ALTHOUGH the incidence of epidemic diseases has

reached historic lows in many parts of the world,

these diseases still causes substantial morbidity globally.

Even where control programs have succeeded in epidemic

diseases locally extinct, unless vaccination coverage is

maintained at extremely high levels, susceptible num-

bers may increase suﬃciently to spark large outbreaks.

Human mobility will drive potentially infectious contacts

and interact with the landscape of susceptibility to deter-

mine the pattern of epidemic diseases outbreaks. These

interactions have proved diﬃcult to characterize empiri-

cally.

So, it is of great interest to explore the degree to

which new sources of data, combined with existing pub-

lic health data, can be used to evaluate the landscape of

immunity and the role of vaccination in the eradication

of epidemic diseases. The understanding of data dynam-

ics of people aﬀected by epidemic diseases from year to

year is important for the management of infectious dis-

ease epidemics. In this context, diﬀerent public health

surveillance systems have been developed to facilitate the

detection of abnormal behavior of infectious diseases and

other adverse health events. To achieve this goal, diﬀer-

ent approaches have been used for assessing and forecast-

ing of infectious disease incidence. The dynamics and

control of infectious diseases in terms of mathematical

models are discussed in, [1], among others. Time series

analysis enjoys of great interest in this ﬁeld. It makes use

of statistical models able to forecast the epidemiological

behavior of the historical surveillance data. Diﬀerent

methods have been reported in the literature. So, ex-

ponential smoothing, [2], and generalized regression, [3],

methods were used to forecast in-hospital infection and

incidence of cryptosporidiosis respectively. Decomposi-

tion methods, [4], and multilevel time series models, [5],

were used to forecast respiratory syncytial virus.

Seasonal autoregressive integrated moving average

(SARIMA) models have been extensively used for epi-

demic time series forecasting including the hemoragic

fever renal syndrome, [6], [7], dengue fever, [8], [9], and

tuberculosis, [10].

Model based on artiﬁcial neural networks were also

used to forecast the incidence of hepatitis A, [4], [11],

and typhoid fever, [12]. The decomposition methods are

the most traditional methods in time series analysis, [13],

[14]. Recently, machine learning based time series models

such as artiﬁcial neural networks have been successfully

applied for modeling infectious disease incidence time se-

ries, [15], [16]. Support vector machines (SVM), a new

type of machine learning methods based on statistical

learning theory, [17], are used for epidemic time series

forecasting, [18].

Two epidemic diseases will make the object of as-

sessing, modeling and forecasting using time series anal-

ysis, in the case studies presented in the paper: se-

vere acute respiratory syndrome and measles infections.

Diﬀerent approaches are used for severe acute respi-

ratory syndrome (SARS) assessing, making the object

of many papers, [19], [20], [21], among others. The

1. Introduction

Time Series Analysis with Application in Public Health

and Biomedical Data

THEODOR D. POPESCU

JSPS Alumni Association Romania

296 Independentei Avenue, 60031 Bucharest

ROMANIA

Abstract: The paper gives a overview of time series modeling and forecasting, using multiplicative SARIMA

models, with application in assessing and forecasting of epidemiological data. After presenting of the main

models and the methodological issues used in Box-Jenkins approach, the paper presents two case studies having

as subject the modeling and forecasting of the cumulative number of individuals infected with severe acute

respiratory syndrome, or SARS, in Singapore, from 24 February to 7 May 2003, and the measles infections, in

Great Britain, 1971-1994, quarterly recorded. For the last series an example of intervention analysis, using as

the exogenous data the measles infections, and as endogenous variable the number of vaccinated persons, in the

same time period, is presented, proved to be a useful approach, when the time series is affected by the effect of

population vaccination.

Keywords- Time series analysis, modeling, forecasting, intervention analysis, Box-Jenkins approach,

epidemiological data, case study.

Received: June 23, 2022. Revised: August 26, 2023. Accepted: September 29, 2023. Published: November 14, 2023.

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

139

Volume 19, 2023

problem of modeling and forecasting of measles infec-

tion is present in many papers. So, in [22] is pro-

vided an early signal of infectious disease epidemics by

analyzing the disease dynamics. The model consisted

of a seasonal autoregressive integrated moving average

SARIM A(3,1,0)(0,1,1)12 model, used in measles dy-

namics analysis in Bangladesh. A mathematical model

of the dynamics of measles in New Zealand, to predict

an epidemic in 1997, which was used in the decision to

carry out an intensive immunization campaign in 1997

is presented in [23]. In [24] is developed a model, the

TSIR (Time-series Susceptible Infected Recovered), that

can capture both endemic cycles and episodic out- breaks

in measles. It is a doubly stochastic model for disease

dynamics, and includes seasonality in the transmission

rates. All parameters of the model are estimated on

the basis of time series data on reported cases and re-

constructed susceptible numbers from a set of cities in

England and Wales in the pre-vaccination era (1944-

1966). A new prediction analysis procedure for measles

epidemics, a combination of nonlinear squares method

with the maximum entropy spectral analysis method, is

presented in [25].

