Predicting the Dynamics of Covid-19 Propagation in Azerbaijan based on

Time Series Models

SAKINA BABASHOVA

Turkish World of Economics Faculty, Department of Economics

Azerbaijan State University of Economics

Baku, Istiqlaliyyat str.6, AZ1001

AZERBAIJAN

Abstract: - The study is dedicated to developing an econometric model that can be used to make medium-term

forecasts about the dynamics of the spread of the coronavirus in different countries, including Azerbaijan. We

examine the number of COVID-19 cases and deaths worldwide to understand the data's intricacies better and make

reliable predictions. Though it’s essential to quickly obtain an acceptable (although not perfect) prediction that

shows the critical trends based on incomplete and inaccurate data, it is practically impossible to use standard SIR

models of the epidemic spread. At the same time the similarity of the dynamics in different countries, including

those which were several weeks ahead of Azerbaijan in the epidemic situation, and the possibility of including the

heterogeneity factors into the model allowed as early as March 2020 to develop the extrapolation working

relatively well on the medium-term horizon. The SARS-CoV-2 virus, which causes COVID-19, has affected

societies worldwide, but the experiences have been vastly different. Countries' health-care and economic systems

differ significantly, making policy responses such as testing, intermittent lockdowns, quarantine, contact tracing,

mask-wearing, and social distancing. The study presented in this paper is based on the Exponential Growth Model

method, which is used in statistical analysis, forecasting, and decision-making in public health and epidemiology.

This model was created to forecast coronavirus spread dynamics under uncertainty over the medium term. The

model predicts future values of the percentage increase in new cases for 1–2 months. Data from previous periods in

the United States, Italy, Spain, France, Germany, and Azerbaijan were used. The simulation results confirmed that

the proposed approach could be used to create medium-term forecasts of coronavirus spread dynamics. The main

finding of this study is that using the proposed approach for Azerbaijan, the deviation of the predicted total number

of confirmed cases from the actual number was within 3-10 percent. Based on March statistics on the spread of the

coronavirus in the US, 4 European countries: Italy, Spain, France, Germany (most susceptible to the epidemic), and

Azerbaijan, it was shown how the trajectory would deviate exponentially from a shape; a trial was carried out to

identify and assess the key factors that characterize countries. One of the unexpected results was the impact of

quarantine restrictions on the number of people infected. We also used the medium-term forecast set by the local

government to assess the adequacy of health systems.

Keywords: - Coronavirus, epidemic spread, regression models, time series analysis, exponential growth model,

prediction, applied econometrics.

Received: September 22, 2021. Revised: June 25, 2022. Accepted: July 6, 2022. Published: July 28, 2022.

1 Introduction

The coronavirus pandemic, which emerged at the end

of 2019 in the Chinese city of Wuhan and spread

almost all over the world in the spring of 2020, has no

close analogs in recent decades, so it is tough to

predict its dynamics and consequences, including the

impact on the economy. The situation with both

forecasting and the fight against the new virus is

aggravated by its properties such as high infectivity.

A rather long incubation period and a high proportion

of asymptomatic carriers make the current official

statistics very inaccurate, and practically exclude the

use of standard models of the spread of epidemics [1],

incl. spatial SIR-models [3] describing the dynamics

of groups of susceptible, infected, and recovered

persons. At the same time, a significant proportion of

carriers with severe symptoms, a high mortality rate

(also not yet accurately determined,) and the

exceptional significance of the impact of the

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pandemic on the global economy and the economy of

individual countries [4] determine the particular

relevance of at least medium-term forecasts of the

dynamics of the number of infected, including, in the

context of developing measures to limit human

contacts, monitoring the effectiveness or

ineffectiveness of their impact on the rate of spread of

the epidemic, and predicting the expected burden on

the health system.

A feature of the new COVID-19 virus epidemic is

the lack of statistics from previous years. In this

regard, there is a problem with good use of the

available information on the parameters of the

developing epidemic, including in other countries of

the world. Many scientific groups in the USA, China,

and Europe are working on developing methods to

predict the spread of a new virus in the short term. On

February 11, 2020, the World Health Organization

announced the Global Research and Innovation

Forum and identified the most critical research goals

in distribution epidemics [5]. In early March 2020,

Science Translation Medicine published an editorial

[6] in which its authors formulated research

directions, the innovative results of which should

contribute to the prevention of the spread of the

epidemic. They also noted that comprehensive

mathematical models that include complex

pathogenic and socially significant variables require

significant time and effort to develop and verify

(often from months to several years). However,

mathematical models predicting the dynamics of

registration of new cases of COVID-19 in real-time

began to appear in journals and online resources [7-

12]. The article [7] provides estimates of the extent of

the epidemic in Wuhan and other cities, including

outside mainland China, to which the virus may have

been exported from Wuhan. The authors predict the

values of the population's domestic and global health

risks from epidemics based on the SEIR (Susceptible-

Exposed-Infectious-Recovered) model, taking into

account possible scenarios for preventive

intervention. The article [9] studied the dynamics of

the spread of coronavirus in India using a system of

differential equations with constant coefficients.

