# COVID-19 Pandemic: How Effective Are Preventive Control Measures and Is a Complete Lockdown Justified? A Comparison of Countries and States

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## Abstract

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^{2}= 0.9480). The average duration of the effective phase was 17.3 ± 10.5 days. The results indicated that lockdown measures are not necessarily superior to relaxed measures, which in turn are not necessarily a recipe for failure. Relaxed measures are, however, more economy-friendly.

## 1. Introduction

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- Several countries have demonstrated that this virus can be suppressed and controlled.
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- The challenge for many countries…is not whether they can do the same—it’s whether they will.
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- We know that these measures are taking a heavy toll on societies and economies…
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- All countries must strike a fine balance between protecting health, minimizing economic and social disruption and respecting human rights.”

_{eff}) [8,9,10,11,12] and the drop of the infected patient’s ratio [13]. Similarly, Haug et al. [14] evaluated and ranked the effectiveness of government interventions based on the amount of R

_{eff}decrease, i.e., ΔR

_{eff}.

_{eff}hinges on the accuracy of the reported number of confirmed cases and on the accurate estimates of the serial interval distribution [15]. The reported number of confirmed cases is affected by various degrees of underreporting [16,17]. The average (or median) of the serial interval varies between different sources, from 3.95 days [18] to 7.5 [19] days. Moreover, R

_{eff}is determined with different equations and methods [20,21,22], which do not necessarily deliver the same results [23].

_{eff}by 25.8%, but without any evidence that compulsory mask-wearing decreases the transmission. Moosa [29] investigated the social distancing, which proves effective. However, it appeared difficult to separate the effect of imposed control measures from the one of voluntary measures. Prakash et al. [30] modeled the impact of social distancing and found that the introduction of strict lockdown policies improves the containment of the pandemic. Further modelling studies on social distancing revealed that early and late interventions delay and flatten the epidemic curve, respectively [31], and that the prevention of within-household transmission is a critical factor for flattening the curve [32].

_{eff}and different methods of assessing the effectiveness of control measures, it is paramount to find a method for calculating the effectiveness that is independent of the number of cases and unaffected by systematic errors.

## 2. Materials and Methods

#### 2.1. Rationale of the Method

#### 2.2. Effectiveness Parameters, Mathematical Derivation and Terminology

_{C}, which approximately follow an S-shaped curve between two constant values, 0 and the maximum number of cases C

_{max}.

_{C}, but rather in C

_{I}, as C

_{C}is a summation. Thus, ${C}_{I}={\displaystyle \underset{{t}_{0}}{\overset{{t}_{\mathrm{max}}}{\int}}{v}^{}\mathrm{d}t}\ne {C}_{C}$, where t is the time in days. This implies that if v is exponential, C

_{I}is exponential too, but C

_{C}is not. Note that v is always positive.

^{2}). The acceleration is positive and negative (= deceleration), if the velocity v increases and decreases, respectively.

^{3}). The jerk j is positive or negative, if the acceleration increases or decreases, respectively. The major decrease of the acceleration (i.e., the major transition from acceleration to deceleration) is denoted by the effective phase or period, T

_{E}(measured in days). During T

_{E}, j is negative on average.

_{max}, the larger the absolute jerk, |j|. This relationship prevents the direct comparison of the j-data of different countries. Therefore, for comparative reasons, j has to be normalised to v. This normalisation process has three advantages, i.e., comparability across different C

_{max}values, independence of the actual case numbers and their associated systematic errors (such as under-reporting) and the definition of an effectiveness parameter, E, where E = −j/v (unit: d

^{−2}).

#### 2.2.1. Mathematical Derivation of Effectiveness Parameters

_{max}), t is the time (in days), m is the day where v reaches its maximum, and s represents the width of the bell curve. The Gaussian function is symmetrical about m. Note that the standard structure of a Gaussian involves a multiplier of two in the denominator of the exponent, which was omitted here for simplification purposes.

_{E1}and t

_{E2}, marks the boundaries of the effective phase. As the bell curve is symmetrical, t

_{E1}+ t

_{E2}= 0, if m = 0.

_{E}is written as:

_{E}by t

_{E}, yielding:

_{min}, and peak velocity, v

_{max}, occur at t = 0. Setting t to zero in Equations (4) and (2) yields:

_{max}.

_{max}or to the average v over the effective phase, ${\overline{v}}_{E}$.

_{E1}to t

_{E2}and divided by 2t

_{E}:

- −
- the effectiveness E in general:$${E}_{t}=-\frac{{j}_{t}}{{v}_{t}}=\frac{2-4{t}^{2}}{{s}^{2}};$$
- −
- the peak effectiveness, E
_{max}:$${E}_{\mathrm{max}}=-\frac{{j}_{\mathrm{min}}}{{v}_{\mathrm{max}}}=\frac{2}{{s}^{2}};$$ - −
- the average effectiveness, $\overline{E}$:$$\overline{E}=-\frac{{\overline{j}}_{E}}{{\overline{v}}_{E}}=\frac{{2}^{}{\mathrm{e}}^{-0.5}}{{s}^{2}{\sqrt{\pi /2}}^{}\mathrm{erf}\sqrt{0.5}}.$$

_{E}is written as:

_{E}in a single parameter.

^{−x}(where x = 1, 2 and 3), which indicates that the narrower (more leptokurtic) the v-peak, the more effective the control measures. This applies to two cardinal parameters, namely $\overline{E}$ and ρ. A third cardinal parameter, T

_{E}, is a function of s

^{+1}according to Equation (6) in a sense that the shorter the effective phase, the more effective are the control measures. “Real-world” data are required for evaluating the three cardinal parameters, applied to daily case data (calculated from cumulative data) of different countries.

_{E}, $\overline{E}$ and E

_{max}.

#### 2.2.2. Relationship between Average Effectiveness, $\overline{E}$, and the Instantaneous Effective Reproduction Number R_{eff}

_{eff}, we took the logarithm of velocity v and used Equation (2) to obtain:

_{eff}can be very well calculated from underestimated data, which stands in contrast to the criticism by Leung et al. [15].

