
|
105393.2
|
30764.85
|
28399.04
|
25771.17
|
23157
|
15540.92
|
|
51
|
32460.8
|
124280.8
|
40738.82
|
31496.42
|
29270.45
|
38955.22
|
17393.48
|
|
52
|
34949.12
|
138060.3
|
59248.99
|
34459.48
|
32563.89
|
69934.14
|
19709.18
|
|
1
|
37698.56
|
151779.8
|
93457.54
|
37703.01
|
40797.49
|
113109.1
|
22590.94
|
|
2
|
41216
|
162918.4
|
132469.9
|
42823.79
|
56698.63
|
142389.8
|
25781.46
|
|
3
|
44661.76
|
169642
|
162121.6
|
46930.3
|
72239.55
|
155666.5
|
28354.46
|
|
4
|
48030.72
|
174107.1
|
179232.1
|
50527.35
|
87008.57
|
162407.8
|
31030.38
|
|
5
|
51456
|
177203.2
|
192917.9
|
53084.92
|
97763.71
|
165649.7
|
33140.24
|
|
6
|
55183.36
|
179746.8
|
203368.4
|
57175.99
|
106203.1
|
168325.7
|
36227.84
|
|
7
|
59361.28
|
182132.7
|
210563
|
61014.9
|
111915.2
|
170075.3
|
38800.84
|
|
8
|
64184.32
|
183786
|
215819.6
|
67087.18
|
116392.2
|
171207.4
|
41270.92
|
|
9
|
68198.4
|
185470
|
220106.8
|
73097.71
|
119325.4
|
172185.2
|
43072.02
|
|
10
|
72576
|
186652.7
|
222397.3
|
78444.4
|
122104.3
|
172802.7
|
45542.1
|
|
11
|
77317.12
|
187955.2
|
224594.6
|
83600.7
|
124780.2
|
173265.8
|
47857.8
|
|
12
|
82150.4
|
189530.6
|
226985.9
|
89348.78
|
127713.4
|
173523.1
|
49658.9
|
|
13
|
86579.2
|
191117.3
|
228600.2
|
94067.66
|
129514.5
|
173986.3
|
50996.86
|
|
14
|
90341.38
|
192132.1
|
229484.3
|
98179.31
|
130955.4
|
174397.9
|
51974.6
|
|
15
|
92318.21
|
193004
|
230396.7
|
101570.5
|
132087.5
|
174706.7
|
52900.88
|
|
16
|
93951.49
|
193682.9
|
231032.8
|
103634.1
|
132910.9
|
175118.4
|
53312.56
|
|
17
|
95152.64
|
194412
|
231993.5
|
105491.8
|
133837.2
|
175324.2
|
53569.86
|
|
18
|
96152.58
|
195476.5
|
232564.7
|
106989.3
|
135483.9
|
175427.1
|
53827.16
|
|
19
|
96593.41
|
195809.8
|
233001.1
|
107622.2
|
136513.1
|
175633
|
53930.08
|
|
20
|
97392.13
|
195993.6
|
233397.9
|
108250
|
137336.4
|
175787.4
|
54033
|
*saisons 99/00 et 00/01 (consultations cumulées
d’après les données du système de surveillance)
également montrées / 99/00 and 00/01 seasons (cumulative
consultations from Spotter data), also shown.
Annex 2. Data and table: Regression line and constituent
values used for model
|
Année / Year
|
Nombre estimé de cas /
Total est.cases
|
Augmentation maximale / Max. increase
|
Limite inférieure de l’IP
/ Lower limit PI
|
Valeur prévue / Predicted
value
|
Limite supérieure de l’IP
/ Upper limit PI
|
|
72
|
199731
|
28313
|
104826.0
|
220514.1
|
336202.2
|
|
73
|
231526
|
45824
|
171721.9
|
290533.8
|
409345.7
|
|
74
|
213606
|
25190
|
92354.9
|
208026.4
|
323698.0
|
|
75
|
466125
|
64972
|
239432.7
|
367099.2
|
494765.7
|
|
76
|
193434
|
14438
|
48155.9
|
165033.4
|
281910.8
|
|
77
|
299162
|
48742
|
182386.6
|
302201.7
|
422016.8
|
|
78
|
244378
|
19610
|
69659.7
|
185714.2
|
301768.7
|
|
79
|
222925
|
20941
|
75121.1
|
191036.3
|
306951.6
|
|
80
|
197274
|
21299
|
76584.9
|
192467.8
|
308350.8
|
|
81
|
396083
|
33945
|
126897.8
|
243034.3
|
359170.7
|
|
82
|
312576
|
28825
|
106854.8
|
222561.4
|
338268.0
|
|
83
|
202035
|
37376
|
140082.2
|
256753.5
|
373424.8
|
|
84
|
162560
|
15514
|
52666.3
|
169335.9
|
286005.4
|
|
85
|
247091
|
24371
|
89056.9
|
204751.6
|
320446.3
|
|
86
|
182989
|
18637
|
65648.4
|
181823.5
|
297998.7
|
|
87
|
134349
|
7424
|
18292.1
|
136987.1
|
255682.1
|
|
88
|
165120
|
18022
|
63104.8
|
179364.4
|
295624.0
|
|
89
|
242125
|
60621
|
224497.0
|
349701.2
|
474905.5
|
|
90
|
103475
|
7015
|
16526.4
|
135351.7
|
254177.0
|
|
91
|
169011
|
20429
|
73023.8
|
188989.1
|
304954.3
|
|
92
|
151589
|
21569
|
77687.5
|
193547.5
|
309407.4
|
|
93
|
243968
|
45210
|
169460.9
|
288078.6
|
406696.4
|
|
94
|
97392
|
4833
|
7062.3
|
126626.7
|
246191.1
|
|
95
|
197433
|
19978
|
71172.7
|
187185.7
|
303198.7
|
|
96
|
232208
|
38815
|
145553.7
|
262507.5
|
379461.4
|
|
97
|
109114
|
6073
|
12449.7
|
131585.0
|
250720.3
|
|
98
|
134420
|
15902
|
54288.1
|
170887.3
|
287486.6
|
Annex 3. Season 1999 / 2000 – Predicted total consultations based
on weekly changes
|
Semaine N° / Week N°
|
Total
|
Total
|
Taux pour / Rate per
|
Total des cas / Total cases
|
Hausse semaine précédente
/
|
*Total prévu pour la saison
|
IC inférieur
|
IC supérieur
|
Cumulé / cumulative
|
|
NO.
