SARIMA Statistical Model to Predict the Consumption of Anxiolytics as

a Treatment for Mental Illnesses

CRISTIAN INCA1, MARÍA BARRERA2, FRANKLIN CORONE3, EVELYN INCA4,a,

JOSEPH GUERRA4,b

1Facultad Departamento de Informática y Electrónica,

Escuela Superior Politécnica de Chimborazo (ESPOCH),

Km 1 1/2 Panamericana Sur, Riobamba, EC060155,

ECUADOR

2Universidad Estatal de Milagro,

Cdla. Universidad "Dr. Rómulo Minchala Murillo" - Km 1.5 vía Milagro - Virgen de Fátima,

ECUADOR

3Departamento Facultad de Ciencias,

Escuela Superior Politécnica de Chimborazo (ESPOCH),

Km 1 1/2 Panamericana Sur, Riobamba, EC060155,

ECUADOR

4Independent Researcher

Riobamba, EC060108,

ECUADOR

aORCiD: https://orcid.org/0000-0001-7055-9019

bORCiD: https://orcid.org/0000-0003-4669-7715

Abstract: - The prevalence of mental health diseases and excessive consumption of anxiolytics has increased in

the world. In this scenario, the need arises to determine a model that describes the behavior of pharmacological

consumption of anxiolytics in Ecuador, in addition to allowing this general behavior to be projected over time.

With a descriptive, exploratory, and non-experimental methodological approach conditioned on obtaining

statistical data from official national and international organizations. The population of interest was generalized

using flow-type temporal data on the effective consumption of anxiolytics, consisting of 144 monthly records in

the period from January 2011 to December 2022. The records represent the proportion of people who consume

anxiolytics in relation to the population total available in the statistics of community health care with mental

illness disorders of the Ministry of Public Health. In this sense, a viable option is the construction of a

temporary SARIMA model. Due to its temporal nature and the management of monthly records, robust

estimation was chosen as an option by applying machine learning that efficiently decomposes and extracts both

the seasonal and trend components present in the data. Determining the pharmacological consumption of

anxiolytics depends on the seasonal factor (months) and the presence of a marked tendency to gradually

increase over time, a situation that must be regulated because it represents a situation of drug dependence and

overdose. Furthermore, the built model presented adequate suitability when quantifying statistical metrics:

RMSE = 5.25% and MAPE = 1%. It is concluded that the proposed model explains the behavior of the

consumption of anxiolytics in Ecuador to mitigate situations that occurred in the affected person (anxiety or

depression) in the last three months, according to the specification of deterministic and random components

identified in the estimated model.

Key-Words: - SARIMA model, prediction, anxiolytics, medical prescription, mental illness, drug

dependence, seasonal factor, overdose.

Received: March 5, 2024. Revised: September 5, 2024. Accepted: October 8, 2024. Published: November 25, 2024.

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1 Introduction

A public health problem worldwide that has been

relevant is associated with the detriment of mental

health conditions in the population. Although it is

true, one of the aspects that define mental health is

related to the study of anxiety and depression levels

in a vulnerable population. Currently, population

indicators exhibit a trend that expresses higher

levels of anxiety and depression.

Anxiety disorders, being a mental health

problem with the highest prevalence worldwide,

tend to affect 6.7% of the population (8.8% of

women and 4.5% of men). Where this percentage

tends to increase to 10.4% when considering the

combination of “anxiety signs or symptoms”, or

alternatively, when evaluating only the behavior of

anxiety, a constant prevalence is described

throughout adulthood, between 10% and 12% for

women. On the other hand, this history of anxiety

disorder occurs between the ages of 35 and 84. In an

expanded way, when symptoms are taken into

account, that number increases between 16% and

18% in prevalence for anxiety, [1].

According to data published by [2], there is a

25% increase worldwide in the population affected

by consuming a greater amount of medications

prescribed to treat depression. In this context, the

worldwide comparative indicator of the high

prevalence of anxiety and depression disorders is

expressed in the consumption of medications that

are greater than 100 daily doses per thousand

inhabitants.

Considering the description of behavior

regarding depression in the context of the Republic

of Ecuador, its characterization turns out to be

multifactorial and difficult for doctors to diagnose.

According to the study by [3], an evaluation of the

attitudes present in Ecuadorian doctors towards

depression was carried out, focusing on a lack of

confidence in the management of this condition and

delimiting the need to implement continuous

training and updating in medical professionals. With

these limitations, the problem increases when

assigning an appropriate treatment to the patient's

clinical condition.