_{eff}was finally calculated as:

_{eff}<< ∞ and as the transition from epidemic to endemic occurs at R

_{eff}= 1, taking the logarithm of Equation (18) puts this transition at log(R

_{eff}) = 0:

_{eff}) decreasing with time is constant. However, Equation (20) is dependent of s

^{2}. The larger the value of s, the wider the Gaussian curve, and the smaller this gradient in Equation (20). This principle establishes the relationship between R

_{eff}, or more precise, the derivative of log(R

_{eff}), with the effectiveness E of preventive control measures, which is also a function of s

^{−2}, according to Equation (14). Thus, the steeper the gradient of log(R

_{eff}), i.e., –2SI s

^{−2}, the more effective the control measures.

_{R}calculated from R

_{eff}, expressed as:

_{R}stands in sharp contrast to the mere (absolute) decrease [14] or decline [8,9,10,11,12] of R

_{eff}, as E

_{R}corresponds to the decrease of log(R

_{eff}) per unit time, across T

_{E}.

_{R}is expressed as:

#### 2.2.3. Data Processing of Real-World Data

_{C}, commonly reported on websites as further specified below. The daily case data, v, were determined from ΔC

_{C}. Due to the noisy nature of the original v-data, they were pre-filtered by subjecting them to a double-running average filter (1

^{st}-order Savitzky–Golay filter) with a window width of 3 data. The major data fit for identifying the trend was performed with a running quadratic filter (2

^{nd}-order Savitzky–Golay filter) over a window of 13 data. This filter method, specifically a window width of 13 data, was obtained from a convergence test. In principle, the absolute peak data (minimum and maximum) of a and j became smaller and may finally asymptote, as the window width widened (e.g., from 5 to 23 data). At smaller windows, the magnitude of the peak data was greater for two reasons, i.e., the slope of the filter data was steeper and the local noise (data fluctuations) was more pronounced. Consequently, the data fluctuations were assessed by means of a randomness index (RI; RI-p-ap method [43]; 0 = perfectly correlated, 0.5 = perfectly random; 1 = perfectly anticorrelated). The smaller the RI, the less the data fluctuate. The RI-data of a and j asymptoted at an average window width of 13 (11–15) data. Using the quadratic filter without the preceding double average filters would require a wider window than 13 data to achieve the same RI effect but resulted in smaller peak data.

- −
- Each filtered v-datum corresponds to the midpoint of a quadratic fit curve over 13 pre-filtered v-data. The residuals between the filtered v-data and the original v-data were used to calculate the confidence interval of each filtered v-datum. The residual standard deviation of each filtered v-datum was divided by √13 to obtain the standard error, which was multiplied by the t-distribution of degrees of freedom of 10 and α = 0.05 to obtain the 95% confidence interval for each filtered v-datum.
- −
- The filtered v-data including their 95% confidence interval data were numerically differentiated twice by calculating the slope over 3 data points to obtain a and j.

_{E}was defined as the time between an a

_{max}and an a

_{min}, where a

_{max}was positive and a

_{min}was negative (Figure 1), and

_{min}− a

_{max}.

_{max}and a

_{min}were determined visually, according to the aforementioned guidelines.

_{j}= Δa.

_{E}, was determined from its boundaries t

_{E1}and t

_{E2}, where t

_{E}corresponded to the intersections of the j-data and the zero line (intersection of a straight line between two consecutive data points, one positive and one negative), one intersection at a

_{max}and one at a

_{min}. Note that T

_{E}is usually non-integer.

_{E}= Δt = t

_{E2}− t

_{E1},

_{E}.

_{E}and $\overline{E}$ data in relation to the Gaussian function (h = 2).

_{E}and $\overline{E}$ data followed a Gaussian function. As such, the s-parameter was determined for T

_{E}and $\overline{E}$ from Equations (5) and (14), respectively:

_{E}and $\overline{E}$ data followed a Gaussian function, then s

_{E}= s

_{T}, and the ratio ς = s

_{E}/s

_{T}must be unity. The ratio ς is defined as:

_{E}:

_{E}is shown in Figure 1, calculated from the simulated daily case numbers (velocity v) and their consecutive time derivatives, acceleration a, and jerk j.

_{eff}, i.e., ${\overline{E}}_{R}$, was calculated as:

_{eff}denotes the decrease of logR

_{eff}during the effective phase.

^{2}) of multiple and single regressions, the combined influence was calculated from the sum of the R

^{2}of the single regressions minus the R

^{2}of the multiple regression. The individual influences (semi-partial correlations) of $\overline{E}$ and logς were calculated from the single regression R

^{2}minus the combined influence. The influences were expressed as a percentage, resulting from 100R

^{2}.

_{eff}, is, however, that the start and the end of the effective phase have to be predetermined from the first method, which are the positive and negative peak data of the acceleration a (Figure 1). Taking the steepest gradient of log(R

_{eff}) or the gradient at R

_{eff}= 1 results only in a local maximum or value of E

_{R}, respectively, instead of an average ${\overline{E}}_{R}$ across the effective phase. Using the amount of R

_{eff}-decrease, i.e., ΔR

_{eff}as a measure [8,9,10,11,14] of the quality or efficiency of a control measure is concerning, as two countries with the same ΔR

_{eff}but achieved over different times clearly showed that the steeper the gradient of the R

_{eff}-curve, the more efficient the control measure. This fact is reflected in Equation (33), with the time window in the denominator.

**Figure 1.**Explanation of the derivation of the effectiveness parameters. Daily case data (velocity v; following a hypothetical Gaussian function of v = 1000 exp[−(t − 20)

^{2}/7

^{2}]) against time t ($\overline{v}$ = average velocity of the effective phase T

_{E}; v

_{max}= peak velocity); acceleration a, first time derivative of the daily case data (a

_{max}= +a

_{E}= maximum acceleration; a

_{min}= −a

_{E}= minimum; Δa = a

_{min}− a

_{max}; Δa/T

_{E}= acceleration gradient across the effective phase); jerk j, second time derivative of the daily case data ($\overline{j}$ = average jerk of the effective phase); and effectiveness E of control measures against time ($\overline{E}$ = average effectiveness of the effective phase); t

_{E}

_{1,2}= start and end of the effective phase.

