Identifying the cycles in COVID-19 infection: the case of Turkey

Abstract

The new coronavirus disease, called COVID-19, has spread extremely quickly to more than 200 countries since its detection in December 2019 in China. COVID-19 marks the return of a very old and familiar enemy. Throughout human history, disasters such as earthquakes, volcanic eruptions and even wars have not caused more human losses than lethal diseases, which are caused by viruses, bacteria and parasites. The first COVID-19 case was detected in Turkey on 12 March 2020 and researchers have since then attempted to examine periodicity in the number of daily new cases. One of the most curious questions in the pandemic process that affects the whole world is whether there will be a second wave. Such questions can be answered by examining any periodicities in the series of daily cases. Periodic series are frequently seen in many disciplines. An important method based on harmonic regression is the focus of the study. The main aim of this study is to identify the hidden periodic structure of the daily infected cases. Infected case of Turkey is analyzed by using periodogram-based methodology. Our results revealed that there are 4, 5 and 62 days cycles in the daily new cases of Turkey.

1. Introduction

Pandemic is a noun used for a disease that spreads over a whole country or the whole world (Oxford Dictionary, 2020). The most undesirable thing in epidemics is uncertainty, in some cases even worse than the disease. Many unknowns make the disease even more mysterious, such as who the disease will target, whether it will be treated, when it will end, how and why it begins. Human beings have experienced many diseases and severe outbreaks throughout history. These losses have been defeated in some way, however, much losses are lost. One of the important issues in outbreaks was the scale and spread of outbreaks. As the world gained a more integrated structure and transportation became easier, such outbreaks gained momentum and increased interaction. The spread of many diseases such as malaria, tuberculosis and influenza started from this stage [9].

Global markets have been heavily affected by the COVID-19 outbreak. The degree of this effect has further increased with the understanding that the coronavirus will not disappear in a short time. World stock exchanges are among the economic elements affected by this situation. Tourism and travel sectors, which are the main sources of income of many countries, have also been affected. Restriction and prohibition of transportation between countries are affecting these industries deeply. According to the statement of the International Air Transport Association, in 2020, the coronavirus will result in an income loss of 63 billion to 113 billion dollars in the global air transportation sector. Similarly, the closure of movie theatres may cause the international film industry to lose more than $5 billion [10].

First, on 31 December, 2019, the World Health Organization (WHO) was informed about some kind of epidemic disease in Wuhan, China. According to the coronavirus status report published by the World Health Organization on 20 July, 2020 worldwide 14,481,365 cases were seen and a total of 605,988 people died due to COVID-19 (Graph 1).

Since the occurrence of the disease in each country coincides with different dates, the first dates of the data received from the WHO and the records of the countries differ, first day of the COVID-19 cases of selected countries are given in Table 1.

Table 1.

First day of the COVID-19 case records in selected countries.

	First day of the		First day of the
Country	record (COVID-19)	Country	record (COVID-19)
Australia	2020-01-25	Italy	2020-01-31
Brazil	2020-02-26	Japan	2020-01-24
Canada	2020-01-26	Russia	2020-03-03
China	2019-12-31	Spain	2020-02-10
France	2020-01-25	Turkey	2020-03-12
Germany	2020-01-28	United Kingdom	2020-02-07
Iran	2020-02-20	United States	2020-01-21

Corona virus, which affected the whole world, has also changed the understanding and conditions of life of people. To protect their own citizens and prevent the spread of the epidemic, many countries have literally hit the border gates. Changes were made in office working conditions, and working from the office has become more widespread. People started to work from their own homes and flexible working practices were introduced in state institutions. The importance of cargo and courier companies has increased. Travels are prohibited, collective events, especially sports, concerts, shows, etc. events have been cancelled. Especially in Europe, where public transportation is used extensively, people have stopped using public transportation and have started to avoid crowded environments (restaurants, shopping malls, etc.) [7]. This emerging virus has once again revealed how sensitive human beings are. Whereas man believes that it is modern and invincible. For this reason, COVID-19 opened an academic field that should be carefully studied.

For the above-mentioned reasons, it is important to understand the spread of the virus. In this study, we aimed to understand the cycles or hidden periodic structure of the data. It may increase the understanding on incubation periods and the peaks or waves in the number of infected people. If there exist significant cycles in the investigated data, it can be used by lawmakers to take necessary measures. Also, if one can understand the reasons behind these cycles, these underlying causes may play an important role in preventing the spread of the virus.

