# Modeling and Forecasting the COVID-19 Temporal Spread in Greece: An Exploratory Approach Based on Complex Network Defined Splines

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Descriptive Analysis of COVID-19 in Greece

## 4. Methodology and Data

_{1}: Day), the COVID-19 cumulative infected cases (variable X

_{2}: Infections) and cumulative deaths (var. X

_{3}: Deaths), the daily infections (var. X

_{4}: Daily Infections), the daily deaths (var. X

_{5}: Daily Deaths), daily recoveries (var. X

_{6}: Daily Recovered), the daily new patients in intensive care units (var. X

_{7}: ICU), and the daily number of tests (var. X

_{8}: Tests). All available variables are shown in Table A1 (Appendix A). Each variable is a time-series x(n) = {x(i)|i = 1, 2, …, n } = {x(1), x(2), …, x(i)}, where each node i = 1, 2, …, n refers to a day since the first infection.

_{2}:X

_{8}. The study is implemented through a double perspective; the first examines the structural dynamics of one variable (X

_{i}, i = 1, …, n) in comparison with the other available variables X

_{j}(analysis between variables, i ≠ j = 1, …, n), whereas the second examines the time-series pattern configured for a variable X

_{i}(analysis within variable X

_{i}, i = 1, …, n). Towards the first direction, Pearson’s bivariate correlation analysis is applied to the set of the available variables (X

_{2}:X

_{8}), and the results are shown in Table 1. In terms of time-series analysis [29], computing Pearson’s bivariate correlation coefficients for variables x, y is equivalent to a cross-correlation analysis with a zero lag (h = 0) applied between variables x

_{t}and y

_{t+h}. Therefore, the first structural perspective does not build on a lagged consideration of the available data since there is neither theoretical evidence in epidemiologic studies [30,31] nor any indication based on data observation that the available time-series have periodical structure. As it can be observed in Table 1, the number of infections (X

_{2}) is significantly correlated with all variables except X

_{6}(daily recovered) and the daily number of infections (X

_{4}) is significantly correlated with the daily number of deaths (X

_{5}) and the patients in ICU (X

_{7}). These significant results imply, on the one hand, that the coevolution of the COVID-19 infection curve with variables X

_{3}:X

_{5}, X

_{7}, X

_{8}is less than 5% likely to be a matter of chance, and, on the other hand, that the coevolution of the daily COVID-19 infections with the daily number of deaths and the patients in ICU is less than 1% likely to be a matter of chance. In general, the correlation analysis indicates that the evolution of the COVID-19 infections in Greece is very likely to submit to causality and less likely to be a matter of chance.

_{6}) is significantly (but not highly) and positively correlated with the number of daily deaths (X

_{5}), expressing a tendency of the Greek health system to get more recoveries when the number of deaths increases. This correlation illustrates the analogy between deaths and recoveries, suggesting a variable for further research. In addition, the number of patients in ICU (X

_{7}) appears significantly and negatively correlated with the number of infections (X

_{2}) and deaths (X

_{3}), implying that the number of patients in ICU tends to decrease when the number of infections and deaths increases. This observation is rationale since cases of death are removed from the ICU. On the other hand, the number of patients in ICU (X

_{7}) is significantly and positively correlated with the number of daily infections (X

_{4}), implying that the number of patients entering in ICU tends to increase when the number of daily infections gets bigger. Next, an interesting observation regards the correlations between the number of tests (X

_{4}) and variables X

_{2}and X

_{3}. Although these correlations r(X

_{4},X

_{2}) and r(X

_{4},X

_{3}) are significant and positive, as expected (implying that the number of tests appears proportional to the number of infections and deaths), the numerical values of these coefficients do not appear considerably high (since r(X

_{4},X

_{2}), r(X

_{4},X

_{2}) < 0.6). Provided that perfect positive linearity between the number of tests and infections (or deaths) implies increasing awareness of the health system proportionally to the spread of the disease, the considerable high distance (>40%) of the correlation coefficients r(X

_{4},X

_{2}) and r(X

_{4},X

_{2}) from perfect (positive) linearity can be seen as an aspect of testing ineffectiveness of the health system in Greece. Overall, the correlation analysis shows that different aspects of the disease in Greece are ruled by nonrandomness, and, therefore, it provides indications that the evolution of the Greek COVID-19 system is driven by short-term linear trends. Therefore, a stochastic analysis is further applied for improvement of the overall system’s determination and, thus, for the better conceptualization of the dynamics ruling the evolution of COVID-19 in Greece.

