Flatten the curve. On a new covid-19 (hit) severity

ABSTRACT

Background

During the health crisis of the COVID-19 pandemic, the adagium was to ‘flatten the curve’. We investigate how well countries succeeded in this aim by constructing an appropriate severity measure. It is able to distinguish between countries that, e.g., experienced identical overall (excess) mortality rates or attained equal case load peaks over a certain period of time. Concretely, this implies that an identical total number of infections or deaths over a certain period is considered relatively worse if there is a higher and/or more peaks. More classical measures (like the total number or the maximum of cases/deaths) neglect this and are therefore inappropriate to assess the resilience of a health care system nor pandemic policy ex post performance.

Methods & Results

We applied our new (hit) severity to a set of 32 countries, and found that the flattening didn’t go equally well. The difference in severity is large, with Norway being consistently the least severely hit by the pandemic (using deaths as indicator) during the whole observation period, while Hungary comes out as eventually being hit the hardest in our sample.

Conclusions

Having constructed a (hit) severity measure that enables to differentiate between countries’ performances in a sound way, further research should now relate these observed differences to the pre-pandemic health care status and the sanitary measures or restrictions imposed during the pandemic; in order to reveal what measures help the most in what type of health care system and society.

KEYWORDS:

• Covid-19
• severity
• measure
• excess persistence

1. Background

The COVID-19 crisis struck hard worldwide and at a pace existing systems could not always take. Governments took different measures, at a different speed and a different intensity. Moreover, they had to operate within their country specific context (despite the pandemic); demographics, urbanisation, climate, style of life, etc. Notwithstanding the dissimilarity, they were all challenged by the same task of preserving human lives. Did they succeed? The answer is undoubtedly yes. According to [1] potentially tens of millions of the population could have succumbed to the virus when no measures (incl. vaccination, though outside of the reference period in this study) would have been taken. The main strategy consisted in slowing down the spread of the virus: flattening the curve. In order to give maximal chances to people who do got infected to be treated, to preserve health care access to non COVID-19 patients and to keep society ‘unlocked’ to a reasonable degree, the reduction of (peak) pressure on the health care system was paramount. Nevertheless, many lives were (prematurely) lost. So, just like put forward in [2], one has all the interest in learning from this disaster for it is reasonable to expect it to happen again. But if reduction of (peak) pressure was (and remains) the goal, shouldn’t we mainly measure exactly that?

Ultimately, we want to know what worked (best) against a given background, e.g. what society and health care system. That is, we turn from ‘Did countries succeed in flattening the curve?’ towards ‘How well did countries succeed in flattening the curve?’. Once we have a metric that captures this performance we can start to look for the reason why certain countries performed differently as they did. This paper aims to construct such a metric in a sound way. We see it as future work to investigate the reasons of these differences using our new metric. One could envisage regression methods or any other modelling using key background variables such as population density and composition, urbanisation level, a stringency index¹ capturing sanitary measures (like school closures, travel bans, …), and the OECD Health at a Glance indicators on population health and health system performance.

Now, what severity measure should one use? A possible metric would be the total amount of COVID-19 deaths per capita over a given period of time. In that case, a country that has had more deaths did worse than any country with less deaths. But clearly that metric totally neglects how the deaths occur during the period. It is not only completely memoryless² but it is also incapable of distinguishing between high vs. low mortality rate periods: it is peak-blind. Flattening the curve is all about avoiding peaks. Imagine two countries (cases C1 and C3 in Table 1) that register the same total number of deaths as there are days in the observation period. However, in one country, this happens via 1 a day (flat curve) and in the other this happens by 0.5 half the days and 1.5 the other half (no flat curve). Then, a well-adapted metric should deem the performance of the second country worse than the first one as the latter succeeds in maintaining a more stable curve. Note that taking the arithmetic average as a candidate severity metric suffers the same drawback of being completely memoryless and peak-blind. Next, we should take the maximum as the severity metric, and then this measure is indeed not peak-blind. It would quantify a country with a higher maximal mortality rate (1.5 for C3) as doing worse than a country with a lower maximal mortality rate (0.5 for C1). However, we consider two countries (cases C4 and C5 in Table 1) that again register the same total number of deaths as there are days in the observation period but have the same maximal value. Using the maximum as the severity metric would not distinguish between the two countries. Yet, we observe one country attains this maximal level two times more frequently as the other. The maximum as a measure does not take into account how long this level prevails.

Table 1. Getting to know the (hit) severity measure.

