A pandemic momentum index to manage the spread of COVID-19

Abstract

Quantifying the spreading power of a pandemic like COVID-19 is important for the early implementation of early restrictions on social mobility and other interventions to slow its spread. This work aims to quantify the power of spread, defining a new indicator, the pandemic momentum index. It is based on the analogy between the kinematics of disease spread and the kinematics of a solid in Newtonian mechanics. This index, ${\mathrm{I}}_{\mathrm{PM}}$, is useful for assessing the risk of spread. Based on the evolution of the pandemic in Spain, a decision-making scheme is proposed that allows early responses to the spread and decreases the incidence of the disease. This index has been calculated retrospectively for the pandemic in Spain, and a counterfactual analysis shows that if the decision-making scheme had been used as a guide, the most significant decisions on restrictions would have been advanced: the total number of confirmed cases of COVID-19 would have been much lower during the period studied, with a significant reduction in the total number of cases: 83 % (sd = 2.6). The results of this paper are consistent with the numerous studies on the pandemic that concluded that the early implementation of restrictions is more important than their severity. Early response slows the spread of the pandemic by applying less severe mobility restrictions, reducing the number of cases and deaths, and doing less damage to the economy.

1. Introduction

After the first cases of COVID-19 were registered at the end of 2019, governments around the world implemented numerous measures to reduce the rate of transmission of the virus. This strategy is known as “flattening” the curve that represents new daily cases against time. In this paper, we propose a model for quantifying the capacity of the disease to spread (spread capacity) that allows us to estimate the most appropriate timing for measures intended to slow down propagation. This objective is crucial to the management of the pandemic since introducing these measures early leads to a significant reduction in the burden of the disease on the health of citizens and on the economy.

Large differences are observed in the COVID-19 burden between countries with very high Human Development Indices. In the countries selected as examples (Fig. 1 ), all with populations over 5 million, differences of up to three magnitude degrees are observed in cumulative COVID-19 cases per 100,000 inhabitants (data source: OurWorldInData (Ritchie et al., n.d.)). These log differences reach values of 3.5 (Czhechia versus Taiwan). Similar logarithmic differences are seen in deaths per 100,000 inhabitants, reaching log differences up to 3.8 (Hungary versus Taiwan).

Fig. 1.

log10 of total COVID-19 cases/100,000 population from pandemic start to 31-3-2021 in 25 countries with very high Human Development Index (>0.8) and population > 5 million.

These differences have been associated with different modes of pandemic management in each country (Narayanan et al., 2021; Pearce et al., 2020; Lee et al., 2020; Detsky and Bogoch, 2020), reflecting policy decisions that largely involved the implementation of such non-pharmaceutical interventions (NPIs) (Lancet COVID-19 Commission, 2020) as quarantine, lockdown and border control. These interventions, with some differences and intensities, have been the same worldwide, but with different timings. Some countries implemented NPIs when COVID-19 incidence was very low, while others took longer.

Paradoxically, as Kochańczyk et al. showed (Kochańczyk and Lipniacki, 2021), the NPIs adopted have been most severe in the most affected countries, which in turn has been reflected in further economic damage. The cumulative reduction in mobility and the cumulative number of deaths associated with COVID-19 (normalized by population) are significantly correlated. This means that the stricter the NPIs limiting mobility, the higher the number of deaths caused by the pandemic. The reason for this paradox is that as long as the goal is to suppress the pandemic, the compromise between the death toll and the economic losses is illusory. The high death rate is correlated with deep and prolonged lockdowns, causing a severe economic recession. The delay in implementing NPIs increases the number of infections and deaths, and this increase must be blocked, ultimately, by a strict and sustained lockdown.

Many studies have highlighted the importance of early response (Bertone et al., 2020; Loewenthal et al., 2020; Dergiades et al., 2020) with examples from less affected countries such as Taiwan, New Zealand, Australia and South Korea (Summers et al., 2020 Nov; Fouda et al., 2020; Park and Chung, 2021; Shakespeare-Finch et al., 2020). In these countries, early adoption of NPIs resulted in better pandemic control, less effort applied and less damage to the health and the economy. Similarly, studies performed through counterfactual analyses conclude that if containment measures had been implemented a week earlier in Spain, when the first wave started, the burden of the pandemic would have been much lower (Orea and Álvarez, 2020; Steinegger et al., 2021).