#### 4.1. Regression Analysis

_{2}= f(t) that best describes its variability through time. The available types of fitting curves examined in the regression analysis are linear, quadratic (2nd order polynomial), cubic (3rd order polynomial), power, and logarithmic. All the available types of fitting curves can be generally described by the general multivariate linear regression model expressed by the formula [32]

_{i}can represent a function of x, namely, x

_{i}= f(x), as it is shown in relation (2).

^{m}, or polynomial f(x) = x

^{m}, or exponential f(x) = (exp{x})

^{m}, or any other. Within this context, the purpose of the regression analysis is to estimate the parameters b

_{i}of Model (2) that best fits the observed data y, so that to minimize the square differences of ${y}_{i}-\widehat{{y}_{i}}$ [32], namely:

_{i}) by using the least-squares linear regression (LSLR) method [32] based on the assumption that the differences e in Relation (3) follow the normal distribution N(0,${\sigma}_{e}^{2}$). In this paper, the time (days since the first infection, variable X

_{1}) is set as an independent variable, and each other available variable is set as a response variable to the models. In all cases, the simplest form of regression model that best fits the data is chosen. The simplicity criterion regards both the number of the used terms ${b}_{i}f\left({x}_{i}\right)$ and the polynomial degree m. That is, the model with the least possible terms and the lowest possible degree m < n−2 (where n is the number of observations in the dataset) is chosen if it best fits the data. The determination ability of each model is expressed by the coefficient of determination R

^{2}, which is given by the formula [32,33]:

_{2}). As it can be observed, the 3rd order polynomial (cubic) fitting curve best describes the data of the Greek COVID-19 cumulative infections. The last (very recent) part of the cubic curve appears convex, implying that the number of cumulative infections tends to saturate.

_{4}). As can be observed, similar to variable X

_{2}, the 3rd order polynomial (cubic) fitting curve best describes the data of the Greek COVID-19 cumulative deaths. The shape of this curve also implies that the number of cumulative infections tends to saturate.

_{7}). Similar to variables X

_{2}and X

_{4}, the 3rd order polynomial (cubic) fitting curve best describes the data of (cumulative) variable X

_{7}, and the shape of the curve implies that the number of cumulative ICU patients also tends to saturate.

_{2}), death (X

_{4}), and ICU patients (X

_{7}) curves in Greece is to the 3rd order polynomial (cubic) pattern than to linear, power, logarithmic, or 2nd order polynomial patterns. As was previously observed, the cubic-shape of the fitting curves (which ends up as a convex area representing the recent past of the time-series) illustrates saturation trends of the COVID-19 evolution in Greece. To improve the accuracy and determination ability of the fittings, we apply next a regression analysis based on the regression splines algorithm.

#### 4.2. Regression Splines

_{o}< t

_{1}< t

_{2}< … < t

_{k}

_{−1}< t

_{n}= b of the interval [a,b], a spline is a multi-polynomial function S(t) defined by the union of functions [34]:

_{o}, t

_{1}, t

_{2}, …, t

_{n}) dividing the interval [a,b] into k−1 convex subintervals. Each function S

_{i}(t), i = 1, …, n, is a polynomial of low (usually square) degree (sometimes can also be linear) that fits to the corresponding interval [t

_{i}

_{−1,}t

_{i}], i = 1, …, n, so that the aggregate spline function is continuous and smooth. The spline algorithm is preferable than that of simple regression in cases when the simple regression generates models of high degree [34]. This piecewise approach yields models of high determination by using low degree polynomial piece-functions. In terms of the bias-variance trade-off dilemma [35], stating that simple (i.e., of low degree) models have small variance and high bias whereas complex models have small bias and high variance, the spline algorithm can generate fittings of both low variance and low bias, and thus it minimizes the expected loss expressed by the sum of square bias, variance, and noise.

_{o}, t

_{1}, t

_{2}, …, t

_{n}) so as to obtain the smoothest and best determination spline model, and, secondly, the selection of the polynomial degree, so that the model is smooth and continuous at the borders of the subintervals. Therefore, this highly effective (in terms of model determination) fitting method is very sensitive to the selection of the knot vector, which is usually being determined either uniformly, or arbitrarily, or intuitively, or based on the researchers’ experience [34,35]. The more sophisticated knot-selection techniques in the literature [36,37] build on heuristics to determine the knot vector, generating the best fitting and smoothening of the spline model. Within this open debate of knot determination, this paper builds on the recent work of [12] and introduces a novel approach for the determination of the spline knot vector, based on complex network analysis. More specifically, the proposed model introduces a novel approach for the determination of the spline knot vector based on complex network analysis based on the COVID-19 infection curve of Greece. According to this approach, the spline is divided into five knots that represent the evolution of the disease in Greece, which went through five stages of declining dynamics [12].