Moreover, the concept of the primary reproduction

number, applying the Pontryagin maximum principle

to solve the problem of optimizing preventive

measures. The work [10] draws attention to the

similarity of the dynamics of the total number of

infected, recovered, and dead people in China and

Italy. It also analyzes the solutions of the system of

differential equations adopted in the SIRD

(Susceptible-Infected-Recovered-Deaths) model. It

notes that although the SIRD model is rather crude,

its use gives a good chance to reflect at least the

general features of the evolution of the epidemic and

predict the dynamics in real-time. The suggested

technique aims to predict the peak in Italy in terms of

the increase in new infections and the number of

deaths over the entire period of the epidemic. The

author's articles [10] hypothesize that any country that

becomes a theater epidemic outbreak can be seen, at

least as a first approximation, as an environment in

which different population groups interact according

to some general rules, regardless of geographical

variations. The authors of the article [11] use the

quantitative picture of the spread of the COVID-19

disease in China as a test case and infection data from

eight countries to assess the evolution of the

epidemiological process in each of these countries.

This approach is based on the Gaussian hypothesis of

virus spread and the basic SIR (Susceptible-Infected-

Recovered) model. Considering difficulties in

applying to predict the dynamics of the spread of

COVID-19 deterministic models such as SIR, SEIR,

and SIRD, which are built on the mechanisms of virus

spread from individual to individual and use estimates

distribution parameters of known viruses, our efforts

will be focused on looking for other methods. An

important observation formulated in [10] about the

similarity parameters of the epidemic in China and

Italy, as well as an analysis of statistics on the spread

of coronavirus in the USA, Italy, Spain, France, and

Germany led us to a hypothesis about the possible

dependence of the growth rate of the leading

indicators recorded in Azerbaijan on similar

indicators in the countries surveyed.

2 Research Methodology

The baseline study was carried out based on March

data on the number of infected in the United States,

and the four European countries most affected by the

epidemic - Italy, Spain, France, Germany, and

Azerbaijan. Its purpose was to build a mid-term

forecast of the dynamics of coronavirus infection in

Azerbaijan, where at that time the epidemic had just

begun, and the number of cases did not exceed the

thousandth mark. It was important to estimate at least

the order of the numbers for the number of infected

and the timing of reaching the plateau since the values

in different sources from the end of March to the

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beginning of April differed not even several times,

but tens and hundreds of times. At the same time, the

publication of this study to a wider audience one year

after the calculations has different aims. First, it is

important to demonstrate that it is possible to obtain a

forecast of acceptable quality based on a simple

extrapolation of data and analogies between countries

before the available statistics allow the use of more

complex and correct tools (while, of course, you need

to understand the limitations of extrapolation and that

it only works before the tendency changes). Secondly,

econometric models make it possible to identify the

significance of certain factors in the context of

influencing the resulting indicator, and these results

may be important, including for taking certain

decisions by the authorities. Thirdly, having real

statistics on the dynamics of the spread of the virus in

different countries, we see its discrepancies with

preliminary estimates, which gives grounds for

adjusting measures aimed both at combating the

pandemic and its economic consequences. The

official statistics on the number of detected cases of

infection, presented on the website

https://www.worldometers.info/coronavirus [17],

were used as the initial data. We realized that these

statistics were incomplete and inaccurate. Probably,

the actual number of infected in asymptomatic and

mild forms exceeds the official figures by several

times.

However, the statistics presented quite accurately

reflected the tendencies occurring in reality, including

the dynamics of the spread of the epidemic, which

means that it was possible to focus on them.

Let's summarize the data for the countries most

prone to the epidemic, as well as for Azerbaijan in

Table 1 (see next page).

2.1 Exponential Growth Model

A growth curve is an empirical model of a quantity's

evolution over time. Growth curves are widely used

in biology to analyze quantities such as population

size in population ecology, demography for

population growth analysis, and individual body

height in physiology for personal growth analysis.

Growth is a fundamental property of many systems,

including economic expansion, epidemic spread,

crystal formation, adolescent growth, and stellar mass

condensation. One of the simple models in which the

population grows at a constant proportional rate over

time is the exponential growth (unlimited population

growth) model [22]. Depending on whether

reproduction is assumed to be continuous or periodic,

the relationship can be expressed in one of two ways

[25]. Exponential growth produces a continuous curve

of increase or decrease, the slope of which varies in

direct proportion to population size.

    (1)

where r is the constant rate of growth,  is the

initial population size, and t and  represent time and

population at time t, respectively (Method 1). Another

type of exponential curve is shown below.

    (2)

where   󰇡

󰇢and thus the growth rate in Eq.

(2) is not a constant growth rate.

In the initial stage, the spread of the virus occurs

by the laws of exponential growth. And an

exponential model of the form of

   (3)

which corresponds to the situation of a constant

daily increase in the number of infected persons, and

can be considered as a benchmark. The differences

between the countries consisted only in the initial

level and growth rates, which were easy to calculate

from the March data, as well as to make a mid-term

approximation.

With the current COVID-19 outbreak, we are

hearing about exponential growth. In this study, an

attempt was made to understand and analyze the data

using an exponential growth curve. The reason for

using an exponential growth curve to study the

pattern of COVID-19 incidence is that

epidemiologists have studied these events. It is well

known that the first period of an epidemic follows

exponential growth. The exponential growth function

is not always a perfect representation of the epidemic.