Name	Description	Total	Max	Mean	Std	SEV
C1	incidence of 1 each day	584	1	1	0	0.50
C2	incidence of 3 each day	1752	3	3	0	4.47
C3	incidence of 0.5 half the days and 1.5 the other half	584	1.5	1	0.5	0.62
C4	three peaks, each of 10 days with incidence of 19.47	584	19.47	1	4.3	9.71
C5	three peaks, of 10/20/30 days with incidence of 19.47/9.73/6.49	584	19.47	1	3.3	5.93
Cx	continuous profile	584	4.2	1	1.32	1.36

Note: The length of the observation period is 584 days in all cases.

We included some other metrics in Table 1 for information purposes and invite the reader to assess their aptness. It is clear, a completely memoryless metric is not apt to assess the performance of curve flattening nor – though already better – does a metric that is just not peak-blind. We therefore require the following two properties of a (hit) severity measure of an indicator:³

(P1) the higher the indicator gets over a certain period, the higher the severity;

(P2) the longer the indicator remains at (or above) a certain level, the higher the severity.

Many papers [3–6] exist on the severity of COVID-19 as a disease on the individual level or on a population level, e.g. looking at excess mortality, cases per million population (CPM), deaths per million population (DPM), infection fatality ratio (IFR) and case fatality ratio (CFR). But neither go as such beyond a classical way of assessing severity. For our goal, without any transformation, they are incomplete and sometimes even deficient. To our knowledge, no metric having the two aforementioned properties has been presented in literature to date. Note that, we have applied our metric to CPM and DPM, but it can be easily applied to any base indicator. It captures second-level information and thus is an enrichment to standard analysis. In that regard, we mention excess mortality as an alternative base indicator and refer to [7] for an in-depth analysis of COVID-19 reporting differences.

Our paper is outlined as follows. We commence with the data to which we are about to apply our method. Next, in Section 3, we introduce the concept of excess persistency used in the definition of our (hit) severity measure. Some mockup examples are given for educational purpose and illustrating the earlier mentioned properties. In Section 4, we present the results on a sample of 32 countries and discuss our findings. We end with a conclusion and some remarks.

2. Data

We based our calculations on reported COVID-19 deaths and cases data, encompassing the period from 15 March 2020 until 20 October 2021, which were retrieved from the COVID-19 Data Repository⁴ by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU) and downloaded on 22 October 2021. In [8] the authors note that the number of cases or deaths reported by any institution (including JHU, the World Health Organisation, the European Centre for Disease Prevention and Control, and others) on a given day does not necessarily represent the actual number on that date. This is because of the long reporting chain that exists between a new case/death and its inclusion in statistics. This also means that negative values in cases and deaths can sometimes appear when a country corrects historical data, because it had previously overestimated the number of cases/deaths.

We argue that reporting chains are similar to all countries, but a daily measurement is too prone to reporting issues. Therefore, we opted for a 7 day running average to mitigate this effect. Should negative values occur before this change to the moving average, we replaced them by zeroes and scaled the entire curve in order to maintain the initial overall sum and shape. The resulting indicator profiles are shown in Figure 1 for a set of 10 out of the 32 countries listed in Table 2.

Figure 1. Indicator profile for 10 selected countries.

Table 2. The (standardised) value of the (hit) severity measure using the 7 day running average of deaths (D) or cases (C) per country.