The paper is organized as follows. In Section 2 is

given a general view on the time series models, regres-

sion and intervention models, to be used in modeling

and forecasting of epidemiological surveillance data. Sec-

tion 3 discusses some methodological aspects of time

series modeling and forecasting, based on Box-Jenkins

methodology, with the emphasis on practical aspects.

Section 4 discusses a case study having as object mod-

eling and forecasting of a time series representing the

cumulative number of individuals infected with severe

acute respiratory syndrome, or SARS, in Singapore, from

24 February to 7 May 2003. Section 5 presents a case

study of modeling and forecasting, using a multiplicative

SARIM A model, for a time series representing the num-

ber of measles infections, in Great Britain in the period

1971-1994, and an example of intervention analysis, us-

ing as the exogenous data the measles infections, and as

endogenous variable the number of vaccinated persons,

in the same time period.

The statistical approaches adopted in time series

modeling and forecasting usually rely on multiplicative

SARIM A (Seasonal Auto Regressive Integrated Moving

Average) model. A such model has the following form

for the time series zt, [26]:

φ(B)Φ(Bs)▽d▽D

szt=θ(B)Θ(Bs)at(1)

where atis a white noise and

φ(B) = 1 + φ1B+φ2B2+···+φpBp;

θ(B) = 1 + θ1B+θ2B2+···+θqBq;

Φ(Bs) = 1 + ΦsBs+ Φ2sB2s+...+ ΦP sBP s;

Θ(Bs) = 1 + ΘsBs+ Θ2sB2s+...+ ΘQsBQs;

with Bthe time delay operator, Bzt=zt−1,▽zt=

(1−B) = zt−zt−1, nonseasonal diﬀerentiating operator,

and ▽szt= (1 −Bs) = zt−zt−s, seasonal diﬀerentiating

operator: dis the nonseasonal diﬀerentiating order, D

is the seasonal diﬀerentiating order and sis the seasonal

period of the series.

The model is deﬁned as SARIMA(p, d, q)(P, D, Q)s

where (p, d, q) denotes nonseasonal orders, and (P, D, Q)

seasonal order of the model. The model is presented in

Fig. 1.

-θ(B)Θ(Bs)

▽dφ(B)▽DΦ(Bs)

Fig. 1: Multiplicative SARIM A(p, d, q)(P, D, Q)s

model

The multiplicative form of the model simpliﬁes the

stationarity and invertibility conditions checking; these

conditions can be separately checked, for seasonal and

nonseasonal coeﬃcients of the model.

Starting from the general model form of the model

SARIM A it can be obtain related models: AR (Auto

Regressive), MA (Moving Average), ARM A (Auto Re-

gressive Moving Average) and ARIM A (Auto Regressive

Integrated Moving Average), with or without seasonal

components. These models are identiﬁed by the mean of

the autocorrelation (ACF ) and the partial autocorrela-

tion functions (P ACF ).

In some situations, it is known that some external

events can aﬀect the variables for which the practitioner

intends to forecast the future time series values. Dy-

namic models, used in this case, include several vari-

ables, as input variables, which are intended to take into

account in the dynamics model, the mentioned excep-

tion events. A special kind of SARIM A model with in-

put series is called an intervention model or interrupted

time series (ITS) model, [27]. In an intervention model,

the input series is an indicator variable that contains dis-

crete values that ﬂag the occurrence of an event aﬀecting

the response series. This event is an intervention in or

an interruption of the normal evolution of the response

time series, which, in the absence of the intervention,

is usually assumed to be a pure SARIM A process. As

examples of practical interventions can be mentioned:

the eﬀect of diﬀerent promotions activities on the sales,

the eﬀect of strikes on the volume of the products and

the price of the products, the eﬀect of medication on

the health of the patient, the eﬀect of the exchange of

the laws in the legislation on the mortalities resulting

from car accidents, etc. In this case, some variables as

step function, consisting of ”zero” values and ”unit” val-

ues, before and after application respectively change pol-

icy, medication, or exchange of laws are included in the

model, as an external variable.