Because the exponential curve only fits the outbreak

at the beginning, we attempted to fit it first and then

studied the logarithmic growth curve. At some point,

recovered people will no longer spread the virus, and

when someone has been infected, the virus's growth

will cease. Logarithmic growth is distinguished by

increasing growth in the early period followed by

decreasing growth after the point of inflection [23]. In

the case of the coronavirus, for example, the

maximum limit would be the total number of exposed

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people in Azerbaijan, because once everyone is

infected, the virus's growth will be halted. Following

that, the increasing rate of the curve begins to decline

and eventually reaches a minimum.

Table 1. The total number of persons infected with COVID-19 by countries, March 2020

(Source: Compiled by the author based on [17])

Date

USA

Italy

Spain

France

Germany

Azerbaijan

1-Mar-20

1701

130

2-Mar-20

100

2036

120

191

165

3-Mar-20

124

2502

165

212

203

4-Mar-20

158

3089

228

285

262

5-Mar-20

221

3858

282

423

545

6-Mar-20

319

4636

401

653

670

7-Mar-20

435

5883

525

949

800

8-Mar-20

541

7375

674

1209

1040

9-Mar-20

704

9172

1231

1412

1224

10-Mar-20

994

10149

1695

1784

1565

11-Mar-20

1301

12462

2128

2281

1966

12-Mar-20

1630

15113

2950

2876

2745

13-Mar-20

2183

17660

4209

3361

3675

14-Mar-20

2771

21157

5753

4499

4599

15-Mar-20

3617

24747

7753

5423

5813

16-Mar-20

4604

27980

9191

6633

7272

17-Mar-20

6357

31506

11178

7730

9367

18-Mar-20

9317

35713

13716

9134

12327

19-Mar-20

13898

41035

17147

10995

15320

20-Mar-20

19551

47021

21571

12612

19848

21-Mar-20

24418

53578

25496

14459

22364

22-Mar-20

33840

59138

29909

16689

24873

23-Mar-20

44189

63927

35480

19856

29056

24-Mar-20

55398

69176

42058

22302

32991

25-Mar-20

68905

74386

50105

25233

37323

26-Mar-20

86379

80589

57786

29155

43938

122

27-Mar-20

105217

86498

65719

32964

50871

165

28-Mar-20

124788

92472

73232

37575

57695

182

29-Mar-20

144980

97689

80110

40174

62435

209

30-Mar-20

168177

101739

87956

44550

66885

277

31-Mar-20

193353

105792

95923

52128

71808

298

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3 Results & Discussion

3.1 Exponential Growth Model Results

Let us summarize in Table 2 the results of calculations

using the exponential growth model, estimated on

March 2020 data for each of the countries.

Table 2. Average growth rates of the number of

infected by countries and approximation

(Source: Compiled by the author)

Country

Growth

rate, %

Growth in

30 days,

times

Approximation

for 30.04,

thousand

people

USA

27.6

1483.3

286800

Italy

14.1

51.6

5463

Spain

24.2

657.8

63097

France

19.5

210.6

10976

Germany

21.7

359.7

25832

Azerbaijan

23.1

513.8

1201

This cannot even be called a forecast - the growth

rates in the first weeks of the spread of the virus can

be very high until a certain critical level is reached,

then they gradually decrease. At the same time, all

European countries, with insignificant specific

features, in contrast to China and other countries of

the Far East, where extremely strict quarantine

measures were introduced, and rapid identification

and localization of foci were carried out, move along

approximately the same trajectory, adjusted for the

time, which is also possible take into account in the

model [24].

Let us make the following clarification: we will

calculate the average initial rate of exponential growth

not according to data for March 1-31, but for a fixed

number of days (for example, 15 or 30 days) from the

moment the country exceeded the threshold of 1000

detected cases (before that, random daily fluctuations

were too large, and data is too sensitive to single

large-scale infections). In different countries, this

happened at different times, as shown in Table 3,

where, along with the average daily rate of increase in

the number of detected cases, the moment of crossing

the threshold is indicated. Also, the last column of

Table 3 shows how much the growth rates decreased

when switching from a two-week to a monthly

modeling horizon.

The reported initial growth rates may serve as a

rough indicator of the rate at which the virus spreads.

In particular, in Azerbaijan, they were slightly lower

than in key European countries. Moreover, another

advantage could be called a temporary difference of 4

weeks - there was a little more time in Azerbaijan to

assess the risks and take measures to localize the foci

of the spread of the virus.

Table 3. The average rate of increase in the number of

infected after crossing the threshold

(Source: Compiled by the author)

Country

Date of

crossing

the

threshold

Rate of

increase

for 15

days, %

Rate of

increase

for 30

days, %

Growth

rate

decrease

, %

USA

11-Mar-20

29.4

20.6

8.8

Italy

29- Feb-20

14.8

5.2

Spain

9-Mar-20

23.5

15.6

7.9

France

8-Mar-20

18.9

14.6

4.3

Germany

8-Mar-20

23.5

15.8

7.7

Azerbaijan

11-Apr-20

17.1

14.6

2.5

Unfortunately, this was not sufficient to prevent

the epidemic (this, in particular, can be seen in the

smallest decrease in the growth rate among all

countries when moving from a 15-day to a 30-day

horizon). The size of the country is also an objective

reason. When the epidemic ends in some regions, an

outbreak may occur in others and the process

continues. At the same time, exponential growth

cannot last forever, and even for a medium-term

forecast, more complex models should be considered.