#### 2.3. Data Sets of Real-World Data

_{E}ending the latest on 15 May 2020 and C

_{C}≥ 250 at this date. We took the cumulative case data and the cumulative death data from several websites that provide databases for different countries, states and provinces [1,2,3,4].

#### 2.4. Classification of Intervention Measures

- (a)
- A nationwide (state-wide/territory-wide) compulsory stay-home order for 24 h per day and at least for 14 days;
- (b)
- Enforced by law, police and by penalties in case of infringement;
- (c)
- With very few exceptions that allow people to leave their home (e.g., essential work and study; shopping for essential goods; medical care; exercise).

#### 2.5. Statistics

_{E}, $\overline{E}$, ρ and ς. Except for ρ and ς, for all the other parameters, the upper and lower confidence interval bounds were determined. As T

_{E}, $\overline{E}$ and ρ data were not normally distributed, we used the Mann–Whitney U-test (MWU-test) for comparing these parameters for countries with and without a lockdown. The threshold for a significant difference was α = 0.05. The effect size r was calculated from the smaller U: r = 1 − 2U/(n

_{1}× n

_{2}), where n

_{1}and n

_{2}are the numbers of data of each of the 2 groups. The effect sizes r were interpreted according to McGrath and Meyer [51]. The effectiveness was visualised as a map in Matlab (Release 2018b, The MathWorks, Natick, MA, USA) for European countries by colour-coding the parameters T

_{E}, $\overline{E}$, ρ and ς.

#### 2.6. Validation

_{E}, $\overline{E}$ and ρ. For the GDP analysis, only 82 out of the 92 countries states and provinces were included due to missing GDP data of dependencies and Chinese provinces. The influence was tested first with a trend analysis from a power law regression, its R

^{2}value (as a percentage: R

^{2}% = R

^{2}× 100 × sgnR, where sgn denotes the sign function, to identify positive and negative correlations) and its p-value (α = 0.1). R

^{2}% explains at which percentage the effectiveness of control measures can be explained from an external factor. For example, we can assume theoretically that the greater the population, the less efficient the same control measures, comparatively applied to a smaller population. If the trend was significant (p < 0.1), then a threshold value (e.g., of the population) was determined, which separates two groups (below and above the threshold) and at which the medians of both groups are significantly different (MWU-test; p < 0.05).

^{2}values of multiple and single regressions, the combined influence, which has to be positive, was calculated from the sum of the R

^{2}of the single regressions minus the R

^{2}of the multiple regression. The individual influences (semi-partial correlations) of the population size and the country GDP were calculated from the single regression R

^{2}minus the combined influence.

**Table 1.**Countries and effectiveness parameters for countries, states and provinces of which the effective phase ended the latest on 15 May 2020. LOCK: lockdown measures (Y = yes, N = no; X: lockdown only in some cities, or a specific country replaced by individual states or provinces (USA and CHN)); T

_{E}, effective phase; CI, 95% confidence interval; E, effectiveness; ρ = E/T

_{E}; ς, shape parameter of the velocity profile); 3-letter country codes according to ISO 3166-1 Alpha-3; Chinese provinces 2-letter codes: ISO 3166-2:CN; 2-letter code of the states of the USA: ISO 3166-2:US; KOS, Kosovo; R

_{eff}, effective reproduction number.

Country | ISO Code | LOCK | T_{E} (d) | T_{E}, Lower CI | T_{E}, Upper CI | Average E (10^{−3} d^{−2}) | E, Lower CI | E, Upper CI | ρ (10^{−3} d^{−3}) | log ς | Average E_{R} (10^{−3}) from R_{eff} |
---|---|---|---|---|---|---|---|---|---|---|---|

Alaska | AK | N | 19.6 | 19.8 | 20.8 | 11.49 | 14.09 | 11.23 | 0.587 | −0.096 | 14.23 |

Albania | ALB | Y | 36.6 | 35.9 | 36.9 | 8.19 | 10.02 | 8.02 | 0.224 | −0.293 | 11.94 |

Andorra | AND | N | 6.9 | 6.3 | 7.7 | 28.63 | 40.99 | 21.39 | 4.168 | 0.161 | 34.08 |

Australia | AUS | Y | 12 | 11.9 | 13.7 | 18.81 | 20.49 | 15.2 | 1.562 | 0.008 | 26.21 |

Austria | AUT | Y | 9.8 | 10.3 | 9.8 | 22.06 | 21.6 | 21.33 | 2.254 | 0.064 | 29.16 |

Belgium | BEL | Y | 35.2 | 33.9 | 36 | 5.67 | 6.01 | 5.78 | 0.161 | −0.197 | 10.04 |

Bosnia and Herzegovina | BIH | N | 37 | 36 | 38.2 | 5.6 | 7.28 | 5.02 | 0.151 | −0.216 | 6.22 |

Bulgaria | BGR | Y | 7.1 | 5.9 | 7.3 | 24.74 | 32.76 | 22.98 | 3.499 | 0.18 | 28.63 |

Canada | CAN | N | 37.2 | 37.5 | 37.2 | 3.67 | 3.85 | 3.55 | 0.099 | −0.127 | 4.65 |

China | CHN | X | 10.1 | 11.1 | 9.8 | 18.93 | 17.24 | 19.93 | 1.865 | 0.081 | 23.87 |

Colorado | CO | N | 8 | 6.9 | 8.9 | 20.59 | 25.76 | 19.91 | 2.581 | 0.168 | 24.05 |

Connecticut | CT | N | 20.3 | 19.8 | 20.3 | 8.91 | 12.57 | 8.11 | 0.439 | −0.056 | 11.06 |

Croatia | HRV | N | 13 | 12.3 | 13.3 | 14.05 | 16.05 | 13.39 | 1.082 | 0.039 | 26.59 |

Cyprus | CYP | N | 19.8 | 18.2 | 20.7 | 13.63 | 16.75 | 12.33 | 0.688 | −0.138 | 20.26 |

Czech Republic | CZE | Y | 20 | 16.7 | 20.2 | 8.95 | 10.63 | 9.6 | 0.447 | −0.051 | 13.46 |

