Journal of Probability and Statistics
Volume 2021 (2021), Article ID 5982784, 9 pages
https://doi.org/10.1155/2021/5982784

Research Article

Forecasting the COVID-19 Diffusion in Italy and the Related Occupancy of Intensive Care Units

Livio Fenga

Italian National Institute of Statistics, ISTAT, Rome 00184, Italy

Correspondence should be addressed to Livio Fenga; lfenga@ucsd.edu

Received 13 April 2020; Accepted 16 September 2020; Published 16 January 2021

Academic Editor: Alessandro De Gregorio

Copyright © 2021 Livio Fenga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper provides a model-based method for the forecast of the total number of currently COVID-19 positive individuals and of the occupancy of the available intensive care units in Italy. The predictions obtained—for a time horizon of 10 days starting from March 29th—will be provided at a national as well as at a more disaggregated level, following a criterion based on the magnitude of the phenomenon. While those regions hit the most by the pandemic have been kept separated, those less affected regions have been aggregated into homogeneous macroareas. Results show that—within the forecast period considered (March 29th–April 7th)—all of the Italian regions will show a decreasing number of COVID-19 positive people. The same will be observed for the number of people who will need to be hospitalized in an intensive care unit. These estimates are valid under constancy of the government’s current containment policies. In this scenario, northern regions will remain the most affected ones, whereas no significant outbreaks are foreseen in the southern regions.

1. Introduction

On March 19th, the death toll paid by Italy for the spread of the virus COVID-19 amounted to 3405 deaths, the highest paid by a single country in the world. Despite hard and relatively timely lockdown policies implemented by the government, on March 26th this figure has risen to 8165 deaths.

In such an emergency situation, a reliable forecast method for the infection development is essential for policy and decision makers to design evidence-based policies and to implement fast actions to curb the spread of the infection. In particular, predicting the number of people currently tested positive for COVID-19 (thereafter “positive cases”) could be useful to draw the epidemiological curve of the infection and therefore to predict its peak. Other than this variable, the forecasting procedure presented in this paper is used to predict the future values of another crucial variable, i.e., the number of people needing hospitalization in an intensive care unit (ICU). The Italian ICU system is at the moment severely stressed due to the spread of the disease; therefore, predictions of future ICU demand could be fruitfully considered in the design and the implementation of operational schemes. The forecast horizon for both the variables is of 10 days starting from March 29th.

Since the Italian regions are affected in different extents by the COVID-19, it has been decided to perform the forecasting exercise for the following geographical areas: Lombardia, Piedmont, Valle d’Aosta, Veneto, Friuli Venezia Giulia, Trentino Alto Adige, Lazio, and Campania. The remaining regions have been grouped in the following macroareas: “Center” (Marche, Umbria, and Toscana) and “South” (Abruzzo, Molise, Puglia Basilicata, Calabria, Sicilia, and Sardegna). At least two other reasons justify such a break down:

(1)The different starting times recorded for the lockdowns(2)The southern regions have been hit less severely and therefore, especially at the beginning of the observation period, show several zeroes or low numbers across the considered time span

In essence, in this study, the available official data, detailed in Section 2, have been employed in a three-step procedure, i.e.:

(1)Data preprocessing, in which data anomalies are identified and corrected according to an approach of the type a Kalman filter(2)Univariate forecasting, based on an autoregressive moving average (ARMA) model for number of positive cases and ICU(3)Bootstrap-based generation of predicted values and confidence intervals

2. The Data

This paper employs the data related to COVID-19, collected and regularly updated by the Italian National Institute of Health (an agency of the Italian Ministry of Health) and by the Italian Civil Protection Department. The whole data set is freely and publicly available in a comprehensive database, accessible on the Internet at the web address https://github.com/pcm-dpc/COVID-19/tree/master/dati-regioni (the file name is dpc-covid19-ita-regioni-20200323.csv). It collects crucial data related to all the persons tested for COVID-19—from the outbreak of the pandemics (February 24th)—and, in particular,

(1)is a collection of 21 data points—representing 19 Italian regions plus the two autonomous provinces of Trento and Bolzano—one for each day starting from the disease’s outbreak,(2)considers crucial variables, such as positive cases, recovered cases, deaths, number of people hospitalized, and number of people admitted to intensive care units (ICUs).