In particular, the growth rate is gradually

decreasing from the initial high level to lower values.

In the simplest version of the model, this decrease can

be linear. The data showed that the base rate of

growth at the time the threshold reached 1000 infected

was 25.7% per day, slightly differing across countries

and decreasing daily by an average of 0.99 percentage

points. However, a decreasing linear function always

sooner or later goes into the negative region -

assuming the same decrease in 40 days in the USA, 37

in Spain, 33 in Italy and Germany, 31 in France, and

26 in Azerbaijan. Moreover, this cannot be the case

for the indicator of the dynamics of cumulative

dependence (we consider as a resulting indicator the

total number of infected people detected since the

beginning of the epidemic and not the number of

patients at the moment), so it is desirable to change

the model specification. As a modified version, we

consider the exponential decrease in the relative

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increase in y from time t. Let us also take into account

the decrease in growth depending on the proportion x

of infected people in the country. The mechanisms of

the effect of this indicator can be very different, but in

general, this corresponds to the negative relationship

of the limit indicators with the current level of the

cumulative value, which is characteristic of many

processes, taking into account a large number of

undetected asymptomatic cases, as well as the high

proportion of cases in the capital and large

metropolitan areas with much lower morbidity in the

regions and, especially, in the outback. We will also

take into account the change in the system for

measuring the number of infected in the United States

during March 17-23, which led to a surge in the

number of registered cases, and the timing of the

introduction of the main quarantine measures (March

21 - in the United States, February 23 - in Italy, March

14 - in Spain, March 17 - in France, March 16 - in

Germany, March 23 - in Azerbaijan) with a time lag

of 5 days (a certain period passes from the moment of

infection to detection). The resulting equation (4)

looks like this:



 

󰇛󰇜

󰇛󰇜 

󰇛󰇜 

󰇛󰇜



󰇛󰇜

󰇛󰇜

󰇛󰇜

󰇛󰇜

󰇛󰇜 

󰇛󰇜

󰇛󰇜



󰇛󰇜

󰇛󰇜 

󰇛󰇜

󰇛󰇜 

󰇛󰇜

󰇛󰇜 󰇛󰇜

here,

~1 ttt yyy

is the relative increase in

the number of infected, t is the day since the threshold

of 1000 infected was exceeded, xt - is the number of

infected per million people, and mt is a dummy

variable for the period of change in the measurement

system in the United States (taking a single value in

the period from 17 to 23 March), qt-5 is a dummy

variable equal to one for the period when quarantine

measures are in effect, with an offset of 5 days (from

March 26 in the US, etc.),

)6()1( ,..., tt zz

is a dummy

variable for the USA, Italy, Spain, France, Germany,

and Azerbaijan, respectively. Under the estimates of

the coefficients, their standard errors are indicated in

parentheses. The equation (4) presented above

suggests that the base (at the time of passing the

thousandth threshold) average daily growth rate of the

number of infected is 35.4% in the USA (e-1.405+0.365 =

0.354), 30.8% - in Italy, 41.2 % - in Spain, 27.0% - in

France, 30.7% - in Germany and 24.5% - in

Azerbaijan. At the same time, every day the growth is

reduced by 2.56% (note, it is a percentage, not a

percentage point!). The proportion of those infected

has a significant negative effect. The control for

changes in the measurement system in the United

States increased the accuracy of the model. At the

same time, contrary to what was expected, the data

provided did not reveal a significant impact of

quarantine measures. The t-statistic, calculated as the

ratio of the coefficient estimate to its standard error,

equal to 0.049/0.063 = 0.784, means that one cannot

trust the negative sign of the coefficient. Any

restrictive measures (prohibition of mass events,

closure of shopping centers, restaurants, cinemas,

sports complexes, and other public places, the

transition of several industries, including the

education system, to online mode, restrictions on

movement, etc.) slow down the spread of the virus,

reduce the maximum number of active cases and

allow to prevent the collapse of the medical system.

On the other hand, they increase the duration of the

epidemic and the economic costs associated with

reduced economic activity. Therefore, a very

important question is how effective the stringency of

the constraints is. It is hypothesized that the

insignificance of the factor of restrictions may be

associated with an inaccurate specification of the

model, for example, an erroneous lag between their

introduction and the slowdown in the spread of the

virus. However, if the lag is increased or decreased,

the significance of the introduction of restrictions not

only increases but typically decreases or even

becomes of the opposite sign (for example, with a lag

of fewer than 2 days). Data on the t-statistics of the

coefficient in restrictive measures depending on the

lag are presented in Table 4.