To maintain controlled levels of anxiety and

depression disorders, the diagnosis leads the

medical specialist to prescribe anxiolytics because

this consumption allows the treatment of anxiety

disorders, generalized anxiety disorders, panic

disorders, social phobia, and depression, [4].

It is a reality that excessive consumption of

anxiolytics tends to produce dependence and

tolerance, implying that treated people may need

increasingly higher doses to obtain the same effect.

Additionally, anti-anxiety medications can interact

with other medications and substances, such as

alcohol, which can increase the risk of serious side

effects, [5], [6].

Furthermore, it is important to highlight that the

consumption of anxiolytics must be supervised by a

mental health professional, since their inappropriate

use can have adverse effects such as dependence

and addiction, drowsiness, problems with

coordination and memory capacity, memory visual

and/or verbal, memory work, confusion, and

disorientation, among others, [5], [6].

Currently, the need to control this type of non-

communicable disease in relation to mental health

and the phenomena inherent to the socioeconomic

aspects derived from drug use that characterize the

current epidemiological profile of Ecuador are

highlighted. Indeed, it represents a key and priority

aspect for improving the health of the population

and the national health system.

For this reason, the general area of this line of

research, based on the machine learning paradigm

[7], has focused on the development of a statistical

model that allows predicting the consumption of

anxiolytics as a direct treatment to mitigate mental

illnesses. This model is based on a univariate time

series methodology applied to data related to the

consumption of anxiolytics and the clinical

resources available so that the patient is treated for a

certain pathological condition based on the control

of mental health. These statistical data are made up

of records from official entities in Ecuador.

Regarding the methodological background for

empirical developments, data mining and time series

modeling are prominent in explaining phenomena

that occur in society, whose beginning was a

relative boom based on the contributions of [8].

Since then, research and development have focused

on various aspects of this field. The research by [8]

led to obtaining concrete answers to fill the gaps

that were registered in the area of time series

modeling.

Research on stochastic methods addressed by

machine learning within artificial intelligence was

founded by [7], and the different aspects of data pre-

processing were consolidated by [9]. These

contributions led to multiple contributions that were

enriched as scientific articles were published.

The problem with real social environment data

sets is that they have complex structures that are

described with different connotations associated

with the way the data is distributed, exhibiting

underlying patterns over short- and long-term time

periods, and even redundant data points and errors

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that create complicated condition for time series

analysis.

This complexity was assumed and adjusted

through the construction of hybrid models in

different investigations, [9], [10], [11]. The

contributions of these authors were what promoted

the empirical applicability of the statistical methods

developed as well as the applicability of the most

recent machine learning methods.

As new models are developed according to the

nature of the variables of interest and data collection

on a scale of appropriate measurements, the

estimates provided under the machine learning

paradigm are increasingly precise and efficient [7],

which will govern a timely decision-making

process.

Within these methodologies are SARIMA-type

time series models, which represent a class of

models that are used to predict future values based

on past values. These models are particularly useful

for predicting anxiolytic consumption, as these data

have shown seasonal and trend patterns.

Added to this scenario is the presence in

Ecuador of limitations in access to data and

information related to mental and brain health

conditions. Many times, these failures at the

national level are attributed to the management of

information according to bioethical criteria that limit

data access in certain sociodemographic regions.

Based on the aforementioned limitations present

in the Republic of Ecuador, a public health problem

with high prevalence in recent years is framed,

considering the main prevalence in the increase in

the consumption of anxiolytics as a consequence of

several factors (stress, anxiety, and depression), [4].

Stress is expressed as an important risk factor

for anxiety and depression. Anxiety is considered a

normal response to stress, but when it becomes

excessive or uncontrollable, it can become an

anxiety disorder, while depression represents a

mood disorder that can cause feelings of sadness,

hopelessness, and loss of interest in things and

activities, [4].

It is necessary to have a good description of the

phenomenon to be able to establish strategies for the

early detection of possible mental disorders and

study the implications of the appropriate use of

anxiolytic medications to channel a significant

impact on people's quality of life, [12].

The above represents alternatives that determine

the construction of a statistical model for the

prediction of the pharmacological consumption of

anxiolytics as a treatment applied to mental illness

disorders in Ecuador during the period 2011-2022.