As already pointed out, in the present study, the variables of interest are the number of people who have been

(1)tested positive for COVID-19 (in what follows denoted by the bold Latin letter V),(2)hospitalized in an ICU (which will be denoted by the bold Latin letter U).

It is worth outlining how, according to the regulations issued by the Italian government, only the people showing moderate to severe symptoms, generally associated with the infection, or who have been in close proximity with at least one positive person, are tested. Therefore, the predictions obtained are to be referred to the sample, as no attempt have been made to carry out inference procedures for the estimation of the variables at the population level.

In order to correctly process the data, all the regions showing no positive cases at the beginning of the recording period and/or low values along the whole time span have been aggregated into macroareas. This has been done to (i) give more meaningful results and (ii) save degrees of freedom (which are always precious in short time series).

In details, the prediction exercise will be performed on the following regions/macroareas:

(A)Nothtern regions

(1)Lombardia(2)Piedmont(3)Valle d’ Aosta(4)Veneto(5)Friuli Venezia Giulia(6)Emilia-Romagna(7)Liguria(8)Macroarea “Trentino Alto Adige” (Trento and Bolzano)

(B)Center regions 

(1)Lazio(2)Macroarea “Center”: Marche, Umbria, and Toscana

(C)Southern regions

(1)Campania(2)Macroarea “South”: Abruzzo, Molise, Puglia, Basilicata, Calabria, Sicilia, and Sardegna

The north Italy regions—at the moment the more severely affected by the pandemic—have been treated separately along with two other regions, i.e., Lazio and Campania, since their major cities—Rome and Naples—deserve special attention for the institutional role played and their population density. On the other hand, the regions showing less worrying figures have been aggregated into macroareas according to their geolocation. The only exception is Valle d’Aosta, which has been left separated as no aggregation options could be found.

To simplify notation, for both the variables of interest V and U, the following convention is introduced:

Here, the upper left superscript (denoted by the upper case Latin letter K) refers to the geographical areas (i.e., North, Center, and South), whereas the subscript is associated with the number the different regions or macroareas are codified with, as above detailed. For example, by the symbols and the number of positive cases for the Emilia-Romagna region and the Center macroarea “Marche-Umbria-Toscana” are respectively identified.

3. Data Preprocessing

Missing data and other anomalies become the first challenge when designing predictive models, as statistical methods, in general, are designed and tested under the assumption of no missing observations [1]. Before delving into the details of the proposed procedure, a word of caution is needed since, unfortunately, a visual inspection of the data suggests the presence of a number of anomalous data, both at a regional and a country level. The detected anomalies might be associated with the biological sample collecting process and the related testing procedures. In fact, the typical lab workflow is governed by a set of rigid protocols which might be critically affected by factors such as the availability of manpower, swabs, reagents, and other laboratory materials. In emergency situations, such a workflow can be disrupted, and temporal inconsistency might appear as a result. For example, a set of samples might be delivered to a laboratory with longer than usual delays with respect to the time of collection, or a given lab can only complete the screening process for a certain number of samples. In both the cases, a shift of one day (or more) in the release of the lab results can be reasonably expected. A further source of anomalies is represented by the data entry and data editing processes, carried out in working environment likely affected by the risk of contagious and under rigid deadlines.

An example of such anomalous data is given in Figure 1, where the series (Lombardia) is depicted. Here, some data points showing values inconsistent with the overall pattern are clearly noticeable. Given the (very small) available sample size, the relative weight of such data is almost surely not negligible and can introduce severe distortions in the model parameter inference procedures and thus in the predicted values.

Figure 1: Number of people tested positive (Lombardia) (original data).

In order to correct those data, a Kalman smoother state-space model [2] has been applied. In particular, the Kalman smoother adopted is of the type fixed-point smoothing. This algorithm is designed to obtain the estimate of a realization (the time is fixed ) of a given random variable , given a set of observations . A thorough explanation of this method goes beyond the scope of the paper; therefore, the interested reader is referred to the excellent paper by Sage and Melsa [3].

In Figure 2, the corrected version of the series —resulting by applying the Kalman smoother—is depicted. Not only this series lends itself to a better visual inspection but, more importantly, is more suitable to be processed by the adopted prediction model.

Figure 2: Number of people tested positive (Lombardia) (data adjusted via Kalman filter).