Estonia | EST | N | 15.2 | 14.5 | 15.5 | 19.38 | 25.08 | 16.34 | 1.278 | −0.098 | 27.38 |

Finland | FIN | N | 5.3 | 5.3 | 5.5 | 26.88 | 36.14 | 19.77 | 5.029 | 0.284 | 29.65 |

Florida | FL | N | 25.1 | 25.4 | 24.8 | 5.93 | 6.65 | 6.16 | 0.236 | −0.059 | 7.89 |

France | FRA | Y | 18.3 | 17.4 | 18.5 | 9.2 | 11.06 | 8.56 | 0.503 | −0.018 | 11.49 |

Georgia | GEO | N | 22.6 | 21.8 | 23.1 | 12.39 | 15.66 | 11.57 | 0.547 | −0.175 | 16.59 |

Germany | DEU | N | 17.5 | 18.5 | 17.1 | 9.39 | 9.57 | 9.39 | 0.536 | −0.004 | 13.07 |

Greece | GRC | Y | 21.8 | 23.5 | 20 | 10.08 | 18.67 | 10 | 0.463 | −0.113 | 17.1 |

Guangdong | GD | N | 9.3 | 8.4 | 10.1 | 27.88 | 31.67 | 24.79 | 2.996 | 0.035 | 39.75 |

Guernsey | GGY | Y | 9.3 | 10.2 | 9.2 | 28.13 | 29.27 | 26.42 | 3.023 | 0.033 | 37.51 |

Hawaii | HI | Y | 9.3 | 9.2 | 9.5 | 19.76 | 23.98 | 17.05 | 2.126 | 0.11 | 23.59 |

Henan | HA | N | 13.3 | 12.5 | 13.6 | 20.14 | 21.85 | 19.52 | 1.51 | −0.051 | 30.08 |

Hong Kong | HKG | N | 8.7 | 9.3 | 8.5 | 21.1 | 20.6 | 22.4 | 2.424 | 0.125 | 24.63 |

Hubei | HB | Y | 9 | 7.6 | 9.1 | 23.22 | 27.06 | 22.18 | 2.577 | 0.089 | 30.35 |

Hunan | HN | N | 13.9 | 13.3 | 14.8 | 17.65 | 20.08 | 15.98 | 1.266 | −0.041 | 25.45 |

Hungary | HUN | N | 7.3 | 6.6 | 7.6 | 33.71 | 40.85 | 30.72 | 4.627 | 0.1 | 44.63 |

Iceland | ISL | N | 21.4 | 19.7 | 21.5 | 11.92 | 13.15 | 12.41 | 0.557 | −0.142 | 20.49 |

Idaho | ID | N | 7.9 | 7.5 | 8 | 42.29 | 46.75 | 39.51 | 5.375 | 0.017 | 59.97 |

Iowa | IA | N | 8.9 | 8 | 12.3 | 17.08 | 23.86 | 12.06 | 1.921 | 0.161 | 20.12 |

Iran | IRN | N | 10.3 | 9 | 11.5 | 15.82 | 19.19 | 14.26 | 1.529 | 0.112 | 20.05 |

Ireland | IRL | Y | 6.8 | 5 | 7.9 | 22.03 | 37.93 | 17.55 | 3.249 | 0.224 | 24.7 |

Isle of Man | IMN | Y | 15.8 | 14.5 | 16.5 | 17.23 | 27.31 | 15.6 | 1.087 | −0.092 | 22.5 |

Israel | ISR | N | 14.7 | 13.4 | 15.3 | 14.96 | 16.5 | 15 | 1.021 | −0.027 | 22.2 |

Italy | ITA | Y | 15.2 | 15.2 | 14.8 | 7.97 | 7.8 | 8.28 | 0.526 | 0.095 | 10.48 |

Jamaica | JAM | N | 18.1 | 17.4 | 19.1 | 16.83 | 24.08 | 13.37 | 0.929 | −0.145 | 28.85 |

Japan | JAP | N | 15.9 | 15.5 | 16 | 10.74 | 10.6 | 10.97 | 0.676 | 0.01 | 13.72 |

Jersey | JEY | Y | 10.5 | 13.7 | 9.5 | 20.93 | 29.55 | 19.4 | 1.988 | 0.044 | 27.56 |

Jordan | JOR | Y | 6.3 | 5 | 6.9 | 33.98 | 52.26 | 30.62 | 5.374 | 0.16 | 39.81 |

Kansas | KS | N | 21.9 | 21.9 | 21.3 | 8.59 | 11.48 | 7.41 | 0.393 | −0.081 | 13.19 |

Kosovo | KOS | N | 19.8 | 18.3 | 21 | 13.55 | 19.34 | 12.82 | 0.683 | −0.137 | 21.13 |

Latvia | LVA | Y | 8.4 | 8.8 | 8.5 | 18.32 | 22.71 | 17.21 | 2.174 | 0.169 | 33.21 |

Lebanon | LBN | Y | 7.2 | 5.7 | 8.9 | 27.46 | 49.3 | 21.48 | 3.825 | 0.151 | 39.68 |

Lithuania | LTU | Y | 32.9 | 31.5 | 34.7 | 9.25 | 11.49 | 8.96 | 0.281 | −0.274 | 23.69 |

Louisiana | LA | Y | 8.3 | 6.6 | 9.2 | 31.36 | 47.2 | 27.4 | 3.775 | 0.059 | 42.69 |

Luxembourg | LUX | N | 19 | 19.9 | 17.8 | 11.66 | 14.08 | 11.87 | 0.613 | −0.086 | 17.12 |

Malaysia | MYS | N | 34 | 33 | 34.6 | 6.82 | 7.58 | 6.47 | 0.2 | −0.222 | 13.92 |

Malta | MLT | Y | 7.6 | 6.7 | 9.7 | 37.16 | 62.48 | 24.99 | 4.879 | 0.06 | 46.91 |

Massachusetts | MA | Y | 8.2 | 7.5 | 8.9 | 19.49 | 21.03 | 18.06 | 2.367 | 0.166 | 22.42 |