Likewise, the identification of the predictive

components and consumption patterns of anxiolytics

helps to better understand the determining

components of mental health in Ecuador and

develop health policies more adapted to the specific

needs that promote the psychological well-being of

the Ecuadorian population, [12].

In essence, the objective of the present study

focused on the construction of a SARIMA model

that assumes a seasonal process. This construction

was carried out based on the perspectives expressed

in research developed with this statistical

methodology, [13], [14], [15], [16]. Assuming

model construction using a machine learning

approach to obtain seasonal time series modeling

under Python 3.11 in a PyCharm 2024.1 integrated

development environment.

2 State of-the-art

Machine learning is used to diagnose mental health

problems, as it allows us to broadly examine data

patterns that indicate certain illnesses. This data can

be collected and curated from various official

sources, including hospital records, brain imaging

scans, and social media posts.

In these modeling scenarios, different

algorithms are designed, including supervised

learning algorithms, which are trained on previously

labeled data. Or, failing that, unsupervised learning

algorithms, can discover patterns in the data without

the need for labeling or prior identification of an

explicit description.

Once the model is built and trained with the data

set of interest, specific predictions are determined.

Indeed, one can consider determining whether a

person has a certain mental health condition based

on their data or, alternatively, studying the behavior

implicit in records related to mental disorders.

Machine learning researchers make predictions

using the patterns learned from new data sets and

the comparative results resulting from the structure

of the identified models to carry out decision-

making processes.

Despite the high prevalence of mental health

illnesses, they are currently misdiagnosed and

undertreated. As a multifactorial problem, it

includes comorbidity with other diseases that

complicate an adequate diagnosis [17], the inability

of doctors to make correct diagnoses as a

consequence of the complexity of overlapping

symptoms [17], or the combination of subjective

dependence on the actions taken by patients.

In other cases, there is abuse in its use that

causes excessive drug consumption. Another

triggering factor is related to the lack of human

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resources for mental health treatment, [18]. All of

these limitations have contributed to underdiagnosis,

preventing people who require help from obtaining

the necessary care.

Due to the need to promote the description and

prediction of behaviors in the consumption of

anxiolytics that cause multiple problems, including

dependence, tolerance, and overdose in patients,

there is a need to specify how the trend and

seasonality components are described in these

disorders that are described with the consumption of

anxiolytics through the application of machine

learning. In essence, build a mathematical model

that expresses the aforementioned behavior in terms

of time dependence by estimating a time series

model with the presence of a marked seasonal

component and heteroskedasticity in the data.

Based on the above, time series prediction

models collect observations over a designated time

period, where each observation represents a specific

time (t), and then predict future outcomes based on

past events. On the other hand, seasonality is

presented as a marked component in the

construction of the model because it is an integral

part that explains the behavior of individuals under

drug consumption. This dictates that the demand for

medicines fluctuates with the seasons and the

inherent conditions of the patient. So it underscores

the need for forecast models to skillfully incorporate

these seasonal nuances, [19].

In this sense, consider guidelines from [20] that

have evaluated the comparative efficiency between

methodologies related to the construction of simple

regression models with respect to time series models

with autoregressive components integrated with

moving averages (ARIMA) to explain and predict

future epidemics. In this context, comparative

advantages were found with the use of ARIMA

models with a seasonal component over simple

regression models. These findings have been valid

when considering including in the models the

periodic seasonal variations, the underlying

changing trends, and the random perturbations that

are inherent characteristics of a time series. In

addition, it was ruled to use associations in

sequentially lagged relationships to predict future

values, [21], [22].

In this context, the optimal use of prediction

techniques under the machine learning paradigm

was reflected as necessary to evaluate

epidemiological studies, marking absolute relevance

in applying these models to research on social well-

being and predicting volatility in social behaviors.

On the other hand, worldwide, the

pharmaceutical industry states that studies on the

evaluation of population behavior in the

consumption of anxiolytics as an indicator of

healthcare policies must be carried out using

predictive models that largely represent forecasting

options and decision-making as fundamental

alternatives in the process of configuring the

management of processes related to the anticipation

of future trends, [23].

With respect to innovation and overcoming the

challenges in the traditional approach, significant

advances have occurred in recent years that have led

to the generation of new advanced algorithms under

the machine learning paradigm in correspondence

with the provision of robust computational resources

provided by the language (Python programming

comes in its different versions).