4. Theoretical Framework

The approach used in this paper relies on (i) the theory of stochastic process and (ii) a resampling method. While the former is necessary to generate the input (predicted values) of the bootstrap algorithm, as well as to justify the employment of the outlier correction method, the latter serves the purpose of

(1)generating the final predictions, which are affected by a reduced amount of uncertainty (with respect to those generated by the stochastic model),(2)yielding the related confidence intervals.

4.1. The Stochastic Processes Paradigm

The approach proposed in the present paper relies on the assumption that the (transformed) time series and are approximately a realization of a process of the type ARMA (autoregressive moving average) [4].

Let be a real order stationary process, and it is said to admit a ARMA(p, q) representation if, for some constant , , will be

under the following conditions:

Here, denotes the sigma algebra induced by , and and are assumed not to have common zero.

The above conditions assure that can be represented as

with decreasing to 0 at geometric rate.

The dynamics of the series under investigations are not suitable for this theoretical framework as it requires 2nd-order stationarity and homoscedasticity; those conditions are simultaneously achieved by preprocessing the series according to the following filter: , being the symbol the difference operator and the exponent indicating the order of the difference. To fully understand the role played by , the backward operator is now introduced. In essence, moves the time index of an observation back by time intervals, i.e., , and thus we have

4.2. The Resampling Method

In order to extract valuable information from our data and, at the same time, decrease the total amount of uncertainty associated with the outcomes of the ARMA model, a resampling procedure has been employed. Among the several resampling methods for dependent data available—many of which are freely and publicly available in the form of powerful routines working under software packages such as Python® or R®—the adopted resampling method is of the type maximum entropy bootstrap (MEB). Proposed by Vinod [5] and subsequently improved (see, e.g., Vinod [6]), it is based on basic assumptions which are different from those usually followed by standard schemes. In more details, while in the classic bootstrap an ensemble represents the population of reference the observed time series is drawn from, in MEB a large number of ensembles (subsets), say becomes the elements belonging to , each of them containing a large number of replicates .

Unlike standard bootstrap schemes, in the MEB case the resample set mimics the observed realization of the underlying stochastic process, in MEB a large number of subsets, say becomes the elements belonging to , each of them containing a large number of replicates . Among the important features of the MEB scheme, it is worth mentioning the consistency of its bootstrap samples with the ergodic theorem (see, e.g., Birkhoff [7]) and with the probabilistic structure of the observed time series. In Figure 3, an example of the application of MEB for the variable is given.

Figure 3: Lombardia: B = 500 bootstrap replications performed via the MEB algorithm on the adjusted, log-transformed, data (in red, the original time series is depicted).

5. The Forecasting Method

In what follows, the proposed procedure is presented in a step-by-step fashion:

(1)Equation (2) is estimated for both and so that the model orders (equation (2)) and become available.(2)For each time series and , the MEB procedure is applied so that the sets and —each containing “bonafide” replications—are available, i.e., and (in Figure 4, the set for the variable is given).(3)For each of the replications stored in , equation (2) is estimated according to the model order selected, i.e., , and the 1- to 10-step-ahead predictions—as well as the 5% and 95% bootstrap confidence interval—are generated.(4)The B predictions and the confidence intervals obtained in the previous step are stored in the matrix , whose columns are lower bootstrap confidence interval, bootstrap prediction, and upper bootstrap confidence interval, respectively, denoted by the symbols , , and .(5)The median value is then extracted along with the 95% confidence intervals and , computed according to the t-percentile method. The explanation of this procedure goes beyond the scope of this paper; therefore, the interested reader is referred to the excellent paper by Berkowitz and Kilian [8].(6) and (the subscript is omitted for brevity) are computed conditional to a subset of , say , made up of the bootstrap replications whose range falls between the minimum and maximum values of the values of the confidence intervals computed for . In symbols,

(7)Steps 1–6 are repeated for , so that a new matrix of prediction of dimension is built, i.e., , whose columns are as in and denoted by the symbols , , and .

Figure 4: Italy, time series data of positive (data corrected via Kalman filter, left side axis) and of the number of people hospitalized in ICUs (right side axis).

Unfortunately, the whole procedure cannot be considered fully automatic since the estimation of equation (2) (step 1) is required.