In pandemic cases since the capacities of the hospitals are limited, there are not enough health employee and vaccine, the governments should make plans according to the predictions of the number of new patients. It is crucial to predict the possible dates for the peaks and waves. The goal of this study is to identify the hidden cycles in the investigated period. To do this, we will use the methodology of [2,3,4,5,11,12].

According to periodogram-based analysis from 12 March 2020 to 16 July 2020, the changes in the number of new cases have hidden cycles of 4, 5 and 62 days. Thus we obtain three different cycle periods for the common cycle from the pour sample period.

The remainder of this paper is organized as follows. In Section 2, we introduce the literature. Section 3 devoted to the methodology. Section 4 represents analysis. Lastly, Section 5 concludes our work with results and comments.

2. Literature

Looking historically, there have been many pandemic cases. Today, academic studies for COVID-19 cases have gained momentum as it has deep and influential effects globally. Besides studies in the field of health, statistical studies are also gaining speed. Statistical studies such as forecasting and making predictions about pandemic cases gained importance. In light of COVID-19 cases, the scientific community has begun to undertake statistical and modelling approaches.

In the COVID-19 cases recorded daily, [13] used the nonlinear least squares (NLS) method and applied a logistic growth model with the estimated parameters. In the discussed model is analyzed the time series data for China, South Korea and Iran. In the study, it was found that there is heteroscedasticity and positive serial correlation in some countries.

Zhou et al. [20] used the number of cases from 23 January to 20 March 2020 to estimate the probable spread size of the eight outbreaks of COVID-19, particularly at high risk, and the time to peak. For the prediction, logistic growth model, basic SEIR model and corrected SEIR model were preferred. Using different model inputs and different model outputs, he presented three spreading scenarios for the epidemic and three forecasting models.

Based on the COVID-19 cases, Hermanowicz [7] used a simple logistic growth model compatible with the data. With the model obtained, three different data sets were used, and real-time predictions were obtained. According to these estimates, the maximum number of cases in the regions examined is 21,000, 28,000 and 35,000, respectively.

Imai et al. [8] used the COVID-19 data in Wuhan. Accordingly, the authors based the updated estimates of the spread scale of the outbreak on the analysis of flight and population data in that city. It has been evaluated that it causes much more moderate or severe respiratory diseases than officially determined for the outbreak of COVID-19.

Yang et al. [19] analyzed the population migration data before and after 23 January 2020. Based on these data, they created the Sensitive-Exposed-Infectious-Removed (SEIR) model to reveal the COVID-19 epidemic curve. The authors also used a trained artificial intelligence (AI) approach to 2003 SARS data to predict the spread of the COVID-19 outbreak.

Wu et al. [18] have formed the logistics growth model, the general logistics growth model, the generalized Richards model and the generalized growth model according to the number of infected cases reported from 29 countries and 19 countries and regions exposed to major epidemics in China. They stated that in the number of outbreaks, Europe and the USA exceeded the point of curl. It has shown that the trajectory is much faster after reaching the vertex. Unlike most places in China, they have been shown to enter a post-summit orbit, anticipated longer than a classic logistics model predicted.

Wu et al. [17] forecasted the impact of quarantines launched in Wuhan and surrounding cities on 23–24 January 2020. National and global spread of 2019-nCoV is also envisaged in the same study.

Shen et al. [14] compared 2019-nCov, SARS and MERS. They examined the transmissibility and mortality rates of these diseases and provided a temporal assessment of the outbreak based on cases in China. They stated that quarantine, coordinated interventions and a rapid diagnosis system will have a major impact on future case numbers.

3. Methodology

Periodograms are often used to investigate latent periodicity in stationary time series. Periodograms can be obtained by trigonometric transformations of the series. The most crucial assumption in periodograms based time series is the stationarity. The investigated time series should be stationary to have meaningful inferences. In the case of non-stationarity, the proper transformation should be made to obtain a stationary time series. If any stationary time series is given, such as $\{Y_1,Y_2,\dots ,Y_n\}$, for testing whether the series contains a periodic component or not, for these series it is assumed that a model in the form of model given below

$$Y_t=\mu +Rcos(w_{k}t+\varphi )+e_t,t=1,2,\dots ,n$$ (1)

is suitable. Here the terms µ, R, ϕ and $w_k$ are the expected value, amplitude, phase and frequency, respectively. The model (1), by using trigonometric properties and by taking $w_k=2\pi k/n,k=1,2,...,[n/2]$ where [.] represents the integer part, can be written as

$$Y_t=\mu +\alpha cos(w_{k}t)+\beta sin(w_{k}t)+e_t,t=1,2,\dots ,n.$$ (2)