Mauritius | MUS | Y | 7.5 | 5.5 | 7.7 | 37.05 | 67.36 | 32.62 | 4.95 | 0.068 | 50.61 |

Michigan | MI | Y | 9.5 | 8.8 | 10.3 | 18.2 | 20.64 | 16.25 | 1.909 | 0.117 | 23.74 |

Montana | MT | Y | 17 | 16.4 | 18 | 15.14 | 16.73 | 13.23 | 0.892 | −0.093 | 22.55 |

Montenegro | MNE | N | 6.4 | 5 | 7.3 | 52.98 | 76.69 | 42.65 | 8.282 | 0.058 | 72.22 |

Morocco | MAR | N | 29.7 | 27.7 | 31.2 | 7.03 | 8.99 | 6.98 | 0.237 | −0.17 | 8.92 |

Netherlands | NLD | N | 26.9 | 26.6 | 28.2 | 5.96 | 6.56 | 6.03 | 0.222 | −0.091 | 7.53 |

New Hampshire | NH | Y | 37.2 | 36.2 | 37.5 | 4.61 | 4.64 | 4.88 | 0.124 | −0.176 | 5.24 |

New Jersey | NJ | Y | 27 | 27.4 | 26.7 | 5.13 | 5.36 | 4.96 | 0.19 | −0.06 | 6.7 |

New York | NY | Y | 38.6 | 38.7 | 38.2 | 4.6 | 4.88 | 4.34 | 0.119 | −0.191 | 9 |

New Zealand | NZL | Y | 15.9 | 15.1 | 16.6 | 18.01 | 19.77 | 17.19 | 1.132 | −0.103 | 29.3 |

North Macedonia | MKD | N | 7 | 5.7 | 8.5 | 32.64 | 53.3 | 24.06 | 4.68 | 0.126 | 39.74 |

Norway | NOR | Y | 15.5 | 15.5 | 15.5 | 14.51 | 15.51 | 13.79 | 0.935 | −0.045 | 19.92 |

Oregon | OR | Y | 49.7 | 48.6 | 49.8 | 3.71 | 4.44 | 3.22 | 0.075 | −0.255 | 5.22 |

Pennsylvania | PA | Y | 9.2 | 8.7 | 12.1 | 14.71 | 18.04 | 10.95 | 1.591 | 0.176 | 19.29 |

Portugal | POR | N | 20.5 | 18.4 | 20.8 | 9.19 | 11.02 | 9.19 | 0.449 | −0.067 | 12.7 |

Reunion | REU | N | 7.4 | 7.1 | 7.5 | 27.28 | 32.45 | 24.44 | 3.681 | 0.138 | 33.45 |

Rhode Island | RI | N | 21.2 | 20.3 | 21.7 | 7.57 | 8.42 | 7.18 | 0.357 | −0.04 | 11.58 |

Romania | ROU | Y | 45.8 | 47.2 | 46.1 | 3.14 | 3.29 | 3.21 | 0.069 | −0.183 | 5.68 |

Russia | RUS | X | 13.5 | 12.6 | 14.2 | 6.63 | 7.41 | 6.37 | 0.489 | 0.184 | 7.74 |

San Marino | SMR | Y | 23.8 | 24.1 | 23.3 | 12.57 | 20.91 | 9.8 | 0.528 | −0.2 | 17.21 |

Serbia | SRB | N | 6.9 | 6 | 7.6 | 25.09 | 29.94 | 22.69 | 3.639 | 0.188 | 29.1 |

Singapore | SGP | Y | 8.4 | 7.7 | 9 | 24.35 | 29.81 | 22.35 | 2.913 | 0.111 | 30.15 |

Slovakia | SVK | N | 7.6 | 6.3 | 8.5 | 32.39 | 59.38 | 26.73 | 4.262 | 0.09 | 39.19 |

Slovenia | SVN | N | 9.2 | 8.8 | 9.6 | 18.49 | 20.65 | 17.03 | 2.015 | 0.13 | 22.05 |

South Korea | KOR | N | 8.3 | 7.2 | 9 | 37.58 | 44.25 | 36.57 | 4.514 | 0.018 | 51.06 |

Spain | ESP | Y | 15.3 | 17.1 | 14.5 | 10.15 | 9.59 | 11.19 | 0.662 | 0.037 | 12.86 |

Sweden | SWE | N | 32 | 33.2 | 33.8 | 4.55 | 5.72 | 3.67 | 0.142 | −0.108 | 5.02 |

Switzerland | CHE | N | 22 | 22.2 | 21.5 | 8.61 | 9.4 | 8.06 | 0.392 | −0.084 | 6.08 |

Taiwan | TWN | N | 11.2 | 10.6 | 12.3 | 20.78 | 24.84 | 18.04 | 1.853 | 0.018 | 29.87 |

Thailand | THA | N | 15.5 | 13.8 | 16.1 | 14.87 | 19.61 | 14.29 | 0.961 | −0.049 | 22.3 |

Tunisia | TUN | N | 26.4 | 25.7 | 27 | 10.91 | 14.25 | 9.5 | 0.413 | −0.215 | 15.12 |

Turkey | TYR | N | 16.3 | 16.1 | 17.3 | 8.27 | 8.82 | 7.72 | 0.508 | 0.055 | 10.35 |

United Kingdom | GBR | Y | 41.8 | 42.8 | 41.8 | 3.07 | 2.94 | 3.32 | 0.073 | −0.138 | 4.5 |

United States | USA | X | 37.9 | 39.2 | 37.5 | 2.78 | 2.66 | 2.96 | 0.073 | −0.074 | 3.68 |

Uruguay | URY | N | 27.5 | 26.5 | 29 | 11.87 | 17.27 | 9.93 | 0.432 | −0.249 | 24.1 |

Uzbekistan | UZB | N | 7.3 | 5.6 | 7.2 | 30.46 | 48.27 | 30.1 | 4.146 | 0.118 | 36.64 |

Vermont | VT | N | 8.5 | 8.4 | 8.7 | 24.63 | 27.65 | 24.24 | 2.892 | 0.1 | 29.57 |