5.1. The Adopted Models

The stochastic model structures identified for both and are almost always of the type ARMA (1, 0), with the exception of Campania (ARMA(0, 1), for both the variables and ) and Emilia-Romagna, for which the best model for the variable is of the type ARMA(1, 1). The most suitable prefilter (equation (5)) has been always of the type d = 3 difference of the natural log of the variables of interest.

6. Empirical Evidences

At the national level (data have been plotted in Figure 4), the peak in the number of COVID-19 positives will be reached on April 2nd, with a number of predicted positive close to 77,000. The maximum forecasted value for the occupied ICU—expected for April 4th—will be 4280. These values have been calculated using an indirect methodology, i.e., by summing up the estimates obtained at a disaggregated level. The results related to COVID-19 positives and the ICU occupancy are reported, respectively, in Tables 1 and 2, where the bootstrap standard deviations of the quantities and —respectively denoted with the symbols and —are reported along with their confidence intervals, i.e., (lower) and (upper). In what follows, the main results reported in these tables are commented:

(i)Lombardia—the most affected region—will reach the peak of positive cases (25963) and of the demand of ICUs (1425), respectively, on April 2nd and 4th.(ii)Emilia-Romagna is the second most affected region by COVID-19 but still shows a very high number of victims. The trend of infected people will reach its peak on April 5th, whereas the number of cases in intensive care will continue to grow at a progressively slower rate over the forecasting period.(iii)Veneto is the third region for number of deaths. Here, the number of positive cases, as well as the number of cases in ICU, will reach the peak on April 3rd.(iv)For Piedmont—the fourth region for number of victims—the predicted positive cases will reach the peak on March 29th (6635), whereas the persons in ICU will be 431 on March 31st, when the peak is predicted.(v)Liguria will begin a process of relative reduction of positive cases as early as March 29th. The number of cases in intensive care, after a period of stability (lasting until March 31st), will start a slow decreasing path.(vi)Positive cases in Trentino Alto Adige—which incorporates the cities of Trento and Bolzano—are projected to be 2158 on March 30th and then a decreasing trend is expected. The ICU beds occupied in this region will reach its peak on around April 3rd.(vii)The positive cases in Friuli Venezia Giulia show a relatively stable trend in the first half of the prediction interval with a peak around April 4th, after that the absolute number of cases will start decreasing. The number of cases in ICU will reach the peak between March 30th and April 1st.(viii)Valle d’Aosta is a small region which has been relatively less impacted by the virus. Here, a downward trend is expected to start on March 31st (for the positive cases) and around March 31st (cases in ICUs).(ix)The upward trend in the number of positive cases of Lazio is estimated to stop on March 31st and to reach the minimum at the end of the forecasting people (1821 cases). The number of ICU cases is estimated to reach its peak on the period 1–3 April.(x)The macroarea Center will reach its peak at the very beginning of the month of April (for the variable ), whereas for the variable the estimated peak day is around 31st March.(xi)Campania will reach the peak of contagions on April 5th, whereas ICU cases will do on the previous day.(xii)The remaining southern regions (Abruzzo, Molise, Puglia, Basilicata, Calabria, Sicily, and Sardinia) will show an upward trend in the number of future positive cases lasting until April 6th, where 6355 cases are predicted. The number of persons requiring an ICU will reach the peak on April 4th (348 is the estimated number of cases).

Table 1: 10-step-ahead predictions for the variable (number of persons tested positive).