Unlike what is presented in the construction of

inferential models, descriptive analysis in the

construction of time series models is assumed in the

findings of [24], which considered essential aspects

such as the use of the simple exponential smoothing

methodology to explain the constant consumption of

medication. If a notable trend in medication

consumption occurs, the implementation of double

exponential smoothing is recommended. For

oscillating consumption that presents a marked

seasonal component, basic and not very robust

descriptive methods must be used, such as triple

exponential smoothing, which is also referred to as

the Holt-Winters equation. In a similar direction,

there are the contributions of [25], concluding on

the benefits of building a Holt-Winters model,

which analyzes trend and seasonality in prediction.

However, the aforementioned descriptive

methodology loses precision by requiring long-term

predictions.

3 Materials and Methods

3.1 Type and Design of the Research

The study focused on the management of secondary

data from records, forming a documentary research

design. For the purposes of model estimation, the

time horizon was managed from January 2011 to

December 2022 as an exogenous variable. All

information on medication dispensing was purified,

transformed, and stored in a database in the Python

3.11 programming language. Due to its temporary

nature and the management of monthly records, the

robust option for construction lies in a mathematical

model: ARIMA(p,d,q)x SARIMA(P,D,Q)s.

This model is ideal for extracting the seasonal

and trend components underlying the data in the

temporal process. In data collection, the inclusion

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criterion was followed, consisting of patients of all

ages and of both sexes who had been prescribed an

anxiolytic during the study period. It should be

noted that no clinical trial was conducted with an

incident factor, which represents a scientific study

that does not tend to violate the principles of

bioethics and confidentiality.

3.2 Research Methods

Descriptive research was carried out at a

documentary level as a result of collecting

information from the repositories Scielo, Scopus,

Google Scholar, and Science Direct to support this

research with content analysis from scientific

articles with temporal relevance within the last 5

years, consisting of publications published from

January 2019 to December 2023.

For the empirical evaluation, an analytical

investigation is defined based on the construction of

a mathematical model on the variable under study in

this investigation, framed in the prescription for the

consumption of anxiolytics represented on a DHD

measurement scale (daily dose defined per 1,000

inhabitants/day) according to data from the

Community Mental Health Network and State

Addiction Recovery Centers.

3.3 Research Focus

Based on the quantitative information recorded on

medication consumption, a database was created

considering the following variables: temporary

coverage from January 2011 to December 2022. On

the other hand, medication consumption is defined

as the observed series (Anxiolytics related to time

defined in monthly records). The resulting model

underwent internal validation, which represents a

statistical technique used to evaluate the

performance of a predictive model using the same

data that was used to train the model.

3.4 Study Population and Sample Selection

The population of interest was generalized into

flow-type temporal data, which represents the

effective consumption of anxiolytics as constituted

by 144 monthly records in the Republic of Ecuador.

Temporal coverage as a sample of interest was

defined as all records of consumption of anxiolytics

for the treatment of mental illnesses within the

period from January 2011 to December 2022.

3.5 Data Collection Techniques and

Instruments

In the statistical data collection phase, the database

of the Project for the Creation and Implementation

of Services of the Community Mental Health

Network and State Addiction Recovery Centers

(PCISRSMCCE) was used to a greater extent, in

line with the contributions of other instances such as

the Ministry of Public Health of Ecuador,

specifically the National Directorate of

Normalization and the National Directorate of

Disabilities, [4]. In relevance, these data have been

collected through the logistics of these official

entities through formal requests to officials

responsible for the mental health component of the

institutions of the Comprehensive Public Health

Network (RPIS) and private institutions with and

without profit.

Statistical data was collected and refined with

other secondary sources within the 2019 Global

Burden of Disease (GBD) study developed by the

University of Washington [26], as well as official

records on the burden of mental disorders in the

Region of the Americas: Profile of Ecuador and

statistics from the Pan American Health

Organization, [27]. In addition, data from the

National Directorate of Disability, Rehabilitation,

and Palliative Care in Health was used, which

established that in Ecuador there are 48,078 records

for the year 2023 of people with intellectual and

mental disabilities, in a universe of 480,776 people

with some type of disability. This represents 10% of

the total number of people with disabilities in

Ecuador, [28].