A such intervention model can be represented like a

2. Time series models

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

140

Volume 19, 2023

transfer function (T F ) model (see Fig. 2), where ztis

the value of the endogenous variable at time t,ut=

[u1t,...,urt]Tis the vector of exogenous variables, and

atis a white noise error.

Ωi(B) = ωi0+ωi1B+ωi2B2+···+ωiniBni;i= 1,2,...,r

∆i(B) = 1+δi1B+θi2B2+···+δinδiBnδi;i= 1,2,...,r

φ(B), θ(B),Φ(Bs) and Θ(Bs) have been described above.

Ωr(B)

∆r(B)

urt

-Ω1(B)

∆1(B)

u1t

.



@@@





...-?

θ(B)Θ(Bs)

▽dφ(B)▽DΦ(Bs)

Fig. 2: Transfer function (T F ) model

The time series model construction usually include

the following stages, [26]:

•Identiﬁcation (speciﬁcation) of the time series model

using some data analysis tools (diﬀerent graphical

representations, autocorrelation functions (ACF )

and partial autocorrelation functions (P ACF )) in

order to determine the types of transformations to

obtain stationarity and to estimate the degree of dif-

ferentiation needed to induce stationarity in data, as

well as the polynomial degrees of autoregressive and

moving average operators in the model.

•Model parameter estimation of the time series im-

plies the use of eﬃcient methods (such as maximum

likelihood, among others) for parameter estimation,

standard errors and their correlations, dispersion of

residuals, etc.

•Model evaluation (validation) aims to establish the

model suitability, or to make some simpliﬁcations

in structure and parameter estimates. Key elements

for model validation refers to residuals which can not

be justiﬁed, these being any residuals of abnormal

value that can not be explained by the action of

known external factors or other variables; also the

correlations and partial correlations of the residuals

prove useful tools in model evaluation.

More explanations of the process, [28], often add a

preliminary stage of data preparation and a ﬁnal stage

of model application, or forecasting.

Visual analysis of series data allows a ﬁrst image on

the series’ non-stationarity and on the presence of a sea-

sonal pattern in the data. The ﬁnal decision on the inclu-

sion of seasonal elements in the time series model will be

taken after the autocorrelation function (ACF ) and par-

tial autocorrelation function (P ACF ) analysis, as well as

after the estimation results analysis; the visual analysis

of the data can provide useful additional information.

Signiﬁcant changes in the mean value of the series

data require non seasonal diﬀerentiation of the ﬁrst or-

der, while the varying of the rate for average value im-

poses the nonseasonal diﬀerentiation of the second order

of the series. Strong seasonal variations usually require,

not more than the seasonal diﬀerentiation of the ﬁrst

order of the series’data. Autocorrelation function of the

series oﬀers information on the nonseasonal and seasonal

degrees to be used to obtain the stationarity of the data.

An ARM A stationary process is characterized by the-

oretical autocorrelation and partial autocorrelation func-

tions tending to zero. The autocorrelation function tends

to zero after the ﬁrst q−pvalues of the delay, following

the evolution of a exponential function or of a damped

sinusoidal function, and the partial autocorrelation func-

tion is canceled after the ﬁrst p−qvalues of the delay,

[29].

An AR or MA seasonal process is characterized by

similar autocorrelation and partial autocorrelation func-

tions, corresponding to nonseasonal processes, but the

coeﬃcients of autocorrelation and partial autocorrelation

functions, signiﬁcant for the seasonal process, appear at

multiple seasonal delay values.

At the stage of model identiﬁcation a special atten-

tion will be given to nonseasonal autocorrelation coeﬃ-

cients with absolute values of the associated tstatistic

test exceeding the value 1.6, [29]. Model parameters, as-

sociated to these coeﬃcients prove to be signiﬁcant from

the statistical point of view, in the estimation stage.

In the identiﬁcation and validation-diagnosis stages,

the attention will be focused on the coeﬃcients of sea-

sonal autocorrelations with the absolute values of the

tstatistic test associated which overcome 1.25 value.

The seasonal parameters estimates AR or MA , asso-

ciated to these coeﬃcients, will appear more signiﬁcant

in the estimation stage. If the residual autocorrelation

function has zeros values, from statistical point of view,

to seasonal delays: s, 2s, . . . , and to the delays of the

form 0.5s, 1.5s, and in the vicinity of seasonal delays:

s+ 1, s −1,2s+ 1,2s−1,..., the same warning level will

be used: 1.25. More information on the methodology

used in this case can be ﬁnd in [29] and [30].