#### 4.3. Complex Network Analysis of Time-Series

_{1}.

## 5. Results and Discussion

_{2}, X

_{4}, and X

_{7}), the proposed complex-network spline models have better determination ability and lower error terms than both the cubic models resulted by the regression analysis and the randomly calibrated splines. In particular, improvements caused by the proposed method range between 0.00–0.20% for the multiple correlation coefficients (R), between 0.10–0.51% for the model determination (R

^{2}), between 0.37–41.32% for the root mean square error (RMSE), and 0.25–34.19% for the relative absolute error (RAE). These improvements are considerable even in the cases of R and R

^{2}, given the already good fitting performance of the cubic and randomly calibrated spline models. Additionally, it provides the area under the ROC curve metric (AUC) in order to assess the model performance [43]. A ROC curve (receiver operating characteristic curve) is a graph showing the performance of a model at all thresholds. AUC measures the entire two-dimensional area underneath the entire ROC curve from (0,0) to (1,1).

## 6. Limitations and Further Research

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Date | Day (X_{1}) | Infections (X_{2}) | Daily Infections (X_{3}) | Deaths (X_{4}) | Daily Deaths (X_{5}) | Daily Recovered (X_{6}) | ICU (X_{7}) | Tests (X_{8}) |
---|---|---|---|---|---|---|---|---|