By using the characteristic of cosine and sine functions of $\alpha =Rcos(\varphi )$ and $\beta =Rsin(\varphi )$. The least-squares estimators of the parameters according to this model is calculated as

$$\hat{\mu }={\overline{Y}}_n,a_k=\frac{2}{n}\sum _{t=1}^n(Y_t-{\overline{Y}}_n)cos(w_{k}t)\text{and }{b}_k=\frac{2}{n}\sum _{t=1}^n(Y_t-{\overline{Y}}_n)sin(w_{k}t)$$

These calculated $a_k$ and $b_k$ values are called Fourier coefficients and they are invariant of mean. $a_k$ and $b_k$ are the least square estimators of $\alpha$ and $\beta$ when $w_k=2\pi k/n$, respectively. Periodograms are used to detect hidden periodicity in the series, as well as to test the stability of the series. For this, [2] defined the statistics below

$$T_n(w_k)=\frac{2(1-\cos (w_k))}{{\sigma }_n^2}{I}_n(w_k)$$ (3)

Critical values are given in the relevant study. Some of the critical values are givenbelow [2].

$$P(T\le 0.034818)=0.01,P(T\le 0.178496)=0.05,P(T\le 0.369089)=0.10$$

Periodograms can be used to investigate hidden periodicity in the series. For this, consider the model given in Equation (2). If the hypothesis $H_0:\alpha =\beta =0$ is rejected, then we conclude that there are periodic components in the time series. Although the standard F test may seem to be used to test this hypothesis, it is not meaningful as w_k frequencies are unknown [15].

To test this hypothesis

$$V=I_n(w_{(1)}){\left[\sum _{k=1}^m{I}_n(w_k)\right]}^{-1}$$ (4)

test statistic is defined in [15]. Here, $I_n(w_{(1)})$ shows the largest periodogram value and $m$ indicates the integer value of the number $n/2(m=[n/2])$. Under the hypothesis $H_0:\alpha =\beta =0$, it can be written as Equation (5).

$$P(V>c_{\alpha })=\alpha \cong m(1-c_{\alpha }{)}^{m-1}$$ (5)

[15] and $c_{\alpha }$ shows the critical value (at α meaning level). Using this equation, critical values are calculated as $c_{\alpha }=1-(\alpha /m{)}^{1/(m-1)}$.

4. Analysis

In this study, data on national government of Turkey and the World Health Organization website COVID-19 daily cases are used and analyzed by using the software SAS. The data set contains the period between the first case in Turkey seen on 11 March 2020 until 12 July 2020, a total of approximately 4 million people get tested. Of these individuals tested, 241,029 people were positively diagnosed. In the same period, 116,010 people were hospitalized and 5363 deaths were observed. Time-series graphs of the number of cases observed with autocorrelations (ACF) and partial autocorrelations (PACF) are given in Figure 1.

Figure 1.

The daily number of cases detected in Turkey (12 March 2020–16 July 2020).

When Figure 1 is considered, the rate of reduction seen in autocorrelations is quite slow and it may exhibit unstable variance. For this reason, the stationarity of the series is suspected. The stationarity of the series was tested by standard Dickey–Fuller (ADF) method and it was observed that the time series is not stationary. The test results of ADF are given in Table 2.

Table 2.

Daily case counts ADF test results (12 March–16 July 2020).

12 March 2020–16 July 2020 daily case numbers
		t-Statistic	Prob.*
Augmented Dickey–Fuller test statistic		−1.834043	0.3626
Test critical values:	1% level	−3.485586
	5% level	−2.885654
	10% level	−2.579708
12 March 2020–16 July 2020 daily case numbers (first difference)
		t-Statistic	Prob.*
Augmented Dickey–Fuller test statistic		−4.465524	0.0004
Test critical values:	1% level	−3.485586
	5% level	−2.885654
	10% level	−2.579708

The stationarity of the level data also investigated by using the unit root test based on periodograms. To do this an appropriate time series model is needed. Investigating the graph of the partial autocorrelation function of the series with Akaike Information Criteria (AIC) of 1785.913

$$Y_t={\alpha }_0+{\alpha }_1{Y}_{t-1}+{\alpha }_4{Y}_{t-4}+e_t,t=1,2,\dots ,125$$ (6)

an ARIMA model as Equation (6) is considered for the series where $e_t\sim WN(0,{\sigma }^2)$.