Table 4. t-statistics of the coefficient in restrictive

measures depending on the lag (days)

(Source: Compiled by the author)

0.168

0.692

-0.058

-0.210

-0.399

-0.777

-0.568

-0.506

-0.936

The second hypothesis is related to possible

inaccuracies in using a dummy variable to take into

account the introduced quarantine measures, which

takes only values of zero or one, as well as in

indicating the timing of the introduction of these

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measures since the indicated dates were selected

solely based on media reports without serious

additional analysis. The site [18] provides an

isolation index reflecting the severity of restrictions

and taking values from zero (no restrictions at all) to

100 (use of all measures simultaneously in the

strongest edition). Its values normalized to the interval

[0; 1] are given in Table 5.

Table 5. Government Response Stringency Index for March 1-31, 2020 by Country

(Source: Compiled by the author based on [18])

Date

USA

Italy

Spain

France

Germany

Azerbaijan

1-Mar-20

0.083

0.699

0.111

0.194

0.250

0.194

2-Mar-20

0.111

0.699

0.111

0.287

0.250

0.194

3-Mar-20

0.111

0.699

0.111

0.287

0.250

0.306

4-Mar-20

0.111

0.745

0.111

0.287

0.250

0.306

5-Mar-20

0.204

0.745

0.111

0.287

0.250

0.306

6-Mar-20

0.204

0.745

0.111

0.287

0.306

7-Mar-20

0.204

0.745

0.111

0.287

0.329

0.306

8-Mar-20

0.204

0.745

0.111

0.287

0.329

0.306

9-Mar-20

0.204

0.745

0.250

0.287

0.329

0.306

10-Mar-20

0.204

0.824

0.458

0.287

0.329

0.361

11-Mar-20

0.218

0.852

0.458

0.287

0.329

0.361

12-Mar-20

0.301

0.852

0.458

0.287

0.329

0.361

13-Mar-20

0.301

0.852

0.458

0.426

0.329

0.361

14-Mar-20

0.357

0.852

0.671

0.482

0.329

0.528

15-Mar-20

0.412

0.852

0.671

0.482

0.329

0.528

16-Mar-20

0.523

0.852

0.690

0.556

0.421

0.528

17-Mar-20

0.551

0.852

0.718

0.907

0.421

0.528

18-Mar-20

0.551

0.852

0.718

0.907

0.523

0.528

19-Mar-20

0.671

0.852

0.718

0.907

0.551

0.611

20-Mar-20

0.671

0.917

0.718

0.907

0.579

0.611

21-Mar-20

0.727

0.917

0.718

0.907

0.681

0.611

22-Mar-20

0.727

0.917

0.718

0.907

0.732

0.611

23-Mar-20

0.727

0.917

0.718

0.907

0.732

0.685

24-Mar-20

0.727

0.917

0.718

0.907

0.732

0.685

25-Mar-20

0.727

0.917

0.718

0.907

0.732

0.685

26-Mar-20

0.727

0.917

0.718

0.907

0.732

0.685

27-Mar-20

0.727

0.917

0.718

0.907

0.732

0.685

28-Mar-20

0.727

0.917

0.718

0.907

0.732

0.685

29-Mar-20

0.727

0.917

0.718

0.907

0.732

0.685

30-Mar-20

0.727

0.917

0.852

0.907

0.732

0.685

31-Mar-20

0.727

0.917

0.852

0.907

0.732

0.685

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However, the use of the isolation index did not

significantly change the significance of the quarantine

measures. Moreover, with a lag of more than 5 days,

the refined model even gave positive values of the

corresponding coefficient. The data on t-statistics for

this indicator depending on the lag (the period from

zero to 8 days between the introduction of restrictions

and the impact on the number of infected was

considered) are summarized in Table 6.

Table 6. t-statistics of the coefficient under

constraints depending on the lag (days) for the model,

taking into account the isolation index (Source:

Compiled by the author)

-0.850

-0.159

-0.442

-0.100

-0.007

0.125

0.059

0.193

0.365

Thus, the available data did not reveal a significant

relationship between the severity of quarantine

measures and the scale of the epidemic. This is

indirectly revealed by the fact that the level of spread

of the virus (the number of detected cases per 1

million people) is approximately the same as in

countries with relatively strict restrictions - Italy (the

maximum index level is 0.935), France (0.907),

Russia (0.870), the average is the United Kingdom

(0.759), USA (0.745), Germany (0.732) and the

lowest - Sweden (0.407) and Belarus (0.194). Taking

into account the fact that even in China not all

restrictions were introduced (the maximum isolation

index was 0.819, although the introduced restrictions

were strictly observed), and in other Asian countries

the values were even lower (Hong Kong - 0.667,

Japan - 0.472), probably a more important factor is

precisely the basic measures - restrictions on holding

mass events, wearing a mask in public places, transfer

to online services, etc. - and their unconditional

implementation everywhere. At the same time, many

of the restrictions introduced, including in Baku, i.e.,

bans on single walks in parks, access control, etc. - do

not lead to a decrease in the scale of the epidemic.

3.2 Forecasting

Let us move on to forecasting. We will demonstrate a

medium-term forecast for each of the countries based

on the base equation (2) with a dummy variable for

quarantine measures and a lag of 5 days. This forecast

was presented on April 1, 2021. Some of its results

(forecasts for April 15, May 1, May 15, and June 1,

2021) are presented in Table 7.