3.6 Seasonal ARIMA Process Methodology

(SARIMA)

The methodology for the construction of ARIMA

stochastic processes with a seasonal component, as

referred to in the scientific literature as SARIMA, is

based on the fulfillment of a series of phases:

identification and specification of the underlying

components, estimation of the model parameters,

statistical validation, and prediction, [7].

The previous treatment of the data is framed by

the use of transformations in the logarithmic series

(Box-Cox) and the application of regular and

seasonal differentiation to normalize the series and

obtain stationary data. Then, using machine learning

algorithms under the PyCharm integrated

development environment and the Python

programming language, the optimal mathematical

structure is determined and the parameters are

estimated according to the appropriate order in the

different stochastic models identified. In the last

procedure, the best model is found after cross-

validation with training and test data to discriminate

suitability in the model.

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If the anxiolytic consumption series 󰇝󰇞 presents a

component marked with period s, it can be

eliminated by applying the seasonal difference

operator with a lag of order s =12, equivalent to the

data collection periods, and thus obtain a series

󰇝

󰇞with a process structure WEAPON.

In addition, if the temporal process exhibits a

regular trend and a marked seasonal component, the

order of regular (d) and seasonal (D) differentiation

is defined. If d and D are non-negative integers, then

󰇝󰇞it represents a model under a multiplicative

seasonal process, which is produced by the

interaction of a regular part 󰇛  󰇜in

conjunction with a seasonal part

󰇛  󰇜with a seasonal period.

The series, differentiated into its regular and

seasonal components, is denoted as an ARMA

process:



   󰇛  󰇜󰇛 󰇜.

Which is defined by:

󰇛󰇜󰇛󰇜

  󰇛󰇜󰇛󰇜 󰇝󰇞 

  󰇛 󰇜

Where:

󰇛󰇜: Represents the polynomial that assumes the

delay operators of the autoregressive coefficients

that make up the regular part of the model.

󰇛󰇜: Represents the polynomial that assumes

the seasonal delay operators of the autoregressive

coefficients that make up the seasonal part of the

model.



: Consumption of anxiolytic medications related

to the time defined in monthly records.

󰇛󰇜: It represents the polynomial that assumes the

delay operators of the moving average coefficients

that make up the regular part of the model.

󰇛󰇜: Represents the polynomial that assumes

the seasonal delay operators of the moving average

coefficients that make up the seasonal part of the

model.

: A random disturbance must be adjusted to

normal behavior with a zero mean and constant

variance.

This type of model is called󰇛  󰇜

󰇛  󰇜. In essence, according to [29], the

seasonal ARIMA model includes autoregressive and

lagged moving average terms.

The ARIMA seasonal process methodology is

used to forecast the future of a time series that

presents a seasonal pattern (s = 12 months). To do

this, data from historical data series are used to

estimate the parameters of the ARIMA model for

the regular part and the seasonal part in their

autoregressive and moving average components.

Once the parameters are estimated, the model can be

used to forecast the future of the time series, [30].

The structure or components that constitute the

specification in this type of multiplicative

component model involve identifying the dynamics

of seasonal processes based on the stochastic

process under study. In this sense, it must be

described as follows:

 The trend component, which represents the

general growth of the time series observed or

under study,

 The seasonal component, which represents the

seasonal fluctuations of the series,

 The non-stationary component, which

represents the random fluctuations of the time

series that represent the pharmacological

consumption of anxiolytics in Ecuador.

Models under a seasonal ARIMA process come

with multiple applications when complex structures

must be implemented under stochastic processes

that exhibit a variety of seasonal patterns, including

additive and multiplicative seasonal patterns. They

can also be used to forecast time series that exhibit a

variety of non-stationary patterns, including linear

and non-linear patterns, [31]. The strong advantage

of using this type of model lies in the advanced

accuracy of short-term prediction results, [32].

3.7 Seasonal ARIMA (SARIMA) Process in

Python

This section describes the commands and

parameters used to build ARIMA models, which are

described by using the statsmodels.tsa.arima_model

module, proceeding to import the data and the

hyperparameters p, d, and q (in that order) using a

machine learning algorithm to decompose a time

series into its seasonal, trend, and random

disturbance components, [33]. Below is the fit() call

in this module, which returns a trained model that is

used for evaluation and inference. Another

alternative is based on defining the ARIMA.fit

command for the specification and estimation of the

model parameters, [34].