Italy

29 March
66504
71586
76647
6911

30 March
69753
73409
79373
12785.3

31 March
70760
75696
82773
16989.7

1 April
70817
76723
86430
20353.4

2 April
69925
77811
90800
28812.7

3 April
69309
76952
96806
35454.1

4 April
68480
75927
102946
42234.7

5 April
66531
74860
109265
58238.4

6 April
64940
75006
117550
72145.3

7 April
60197
74875
126205
91586.4

Lombardia

29 March
22538
25214
26384
5001.4

30 March
23296
25456
26790
5010.8

31 March
23969
25792
27565
5167.0

1 April
23864
25842
27634
5301.3

2 April
23662
25963
28230
6248.7

3 April
23209
25675
28342
7123.5

4 April
22709
25717
28781
8307.2

5 April
21964
25067
29524
9972.6

6 April
20802
24431
30131
12368.5

7 April
19619
23752
30628
14375.8

Piedmont

29 March
6115
6635
7017
1136.4

30 March
6264
6610
7193
1253.0

31 March
6209
6568
7481
1759.2

1 April
6079
6401
7942
2531.6

2 April
5837
6152
8271
3357.0

3 April
5502
5898
8590
4025.1

4 April
5134
5566
8753
4985.6

5 April
4666
5135
8995
5893.4

6 April
4178
4732
9279
6809.7

7 April
3707
4289
9417
7986.3

Liguria

29 March
2026
2062
2217
220.9

30 March
2007
2048
2242
299.7

31 March
1955
2019
2361
476.0

1 April
1891
1969
2416
634.1

2 April
1804
1900
2515
803.5

3 April
1701
1805
2580
988.9

4 April
1589
1709
2645
1178.1

5 April
1470
1590
2748
1445.7

6 April
1344
1440
2917
1878.3

7 April
1214
1311
3090
2158.2

Valle d’Aosta

29 March
404
480
512
111.7

30 March
410
491
524
121.6

31 March
417
501
546
142.2

1 April
420
500
551
146.0

2 April
411
494
551
152.3

3 April
382
483
542
179.3

4 April
359
462
526
188.9

5 April
339
443
541
222.6

6 April
301
412
530
254.5

7 April
267
383
516
280.3

Veneto

29 March
6346
7145
7559
1730.7

30 March
6569
7213
7749
1745.4

31 March
6706
7431
8067
1888.1

1 April
6721
7463
8144
2031.5

2 April
6535
7539
8415
2443.8

3 April
6414
7469
8758
3121.9

4 April
6073
7374
9168
4423.1

5 April
5703
7182
9214
4957.7

6 April
5303
6842
9593
6023.7

7 April
4798
6537
10203
7743.4

Friuli Venezia Giulia

29 March
1029
1124
1167
180.2

30 March
1083
1138
1195
141.6

31 March
1100
1169
1226
153.6

1 April
1104
1189
1269
211.4

2 April
1109
1212
1267
201.2

3 April
1097
1209
1272
207.6

4 April
1060
1230
1287
287.8

5 April
1035
1203
1305
321.4

6 April
1005
1202
1333
413.2

7 April
976
1208
1343
445.7

Emilia-Romagna

29 March
9163
10500
11265
2912.4

30 March
9464
10975
11717
3323.8

31 March
9535
11360
12176
3591.6

1 April
9587
11613
12562
4123.3

2 April
9476
11815
12954
4783.0

3 April
9276
11943
13291
5547.1

4 April
9002
12132
13539
6471.6

5 April
8528
12244
13720
7240.6

6 April
8034
12125
13995
8173.0

7 April
7311
11966
14189
9017.2

Trentino Alto Adige

29 March
2021
2147
2324
325.1

30 March
2076
2158
2488
426.1

31 March
2114
2124
2617
581.6

1 April
2110
2083
2707
685.9

2 April
2097
1988
2787
761.4

3 April
2064
1868
2764
817.6

4 April
2023
1725
2742
827.9

5 April
1944
1548
2751
902.1

6 April
1851
1373
2718
991.0

7 April
1707
1200
2641
1041.3

Lazio

29 March
2013
2208
2355
429.0

30 March
2160
2244
2436
348.2

31 March
2182
2298
2553
413.7

1 April
2153
2279
2666
612.4

2 April
2161
2256
2785
723.5

3 April
2140
2222
2991
1016.0

4 April
2100
2154
3132
1278.3

5 April
2041
2046
3233
1480.3

6 April
1908
1956
3485
1817.4

7 April
1768
1821
3782
2128.3

Macroarea Center

29 March
6822
7499
7911
1459.1

30 March
7104
7671
8087
1403.7

31 March
7130
7864
8396
1740.8

1 April
7001
7896
8741
2316.0

2 April
6758
7889
8987
3019.6

3 April
6424
7851
9156
3800.7

4 April
6078
7813
9695
5015.6

5 April
5512
7718
10081
5917.4

6 April
4925
7494
10580
7727.4

7 April
4397
7238
10941
9036.2

Campania

29 March
1169
1453
1549
499.7

30 March
1267
1494
1662
513.7

31 March
1293
1527
1747
600.4

1 April
1287
1548
1835
709.3

2 April
1280
1582
1898
817.0

3 April
1252
1619
1953
901.6

4 April
1222
1640
1972
998.3

5 April
1160
1649
2020
1047.3

6 April
1090
1648
2063
1218.5

7 April
994
1645
2145
1418.8

Macroarea South

29 March
4380
5226
5662
1501.9

30 March
4646
5406
5846
1504.8

31 March
4689
5648
6246
1981.0

1 April
4705
5791
6528
2312.2

2 April
4496
5931
6902
3100.9

3 April
4122
6129
7325
3973.6

4 April
3841
6169
7695
4984.0

5 April
3445
6346
8141
6118.4

6 April
2981
6355
8787
7459.2

7 April
2512
6345
9397
8918.4

Table 2: 10-step-ahead predictions for the variable Ut (people in ICUs).