Vietnam | VNM | N | 14.4 | 13.9 | 14.8 | 15.51 | 19.92 | 12.83 | 1.076 | −0.028 | 20.59 |

Washington | WA | N | 16.9 | 15.7 | 17.5 | 9.38 | 10.35 | 9.35 | 0.555 | 0.012 | 13.15 |

Zhejiang | ZJ | N | 8.8 | 8.6 | 8.9 | 35.46 | 36.22 | 34.49 | 4.049 | 0.009 | 47.49 |

## 3. Results

#### 3.1. Practical Explanation of Effectiveness Parameters

_{E}and a great $\overline{E}$ and ρ. Table 1 shows the data of countries and states of which the effective phases ended before 16 May 2020. From this table, the average duration of the effective phase T

_{E}was 17.3 ± 10.5 d (5.3–49.7; range: 44.4 d). The average effectiveness $\overline{E}$ was 17.0 × 10

^{−3}± 10.3 × 10

^{−3}d

^{−2}(2.8 × 10

^{−3}– 53.0 × 10

^{−3}; range: 50.2 × 10

^{−3}d

^{−2}). The average of the ratio ρ was 1.73 × 10

^{−3}± 1.70 × 10

^{−3}d

^{−3}(0.07 × 10

^{−3}– 8.28 × 10

^{−3}; range: 8.21 × 10

^{−3}d

^{−3}).

^{−3}d

^{−2}, close to the overall country average), a medium duration of the effective phase of 12.0 d (still shorter than the overall country average) and ρ of 1.56 × 10

^{−3}d

^{−3}(close to the overall country average).

^{−3}d

^{−2}), a short duration of the effective phase of 6.8 d and a high ρ of 3.25 × 10

^{−3}(twice as high as Australia).

^{−3}d

^{−2}, the same as Australia, close to the overall country average), a medium duration of 15.9 d (close to the overall country average) and a smaller ρ of 1.13 × 10

^{−3}.

^{−3}d

^{−2}), a long duration of 34.0 d (twice the overall country average) and a marginally effective ρ of 0.20 × 10

^{−3}.

#### 3.2. Interrelationship of Effectiveness Parameters

_{E}, with respect to the hypothetical data of a Gaussian function (separating triangular and trapezoidal v-profiles). The point map and its power-law fit function (R

^{2}= 0.8337) deviated from and crossed over the hypothetical Gaussian function data. The data can be divided into three areas: velocity data profiles ranging between Gaussian and triangular with high effectiveness (green area); profiles ranging from triangular over Gaussian to trapezoidal with the medium effectiveness (yellow area); and profiles ranging from Gaussian to trapezoidal with the low effectiveness (pink area). Figure 3 shows that there were no marginally effective triangular velocity profiles and no highly effective trapezoidal profiles. Figure 4 shows $\overline{E}$, T

_{E}, ρ and logς colour-coded on the country map of Europe. Finland had the shortest effective phase (5.3 d), and Romania had the longest effective phase (45.8 d). The least and most effective countries ($\overline{E}$) were Great Britain (3.1 × 10

^{−3}) and Montenegro (53.0 × 10

^{−3}), respectively (Figure 4).

_{eff}, as shown in Figure 5a. The slope of the regression line is 1.2656 and is not 1.41 according to Equation (22), which is applicable to Gaussian functions only. The intercept of the regression function is very close to 0. In Figure 5b, the regression slope of ${\overline{E}}_{R}$ vs. $\overline{E}$ is plotted against the averages of logς, showing that the slope decreases as logς increases. The intercept of the regression function is 1.4003, which corresponds to the slope predicted at logς = 0 and is close to the predicted multiplier of 1.41 according to Equation (22). Figure 5 proves that $\overline{E}$ and ${\overline{E}}_{R}$ are comparable and complementary measures.

^{2}; Figure 5a) and 22.61% from logς. The multiple regression dependency of ${\overline{E}}_{R}$ on $\overline{E}$ and logς was 96.19%. The dependency of ${\overline{E}}_{R}$ on $\overline{E}$ and logς of 3.81% remained unexplained. The individual influences (semi-partial correlations) of $\overline{E}$ and logς on ${\overline{E}}_{R}$ were 73.58% and 1.39%, respectively, and the combined influence of $\overline{E}$ and logς on ${\overline{E}}_{R}$ was 21.11%. The semi-partial correlations revealed that any influence of logς on ${\overline{E}}_{R}$ happened only in combination with $\overline{E}$. The reason for this could be explained from the fact that logς was 43.41% influenced by $\overline{E}$ and even 44.66% by $\overline{E}$ of <0.03. More efficient countries tended to have a triangular v-profile, whereas less efficient countries were characterised by a more trapezoidal v-profile.

**Figure 3.**(

**a**) Average effectiveness ($\overline{E}$) against the duration of the effective phase (T

_{E}). The blue curve represents a Gaussian function; the green area indicates the velocity profiles between triangular and Gaussian; the yellow area represents the velocity profile transition from triangular across Gaussian to trapezoidal; the pink area represents the velocity profiles between Gaussian and trapezoidal; the dark-green dashed lines denote isolines of the $\overline{E}$ /T

_{E}ratio (ρ); the dashed grey curve represents the power function fit of all data; note that the data located on the blue curve (Gaussian function) are not necessarily Gaussian but can be pseudo-Gaussian, as a transition from triangular to trapezoid velocity profile (as shown in (

**b**)) can be a very short trapezoid plateau (shorter than the one of New Zealand shown in Figure 2b). (

**b**) Average effectiveness ($\overline{E}$ ) against the duration of the effective phase (T

_{E}) on a double-logarithmic graph. The pink lines are isolines of the shape parameter s, associated with the width of the velocity profile (the smaller s, the greater the effectiveness); The light-blue lines are isolines of shape parameter h, associated with the shape of the velocity profile, indicating the transition from a triangular velocity profile over Gaussian and trapezoidal to an extreme and hypothetical rectangular profile. (

**c**) Average effectiveness ($\overline{E}$ ) against the duration of the effective phase (T

_{E}) on a double-logarithmic graph. Parameter ρ is the ratio of $\overline{E}$ to T