It is, therefore, clear that the rigor and scope of the NPIs do not by themselves guarantee control of the pandemic. What really matters is implementing the containment measures at the right time, which highlights the importance of the two objectives of this work. The first is to define quantitative indicators of the pandemic's ability to spread in order to facilitate early implementation of responses. The second is to determine the optimal times at which to apply NPIs.

There are several reasons, however, why it is difficult to find these optimal times:

• -Much of the information available was unreliable because of biases, deficient registration systems, errors, etc.
• -There were differences between the official number of cases and the actual number of infections. Recorded cases are a proxy of real cases, since not all cases appear in official records, due to asymptomatic cases and limited testing capacity.
• -There is a time lag between when an infection occurs and when it is officially registered as a confirmed case, which means that the policymakers do not have a real-time picture of the developing pandemic and, therefore, interventions are usually delayed (Steinegger et al., 2021).

Infection-registration lag time is perhaps the main difficulty. This lag time includes several days between infection and onset of symptoms and between diagnosis and laboratory confirmation and further delays in communication and inclusion in the official records (Huang et al., 2020; Wang et al., 2020; Backer et al., 2020; Baud et al., 2020; Kretzschmar et al., 2020). Evaluating this lag time is complex, since it fluctuates according to the pandemic situation and between countries. However, in the absence of a more accurate evaluation, this delay can be taken to be 14 ± 2 days. Steignegger et al. (Steinegger et al., 2021) estimated an average delay of 15 days between infection and registration during the 1st wave in Spain, with delays of up to 20 days in some regions. This lag time emphasizes the need for early interventions to block the spread because the effectiveness of interventions will not be reflected in the records until two weeks later.

In this paper, a new variable of the pandemic's ability to spread at a given time, which we have called $\mathit{PSM}$ Pandemic Spread Momentum, has been defined. The decimal logarithm of the $\mathit{PSM}$is used as an indicator of the risk of spread and is symbolized as $I_{\mathit{PM}}$, Pandemic Momentum Index. These variables represent the ability of the pandemic to spread in the form of new cases, regardless of its effects on health, and are calculated from the series of cumulative total cases since the start of the pandemic. In the next section, pandemic momentum is described as a concept analogous to the momentum of a moving body in Newtonian mechanics.

As a case study, the model based on $\mathit{PSM}$ was applied to the pandemic in Spain during the first three waves. A qualitative calibration of the index $I_{\mathit{PM}}$ was conducted, establishing a risk scale with five levels depending on the $I_{\mathit{PM}}$ values. A decision-making scheme for management oriented to early response has been proposed as a guide to imposing or easing NPIs.

The feasibility of the decision-making scheme was corroborated by counterfactual analysis. For this, we calculated the decrease in the pandemic incidence that would have occurred during the first three waves in Spain if significant restrictions had been implemented earlier, in accordance with the proposed decision-making scheme.

A novel graphical representation of the epidemic curves has been produced, based on the components of the $\mathit{PSM}$. This graph visualizes the pandemic waves as hysteresis loops, with different trajectories in the ascending and descending phases.

2. Theoretical framework

The most widely used models in epidemiology to predict the number and distribution of cases of an infectious disease transmitted through a population are based on the “compartment model”, introduced by Kermack and McKendrick in 1927 (Kermack and McKendrick, 1927). Later approaches, such as the SEIR model, assume that a given population is divided into four compartments, each representing a fraction of the population: S (susceptible), E (exposed), I (infectious) and R (recovered), whose sum is the total population. Although this model describes the exponential spread of the disease very well and provides a basic model for investigating different types of epidemics, its practical application is complex due to the disturbances produced by interventions to stop the spread (He et al., 2020). In situations such as the one experienced in 2020, when the nature and behavior of SARS-CoV-2 were almost completely unknown, and treatments and vaccines were not available, it is reasonable to resort to empirical models to predict how the disease will spread. In this context, the use of indicators of the capacity of spread is relevant.