	D-severity		C-severity
country	value	standardised	value	standardised
Austria	6.64	− 0.47	23,957.80	− 0.37
Belgium	21.37	1.00	43,321.00	0.33
Bulgaria	33.12	2.18	21,004.20	− 0.47
Canada	1.58	− 0.98	4,908.86	− 1.05
Croatia	17.80	0.65	39,928.09	0.21
Cyprus	1.25	− 1.01	64,340.32	1.09
Czechia	32.48	2.12	101,172.23	2.42
Denmark	1.03	− 1.03	11,766.64	− 0.81
Estonia	4.49	− 0.69	62,084.18	1.01
Finland	0.13	− 1.12	1,809.57	− 1.17
France	8.75	− 0.26	33,268.83	− 0.03
Germany	5.03	− 0.63	7,517.84	− 0.96
Greece	6.67	− 0.47	11,877.80	− 0.80
Hungary	41.52	3.02	34,708.08	0.02
Iceland	0.13	− 1.12	5,409.15	− 1.04
Ireland	5.37	− 0.60	25,682.77	− 0.30
Italy	13.95	0.26	18,916.22	− 0.55
Latvia	8.46	− 0.29	33,955.80	− 0.00
Lithuania	15.09	0.38	66,049.98	1.16
Luxembourg	6.71	− 0.46	50,511.31	0.59
Malta	2.86	− 0.85	16,865.44	− 0.62
Mexico	8.97	− 0.24	1,784.23	− 1.17
Moldova	7.49	− 0.39	15,299.14	− 0.68
Monaco	7.56	− 0.38	22,233.90	− 0.43
Montenegro	26.62	1.53	127,566.51	3.38
Netherlands	3.86	− 0.75	37,487.73	0.12
Norway	0.09	− 1.13	3,414.16	− 1.11
Poland	14.84	0.35	27,087.69	− 0.25
Portugal	18.61	0.73	46,027.58	0.43
Romania	12.35	0.10	20,463.84	− 0.49
Russia	4.83	− 0.65	5,736.96	− 1.02
Serbia	5.34	− 0.60	84,257.72	1.81
Slovakia	25.39	1.41	22,844.75	− 0.41
Slovenia	25.26	1.39	67,385.29	1.20
Spain	11.05	− 0.03	30,491.22	− 0.13
Sweden	7.30	− 0.40	39,394.32	0.19
Switzerland	7.69	− 0.37	32,127.42	− 0.07
Ukraine	6.09	− 0.53	11,334.87	− 0.82
United Kingdom	16.01	0.47	46,326.33	0.44
United States	9.89	− 0.15	43,370.65	0.34
	mean	st.deviation	mean	st.deviation
	11.34	9.99	34,092.26	27,667.89

Note: Severity calculated over the period 15 March 2020 till 20 October 2021.

We point out that no other manipulations of the data took place. One could consider an alteration of the cases indicator to take into account, e.g., the sort of testing strategy used in each of the countries, or the death indicator to adjust for, e.g., the difference between ‘died because of Covid-19’ and ‘died with Covid-19’. Adjusting for these effects or including a stability analysis regarding different data sources⁵ would entail a study on its own and goes beyond the scope of our aim. We refer to [5] for an in depth analysis in the case of COVID-19 mortality in Belgium during the first month of the pandemic.

3. Method

Given a discrete⁶ time series $\mathrm{T}=\{({\mathrm{t}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{i}})|1\le \mathrm{i}\le \mathrm{d}\}$ with values v_i for different times t_i, let us define its excess persistency p_T by the proportion of the observation period the values are above a certain level, that is

$p_T\left(\mathit{h}\right)=\frac{\mathrm{\#}\left\{t_i|v_i\ge \mathit{h}\right\}}{\mathit{d}}$

Note that the excess persistency is a decreasing function by construction. Moreover, at level h = 0 one has p_T(0) = 1 when all values v_i are nonnegative and at levels $\mathit{h}>\underset{i}{max}\left\{v_i\right\}$ one has p_T(h) = 1. The (hit) severity SEV_T of a time series T is obtained by summing up, over all levels, its excess persistency multiplied with the corresponding level. Under the above conditions, the following integral is indeed well defined⁷

$\mathrm{SE}{\mathrm{V}}_{\mathrm{T}}=\underset{0}{\overset{+\mathrm{\infty }}{\int }}\mathrm{h}.{\mathrm{p}}_{\mathrm{T}}\left(\mathrm{h}\right)\mathrm{dh}\left(\ast \right)$

The reader is invited to check that this definition of (hit) severity has the aforementioned two required properties (we provide a sketch of proof in Appendix). In Figure 2 we depict the indicator profile, as well as the corresponding excess persistency and severity integrand for the six examples described in Table 1. One can compute the (hit) severity of a time series with constant value c to be c²/2. Also, if two time series are proportionate with factor f then their (hit) severities are proportionate with factor f². In practice, the integral in (*) is computed numerically using a quadrature rule.

Figure 2. Severity measure build-up mokup.

When we assume furthermore the time series is ordered, that is $t_i<t_j$ when i<j, and denote $\mathit{T}\left(\mathit{t}\right)=\{(t_i,v_i)|1\le \mathit{i}\le \mathit{t}\}$ the part of T up to time t<d, we can define the time evolution of the (hit) severity SEV_T(t) at time t of the time series T to be the (hit) severity SEV_T(t) of the time series T(t). One can envisage further extensions derived from our measure, e.g. a -time frame evolution of the (hit) severity $\mathit{SE}{V}_{\mathit{T},}\left(\mathit{t}\right)$ at time t of the time series T to be the (hit) severity $\mathit{SE}{V}_{\mathit{T}\left(\mathit{t},\right)}$ of the time series $\mathit{T}\left(\mathit{t};\mathrm{\Delta }\right)=\{(t_i,v_i)|\mathit{t}\le t_i\le \mathit{t}+\mathrm{\Delta }\}$.