26-Feb-20 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

27-Feb-20 | 2 | 3 | 2 | 0 | 0 | 0 | 0 | 0 |

28-Feb-20 | 3 | 4 | 1 | 0 | 0 | 0 | 0 | 0 |

29-Feb-20 | 4 | 7 | 3 | 0 | 0 | 0 | 0 | 0 |

1-Mar-20 | 5 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |

2-Mar-20 | 6 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |

3-Mar-20 | 7 | 9 | 0 | 0 | 0 | 0 | 0 | 270 |

4-Mar-20 | 8 | 11 | 2 | 0 | 0 | 0 | 1 | 0 |

5-Mar-20 | 9 | 31 | 22 | 0 | 0 | 0 | 0 | 0 |

6-Mar-20 | 10 | 46 | 14 | 0 | 0 | 0 | 1 | 0 |

7-Mar-20 | 11 | 66 | 21 | 0 | 0 | 0 | 0 | 0 |

8-Mar-20 | 12 | 73 | 7 | 0 | 0 | 0 | 0 | 0 |

9-Mar-20 | 13 | 84 | 11 | 0 | 0 | 0 | 1 | 0 |

10-Mar-20 | 14 | 89 | 5 | 0 | 0 | 0 | 1 | 0 |

11-Mar-20 | 15 | 99 | 10 | 0 | 0 | 0 | 2 | 0 |

12-Mar-20 | 16 | 117 | 18 | 0 | 1 | 2 | 0 | 1910 |

13-Mar-20 | 17 | 190 | 73 | 1 | 0 | 0 | 3 | 520 |

14-Mar-20 | 18 | 228 | 38 | 1 | 2 | 6 | 0 | 700 |

15-Mar-20 | 19 | 331 | 103 | 3 | 1 | 2 | 2 | 0 |

16-Mar-20 | 20 | 352 | 21 | 4 | 0 | 0 | 5 | 920 |

17-Mar-20 | 21 | 387 | 35 | 4 | 1 | 4 | 2 | 580 |

18-Mar-20 | 22 | 418 | 31 | 5 | 0 | 0 | 6 | 1100 |

19-Mar-20 | 23 | 464 | 46 | 5 | 1 | 5 | 5 | 300 |

20-Mar-20 | 24 | 495 | 31 | 6 | 3 | 0 | 4 | 872 |

21-Mar-20 | 25 | 530 | 35 | 9 | 4 | 0 | 3 | 658 |

22-Mar-20 | 26 | 624 | 94 | 13 | 2 | 0 | 9 | 176 |

23-Mar-20 | 27 | 695 | 71 | 15 | 2 | 10 | 4 | 638 |

24-Mar-20 | 28 | 743 | 48 | 17 | 3 | 3 | 5 | 427 |

25-Mar-20 | 29 | 821 | 78 | 20 | 2 | 4 | 5 | 1424 |

26-Mar-20 | 30 | 892 | 71 | 22 | 4 | 6 | 2 | 0 |

27-Mar-20 | 31 | 966 | 74 | 26 | 2 | 10 | 9 | 2982 |

28-Mar-20 | 32 | 1061 | 95 | 28 | 4 | 0 | 1 | 886 |

29-Mar-20 | 33 | 1156 | 95 | 32 | 6 | 0 | −2 | 788 |

30-Mar-20 | 34 | 1212 | 56 | 38 | 5 | 0 | 3 | 810 |

31-Mar-20 | 35 | 1314 | 102 | 43 | 6 | 0 | 13 | 771 |

1-Apr-20 | 36 | 1415 | 101 | 49 | 1 | 0 | 5 | 618 |

2-Apr-20 | 37 | 1544 | 129 | 50 | 3 | 9 | 1 | 1494 |

3-Apr-20 | 38 | 1613 | 69 | 53 | 6 | 17 | 1 | 3593 |

4-Apr-20 | 39 | 1673 | 60 | 59 | 9 | 0 | 0 | 896 |

5-Apr-20 | 40 | 1735 | 62 | 68 | 5 | 0 | 1 | 2120 |

6-Apr-20 | 41 | 1755 | 20 | 73 | 6 | 191 | −3 | 740 |

7-Apr-20 | 42 | 1832 | 77 | 79 | 2 | 0 | 0 | 2391 |

8-Apr-20 | 43 | 1884 | 52 | 81 | 2 | 0 | −6 | 3944 |

9-Apr-20 | 44 | 1955 | 71 | 83 | 3 | 0 | −5 | 1106 |

10-Apr-20 | 45 | 2009 | 56 | 86 | 4 | 0 | −2 | 1798 |

11-Apr-20 | 46 | 2081 | 70 | 90 | 3 | 0 | −2 | 1912 |

12-Apr-20 | 47 | 2114 | 33 | 93 | 5 | 0 | 1 | 4917 |

13-Apr-20 | 48 | 2145 | 31 | 98 | 1 | 0 | −3 | 1156 |

14-Apr-20 | 49 | 2170 | 25 | 99 | 2 | 0 | 3 | 5381 |

15-Apr-20 | 50 | 2192 | 22 | 101 | 1 | 0 | −4 | 1973 |

16-Apr-20 | 51 | 2207 | 15 | 102 | 3 | 0 | −3 | 0 |

17-Apr-20 | 52 | 2224 | 17 | 105 | 3 | 0 | 2 | 0 |

18-Apr-20 | 53 | 2235 | 11 | 108 | 2 | 0 | −4 | 2519 |

19-Apr-20 | 54 | 2235 | 0 | 110 | 3 | 0 | −4 | 0 |

Sum | 2235 | 2235 | 113 | 113 | 269 | 69 | 53,290 |

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**Figure 1.**The time-series of the COVID-19 infection curve in Greece for the period 15 February 2020 to 18 April 2020. The first infection emerged on 26 February 2020 (and was recorded on 27 February 2020; data source [15]). (

**a**) Total Coronavirus Cases; (

**b**) Novel Coronavirus Dairy Cases.

**Figure 2.**Infected cases per million in Greece (source: [27]).

**Figure 3.**The time-series of the COVID-19 death curve in Greece for the period 15 February 2020–18 April 2020. The first death was recorded on 12 March 2020 (data source: [15]). (

**a**) Total Coronavirus Deaths; (

**b**) Novel Coronavirus Dairy Deaths.

**Figure 4.**Comparative diagram with the time-series of the COVID-19 infection cases versus the recorded deaths in Greece (data source: [27]).

**Figure 5.**The aggregate time-series of the COVID-19 in Greece, showing the number of confirmed (total) cases, new infections, deaths, and recoveries, for the period 15 February 2020–18 April 2020 (data source: [15]).

**Figure 6.**Available types of fitting curves applied to the cumulative COVID-19 infection curve (variable X

_{2}) of Greece. Time-series data of the variable are shown in dots.

**Figure 7.**The available types of fitting curves applied to the cumulative COVID-19 death curve (variable X

_{4}) of Greece. Time-series data of the variable are shown in dots.

**Figure 8.**The available types of fitting curves applied to the cumulative COVID-19 ICU patients (variable: cumulative X

_{7}) of Greece. Time-series data of the variable are shown in dots.

**Figure 9.**Community detection of the Greek COVID-19 infection visibility graph based on the modularity optimization algorithm of [40]. Node size in the network is proportional to node degree.