The value of the test statistics given in Equation (3) is calculated by using the variance of the white noise of the model given in Equation (6) as $I_n(w_1)=84805906.7$ and ${\hat{\sigma }}_n^2=91648.93$ where the value of the test statistics is $t_n(w_1)=2.33747$. According to these results, it can be said that a number of daily new cases are not stationary either according with ADF test results or periodogram-based stationary test results. The time series is transformed by taking the first differences. It is seen that the first difference is stationary. For this reason, the first-order differences of the series were used to investigate the cycles in the daily number of cases.

Periodicity was investigated for the $\mathrm{\nabla }{Y}_t=Y_t-Y_{t-1}$ series where the first-order difference is stationary. Now on we are working on the first difference of the time series throughout the study which is naturally interpreted as the changes in the daily new cases. Time-series graphs related to the first-order difference series are given in Figure 2.

Figure 2.

Time-series graphs of first-order difference series.

The highest 5 periodogram values and their corresponding periods and statistics with p-values are given in Table 3.

Table 3.

Five highest periodogram values and corresponding periods.

i	In(w(i))	Vi	Period	p-values	Result
1	986578	0.0853	4.133	0.0021	Significant
2	655764	0.0620	4.960	0.0472	Significant
3	601557	0.0607	62.000	0.0560	Significant
4	515296	0.0553	4.276	0.1134	Not Significant
5	512716	0.0583	4.769	0.0767	Not Significant

However, according to the formula given in Equation (5), critical values (where $m=n$ is taken instead of $m=[n/2]$ due to lack of COVID-19 case data spreading in a longer period at the daily level) are calculated and given in Table 4.

Table 4.

Critical values.

α	0.01	0.02	0.03	0.04	0.05
cα	0.0738	0.0685	0.0654	0.0633	0.0616
α	0.06	0.07	0.08	0.09	0.10
cα	0.0602	0.0589	0.0580	0.0571	0.0563

Considering Table 3, according to the p-values, 4 and 5 days periods seem to be significant at 5% level and 62 day period is significant at the 6% level for the first-order differences of the daily case numbers. According to these results, harmonic regression model given in Equation (7) was taken into consideration for the first-order difference series of daily new cases. Parameter estimates for this model are given in Table 5.

$$\begin{array}{rl}\mathrm{\nabla }{Y}_t& =\mu +A_{1}cos\left(\frac{2\pi t}{4}\right)+B_{1}sin\left(\frac{2\pi t}{4}\right)+A_{2}cos\left(\frac{2\pi t}{5}\right)+B_{2}sin\left(\frac{2\pi t}{5}\right)\\ & +A_{3}cos\left(\frac{2\pi t}{62}\right)+B_{3}sin\left(\frac{2\pi t}{62}\right)+e_t,t=1,2,\dots ,n\end{array}$$ (7)

Considering these table values, it is seen that some parameters are not significant. For this reason,

$$\mathrm{\nabla }{Y}_t=B_{1}sin\left(\frac{2\pi t}{4}\right)+B_{2}sin\left(\frac{2\pi t}{5}\right)+B_{3}sin\left(\frac{2\pi t}{62}\right)+e_t,t=1,2,\dots ,n$$ (8)

a regression model in the form of Equation (8) is assumed to be suitable. Parameter estimates for this model are given in Table 6. Equation (8) is sinusoidal. This means that the absence of the intercept term in Model given in Equation (8) indicates that there is zero observation at the beginning since $\sin (0)=0$. If the model contains the term cosine, then the intercept term would not be zero since $\cos (0)=1$ which shows the consistency of the model given in Equation 8.

Table 5.

Parameter estimates of model (7).

Parameter estimates
Variable	DF	Parameter estimate	Standard error	t Value	Pr > |t|
Intercept	1	6.89517	26.01156	0.27	0.7914
A1	1	−48.55029	36.80077	−1.32	0.1897
B1	1	−75.20387	36.80077	−2.04	0.0432
A2	1	−2.54641	36.68664	−0.07	0.9448
B2	1	−94.85392	36.92422	−2.57	0.0115
A3	1	−36.55583	36.78827	−0.99	0.3224
B3	1	91.78146	36.78362	2.50	0.0140

Table 6.