We are aware that our definition of (hit) severity remains to a degree memoryless; in the way that, it is blind to ordering. If one were to shuffle the time series values, then the (hit) severity would not change. However, taking ordering into account would mean the introduction of (a sort of) compensation which is itself debatable. Incoming patients facing the health care system under high pressure cannot be contented by arguing that at a later or earlier moment the system was under less pressure. It is nevertheless instructive to also look at the time evolution of the (hit) severity.

4. Results and discussion

Using the methods described in the previous section, we determine the excess persistency and severity integrand for the 32 countries listed in Table 2. This is visualised for a subset of 10 selected countries in Figure 3. The speed of decrease of the excess persistency gives already a first view of a country’s performance.

Figure 3. Severity measure build-up for 10 selected countries.

The calculated (hit) severity of the 7 day running average of deaths (which we will refer to as D-severity) resp. cases (referral: C-severity) is presented in Table 2. The mean resp. standard deviation of the D-severity is 11.34 resp. 9.99, whereas 34,092.26 resp. 27667.89 for the C-severity. The D-severity is therefore slightly more dispersed⁸ than the C-severity. Both are positively skewed.

Standardisation of the severities enables a country by country comparison. We show this in two panels. In Figure 4(a), countries are ordered according to standardised C-severity, whereas in Figure 4(b), countries are ordered according to standardised D-severity. Note that a bar pointing towards the country’s name means performing better than average (negative standardised value), and away from it worse than average (positive standardised value). We see that, in general, Slavic countries were severely hit, whereas Scandinavian countries (except Sweden) were hit least. The link between the performance based on deaths and one based on cases is positive, but there are exceptions. In this respect, Hungary is an interesting case: using deaths as indicator it performed the worst, but based on cases it did average. Two future lines of research emerge. Compare countries (and try to explain the difference) using the same indicator; e.g. why is the deaths based performance of Belgium so much worse than that of its neighbour, the Netherlands. Or, compare the performance of the same country using different indicators; e.g. why did Cyprus’ bad cases based performance not result in a bad deaths based performance. Regarding the latter, refer also to Figure 5(a). We recall that we did not alter the underlying indicators (besides moving to a 7 day running average). This means that the observed difference may be due (in part) to a different way of reporting.

Figure 4. Standardised severity calculated over the period 15 March 2020 till 20 October 2021.

Figure 5. Standardised D-severity compared to other standardised indicators. (Grey area represents the 95% pointwise confidence interval around the regression line. Severity calculated over the period 15 March 2020 till 20 October 2021.

Next, the D-severity is compared to other indicators: C-severity, total number of deaths, excess mortality and total number of cases. Regression plots of the standardised values are shown in Figure 5. The graph in Figure 5(b) demonstrates that our approach gives additional (and more precise) information. Using total deaths as a performance measure, one would deem Belgium, Italy and the United States as equal. They did, however, not experience this ‘same death toll’ in the ‘same way’. Thanks to the (hit) severity based on deaths we can say Belgium came under much higher pressure when viewed over the same total observation period.

In all, we can conclude that, based on the reported figures, Portugal, Belgium, Slovenia, Slovakia, Czechia and Hungary didn’t succeed as well as other countries did with the same mortality at flattening the curve. Likewise, Figure 5(d), shows the same total case load presented itself differently in France as it did in Belgium. When we turn to excess mortality (taken from [9] Table 1]), we see the same set of countries popping up outside the grey 95% pointwise confidence zone. We also refer to [10] and references therein for a number of comparisons in terms of excess mortality.

We end our results with the time evolution of the standardised severities during the observation period, see Figure 6. Coming back to Belgium and Italy that have the same total COVID-19 deaths over the observation period but ended up with a different (hit) severity based on deaths, we see this is due to Italy’s ability (severity plateau around 1.4) to keep the peak of its initial wave much lower than Belgium (severity plateau around 4.8). A final caveat should again be made that the time series were not adjusted for possible differences in COVID-19 reporting practice. When in the future, turning to causal grounds investigation, one could either opt to preemptively correct time series or to include this reporting practice as an explanatory factor as such.