X_{3} | X_{4} | X_{5} | X_{6} | X_{7} | X_{8} | ||
---|---|---|---|---|---|---|---|

Infections (X_{2}) | r(x,y) | 0.979 ** | 0.317 * | 0.612 ** | 0.138 | −0.316 * | 0.585 ** |

Sig. (2-tailed) | 0.000 | 0.020 | 0.000 | 0.321 | 0.020 | 0.000 | |

Deaths (X_{3}) | r(x,y) | 1 | 0.153 | 0.496 ** | 0.123 | −0.421 ** | 0.552 ** |

Sig. (2-tailed) | 0.268 | 0.000 | 0.376 | 0.002 | 0.000 | ||

Daily Infections (X_{4}) | r(x,y) | 1 | 0.487 ** | −0.038 | 0.358 ** | 0.232 | |

Sig. (2-tailed) | 0.000 | 0.785 | 0.008 | 0.091 | |||

Daily Deaths (X_{5}) | r(x,y) | 1 | 0.277 * | 0.010 | 0.339 * | ||

Sig. (2-tailed) | 0.042 | 0.941 | 0.012 | ||||

Recovered (X_{6}) | r(x,y) | 1 | −0.133 | 0.003 | |||

Sig. (2-tailed) | 0.339 | 0.982 | |||||

ICU (X_{7}) | r(x,y) | 1 | −0.077 | ||||

Sig. (2-tailed) | 0.582 |

^{a}This analysis equivalents to a cross-correlation analysis with a zero lag (h = 0) applied between variables x

_{t}and y

_{t+h.}* Coefficient is significant at the 0.05 level. ** Coefficient is significant at the 0.01 level.

Model | R | R^{2} | RMSE * | RAE ** | AUC |
---|---|---|---|---|---|

Dependent Variable: Infections (X_{2}) | |||||

Cubic | 0.998 | 0.996 | 2.229 | 4.182% | 0.9962 |

Regression Splines with 3 random knots | 0.999 | 0.998 | 1.805 | 3.798% | 0.9979 |

Regression Splines with 4 random knots | 0.999 | 0.998 | 1.621 | 3.277% | 0.9984 |

Regression Splines with 5 random knots | 0.999 | 0.998 | 1.420 | 2.986% | 0.9985 |

Complex-Network Regression Splines | 1.000 | 1.000 | 1.308 | 2.752% | 0.9998 |

Dependent Variable: Deaths (X_{4}) | |||||

Cubic | 0.995 | 0.990 | 2.423 | 4.981% | 0.9903 |

Regression Splines with 3 random knots | 0.995 | 0.990 | 2.584 | 5.013% | 0.9904 |

Regression Splines with 4 random knots | 0.995 | 0.990 | 2.423 | 4.798% | 0.9903 |

Regression Splines with 5 random knots | 0.995 | 0.990 | 2.410 | 4.732% | 0.9905 |

Complex-Network Regression Splines | 0.995 | 0.991 | 2.401 | 4.720% | 0.9912 |

Dependent Variable: ICU Patients (X_{7}) | |||||

Cubic | 0.987 | 0.974 | 6.300 | 12.659% | 0.9743 |

Regression Splines with 3 random knots | 0.987 | 0.974 | 6.287 | 12.648% | 0.9744 |

Regression Splines with 4 random knots | 0.988 | 0.976 | 6.186 | 12.114% | 0.9762 |

Regression Splines with 5 random knots | 0.989 | 0.979 | 6.204 | 12.007% | 0.9793 |

Complex-Network Regression Splines | 0.989 | 0.979 | 6.119 | 11.731% | 0.9795 |

**bold**font indicate best determination models.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Demertzis, K.; Tsiotas, D.; Magafas, L. Modeling and Forecasting the COVID-19 Temporal Spread in Greece: An Exploratory Approach Based on Complex Network Defined Splines. *Int. J. Environ. Res. Public Health* **2020**, *17*, 4693.
https://doi.org/10.3390/ijerph17134693

**AMA Style**

Demertzis K, Tsiotas D, Magafas L. Modeling and Forecasting the COVID-19 Temporal Spread in Greece: An Exploratory Approach Based on Complex Network Defined Splines. *International Journal of Environmental Research and Public Health*. 2020; 17(13):4693.
https://doi.org/10.3390/ijerph17134693

**Chicago/Turabian Style**

Demertzis, Konstantinos, Dimitrios Tsiotas, and Lykourgos Magafas. 2020. "Modeling and Forecasting the COVID-19 Temporal Spread in Greece: An Exploratory Approach Based on Complex Network Defined Splines" *International Journal of Environmental Research and Public Health* 17, no. 13: 4693.
https://doi.org/10.3390/ijerph17134693