This model overview provides several statistical

measures to evaluate the performance of the

ARIMA model in the Python programming

language, based on scores given by the criteria AIC

(Akaike Information Criterion), BIC (Bayesian

Information Criterion), HQIC (Hannan-Quinn

Information Criterion), and the standard deviation of

innovations (innovations are the difference between

the actual value at time t and the expected value at

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that moment of time). However, measures such as

AIC, BIC, and HQIC depend significantly on the

probability learned from the training data.

Additionally, to check the percentage fit of the

trained model to the collected data that defines the

time series, the plot_predict command of the

ARIMA forecasting Python environment is used,

which is obtained by training the actual and

predicted values on top of each other by plots. This

line graph is calculated from the weights learned

and trained by the model. The above allows us to

check how well the prediction works based on the

learned coefficients, [33].

3.8 Evaluation of the Quality of the

Proposed Model

In consideration of evaluating the adequacy of the

predictions with the estimated model, available error

metrics, or Key Performance Indicators (KPI), were

used: root mean square (RMSE), mean absolute

error (MAE), mean percentage error (MPE), mean

absolute percentage error (MAPE), the correlation

coefficient (to measure similarity), and the

minimum allowed error.

4 Results

4.1 Seasonal ARIMA (SARIMA) Process

Specification

In this context, the performance of the different

models that have been identified under the seasonal

decomposition machine learning algorithm was

adjusted step by step through routines developed in

the Python 3.11 programming language under the

PyCharm 2024.1 integrated development

environment. These performances summarize the

best mathematical structure to identify, as shown in

Table 1.

Table 1. Specification and estimation of parameters

for the Model󰇛  󰇜󰇛  󰇜

Variable Dep.: Y

No. Observations: 119

Model :

SARIMA(3, 1, 1)x(1, 0, 1, 12)

Log Likelihood 235.108

Date: Fri, 23 Feb 2024

AIC -454,215

Time: 13:01:22

BIC -432,050

Sample: 01-01-2011

HQIC -445.215

-12-01-22

Source: Results generated by Python 3.11

This package, Python 3.11, adapts series to

models󰇛  󰇜󰇛  󰇜, that is, the

autoregressive and moving average parts for the

regular and seasonal components in the series under

study. In this sense, the estimation of the parameters

in the model that was determined through the

package is shown (Table 2) in this case a

multiplicative model:

󰇛󰇜󰇛󰇜.

Table 2. Estimation of parameters for the

Model󰇛  󰇜󰇛  󰇜

coef

std err

P>|z|

[0.025

0.975]

intercept

9.72E-

3.89E-

0.25

0.803

-6.67E-05

8.60E-05

ar.L1

-0.2973

0.115

-2,585

0.010

-0.523

-0.072

ar.L2

0.0354

0.139

0.256

0.798

-0.236

0.307

ar.L3

0.4628

0.135

3,419

0.001

0.198

0.728

ma.L1

-0.8993

0.103

-8,721

0.000

-1,101

-0.697

ar.S.L12

0.9601

0.096

10,041

0.000

0.773

1,148

ma.S.L12

-0.7909

0.255

-3,099

0.002

-1,291

-0.291

sigma2

0.0010

0.000

6,132

0.000

0.001

Ljung -Box (L1) (Q): 0.80

Jarque-Bera (JB): 125.70

Prob (Q): 0.37

Prob (JB): 0.00

Heteroskedasticity (H): 0.34

Skew : -1.22

Prob (H) ( two-sided ): 0.00

Kurtosis : 7.43

Source: Results generated by Python 3.11

Where the mathematical structure is built by

developing the following expression:

󰇛     󰇜󰇛

 󰇜󰇛󰇜



 󰇛  󰇜󰇛  󰇜

The previous mathematical formulation is

simplified to a non-stationary seasonal

multiplicative model ARIMA(3,1,1)x(1,0,1) 12

defined as follows:

             

        

 

This model is used to analyze time series that

present a non-stationary trend and seasonality, the

graphical representation of which is shown in Figure

1. The non-stationary trend is adjusted by specifying

the significant autoregressive and moving average

coefficients in its regular part, while the seasonal

part is adjusted by specifying the seasonal

autoregressive and moving average coefficients.

Moving average coefficients are used to remove

random noise from the time series.

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          

       

       

 

In essence, the estimated model suggests that

the explanation of anxiolytic consumption in

Ecuador, at a deterministic level, is defined in its

regular component in terms of the events that

occurred in the population within the last three

months with the consideration of random events that

occurred. The last month and in its seasonal

component, both at a deterministic and random

level, it depends on what happened in the previous

month.