Figure 6. Time evolution of the standardised severity during the observation period.

5. Conclusion

Flatten the curve is indeed paramount if one e.g. is to ensure that the health care system keeps coping and that society can remain functioning at some level; high peaks are extremely disruptive, and need to be avoided. In that respect, what health care system has been more resilient during the COVID-19 pandemic? What pandemic policy was best adapted? In order to answer these questions and to learn from the past, one needs first to objectify in a sound way; to construct an apt measure of comparison. When only looking at a simple aggregation (be it the total number, the maximum, the average, …) of COVID-19 deaths or cases, however, too much information is lost in the process of assessing the severity by which COVID-19 struck populations. We argued what prerequisites a sound severity measure should entail, fit for the purpose of quantifying ‘curve flattening performance’. Next, we constructed such a (hit) severity measure that we believe enables further investigation into what accounts for its observed different values; be it the type of (sanitary) measures that have been taken, the overall state (health, income, rurality, …) of the society, or a combination of these.

We end with two remarks. First, one can argue that excess mortality is better suited to assess the performance then COVID-19 reported mortality. However, that is not the point of our paper. We believe and stress that the aggregation measure that is applied to the underlying indicator curve should encompass the right properties – regardless the indicator being (excess) mortality, a possible modification of the cases or deaths indicator, or something completely else.

Secondly, one can believe that our metric can be only sensibly used on cases but not on deaths. For instance, because of a (ethical) reasoning that (lost) lives count or should be valued the same no matter when they occur. Also here, this is not the point of our paper. We argue that mortality can be treated by the same methods. Indeed, when the health care system is under such pressure (e.g. by an unsurmountable influx of patients) that care isn’t accessible any longer in a normal fashion, it is to be expected to show up in mortality figures.

Authors’ contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by John Crombez and Rob De Staelen. The first draft of the manuscript was written by Rob De Staelen and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Availability of data and material

The datasets generated and/or analysed during the current study are available in the COVID-19 Data repository, https://github.com/CSSEGISandData/Covid-19.

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Acknowledgments

The authors would like to thank Kristof Eecklo for his valuable comments.

Disclosure statement

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

Additional information

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Notes

1. Like the Oxford COVID-19 Government Response Tracker, see [11].

2. By memoryless, we mean invariant to every change to the time series’ values that does not alter its overall sum.

3. Consider two countries that have exactly the same profile of incidence; the country with the higher total incidence will have the higher severity. On the other hand, if two countries have the same total incidence, the country with the higher or longer peaks will have the higher severity.

4. See [9, https://github.com/CSSEGISandData/Covid-19].

5. e.g. Our World in Data (OWID) or the European Respiratory Virus Surveillance Summary (ERVISS).

6. The concepts naturally transpose to a continuous setting by replacing the cardinality with the Lebesgue measure.

7. In fact, the severity is proportionate to the mean of the continuous random variable $\mathit{H}$ with probability density proportionate to $p_{\mathrm{T}}$.

8. Measured by the coefficient of variation.

Appendix

Sketch of proof

(P1) the higher the indicator gets over a certain period, the higher the severity;

(P2) the longer the indicator remains at (or above) a certain level, the higher the severity.

For (P1): Consider two time series $T^{\left(\mathit{k}\right)}=\{(t_i,v_i^{\left(k\right)})|1\le \mathit{i}\le \mathit{d}\}$ for k = 1,2; where $v_i^{\left(1\right)}\le v_i^{\left(2\right)}$. Define $h^{\left(\mathit{k}\right)}=\mathrm{ma}{\mathrm{x}}_{\mathrm{i}}{\mathrm{v}}_{\mathrm{i}}^{\left(\mathrm{k}\right)}$ then for h < h⁽¹⁾ one has $\left\{t_i|v_i^{\left(1\right)}\ge \mathit{h}\}=\{t_i|v_i^{\left(2\right)}\ge \mathit{h}\right\}$. Now