‘Pandemic momentum’ is a concept from Newtonian kinematics that describes the movement of solid objects and studies their trajectories as a function of time. The justification for the application of kinematics to assess the evolution of a pandemic is empirical since a previous work (Orihuel et al., 2021) demonstrated the ability of the cubic function to adjust the series of the cumulative number of COVID-19 cases ($y$) versus the number of days since the start of the pandemic ($x$).

The cubic function describes the rectilinear movement of an object whose acceleration grows linearly with time. Working with data from the 2nd pandemic wave in Spain, it was found that cubic function fitted data in sections of 14 days with determination coefficients R² close to one. The first derivative of the cubic equation is the velocity, and the second derivative is acceleration. To avoid biases and case registration errors, the seven-day moving average of the cumulative number of COVID-19 cases ($y^{\ast }$) was taken as the independent variable:

$$y^{\ast }=\mathrm{a}{x}^3+\mathrm{b}{x}^2+\mathrm{c}x+\mathrm{d}$$ (1)

$y^{\ast }=$seven-day moving average of the cumulative number of cases.

$x=$number of days from pandemic start

$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}=\text{constants}$$

This empirical similarity suggests the use of the Newtonian concept of momentum (mass in movement) to assess the ability of a pandemic to spread. The momentum in Newtonian mechanics is the product of the mass ($m$) and velocity ($u$) of an object (Encyclopaedia Britannica, n.d.-a). A visualization of this concept is a snowball picking up both mass and speed as it rolls down a hill (Scott, 2019). Although the snowball can easily be stopped by hand when its mass is small, this is impossible when it is large, even with low velocity. Following the analogy, the concept of momentum can be used to assess the pandemic's ability to progress and, consequently, the effort required to stop its progression.

We can imagine a pandemic as a snowball and define its momentum as the product of the “pandemic mass” and its “velocity of spread” at a given time. The greater the momentum, the greater the risk (spread power) and severe restrictions will be necessary to prevent it from spreading further. We consider pandemic mass on day $x$ as the number of contagious persons, for which we take as a proxy the cumulative rate of cases of the last 14 days per 100,000 inhabitants $\left({\mathit{IA}}_{14}\right)$, the period that is usually considered quarantine time. The daily increase of total cumulated cases is a proxy of the spread velocity.

This empirical model is simple and practical, it uses only the series of accumulated cases as input, and obviously, it has limitations. However, it allows the authorities to make short-term forecasts of the evolution of the pandemic (Orihuel et al., 2021) and to evaluate, albeit roughly, the ability of the pandemic to spread, offering useful information for the decision-making process on implementing or relaxing NPIs.

3. Methodology

3.1. Calculation of $\mathit{PSM}$ and $I_{\mathit{PM}}$

The input for the calculation of $\mathit{PSM}$ and $I_{\mathit{PM}}$ is the time series $y\left(x\right)$ that represents the total number of accumulated cases, where $x$ is the number of days that have elapsed since the start of the pandemic in Spain. These series have been obtained from the Spanish Ministry of Health (Ministerio de Sanidad- Gobierno de España, n.d.), which has published these data daily. Since the data for Saturdays, Sundays and holidays ceased to be published in July 2020, linear interpolation has been used to bridge discontinuities. The smoothed series $y^{\ast }\left(x\right)$ was calculated as the arithmetic mean of the last seven days in order to minimize biases, errors and corrections (Orihuel et al., 2021; Ministerio de Sanidad - Gobierno de España, 2020).