Table 7. Forecast of the number of infected people for

the specified date in the equation (2), persons

(Source: Compiled by the author)

Country/Date

15.04.21

1.05.21

15.05.21

1.06.21

USA

695863

1096024

1299450

1445955

Italy

174274

217809

240786

258072

Spain

179861

224934

246820

262629

France

126103

184489

216504

240651

Germany

181223

262127

304735

336261

Azerbaijan

1222

1824

2808

6845

Since a year has passed since the forecast, it is

possible to assess its accuracy. Table 8 demonstrates

the percentage deviation of actual values from the

forecast for the specified dates.

Table 8. Deviation of the actual number of infected

persons from the forecast in the equation (2), %

(Source: Compiled by the author)

Country/Date

15.04.21

1.05.21

15.05.21

1.06.21

USA

-5.9

3.5

14.8

29.6

Italy

-5.2

-4.8

-7

-9.6

Spain

0.4

11.2

9.2

France

-15.8

-29.4

-34.4

-36.8

Germany

-25.6

-37.4

-42.3

-45.4

Azerbaijan

2.5

1.6

5.8

-20.9

Taking into account the replacement of the dummy

variable qt of the quarantine measures by the isolation

index

, the equation (5) will be as follows:



 

󰇛󰇜

󰇛󰇜 

󰇛󰇜



󰇛󰇜 

󰇛󰇜 

󰇛󰇜

󰇛󰇜



󰇛󰇜

󰇛󰇜 

󰇛󰇜

󰇛󰇜 

󰇛󰇜

󰇛󰇜



󰇛󰇜

󰇛󰇜 

󰇛󰇜

󰇛󰇜  (5)

At the same time, there is no significant difference

between equations (4) and (5). In particular, the

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increase in the number of infected people is

decreasing daily not by 2.56%, but by 2.85%. There

are other small quantitative differences as well. As

Table 9 shows, for some countries the forecast is

getting slightly better, for others, on the contrary, it is

slightly worsening.

Table 9. Deviation of the actual number of infected

persons from the forecast in the equation (5), %

(Source: Compiled by the author)

Country/Date

15.04.21

1.05.21

15.05.21

1.06.21

USA

-5.9

4.5

16.7

32.5

Italy

-3.7

-2.3

-4

-6.1

Spain

0.4

8.5

12.1

10.6

France

-15.3

-28.3

-32.9

-34.8

Germany

-25.2

-36.5

-41.1

-43.8

Azerbaijan

2.7

2.4

3.7

-15.8

At the same time, if the forecast for April can be

considered acceptable, then in the forecast for May,

and even more so in the longer-term forecast,

significant systematic biases are found. In Germany

and partly in France, the epidemic began to come to

an end faster than it was seen in March. At the same

time, in the United States and especially in

Azerbaijan, the departure from the trajectory of

exponential growth is slower than expected.

Moreover, although the growth has slowed down (and

has practically become linear), it continues, and to

date, the lag behind most European countries in terms

of tendency is not 2-3 weeks, but more than 1.5

months.

The 1.5 million forecasted figures for the USA for

June, which seemed to be significantly overestimated

in March, were exceeded. Brazil, which at the end of

May took second place in the world, and several other

countries follow the same trajectory.

What is the basis for clustering countries with a

faster and slower recovery from a pandemic? The size

of the country can be suggested as a hypothesis.

There is a meaningful explanation for this. Large

countries are very heterogeneous, so while in some

parts (for example, in the capital or several major

metropolitan areas) reaching the plateau has already

occurred, in other parts the outbreak is just beginning.

On the contrary, at the initial stage, the number of

cases is reduced, since the epidemic has not yet

affected a significant part of the country. Open

borders between regions with different levels of

morbidity aggravate the situation.

In the end, it turns out that size matters, and in

large countries, the decrease in the growth of the

number of infected persons is slower. For example,

this can be modeled by dividing the coefficient at t by

the area of the country to some small degree α. If we

set the parameter α equal to 0.1 (this means that a 10

times larger country will be characterized by a 20%

slower decrease in the growth rate of the number of

infected: 0,10.1 ≈ 0,7943), then the modified model

specification will look like this equation (6):



 

󰇛󰇜

󰇛󰇜  

󰇛󰇜



󰇛󰇜 

󰇛󰇜  

󰇛󰇜

󰇛󰇜  

󰇛󰇜

󰇛󰇜



󰇛󰇜

󰇛󰇜 

󰇛󰇜

󰇛󰇜  

󰇛󰇜

󰇛󰇜



󰇛󰇜

󰇛󰇜  (6)

Here Si is the area of the i-country. By varying, the

value of the parameter α, it is possible to some extent

to enhance or weaken the influence of the size of the

country. With α = 0, we get the original equation (2).

If α = 0.2, the model will predict large numbers of

people infected in the United States and accelerate the

end of the epidemic in Germany and Azerbaijan. The

forecast for α = 0.1 and the deviation from it are

presented in Tables 10 and 11.