_{E}, (the greater ρ, the greater the effectiveness); parameter ρ is another parameter associated with the shape of the velocity profile, which indicates the transition from a triangular velocity profile over Gaussian to a trapezoidal profile. (

**d**) ρ against logς on a single-logarithmic graph. The green, yellow and pink areas correspond to the areas of the same colours shown in (

**a**).

**Figure 4.**Maps of Europe, showing the effectiveness of control measures of each country, for countries whose effective phase T

_{E}ended the latest on 15 May 2020; upper row: (

**a**): duration of effective phase T

_{E}; (

**b**) average effectiveness $\overline{E}$; (

**c**) ρ ($\overline{E}$ /T

_{E}ratio); (

**d**) logς (shape parameter: blue: Gaussian velocity profile, green: triangular velocity profile, red: trapezoidal velocity profile); for subfigures (

**a**–

**c**): the darker the more effective.

**Figure 5.**Average effectiveness ${\overline{E}}_{R}$ calculated from the effective reproductive number R

_{eff}against the average effectiveness $\overline{E}$ (

**a**) and the slope of the regression of ${\overline{E}}_{R}$ vs. $\overline{E}$ (

**b**). In (

**a**), the dashed grey line represents the linear fit function of the regression, whereas the dashed blue line represents the function expected from a Gaussian model; in (

**b**), to assess the dependency of the regression slope on the shape parameter logς, the data of $\overline{E}$, ${\overline{E}}_{R}$ and logς were sorted with respect to logς; and the averages of logς and the regression slope of ${\overline{E}}_{R}$ vs. $\overline{E}$ were calculated across a running window of 15 data; the dashed green line indicates the slope value expected from a Gaussian model; the dashed blue line represents Gaussian model data at logς = 0.

#### 3.3. Timeline Graphs of the Effectiveness

#### 3.4. Comparison of the Effectiveness of Control Measures

**Table 2.**Medians and significance testing (Mann–Whitney U-test (MWU-test)) of the effectiveness parameters, comparing countries/states/provinces with a lockdown to without a lockdown listed in Table 1. Effect size r interpretation was according to McGrath and Meyer [51]. IQR, interquartile range; T

_{E}, duration of effective phase; $\overline{E}$ , effectiveness; ρ = $\overline{E}$ /T

_{E}.

Statistical Parameters | T_{E} (d) | $\overline{\mathit{E}}$ | ρ (d^{−3} 10^{−3}) |
---|---|---|---|

Median (IQR) no lockdown (n = 52) | 15.32 (12.61) | 14.9 (15.4) | 0.99 (2.47) |

Median (IQR) lockdown (n = 37) | 15.16 (15.39) | 17.2 (13.1) | 1.13 (2.13) |

MWU p-value (α = 0.05) | 0.8415 | 0.7642 | 0.7642 |

U | 986 | 999 | 998 |

Effect size r | 0.0249 | 0.0385 | 0.0374 |

r interpretation | very small (r < 0.1) | very small (r < 0.1) | very small (r < 0.1) |

**Figure 7.**Comparison of countries with and without lockdown measures for T

_{E}(

**a**), $\overline{E}$ (

**b**) and ρ (

**c**) by means of box–whisker plots.

#### 3.5. Mortality Rate

#### 3.6. Influences of the Population Size, the Land Area, the Population Density and the GDP on T_{E}, $\overline{E}$ and ρ

^{2}value and indicated as a percentage (R

^{2}% = R

^{2}×100×sgn(R), where sgn denotes the sign function).

_{E}.

^{−3}d

^{−2}and 9.7 × 10

^{−3}d

^{−2}, respectively, and significantly different (p = 0.0041; U = 649; r = 0.356; medium effect size). The medians of ρ below and above 10 million inhabitants were 1.59 × 10

^{−3}d

^{−3}and 0.53 × 10

^{−3}d

^{−3}, respectively, and significantly different (p = 0.0076; U = 674; r = 0.331; medium effect size).

^{2}. The medians of $\overline{E}$ below and above 115,000 km

^{2}were 18.5 × 10

^{−3}d

^{−2}and 11.9 × 10

^{−3}d

^{−2}, respectively, and significantly different (p = 0.0093; U = 721; r = 0.316; medium effect size). The medians of ρ below and above 115,000 km

^{2}were 1.85 × 10

^{−3}d

^{−3}and 0.68 × 10

^{−3}d

^{−3}, respectively, and significantly different (p = 0.0293; U = 774; r = 0.265; medium effect size).

_{E}, $\overline{E}$ and ρ of eight small islands (Singapore, Hong Kong, Mauritius, Reunion, Guernsey, Jersey, Isle of Man and Malta) to the data of the remaining 84 countries and states. T

_{E}, $\overline{E}$ and ρ of both cohorts were significantly different as following: T

_{E}: 8.5 d (islands) vs. 15.7 d (rest), p = 0.0203, U = 168, r = 0.500 (large effect size); $\overline{E}$: 25.8∙× 10

^{−3}d

^{−2}(islands) vs. 14.3∙× 10

^{−3}d

^{−2}(rest), p = 0.0028, U = 120, r = 0.643 (large effect size); ρ: 3.97 × 10

^{−3}d

^{−3}(islands) vs. 0.91 × 10

^{−3}d

^{−3}(rest), p = 0.0054, U = 135, r = 0.598 (large effect size). Islands had the advantage of natural boundaries which further improved the controllability.

^{−3}d

^{−2}and 8.3 × 10

^{−3}d

^{−2}, respectively, and significantly different (p = 0.0006; U = 287; r = 0.520; large effect size). The medians of ρ below and above 600,000 USD million (country GDP) were 1.08 × 10

^{−3}d

^{−3}and 0.50 × 10

^{−3}d

^{−3}, respectively, and significantly different (p = 0.0048; U = 341; r = 0.430; large effect size).