Italy

29 March
3681
3960
4086
570.4

30 March
3751
4009
4170
601.3

31 March
3730
4075
4264
784.2

1 April
3732
4141
4350
880.0

2 April
3707
4185
4386
945.2

3 April
3667
4262
4493
1012.7

4 April
3608
4280
4610
1402.6

5 April
3492
4275
4701
1611.8

6 April
3347
4249
4814
2097.0

7 April
3143
4243
4899
2418.4

Lombardia

29 March
1272
1369
1426
202.6

30 March
1286
1389
1458
237.4

31 March
1287
1402
1482
242.4

1 April
1291
1412
1490
274.7

2 April
1289
1420
1509
304.6

3 April
1286
1427
1526
325.6

4 April
1270
1425
1537
386.4

5 April
1252
1421
1558
423.7

6 April
1227
1416
1578
501.5

7 April
1191
1412
1599
580.3

Piedmont

29 March
422
433
451
32.7

30 March
428
436
457
40.1

31 March
428
438
470
58.3

1 April
427
436
478
70.2

2 April
424
434
491
92.0

3 April
419
427
498
111.7

4 April
409
417
508
134.0

5 April
395
406
512
163.1

6 April
373
396
523
204.5

7 April
354
383
535
254.3

Liguria

29 March
156
163
168
15.1

30 March
159
163
171
16.1

31 March
158
163
174
21.4

1 April
158
162
178
26.8

2 April
158
161
181
30.9

3 April
156
159
185
39.5

4 April
153
157
187
47.8

5 April
148
153
191
60.0

6 April
143
151
193
70.6

7 April
137
145
195
82.4

Valle Aosta

29 March
24
25
27
2.4

30 March
25
25
29
3.1

31 March
24
25
30
4.0

1 April
24
24
31
4.3

2 April
23
22
32
5.5

3 April
22
21
33
6.8

4 April
21
19
34
8.2

5 April
19
16
36
12.3

6 April
18
14
36
13.3

7 April
15
12
37
18.4

Veneto

29 March
331
357
367
48.5

30 March
335
360
375
53.7

31 March
335
365
382
65.3

1 April
334
370
391
80.6

2 April
332
375
401
96.7

3 April
324
378
409
119.4

4 April
312
375
424
156.7

5 April
298
376
440
200.4

6 April
286
370
448
231.1

7 April
267
369
461
268.0

Friuli Venezia Giulia

29 March
56
57
60
4.3

30 March
56
58
60
4.8

31 March
56
58
63
4.5

1 April
55
58
63
5.2

2 April
54
57
65
7.0

3 April
52
56
66
8.3

4 April
51
56
67
10.2

5 April
49
55
69
11.4

6 April
48
53
71
12.0

7 April
44
52
72
15.3

Emilia-Romagna

29 March
306
349
379
100.5

30 March
309
365
394
116.7

31 March
309
369
409
136.5

1 April
312
372.
421
148.5

2 April
313
379
433
152.7

3 April
314
386
439
174.1

4 April
311
388
442
180.7

5 April
307
392
442
188.6

6 April
298
399
447
212.6

7 April
288
401
444
212.5

Trentino Alto Adige

29 March
110
124
135
20.4

30 March
113
127
144
27.4

31 March
114
130
150
33.5

1 April
113
132
155
40.5

2 April
113
133
158
42.6

3 April
111
133
160
48.5

4 April
107
131
161
54.8

5 April
101
130
164
60.3

6 April
94
126
171
76.8

7 April
87
123
175
87.3

Lazio

29 March
120
136
146
29.8

30 March
125
139
157
35.4

31 March
123
141
171
58.6

1 April
118
144
179
71.0

2 April
111
145
200
109.5

3 April
104
145
219
135.3

4 April
95
141
235
160.3

5 April
84
139
264
196.8

6 April
73
134
294
253.8

7 April
61
131
343
321.5

Macroarea Center

29 March
476
491
509
40.6

30 March
485
492
519
45.3

31 March
484
492
534
64.9

1 April
482
489
547
85.6

2 April
479
484
565
116.2

3 April
475
478
586
142.0

4 April
468
469
608
193.6

5 April
460
460
624
224.0

6 April
448
448
649
273.3

7 April
438
438
681
331.9

Campania

29 March
126
142
167
41.2

30 March
133
159
197
68.5

31 March
138
176
228
97.4

1 April
144
189
257
114.9

2 April
147
201
293
150.8

3 April
148
206
326
197.8

4 April
143
211
348
212.4

5 April
134
203
382
246.2

6 April
120
197
411
311.7

7 April
109
192
447
370.1

Macroarea South

29 March
296
319
342
52.2

30 March
303
326
348
55.4

31 March