Table 10. Forecast of the number of infected persons

for the specified date in the equation (4) with α = 0.1,

persons

(Source: Compiled by the author)

Country/Date

15.04.21

1.05.21

15.0521

1.06.21

USA

787348

1290333

1550012

1740681

Italy

170778

210693

230700

244800

Spain

184247

231092

252931

267925

France

125692

183196

213884

236126

Germany

178494

256129

295553

323208

Azerbaijan

1091

1485

2586

6653

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Table 11. Deviation of the actual number of infected

people from the forecast in the equation (4), %

(Source: Compiled by the author)

Country/Date

15.04.21

1.05.21

15.0521

1.06.21

USA

-16.8

-12.1

-3.7

7.7

Italy

-3.3

-1.5

-3

-4.7

Spain

-1.9

5.1

8.5

France

-15.5

-28.9

-33.6

-35.6

Germany

-24.5

-35.9

-40.6

-43.1

Azerbaijan

-10.7

-18.6

-7.9

-2.8

In addition, for greater clarity, we will present

these data in the graphs (see Fig.1-6).

Fig. 1: Comparison of predicted and actual data of the

USA (Source: Compiled by the author)

Fig. 2: Comparison of predicted and actual data of

Italy (Source: Compiled by the author)

Fig. 3: Comparison of predicted and actual data of

Germany (Source: Compiled by the author)

Fig. 4: Comparison of predicted and actual data of

Spain (Source: Compiled by the author)

Fig. 5: Comparison of predicted and actual data of

France (Source: Compiled by the author)

500000

1000000

1500000

2000000

1-Mar 1-Apr 1-May 1-Jun

USA

data real data forecast

100000

200000

300000

1-Mar 1-Apr 1-May 1-Jun

Italy

data real data forecast

100000

200000

300000

400000

Germany

data real data forecast

100000

200000

300000

Spain

data real data forecast

100000

200000

300000

1-Mar 1-Apr 1-May 1-Jun

France

data real data forecast

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Fig. 6: Comparison of predicted and actual data of

Azerbaijan (Source: Compiled by the author)

In general, the graphs demonstrate a sufficiently

high accuracy for forecasting based on March data.

Suffice it to say that as of March 31, the number of

detected cases in the United States was 10 times

fewer and in Azerbaijan 18 times fewer than at the

beginning of June. Forecasts for Germany and France

turned out to be slightly less accurate - two countries

that at the end of March were supposed to be the next

leaders after Italy and Spain in terms of the number of

infected, but according to official statistics, at the

beginning of June they are respectively in 9th and

12th places. The process of reaching a plateau in

these countries occurred much faster and at a lower

level than it was predicted. At the same time, the

current statistics are also not final, since it contains

such artifacts as the impossible in reality reduction in

the cumulative number of detected cases on April 29

and June 2, as well as unlikely sharp fluctuations in

the levels of the series. In addition, if the current

method of tracking the infected is recognized as more

correct, most likely the March data should also be

retrospectively corrected, which will change the

forecast. Unlike four European countries, where the

model very accurately revealed the shape of the curve

(and for Italy and Spain also the quantity), a different

scenario was realized in Azerbaijan. The exponential

tendency changed to a linear one (which means

reaching a plateau - the number of new cases

coincides with the number of recovered patients).

Among other things, this means a significant delay in

overcoming the epidemic compared to European

countries and the need for significant measures to

support the economy. Indeed, the depth of the

emerging problems and the speed of economic

recovery depends on the duration of the epidemic and

the actions of the state. If restrictive measures

continue for 1-2 months or the state, through fiscal

and monetary policy, does not allow a downward

spiral to unfold, the crisis can have a V-shape with

fairly rapid recovery after the restrictions are lifted.

Otherwise, especially if the crisis causes significant

problems in the banking sector, it may take an L-

shape and turn into a prolonged depression. The

situation is complicated by the costs of restrictions

imposed by the state during the pandemic which are

unevenly distributed, and among the most affected

companies, there is a very high proportion of small

and medium-sized businesses, which are usually not

included in the lists of systemically important

industries and at the same time do not have a financial

cushion, which means a high probability of their

bankruptcy during a prolonged (even 3-4 months)

suspension of activities.

3.3 Discussion

The author of this study provided a brief forecast of

the possible cumulative number of COVID-19

confirmed cases of this epidemic worldwide. Because

this epidemic is widespread, the author published two

months ahead of the forecast in the time series model.

The author forecasted a total number of confirmed

cases from April 1, 2020, to June 1, 2020, based on

the data model until April 1, 2020. This prediction

had a confidence level of around 95%, which was

adequate for the prediction. The outcome

demonstrated that prediction accuracies and, as a

result, multiple-step forecasting were high. Our

research found that the longer the training time, the

better the forecasting. The model revealed that the

width of the prediction intervals decreased on average

as more data was included for forecasts. However, if

the data were reliable and there was no second

transmission, the time series model predicted that the

COVID-19 outbreak would have the same number of

confirmed cases worldwide. The findings of the study

exceeded our expectations. The hypothesis about the

effect of countries' percentage growth dynamics on

the future dynamics of the total number of confirmed

cases in country-followers was confirmed. This

problem allowed us to make 4–8-week forecasts for

the spread of the epidemic in Azerbaijan, with three

predicting intervals. Similar modeling in terms of

percentage growth dynamics could be done for other

countries with a long enough lead time.