In the estimation stage, the use of the initial esti-

mates of the model parameters of the value of 0.1 leads

to good results in most cases; better initial estimates

3. Methodological Aspects

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

141

Volume 19, 2023

for model parameters can be obtained based on the au-

tocorrelation and partial autocorrelation functions, used

to determine the structure of the model. In this stage as

model parameters will be retain those for which |t| ≥ 2,

[29]. The criteria Akaike Information Criterion (AIC),

Bayesian information criterion (BIC) or Schwarz infor-

mation criterion (also SIC, SBC, SBIC), [31], Adjusted

Root Mean Square Error (ARMSE) and Absolute Mean

Percent Error (AMPE), [29], oﬀer information on the

parameter estimation quality.

Forecasting is what the whole procedure is designed

to accomplish. Once the model has been selected, esti-

mated and checked, it is usually a straight forward task

to compute forecasts. The forecasting problem can be

solved, in the most direct way, using the multiplicative

ARIMA model of the form (1). The description of the

model by an inﬁnitely weighted sum of current values

and the earlier noise is proving useful, in particular, to

estimate the variance of forecasting values, as well as

to determine their conﬁdence intervals. Standards and

practices for time series forecasting are given in, [32].

The time series making the object of the case study

represents the cumulative number of individuals infected

with severe acute respiratory syndrome (SARS) in Sin-

gapore 24.02.2003-8.05.2003, [19], and is given in Fig.

0 10 20 30 40 50 60 70 80

100

150

200

250

SARS in Singapore: 24.02.2003−8.05.2003

Days

Cumulative number of individuals infected with SARS

Fig. 3: SARS series 24.02.2003-8.05.2003.

We present in Fig. 4 the autocorrelation (ACF ) and

partial autocorrelation (P ACF ) functions of the original

data, and the Ljung-Box-Q (LBQ) test.

It can be noted, from the data analysis, the non-

stationary character of the series, due to presence of a

trend component in the data. The series of diﬀerences

of the original series is given in Fig. 5 and the ACF and

P ACF functions are presented in Fig. 6.

The results mentioned above suggested the following

model of the original SARS series:

(1 + φ1B)zt= (1 + θ1B+θ2B2+θ3B3)at, v[at] = σ2

(2)

1 2 3 4 5 6 7 8 9 10

−1

−0.5

0.5

A.C.F. of SARS series, LBQ = 598.75

1 2 3 4 5 6 7 8 9 10

−1

−0.5

0.5

P.A.C.F. of SARS series

Fig. 4: ACF andP ACF functions of SARS series.

0 10 20 30 40 50 60 70

Diff. of SARS in Singapore: 24.02.2003−8.05.2003

Diff. of SARS series

Fig. 5: Diﬀerences of SARS series.

1 2 3 4 5 6 7 8 9 10

−1

−0.5

0.5

A.C.F. of Diff. of SARS series, LBQ = 63.24

1 2 3 4 5 6 7 8 9 10

−1

−0.5

0.5

P.A.C.F. of Diff. of SARS series

Fig. 6: ACF andP ACF functions of SARS series dif-

ferences.

4. Modeling and forecasting of cumulative

number of individuals infected with severe

acute respiratory syndrome (SARS)

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

142

Volume 19, 2023

The model parameters: φ1, θ1, θ2, θ3and σ2have been

initialized with 0.1 value. The model parameter esti-

mation has been performed using the Broyden-Fletcher-

Goldfarb-Shanno (BFGS) optimization algorithm, [33].

The results are presented in Table 1 and Table 2, with

the objective function = 168.3458, nr. of iterations =

120 and information criteria: AIC = 4.883 and SBC =

5.0423.

Table 1: Results for ARIM A model parameter estima-

tion

Parameter Estimate Appr.Std.Dev. t-test

φ1-1.0014 0.0050 -202.1841

θ10.6421 0.1137 5.6490

θ20.5229 0.1060 4.9349

θ30.2453 0.1145 2.1422

v[at] 6.8256 1.1011 6.1986

Table 2: Correlation matrix of ARIM Amodel parame-

ter estimates

φ1θ1θ2θ3v[at]

φ11.00

θ10.04 1.00

θ20.04 0.52 1.00

θ30.03 0.09 0.05 1.00

v[at] 0.03 -0.05 -0.04 -0.02 1.00

The model residuals are presented in Fig. 7, and the

residual ACF ,P ACF , with Ljung-Box-Q test, are given

in Fig. 8.

10 20 30 40 50 60 70

−2.5

−2

−1.5

−1

−0.5

0.5

1.5

2.5

Standardized plot of Residuals

Fig. 7: Model residuals.