Fig. 1: Internal validation of the pharmacological

consumption of anxiolytics using the mathematical

model󰇛󰇜󰇛󰇜

Source: Results generated by Python 3.11.

When evaluating Figure 1, consistency is

reflected in the projection achieved between the

observed and fitted values through the structure of

the estimated statistical model. Therefore, to delve

into the benefits of the model, a more precise

evaluation must be carried out in terms of the

metrics obtained in the quantification of forecast

errors. The ultimate goal is to verify the suitability

of the specified structure according to the temporal

coverage of the data.

4.2 Metrics to Evaluate the Suitability of the

Specified Model

In this section, Table 3 establishes the metrics to

evaluate the adequacy of the prediction errors in the

identified model, for which the mean absolute error

(MAE = 4.51%) and the mean absolute percentage

error (MAPE) are used. = 1.01%), the root mean

square error (MSE = 2.76%), and the root mean

square error (RMSE = 5.25%).Which exhibit values

close to zero, an ideal situation to argue that the

identified model represents a solid structure to make

predictions regarding the pharmacological

consumption of anxiolytics for the treatment of

mental disorders in patients in Ecuador, 2020-2022.

The results shown in Table 4 describe the

evolution of the predictions with a confidence level

of 95%, which allows us to infer a gradual increase

in the prescription for the consumption of

anxiolytics on a DHD measurement scale (daily

dose defined by 1,000 inhabitants/day). The

prediction limits of 95.0% for the forecasts are

decisive to emphasize an increase in the

consumption of medications for the treatment of

mental health disorders in Ecuador. In short,

regarding the consumption of anxiolytics, according

to the model's predictions, this DHD should increase

to 95.77 in the month of December 2023.

Table 3. Adequacy metrics in model

forecasts󰇛󰇜󰇛󰇜

Metrics

Mistake

MAE

0.045103

MAP

0.010067

MSE

0.002765

RMSE

0.052582

Source: Results generated by Python 3.11

Table 4. Predictions for the year 2023 on the

pharmacological consumption of anxiolytics using

the mathematical

model󰇛󰇜󰇛󰇜

Period

Forecast

Limit Lower

95.0%

Limit Top

95.0%

Jan-23

4.3107

88.9741

99.6473

Febr-23

89.6427

84.2977

94.9877

Mar-23

98.9746

93.5747

104,375

Apr 23

94.5173

88,554

100,481

May 23

96.2622

90.2949

102,229

Jun-23

94.9075

88.8527

100,962

July 23

96.1003

89.8572

102,344

Aug-23

92.7024

86.4296

98.9752

Sept-23

93.9671

87.6046

100.33

Oct-23

97.4577

90.9897

103,926

Nov-23

93.7357

87,216

100,255

Dec-23

95.7761

89.1733

102,379

Source: Results generated by Python 3.11

5 Discussion

As an initial aspect of discussion, the importance of

taking into account that prediction using

mathematical models for the consumption of

anxiolytics as a mitigation measure for mental

illness represents a complex and constantly evolving

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Volume 12, 2024

area. In this scenario, it is necessary to apply a

multidisciplinary approach that encompasses

psychology, psychiatry, neuroscience, and computer

science to offer more robust findings, [35].

Mathematical models can be a promising tool to

understand the implications of the evolution of

mental disorders that are explained by the increase

in pharmacological consumption, as well as focus

on public policy formulation to dictate

improvements in treatment strategies.

In this line of discussion, the benefits

determined in the research of [19] consolidate

through their study that using the mathematical

structure of an ARIMA time series model is vital to

analyzing past data in order to predict future trends,

taking advantage of the ability to use the random

component of lagged moving averages to smooth

time series data, leading to easy interpretation of

chance behavior. These models are suitable for

predictions of inherent behavior and the

development of technical analysis.

To continue highlighting other benefits of the

ARIMA methodology with seasonal components,

the study by [36] determines important information

about how ARIMA, exponential smoothing models,

and the ANN artificial neural network methodology

compare, including the use of combined models

aimed at consolidating research that aims to

establish interesting conclusions. Although the

combination of techniques is not widely used, it

leads to better predictions.