^{2}of close to zero and p-values greater than 0.1. Nevertheless, this result questions the significant influence of the country GDP on the effectiveness. Correlating the country GDP and the population size of the countries and states listed in Table 1 with a power-law regression results in a positive trend with an R

^{2}value of 0.7076. It is therefore possible that the influence of the county GDP was only an indirect one and the direct influence comes from the population size, as already explained above. To obtain clarity on this despite the missing influence from the per-capita GDP (country GDP normalised to the population size), we investigated the individual and combined influences of both the country GDP and the population size on the effectiveness $\overline{E}$. The individual influences (semi-partial correlations) of the population size and the country GDP on $\overline{E}$ were 0.6% and 7.6%, respectively, and their combined influence on $\overline{E}$ was 7.6%. These data explained that the population influence occurred only combined with GDP, whereas the GDP influence had two components, individual and combined, both of which were at 7.6%. Although the total influence on $\overline{E}$ was only 15.7%, the country GDP was the dominating and primary factor, whereas the population size had only an indirect influence.

## 4. Discussion

^{2}= 0.0109, p = 0.1654). The individual influence of the population size on the GDP is approximately three times greater than the one of the educational index (power law regression: R

^{2}= 0.6066 and 0.1892, respectively; p < 0.0001). This result supports that countries with a higher GDP are more educated on average. We therefore hypothesised that a better education leads to less acceptance of and compliance with government rules related to control measures. This hypothesis is supported by the results of Hall et al. [64], who found the correlations between education and protest attitudes and concluded that “education increases opposition to government repression”. Lockdowns are evidently not related to political repression; nevertheless, political repressions and medical control measures during epidemics/pandemics have one thing in common: controlling citizens. However, the result that control measures of “richer” countries are less effective and stands in sharp contrast to the conclusions of Pincombe et al. [25], namely that “containment and closure policies were more effective in high-income countries” (the GNI / gross national income [65] is related to the GDP [54] at R

^{2}= 0.99). Pincombe et al. [25], however, defined their effectiveness measure in a different way we did, namely from “larger decreases in mobility” and ”smaller COVID-19 case and death growth rates”.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Velocity, acceleration, jerk and effectiveness of the spreading virus against time (day 1 = 01/01/2020) for 4 different velocity profiles: (

**a**) Gaussian (Australia, logς = +0.01, medium effective, ρ = 1.56 × 10

^{−3}); (

**b**) triangular (Ireland, logς = +0.22, very effective, ρ = 3.25 × 10

^{−3}); (

**c**) short trapezoidal (New Zealand, logς = −0.10, medium effective, ρ = 1.13 × 10

^{−3}); (

**d**) long trapezoidal (Malaysia, logς = −0.22, marginally effective, ρ = 0.20 × 10

^{−3}). The green curve represents filtered velocity data and their time derivatives; the light blue and pink curves denote upper and lower confidence intervals (note that after differentiation, the upper and lower boundaries can switch their positions); t

_{E1,2}, start and end of the effective phase; a, acceleration.

**Figure 6.**Instantaneous effectiveness E (

**a**) and average effectiveness $\overline{E}$ (

**b**) against time in days (day 1 = 01/01/2020) for countries, states and provinces of which the effective phase ended the latest on 15 May 2020 (day 136). Three-letter country codes according to ISO 3166-1 Alpha-3; Chinese provinces 2-letter codes: ISO 3166-2:CN; 2-letter code of the states of the USA: ISO 3166-2:US; KOS, Kosovo; note that countries can exhibit multiple effectiveness peaks, as seen in (

**a**) for Uruguay and some Chinese provinces and in Figure 2.

**Table 3.**Mortality (no. of deaths per million population), medians and significance testing (MWU-test) comparing countries/states/provinces with lockdowns to those without lockdowns; and with mortality in the middle of the effective phase of greater and smaller 50 deaths per million population; the interpretation of the effect size r was according to McGrath and Meyer [51]).

Mortality (Deaths per Million Population) | Median (IQR)—Lockdown | Median (IQR)—no Lockdown | n_{1}—Lockdown | n_{2}—no Lockdown | MWU p-Value (α = 0.05) | U | Effect Size r | Inter- pretation |
---|---|---|---|---|---|---|---|---|

At the beginning of the effective phase | 2.3 (18.2) | 2.7 (10.1) | 37 | 47 | 0.7039 | 826.5 | 0.049454 | very small (r < 0.1) |

In the middle of the effective phase | 12 (74.4) | 6.7 (32.6) | 37 | 47 | 0.1031 | 688 | 0.208741 | small (0.1 < r < 0.24) |

At the end of the effective phase | 24 (116.3) | 15.8 (41.1) | 37 | 47 | 0.1052 | 689 | 0.207591 | small (0.1 < r < 0.24) |

at 26 June 2020 | 80.6 (580.3) | 37.3 (128.6) | 37 | 52 | 0.0085 | 645 | 0.329522 | medium (0.24 < r < 0.37) |

Mortality (deaths per million population) | Median (IQR)—mortal- ity in the middle of effective phase > 50/M | Median (IQR)—mortal-ity in the middle of effective phase < 50/M | n1—mortality in the middle of the effective phase > 50/M | n2—mortality in the middle of the effective phase < 50/M | p | U | r | Inter-pretation |

at 26 June 2020 | 614.5 (525.2) | 32.3 (78.3) | 67 | 19 | <0.0001 | 25 | 0.960723 | large (r > 0.37) |

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Fuss, F.K.; Weizman, Y.; Tan, A.M.
COVID-19 Pandemic: How Effective Are Preventive Control Measures and Is a Complete Lockdown Justified? A Comparison of Countries and States. *COVID* **2022**, *2*, 18-46.
https://doi.org/10.3390/covid2010003

**AMA Style**

Fuss FK, Weizman Y, Tan AM.
COVID-19 Pandemic: How Effective Are Preventive Control Measures and Is a Complete Lockdown Justified? A Comparison of Countries and States. *COVID*. 2022; 2(1):18-46.
https://doi.org/10.3390/covid2010003

**Chicago/Turabian Style**

Fuss, Franz Konstantin, Yehuda Weizman, and Adin Ming Tan.
2022. "COVID-19 Pandemic: How Effective Are Preventive Control Measures and Is a Complete Lockdown Justified? A Comparison of Countries and States" *COVID* 2, no. 1: 18-46.
https://doi.org/10.3390/covid2010003