The forecasting, for the resulted model, has been per-

formed, started from the 64 day, for a horizon time of 7

days, and 95% conﬁdence limits, to compare the original

data with the forecasting results. It can be noted that

the forecasting results follow the evolution trend of the

original time series, and are in the conﬁdence limits 95%.

The forecasting results and conﬁdence limits are given in

Fig. 9.

1 2 3 4 5 6 7 8 9 10

−1

−0.5

0.5

A.C.F. of Residuals, LBQ = 10.00

1 2 3 4 5 6 7 8 9 10

−1

−0.5

0.5

P.A.C.F. of Residuals

Fig. 8: ACF and PACF of model residuals.

0 10 20 30 40 50 60 70 80

100

150

200

250

Forecasting results and confidence limits 95%

Days

Cumulative number of individuals infected with SARS

Fig. 9: Forecasting results and conﬁdence limits 95% for

7 days using the resulted model.

The case study making the object of this section has

as subject the modeling and forecasting of a time se-

ries representing the measles infections, in Great Britain

in the period 1971-1994, quarterly recorded, and an ex-

ample of intervention analysis, using as the exogenous

data the number of measles infections, and as endoge-

nous variable the number of vaccinated persons, in the

same time period, using a transfer function (T F ) model.

The time series representing the measles infections,

in Great Britain in the period 1971-1994, quarterly

recorded, is presented in Fig. 10.

We present in Fig. 11 the autocorrelation (ACF ) and

partial autocorrelation (P ACF ) functions of the original

data, and the Ljung-Box-Q (LBQ) test.

It can be noted, from the data analysis, the non-

stationary and seasonal character of the series. Because

the data are quarterly recorded, it can be supposed the

presence in the data series of a seasonal component of

period s= 4 (yearly); it is also conﬁrmed by the auto-

correlation function ACF . So, the original time series

has been seasonal diﬀerentiated with period s= 4, and

it is presented in Fig. 12.

The ACF and P ACF of diﬀerentiated series, and

Ljung-Box-Q test, are given in Fig. 13.

Starting from these functions, the following

5. Modeling and forecasting of

measles Infections

5.1. Modeling and forecasting of measles

infections with a regression model

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

143

Volume 19, 2023

0 10 20 30 40 50 60 70 80 90 100

Measles patients/1000 inhabitans in Great Britain 1971−1994

Quarters

Measles patients

Fig. 10: Number of measles infections/1000 inhabitants,

Great Britain, 1971-1994.

2 4 6 8 10 12 14 16

−1

−0.5

0.5

A.C.F. of Measles patients, LBQ = 284.67

2 4 6 8 10 12 14 16

−1

−0.5

0.5

P.A.C.F. of Measles patients

Fig. 11: ACF and PACF functions of measles infec-

tions/1000 inhabitants, Great Britain, 1971-1994.

0 10 20 30 40 50 60 70 80 90 100

−40

−30

−20

−10

Diff. of measles patients/1000 inhabitans in Great Britain 1971−1994

Diff. of measles patients

Fig. 12: Diﬀerentiated series with s = 4.

2 4 6 8 10 12 14 16

−1

−0.5

0.5

A.C.F. of Diff. of measles patients, LBQ = 457.90

2 4 6 8 10 12 14 16

−1

−0.5

0.5

P.A.C.F. of Diff. of measles patients

Fig. 13: ACF and P ACF of diﬀerentiated series with s

= 4.

SARIM A model structure resulted:

(1+Φ4B4+Φ8B8)(1−B4)zt= (1+θ1B+θ2B2)(1+Θ4B4)at

(3)

and v[at] = σ2.

The model parameter estimation has been performed

using the Broyden-Fletcher-Goldfarb-Shanno (BFGS)

optimization algorithm, [33]. The results are presented

in Table 3 and Table 4, with the objective function =

315.7083, nr. of iterations = 24 and information crite-

ria: AIC = 6.7728 and SBC = 6.9341.

Table 3: Results for SARIM A model parameter estima-

tion

Parameter Estimate Appr.Std.Dev. t-test

Φ4-0.4614 0.1040 -4.4357

Φ8-0.5098 0.1003 -5.0846

θ11.0436 0.1062 9.8243

θ20.5074 0.0882 5.7555

Θ4-0.4927 0.0964 -5.1107

v[at] 40.6531 5.9866 6.7907

Table 4: Correlation matrix of SARIMA model param-

eter estimates

Φ4Φ8θ1θ2Θ4v[at]

Φ41.00

Φ8-0.88 1.00

θ1-0.01 0.04 1.00

θ20.13 -0.11 0.78 1.00

Θ40.53 -0.48 -0.18 -0.21 1.00

v[at] -0.03 0.04 0.01 -0.04 -0.00 1.00

The model residuals are presented in Fig. 14, and

residual ACF ,P ACF , Ljung-Box-Q test, are given in

Fig. 15.