2000

4000

6000

8000

Azerbaijan

data real data forecast

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4 Conclusion

Active steps taken by the state are especially

important, including the allocation of unconditional

transfers (allowing the most affected segments of the

population to survive and at the same time creating

consumer demand, preventing the crisis from

spreading to industries not affected by restrictions),

the abolition (or at least a reduction) of taxes payable

by small and medium-sized businesses, subject to

several conditions, first of all, maintaining

employment and paying salaries (which increases the

number of those who work during the crisis), the

implementation of measures that minimize the costs

of companies forced to stop their economic activity

during the crisis, for the fastest and full launch of

production after the end of restrictive measures.

Perhaps, the basic set of measures presented is not

ideal in the presence of complete information and

sufficient time to make decisions. At the same time, it

is incomplete. In particular, it does not include

measures already implemented, incl. in the field of

medicine, to expand the capacity of the health care

system, or to support specific industries such as

aviation or tourism [20], [21]. At the same time, in a

real situation of a lasting pandemic (as shown by the

study), with severe time pressure and existing

imperfect institutions, the proposed measures, despite

their costly characteristic, will allow accelerating the

recovery of the economy and by the end of 2021 to

approach pre-crisis monthly production levels,

avoiding bankruptcy in a significant share of the

business, which threatens much higher costs from the

state.

A better understanding of the progress of the

epidemic in the country can be obtained by analyzing

the progress of the epidemic at the regional level. In

conclusion, if the current mathematical model results

can be validated within the range provided here, then

the social distancing and other prevention and

treatment policies that the central and various state

governments and people are currently implementing

should be continued until no new cases are seen. The

migration of urban to rural and rich to poor

populations should be closely monitored and

controlled. There are many assumptions about

population homogeneity in terms of urban/rural or

rich/poor that do not capture variations in population

density in mathematical models. If several protective

measures are not implemented effectively, this rate

may be altered. However, the government of

Azerbaijan has already taken various protective

measures, including the establishment of a quarantine

facility, to slow the spread of COVID-19, and we can

hope that the country will be successful in slowing the

spread of this pandemic.

The study looked into the COVID-19 growth rate

in detail and forecasted the number of confirmed

cases, intending to inform the public about the

situation. The author discovered that the COVID-19

confirmed cases curve would continue to rise, urging

everyone to be more aware of the virus. Finally, our

most recent data-driven estimates have remained

reasonably constant. The time series model predicted

the global stage of the outbreak. It was most likely

due to the epidemic's wide-ranging influence. The

projection was predicated on the assumption that

current mitigating efforts would be maintained. Many

studies have been conducted for short-term

forecasting periods such as 5, 10, and 15 days. In this

study, the author used data from the previous month

to forecast the next two months. If the data set is

large, it can accurately predict long periods. Both the

short- and medium-term forecasts capture well the

epidemic trajectory across different waves of

COVID-19 infections with small relative errors over

the forecast horizon. The medium-term forecasts of

COVID-19 mortality can be used in conjunction with

the short-term forecasts as a useful planning tool as

countries continue to relax stringent public health

measures implemented to contain the pandemic.

Furthermore, the exponential growth model

demonstrated excellent accuracy in time series

analysis prediction, which previous models could not

achieve.

As a result, this model should be used to forecast

future analysis of any dataset. The limitations

observed during the prediction were a relatively small

dataset, and the prediction was based on a pandemic

with a high variation in the data set. The output would

be more accurate if the dataset were more extensive

and less variable. Future researchers can use COVID-

19 to investigate prediction models such as artificial

neural networks (ANN), Bayesian networks, and

Support Vector Machines (SVM). This model can

also predict future pandemics and patients with any

disease.

References:

[1] Brauer F., Compartmental models in epidemiology.

Mathematical epidemiology, Springer, Berlin,

Heidelberg, 2008, pp. 19−79.

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Volume 18, 2022

[2] Bailey, Norman T.J., The mathematical theory of

epidemics. Griffin, 1957, pp. 80-170.

[3] Kermack W., McKendrick A., A contribution to the

mathematical theory of epidemics, Proceedings of

the royal society of London. Series A, Containing

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[4] Anderson R.M, May R.M., Infectious diseases of

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[5] Novel Coronavirus Global Research and Innovation

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Available at:

www.who.int/emergencies/diseases/novel-

coronavirus-2019/global-research-on-

novelcoronavirus-2019-ncov (accessed: June 15,

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[6] Layne S. P., Hyman J. M., Morens D. M.,

Taubenberger J. K., New coronavirus outbreak:

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Transl. Med., 2020, vol. 12, Iss. 534, no. eabb1469.

https://doi.org/10.1126/scitranslmed.abb1469

[7] Wu J.T., Leung K., Leung G. M., Nowcasting and

forecasting the potential domestic and international

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Wuhan, China: a modeling study. Lancet, 2020, vol.

395, Iss. 10225, pp. 689–697.

https://doi.org/10.1016/S0140-6736(20)30260-9

[8] Models of Infectious Disease Agent Study. MIDAS

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WSEAS TRANSACTIONS on ENVIRONMENT and DEVELOPMENT

DOI: 10.37394/232015.2022.18.99

Sakina Babashova

E-ISSN: 2224-3496

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Volume 18, 2022