|
Cas / cases
|
Denom.
|
100,000
|
Taux x 52 /
Rate x 52
|
Increase over prev. week
|
*Pred. total for season
|
Lower CI
|
Upper CI
|
À ce jour / to date
|
|
40
|
48
|
456278
|
10.5199
|
547
|
|
|
|
|
547
|
|
41
|
68
|
430612
|
15.79148
|
821
|
274
|
25531
|
17258
|
37769
|
1368
|
|
42
|
72
|
427399
|
16.84609
|
876
|
55
|
25531
|
17258
|
37769
|
2244
|
|
43
|
86
|
434944
|
19.77266
|
1028
|
152
|
25531
|
17258
|
37769
|
3272
|
|
44
|
118
|
448722
|
26.29691
|
1367
|
339
|
28213
|
19071
|
41737
|
4639
|
|
45
|
117
|
441591
|
26.49511
|
1378
|
10
|
28213
|
19071
|
41737
|
6017
|
|
46
|
144
|
440840
|
32.66491
|
1699
|
321
|
28213
|
19071
|
41737
|
7716
|
|
47
|
127
|
437530
|
29.02658
|
1509
|
-189
|
28213
|
19071
|
41737
|
9225
|
|
48
|
164
|
430929
|
38.05731
|
1979
|
470
|
32888
|
22232
|
48654
|
11204
|
|
49
|
263
|
427391
|
61.53616
|
3200
|
1221
|
51478
|
34798
|
76155
|
14404
|
|
50
|
272
|
436715
|
62.28318
|
3239
|
39
|
51478
|
34798
|
76155
|
17643
|
|
51
|
536
|
447541
|
119.7656
|
6228
|
2989
|
78360
|
52969
|
115922
|
23871
|
|
52
|
1394
|
454703
|
306.5737
|
15942
|
9714
|
136243
|
92096
|
201552
|
39813
|
|
53
|
2291
|
380284
|
602.4445
|
31327
|
15385
|
169057
|
114277
|
250096
|
71140
|
|
1
|
3478
|
410038
|
848.2141
|
44107
|
12780
|
169057
|
114277
|
250096
|
115247
|
|
2
|
2501
|
439569
|
568.9664
|
29586
|
-14521
|
169057
|
114277
|
250096
|
144833
|
|
3
|
1134
|
440012
|
257.7202
|
13401
|
-16185
|
169057
|
114277
|
250096
|
158234
|
|
4
|
570
|
435466
|
130.8943
|
6807
|
-6595
|
169057
|
114277
|
250096
|
165041
|
|
5
|
276
|
435990
|
63.3042
|
3292
|
-3515
|
169057
|
114277
|
250096
|
168333
|
|
6
|
228
|
440821
|
51.72167
|
2690
|
-602
|
169057
|
114277
|
250096
|
171023
|
|
7
|
148
|
440627
|
33.5885
|
1747
|
-943
|
169057
|
114277
|
250096
|
172770
|
|
8
|
96
|
432945
|
22.17372
|
1153
|
-594
|
169057
|
114277
|
250096
|
173923
|
|
9
|
79
|
424330
|
18.61759
|
968
|
-185
|
169057
|
114277
|
250096
|
174891
|
|
10
|
54
|
435489
|
12.39985
|
645
|
-323
|
169057
|
114277
|
250096
|
175536
|
|
11
|
40
|
449283
|
8.903074
|
463
|
-182
|
169057
|
114277
|
250096
|
175999
|
|
12
|
23
|
427132
|
5.384752
|
280
|
-183
|
169057
|
114277
|
250096
|
176279
|
|
13
|
39
|
420583
|
9.272843
|
482
|
202
|
169057
|
114277
|
250096
|
176761
|
|
14
|
33
|
436850
|
7.55408
|
393
|
-89
|
169057
|
114277
|
250096
|
177154
|
|
15
|
25
|
430433
|
5.808105
|
302
|
-91
|
169057
|
114277
|
250096
|
177456
|
|
16
|
36
|
429079
|
8.390063
|
436
|
134
|
169057
|
114277
|
250096
|
177892
|
|
17
|
15
|
346809
|
4.325147
|
225
|
-211
|
169057
|
114277
|
250096
|
178117
|
|
18
|
10
|
411464
|
2.430346
|
126
|
-99
|
169057
|
114277
|
250096
|
178243
|
|
19
|
12
|
334341
|
3.58915
|
187
|
60
|
169057
|
114277
|
250096
|
178430
|
|
20
|
10
|
336056
|
2.975695
|
155
|
-32
|
169057
|
114277
|
250096
|
178585
|
*Total prévu d’après l’augmentation maximale
à ce jour.
|