The increases observed in the trend of anxiolytic

consumption in Ecuador during the temporary

coverage from 2011 to 2022 can probably be

explained by a higher prevalence of depression and

risks associated with consumption associated with

adverse effects and dependence that are evident as a

common situation in other countries in clinical

practices, [37], [38], [39], [40]. This situation may

be due, as established by [41], to increases in the

diagnosis and treatment of depressive disorders and

changes in the structure of the population, since

depression in Ecuador represents a multifactorial

public health problem.

The determining findings of the present study

are compared with similarity to those reported in

similar studies where the tendency to consume has

increased due to the effect of the COVID-19

pandemic, in which there has been an increase in the

prevalence of depressive and anxiety disorders, [42],

[43], [44]. In fact, the consumption of anxiolytics is

associated with the relative increase in the

prescription of medications by doctors, which

causes greater demand in the population. These

results coincide with those exhibited by a study

carried out in Spain that evaluated both the

prescription and sale of anxiolytics and

antidepressants, [45].

According to the approaches of the studies

consulted, they indicate that the consumption levels

characterized as low analyzed by the OECD

between the years 2020 and 2021 have been below

40 daily doses per thousand inhabitants (DHD).

Among the countries that are configured at these

levels are: Costa Rica, Estonia, Lithuania, Hungary,

South Korea, and Latvia, [46].

However, the scenario presented by the

Republic of Ecuador is framed at average levels of

87 DHD. This leads us to affirm that not only

depression is the cause of these discomforts, but

unprecedented stress represents another determining

factor. Furthermore, according to [46], other

conditions that increase the pharmacological

consumption of anxiolytics are work limitations,

managing ample support from family or loved ones,

and the environment of community participation.

6 Conclusions

The relatively high prevalence of mental health

disorders occurs in the Republic of Ecuador, which

translates into a significant impact on

pharmacological consumption, causing a

deterioration of the clinical situation in civil society

and a negative impact on the economy.

In this context, statistical adequacy is presented

in the SARIMA model (RMSE = 5.25% and MAPE

= 1%), which was estimated under a machine

learning paradigm. of seasonal decomposition to

offer an interpretation of anxiolytic consumption in

Ecuador. At a deterministic level, this model was

defined by a structure underlying the events of

anxiety and depression that the Ecuadorian

population has experienced that triggered the

consumption of anxiolytics within the last three

months, coupled with the effect presented by

random events related to these disorders that arise in

the previous month. Regarding the existence of a

repetitive pattern consistent with the seasonal

component, both at a deterministic and random

level, the consumption of anxiolytics in Ecuador is

explained by the level of anxiety and depression that

occurred in the previous month.

These tendentious components originate from

the modeling of behavior that translates into the

combination of deterministic and random effects

that formalize a behavior of increased

pharmacological consumption of anxiolytics in

Ecuador above average levels of 87 daily doses per

thousand inhabitants (DHD).

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This situation generates the need to implement

public policies to expand services and resources that

lead to mitigating mental health problems,

considering the significant effects that occur within

one to three months in the pharmacological

consumption of anxiolytics in the Ecuadorian

population. In this scenario, it is recommended to

promote monitoring with the establishment of

mental health clinics and management for

prevention, control of prescriptions, and regulation

of drug consumption, as strategies integrated with

primary care services in mental health.

As future lines of research develop studies that

consider respecting ethical and social principles are

emerging, an approach to the ethical and social

implications of the use of medications to treat

mental illnesses, including the associated stigma and

existing deferences in access to the treatments and

implications that lead to the reduction of the risk of

additions and side effects due to excessive

consumption of unnecessary medications.

Addressing these principles opens a range of

possibilities to overcome the limitations that

currently arise related to the availability of clinical

data to undertake larger studies. Where the

combined construction of models based on the

machine learning methodology prevails (Artificial

Neural Networks, Support Vector Machine,

Random Forest, Logistic Regression, Decision Tree)

that allows explaining, based on randomized clinical

trials, the factors incident to the phenomenon under

study.

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Contribution of individual authors to the

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policy)

The authors contributed equally to the present

research, at all stages from problem formulation to

final findings and solution.

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Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of interests

The authors have no conflicts of interest to declare.

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DOI: 10.37394/232018.2024.12.49

Cristian Inca, María Barrera,

Franklin Corone, Evelyn Inca, Joseph Guerra

E-ISSN: 2415-1521

515

Volume 12, 2024