10 20 30 40 50 60 70 80 90

−3

−2

−1

Standardized plot of Residuals

Fig. 14: Model residuals

The estimation results conﬁrm the model quality, ac-

cording with the Box-Jenkins methodology used in time

series analysis, [29].

The forecasting, for the resulted model, has been per-

formed, started from the 92 quarter, for a horizon time

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

144

Volume 19, 2023

12345678

−1

−0.5

0.5

A.C.F. of Residuals, LBQ = 16.02

12345678

−1

−0.5

0.5

P.A.C.F. of Residuals

Fig. 15: ACF andP ACF of model residuals.

of 4 quarters, and 95% conﬁdence limits, to compare the

original data with the forecasting results. It can be noted

that the forecasting results follow the evolution trend of

the original time series, and are in the conﬁdence limits

95%. The forecasting results and conﬁdence limits are

given in Fig. 16.

0 10 20 30 40 50 60 70 80 90 100

−20

−10

Forecasting and 95% confidence intervals

Quarters

Measles patients

Fig. 16: Forecasting results and conﬁdence limits 95%

for 4 quarters.

In this case an intervention model, a transfer func-

tion (T F ) model, has been used, with the exogenous

variable the number of measles infections, zt,and with

endogenous variable the percent of vaccinated persons,

ut, in the time period making the object of the analy-

sis. The percent of measles vaccinations, Great Britain,

1971-1994 is presented in Fig. 17.

After preliminary analysis of the data, and diﬀerent

model structures, resulted the following structure of the

transfer function model, representing the intervention

model:

(1 −B4)zt=ω1

1 + δ1But+(1 + θ1B+θ2B2)(1 + Θ4

1+Φ4B4+ Φ8B8at;

(4)

with v[at] = σ2and s= 4, due to the nostationarity of

the data. For the model parameters and variance, σ2,

have been used as initial values 0.1. Broyden-Fletcher-

Goldfarb-Shanno (BFGS) optimization algorithm, [33],

was used for parameter estimation, resulting the follow-

0 10 20 30 40 50 60 70 80 90 100

Percent of vaccinated persons in Great Britain 1971−1994

Quarters

Vaccinated percent

Fig. 17: Percent of measles vaccinations, Great Britain

1971-1994.

ing values for model parameters and correlation matrix

(see Table 5 and Table 6, respectively):

Table 5: Results for T F model parameter estimation

Parameter Estimate Appr.Std.Dev. t-test

Φ4-0.5800 0.0907 -6.3979

Φ8-0.3495 0.0860 -4.0657

θ11.0556 0.0888 11.8939

θ20.5293 0.0792 6.6838

Θ4-1.0000 0.0379 -26.3829

ω1-0.2891 0.1183 -2.4435

δ10.8736 0.0637 13.7216

v[at] 25.6828 4.1370 6.2081

for an objective function = 290.7013, nr. of iterations =

50 and information criteria: AIC = 6.2884, and SBC =

6.5035.

Table 6: Correlation matrix of T F model parameter es-

timates

Φ4Φ8θ1θ2Θ4ω1δ1v[at]

Φ41.00

Φ8-0.89 1.00

θ1-0.14 0.14 1.00

θ20.13 -0.12 0.72 1.00

Θ40.34 -0.33 0.11 0.07 1.00

ω10.22 -0.13 -0.05 0.04 0.21 1.00

δ10.13 -0.07 -0.15 -0.07 0.27 0.59 1.00

v[at] 0.43 -0.40 0.14 0.15 0.40 0.31 0.38 1.00

The model residuals are presented in Fig. 18, and the

residual ACF and P ACF , with Ljung-Box-Q test, are

given in Fig. 19.

The results conﬁrm the model quality, according with

the Box-Jenkins methodology used, [29]. The forecasting

results, for the transfer model resulted, started from the

92 quarter for a horizon time of 4 quarters and the 95%

conﬁdence limits are given in Fig. 20; the values used,

as percent of vaccinations for the forecasting measles in-

fections, in forecasting, represent the values recorded for

the last 4 quarters of the original series. It can be noted

5.2. Modeling and forecasting of measles

infections with an intervention model

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

145

Volume 19, 2023

10 20 30 40 50 60 70 80 90

−3

−2

−1

Standardized plot of Residuals

Fig. 18: Transfer function model residuals.