$\mathrm{SE}{\mathrm{V}}_{{\mathrm{T}}^{\left(2\right)}}=\underset{0}{\overset{+\mathrm{\infty }}{\int }}\mathrm{h}\cdot {\mathrm{p}}_{{\mathrm{T}}^{\left(2\right)}}\left(\mathrm{h}\right)\mathrm{dh}=\underset{0}{\overset{{\mathrm{h}}^{\left(1\right)}}{\int }}\mathrm{h}\cdot {\mathrm{p}}_{{\mathrm{T}}^{\left(2\right)}}\left(\mathrm{h}\right)\mathrm{dh}+\underset{{\mathrm{h}}^{\left(1\right)}}{\overset{+\mathrm{\infty }}{\int }}\mathrm{h}\cdot {\mathrm{p}}_{{\mathrm{T}}^{\left(2\right)}}\left(\mathrm{h}\right)\mathrm{dh},$

but for h < h⁽¹⁾ we have $p_{{\mathit{T}}^{(2)}}\left(\mathit{h}\right)=p_{{\mathit{T}}^{\left(1\right)}}\left(\mathit{h}\right)$ meaning that

$\mathrm{SE}{\mathrm{V}}_{{\mathrm{T}}^{\left(2\right)}}=\underset{\mathit{\hspace{0.17em}}}{\overset{\mathit{\hspace{0.17em}}}{\underset{0}{\overset{{\mathit{h}}^{\left(1\right)}}{\int }}\mathit{h}\cdot p_{{\mathrm{T}}^{\left(1\right)}}}}\left(\mathit{h}\right)\mathrm{d}\mathit{h}+\underset{{\mathit{h}}^{\left(1\right)}}{\overset{+\mathrm{\infty }}{\int }}\mathit{h}\cdot p_{{\mathrm{T}}^{\left(2\right)}}\left(\mathit{h}\right)\mathrm{d}\mathit{h}=\mathrm{SE}{\mathrm{V}}_{{\mathrm{T}}^{\left(1\right)}}+\underset{\ge 0}{\underbrace{\underset{{\mathit{h}}^{\left(1\right)}}{\overset{+\mathrm{\infty }}{\int }}\mathit{h}\cdot p_{{\mathrm{T}}^{\left(2\right)}}\left(\mathit{h}\right)\mathrm{d}\mathit{h}}}\ge \mathrm{SE}{\mathrm{V}}_{{\mathrm{T}}^{\left(1\right)},}$

as ${\mathrm{p}}_{{\mathrm{T}}^{\left(1\right)}}\left(\mathrm{h}\right)=0$ for h>h⁽¹⁾. Clearly $\left\{t_i|v_i^{\left(1\right)}\ge \mathit{h}\right\}\subseteq \left\{t_i|v_i^{\left(2\right)}\ge \mathit{h}\right\}$, so

${\mathrm{p}}_{{\mathrm{T}}^{\left(2\right)}}\left(\mathrm{h}\right)=\frac{\left\{{\mathrm{t}}_{\mathrm{i}}|{\mathrm{v}}_{\mathrm{i}}^{\left(2\right)}\ge \mathrm{h}\right\}}{\mathrm{d}}$

For (P2): Alternatively, suppose we do not know that for the same i we have $v_i^{\left(1\right)}\le v_i^{\left(2\right)}$, but only that for a given level h the set $\left\{t_i|v_i^{\left(1\right)}\ge \mathit{h}\right\}$ is of the same size or larger than the set $t_j|v_j^{\left(1\right)}\ge \mathit{h}$, meaning the indicator remains longer at (or above) a certain level. Then

${\mathrm{p}}_{{\mathrm{T}}^{\left(1\right)}}\left(\mathrm{h}\right)=\frac{\left\{{\mathrm{t}}_{\mathrm{i}}|{\mathrm{v}}_{\mathrm{i}}^{\left(1\right)}\ge \mathrm{h}\right\}}{\mathrm{d}}\le \frac{\left\{{\mathrm{t}}_{\mathrm{j}}|{\mathrm{v}}_{\mathrm{j}}^{\left(2\right)}\ge \mathrm{h}\right\}}{\mathrm{d}}={\mathrm{p}}_{{\mathrm{T}}^{\left(2\right)}}\left(\mathrm{h}\right)$

As h and ${\mathrm{p}}_{{\mathrm{T}}^{\left(\mathrm{k}\right)}}\left(\mathit{h}\right)$ are non-negative for h ≥ 0 we have $\mathit{h}\cdot {\mathrm{p}}_{{\mathrm{T}}^{\left(1\right)}}\left(\mathit{h}\right)\le \mathit{h}\cdot {\mathrm{p}}_{{\mathrm{T}}^{\left(2\right)}}\left(\mathit{h}\right)$. Integration being a linear operation it readily follows that $\mathrm{SE}{\mathrm{V}}_{{\mathrm{T}}^{\left(1\right)}}\le \mathrm{SE}{\mathrm{V}}_{{\mathrm{T}}^{\left(2\right)}}$.