Model (8) parameter estimates.

Parameter estimates
Variable	DF	Parameter estimate	Standard error	t Value	Pr > |t|
B1	1	−75.29408	36.60314	−2.06	0.0418
B2	1	−93.17501	36.72326	−2.54	0.0124
B3	1	91.78519	36.60161	2.51	0.0135

Considering Table 6, it is seen that all parameters are significant. Therefore, the model given in Equation (8) for the first-order differences of the daily case numbers can be considered as a suitable prediction model. From now on, we will consider the model given in Equation (8).

5. Results and comments

The graph of the estimated values for the last 45 days and observation values calculated according to the model are given in Figure 3.

Figure 3.

Predicted Values and Observation Values in the last 45 days (blue observed values, black (*) predicted values).

According to results, these cycles; the 4, 5 and 62-day cycles are significant at the 5% significance level for the first-order differences of the daily case numbers in Turkey. As seen in Figure 4, the first wave of COVID-19 starts on March 12 and reaches the peak point on April 12 then goes until May 15. This process is shown in Figure 4.

Figure 4.

Daily COVID-19 cases and periods.

Considering the number of cases in Turkey given in Figure 4, the COVID-19 cases reached the first peak in 12 April 2020 (number of cases in April 12 was 5138) which corresponds to the 62-day cycle in our analysis. Thus the 62-day cycles indicate the second wave which may occur from the new decisions and restrictions about the pandemics in Turkey taken by the government. These decisions such as curfew, flexible working time in public, online education and to wear surgical masks are taken approximately in every two months. These 62 days correspond to the changes in regulations and the effect of these regulations.

A 4-day cycle corresponds to the incubations period of the virus. Finally, 5-day cycle may occur because of the weekend's affects. On weekends social gathering of families increases in closed environments triggers an increase in the number of new cases. Also, the number of PCR tests on weekend is lower than the weekday this can be shown as the reason for a 5-day cycle.

The number of cases decreased as a result of the measures taken in the following period, such as curfews, restriction of travel between provinces, closure of flights, quarantine applications for foreigners, closure of schools, places of worship, and opening pandemic hospitals. In subsequent periods, partial increases occurred due to the reduction of restrictions, opening of the tourism season, and removal of travel restrictions.

Thus the findings of this study may be used by policy-makers to update their plans to fight against the virus. Also 62-day periods imply that the probable wave period is expected to be 60–70 days which may enlighten the expected time to the second wave. If the rules are not followed and necessary care is not taken, second wave in Turkey is likely to happen in October and November 2020 period. There will be an increase in cases as a result of non-compliance with the rules of cleaning, social distance and wearing a mask under the ‘new normal life’ conditions announced.

Another important issue raised by the COVID-19 outbreak is: This outbreak is not just a health crisis. This epidemic is also a development crisis. Outbreaks not only threaten human life. It also weakens global economies. It is not known when to return to life before the epidemic or whether it could be returned. [1]. It is quite obvious that COVID-19 threatens the global economy as well as threatens human health. The economic wheels of many countries have been affected. Evidence from different markets confirms this situation. The operation of the global supply chains of the world, which is more integrated than before, has deteriorated due to the current crisis. This situation caused different effects at all levels. For example, automobile manufacturers had to stop their production from parts manufacturers. To put it another way, the epidemic in the production sector has created a butterfly effect. [6]. Pandemic struck the face of mankind that the development of digital security technologies will not be not enough for humans there are vulnerabilities in many areas of the world. Among these health system deficits come first [16].

Combating COVID-19 will be a long-term struggle. The most important and greatest loss given to pandemic is human life. From a global point of view, the spiritual weight of the situation of the families who lost their lives due to pandemic is much more than the weight of all the economic effects. At some point, our life will somehow return to normal. We do not know whether this normal will be as before, but there are many evaluations as to whether it will be as before. As a result, human beings will have to change many habits and prior needs. We must be prepared for the recurrence of pandemics such as COVID-19. Thus, instead of living in panic, we can live prepared for this type of outbreak.

Disclosure statement

No potential conflict of interest was reported by the author(s).