12345678

−1

−0.5

0.5

A.C.F. of Residuals, LBQ = 9.29

12345678

−1

−0.5

0.5

P.A.C.F. of Residuals

Fig. 19: ACF andP ACF transfer function residuals.

that the forecasting results follow the evolution trend of

the time series of measles infections, and are in the con-

ﬁdence limits 95%.

0 10 20 30 40 50 60 70 80 90 100

−10

Forecasting measles patients anfd confidence limits (95%)

Quarters

Measles patients

Fig. 20: Forecasting results and conﬁdence limits 95%

for 4 quarters using transfer function model.

The time series modeling and forecasting of epidemi-

ological surveillance data using seasonal multiplicative

SARIM A models and the attractive features of the Box-

Jenkins approach provide an adequate description to the

data in this ﬁeld. The SARIM A processes are a very

rich class of possible models and it is usually possible

to ﬁnd a process which provides an adequate description

to the data. Also, the intervention analysis proved to

be a useful approach to model interrupted time series,

in this case, when such time series are aﬀected by the

eﬀect of medication on the health of the patient, popu-

lation vaccination policies, some economical constraints,

etc. The case studies presented in the paper proved the

eﬃciency of the approach. The underlying strategy of

Box and Jenkins is applicable to a wide variety of sta-

tistical modeling situations in assessing and forecasting

of epidemiological data series. It provides a convenient

framework which allows an analyst to think about the

data, and to ﬁnd an appropriate statistical model which

can be used to help answer relevant questions about the

data.

[1] R. M. Anderson, R. M. May, Infectious Diseases of

Humans: Dynamics and Ccontrol, Oxford Univer-

sity Press, London, 1991.

[2] C. Farrington, N. Andrews, ”Outbreak detection:

application to infectious disease surveillance”, in

Monitoring the Health of Populations: Statistical

Principles and Methods for Public Health Surveil-

lance, Ron Brookmeyer and Donna F. Stroup (Edi-

tors), 2003, pp. 203-231.

[3] D. Chadwick, B. Arch, A. Wilder-Smith, N. Paton

”Distinguishing dengue fever from other infections

on the basis of simple clinical and laboratory fea-

tures: application of logistic regression analysis”,

Journal of Clinical Virology, 2006, pp. 147-153.

[4] G. Gonzalez-Parra, A. J. Arenas, L. Jodar, ”Piece-

wise ﬁnite series solutions of seasonal diseases mod-

els using multistage Adomian method”, Communi-

cations in Nonlinear Science and Numerical Simula-

tion”, 2009, pp. 3967-3977.

[5] M. C. Spaeder, J. C. Fackler, ”A multi-tiered time-

series modelling approach to forecasting respiratory

syncytial virus incidence at the local level”, Epi-

demiology and Infection, 2012, pp. 602-607.

[6] Q. Li, N. N. Guo, Z. Y. Han, Y. B. Zhang, S. X. Qi,

Y. G. Xu, Y. M. Wei, X. Han, Y. Y. Liuet, ”Applica-

tion of an autoregressive integrated moving average

model for predicting the incidence of hemorrhagic

fever with renal syndrome”, The American journal

of Tropical Medicine and Hygiene, 2012, pp. 364-

370.

[7] Q. Liu, X. Liu, B. Jiang, W. Yang, ”Forecasting

incidence of hemorrhagic fever with renal syndrome

in China using ARIMA model”, Bmc Infectious Dis-

eases, 2011.

[8] S. Wongkoon, M. Jaroensutasinee, K. Jaroensutasi-

nee ”Development of temporal modeling for predic-

tion of dengue infection in Northeastern Thailand”,

Asian Paciﬁc Journal of Tropical Medicine, 2012,

pp. 249-252.

[9] P. L. Luz, B. V. M. Mendes, C. T. Codeco,

C. J. Struchiner, A. P. Galvani, ”Time series snal-

ysis of dengue incidence in Rio de Janeiro, Brazil”,

American Journal of Tropical Medicine and Hy-

giene, 2008, pp. 933-939.

[10] M. Rios, J. M. Garcia, J. A. Sanchez, D. Perez, ”A

statistical analysis of the seasonality in pulmonary

6. Conclusions

References

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

146

Volume 19, 2023

tuberculosis”, European Journal of Epidemiology,

2000, pp. 483-488.