312
335
367
64.8

1 April
306
340
367
70.3

2 April
302
343
383
100.4

3 April
298
346
394
118.5

4 April
289
348
408
130.7

5 April
278
342
421
171.6

6 April
264
334
441
212.7

7 April
244
318
462
270.5

7. Conclusion

A forecasting method for two variables typically crucial during a pandemic—i.e., the number of positives and the future occupancy of ICU beds—has been proposed. The whole procedure has been designed to fulfill such a need-to-know goal using a minimal set of data; that is, the time series related to the positives and the ICU occupancy. This is a point of strength, as, especially in the initial stages of a pandemic, the available time series are limited to only basic variables (such as the ones considered in this paper) and are necessarily short, a fact that in general rules out multivariate approaches. In addition to that, this procedure uses two powerful tools, i.e., ARIMA models and the MEB resampling scheme, to generate estimates and confidence intervals of those quantities which are less affected by uncertainty components than it would be without using the bootstrap step. Finally, the procedure includes a filter of the type Kalman, which proved to be effective in correcting irregularities and anomalies (e.g., outliers) typically found in this type of data. At least two are the points of weakness of the proposed method: firstly, the assumption that both the time series of interest are both realizations of (unknown) data generating processes of the type ARIMA is arbitrary and entails the introduction of not negligible amount of uncertainty into the analysis (order selection uncertainty). Secondly, once the “best” model order is found, the inference process inevitably leads to the loss of precious degrees of freedom. Future research directions include exploring different prediction models (e.g., of the type exponential smoothing) and combining the predictions generated by them.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of the Italian National Institute of Statistics.

Conflicts of Interest

The author has no conflicts of interest to declare. In particular, he has no affiliations with or involvement in any organization or entity with any financial interest or nonfinancial interest in the subject matter or materials discussed in this manuscript.

References

S. Moritz, A. Sarda, T. Bartz-Beielstein, M. Zaefferer, and J. Stork, “Comparison of different methods for univariate time series imputation in R,” 2015, http://arxiv.org/abs/1510.03924.

D. Simon, “Kalman filtering,” Embedded Systems Programming, vol. 14, no. 6, pp. 72–79, 2001.

A. P. Sage and J. L. Melsa, Estimation Theory with Applications to Communications and Control, vol. 496, McGraw-Hill, New York, NY, USA, 1971.

S. Makridakis and M. Hibon, “ARMA models and the Box-Jenkins methodology,” Journal of Forecasting, vol. 16, no. 3, pp. 147–163, 1997.

H. D. Vinod, “Maximum entropy ensembles for time series inference in economics,” Journal of Asian Economics, vol. 17, no. 6, pp. 955–978, 2006.

H. D. Vinod, “New bootstrap inference for spurious regression problems,” Journal of Applied Statistics, vol. 43, no. 2, pp. 317–335, 2016.

G. D. Birkhoff, “Proof of the ergodic theorem,” Proceedings of the National Academy of Sciences, vol. 17, no. 12, pp. 656–660, 1931.

J. Berkowitz and L. Kilian, “Recent developments in bootstrapping time series,” Econometric Reviews, vol. 19, no. 1, pp. 1–48, 2000.