[11] M. Ture, I. Kurt, ”Comparison of four diﬀerent time

series methods to forecast hepatitis A virus infec-

tion”, Expert Systems with Applications, 2006, pp.

41-46.

[12] X. Zhang, Y. Liu, M. Yang, T. Zhang, A. A. Young

,X. Li, ”Comparative study of four time series meth-

ods in forecasting typhoid fever incidence in China”,

PloS One, 2013, e63116.

[13] B. L. Bowerman, R. T. O’Connell, T. Richard, Fore-

casting and Time Series: An Applied Aproach, Bel-

mont CA Wadsworth, 1993.

[14] J. D. Hamilton, Time Series Analysis, Cambridge

Univ Press, 1994.

[15] G. P. Zhang, ”Time series forecasting using a hybrid

ARIMA and neural network model”, Neurocomput-

ing, 2003, pp. 159-175.

[16] C. C. Chang, C. J. Lin, ”LIBSVM: a library for sup-

port vector machines”, ACM Transactions on Intel-

ligent Systems and Technology (TIST), 2011.

[17] U. Thissen, R. Van Brakel, A. De Weijer,

W. Melssen, L. Buydens, ”Using support vector

machines for time series prediction”, Chemometrics

and intelligent laboratory systems, 2003, pp. 35-49.

[18] X. Zhang, T. Zhang, A. A. Young, X. Li, ”Applica-

tions and comparisons of four time series models in

epidemiological surveillance data”, PLoS One, 2014,

e88075.

[19] B. H. Heng, S. W. Lim, ”Epidemiology and con-

trol of SARS in Singapore”, Epidemiological News

Bulletin, 2003, pp. 42-47.

[20] K. C. Ang, ”A simple model for a SARS epidemic”,

Teaching Mathematics and Its Applications, 2004,

pp. 181-188.

[21] G. M. Leung, A. J. Hedley, L. M. Ho, P. Chau, ”The

epidemiology of severe acute respiratory syndrome

in the 2003 Hong Kong epidemic: an analysis of all

1755 patients”, Ann Intern Med., 2004, pp. 662-73.

[22] S. Sharmin, I. Rayhan, ”Modelling of infectious dis-

eases for providing signal of epidemics: A measles

case study in Bangladesh”, J Health Popul Nutr.,

2011, pp. 567-573.

[23] M. G. Roberts, M. I. Tobias, ”Predicting and pre-

venting measles epidemics in New Zealand : applica-

tion of a mathematical model”, Epidemiol. Infect.,

2000, pp. 279-287.

[24] O. N. Bjrnstad, B. F. Finkenstdt, B. T. Grenfellet,

”Dynamics of measles epidemics: estimating scaling

of transmission rates using a time series SIR model,

Ecological Monographs, 2002, pp. 169-184.

[25] A. Sumi, N. Ohtomo, Y. Tanaka, S. Sawamura,

L. F. Olsen, N. Kobayashi1, ”Prediction analysis

for measles epidemics, Jpn. J. Appl. Phys.”, 2003.

[26] G. E. P. Box, G. M. Jenkins, Time Series Analysis:

Forecasting and Control, 2-nd Edition, Holden Day,

San Francisco, 1976.

[27] G. E. P. Box, G. C. Tiao, ”Intervention analysis

with applications to economic and environmental

problems, Journal of the American Statistical As-

sociation, 1975, pp. 70-79.

[28] S. Makridakis, S. C. Wheelwright, R. J. Hyndman,

Forecasting: Methods and Applications, New York:

John Wiley & Sons, 1998.

[29] A. Pankratz, Forecasting with Univariate Box-

Jenkins Models, Wiley, New York, 1983.

[30] P. J. Brockwell, R. A. Davis, Introduction to Time

Series and Forecasting, Springer-Verlag, New York,

1996.

[31] S. Konishi, G. Kitagawa, Information Criteria and

Statistical Modeling, Springer, 2008.

[32] J. Scott Armstrong, Standards and practices for

forecasting, Principles of Forecasting: A Hand-

book for Researchers and Practitioners, J. Scott

Armstrong (ed.), MA: Kluver Academic Publishers,

2001, pp. 1-46.

[33] J. Casals, A. G. ,Hiernaux, M. Jerez,S. Sotoca,

A. Trindade, State-Space Methods for Time Se-

ries Analysis: Theory, Applications and Software,

Chapman and Hall/CRC, 2016.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on SIGNAL PROCESSING

DOI: 10.37394/232014.2023.19.15

Theodor D. Popescu

E-ISSN: 2224-3488

147

Volume 19, 2023