The values of the pandemic mass and velocity of spread proxies, whose product is the $\mathit{PSM}$, are normalized by population and are calculated according to the equations:

$$\text{Pandemic mass proxy}:{\mathit{IA}}_{14}=\frac{y\left(x\right)-y\left(x-14\right)}{\mathit{Population}}{10}^5$$ (2)

$$\text{Velocity of spread proxy}:u=\frac{y\left(x\right)-y\left(x-1\right)}{\mathit{Population}}{10}^5$$ (3)

$\mathit{PSM}$ and the $I_{\mathit{PM}}$ are cal culated with equations:

$$\text{Pandemic Spread Momentum}:\mathit{PSM}\left(x\right)={\mathit{IA}}_{14}u=\frac{\left[y\left(x\right)-y\left(x-14\right)\right]\left[y\left(x\right)-y\left(x-1\right)\right]}{\mathit{Population}}{10}^5$$ (4)

$$\text{Pandemic Momentum Index}:I_{\mathit{PM}}\left(x\right)={\mathit{log}}_{10}\mathit{PSM}={\mathit{log}}_{10}\left({\mathit{IA}}_{14}u\right)$$ (5)

These variables, when computed from the smoothed series$y^{\ast }\left(x\right)$, are symbolized using an asterisk as a superscript: ${{\mathit{IA}}^{\ast }}_{14}$, $u^{\ast }$, ${\mathit{PSM}}^{\ast }$ and $I_{\mathit{PM}}^{\ast }$. Only population and $y$ values corresponding to days $x$, ($x-1)$ and ($x-14$) are used as inputs.

The $I_{\mathit{PM}}$ takes values between −10 and 10, since the maximum $\mathit{PSM}$ value corresponds to a situation with the entire population of a territory infected on the same day. In this case the ${\mathit{IA}}_{14}$ the value would be 10⁵, and the velocity of spread would also be 10⁵, so the $\mathit{PSM}$ value would be 10¹⁰. The minimum $\mathit{PSM}$ corresponds with the first and only contagion in a territory of 10⁴ million people (more than the current world population), and this corresponds to a value of both ${\mathit{IA}}_{14}$and $u$of 10⁻⁵, this is, a $\mathit{PSM}$ value of 10⁻¹⁰.

3.2. Graph of $u$ versus ${\mathit{IA}}_{14}$

In the next section, a graphical representation of the pandemic plots the two $\mathit{PSM}$ factors: $u$ versus ${\mathit{IA}}_{14}$. In this graph, a theoretical line of symmetry can be plotted that passes through the origin with a slope equal to 0.0714 = 1/14. The equation is:

$$u={\mathit{IA}}_{14}/14$$ (6)

This line divides the plane into two sections, growth and decrease, and represents the evolution of an epidemic from fourteen days after the first case, with the number of cases evolving with constant acceleration: $\mathit{du}/\mathit{dx}=\text{constant}$. Epidemics evolve like a solid in motion with variable acceleration. Thus, during the growth phase of pandemics the slope of the line from the origin of the coordinates is >0.0714, and during the decrease phase the slope is <0.0714.

The plane can also be divided into sectors delimited by curves that represent constant values of $I_{\mathit{PM}}$, corresponding to different levels of the pandemic's ability to spread. The equation of these curves is:

$$u={10}^{I_{\mathit{PM}}}/{\mathit{IA}}_{14};I_{\mathit{PM}}=1,2,3\dots$$ (7)

Later we will see the usefulness of this type of graph in interpreting the pandemic situation.

3.3. Counterfactual analysis

The suitability of the “decision-making scheme” proposed in the next section is verified through a counterfactual analysis applied retrospectively to the three waves in Spain (the case study of this paper). This analysis estimates roughly the decrease in total cases that would have occurred if NPIs had been implemented in accordance with the decision-making scheme. To carry out this analysis, the following assumptions are made:

• -The most significant interventions aimed at slowing the spread of COVID-19 had been advanced according to the decision-making scheme.
• -Due to the infection-registration delay time, pandemic peaks occur 14 ± 2 days after the implementation of significant sets of NPIs.
• -There is a geometric similarity between the descending leg of the counterfactual epidemic curve ($u^{\ast }$ versus $x$) and the descending leg of the actual curve, with the same ratio between each day and the previous one.
• -The waves are independent events: that is, the development of one wave does not affect the next, and therefore the same starting dates of each of them are considered.

This set of assumptions is conservative, at least in the case of Spain, since, as described below, it is obvious that effective control of the 2nd wave would have meant delaying the 3rd and reducing its incidence.

4. Results and discussion

4.1. Pandemic Momentum Index $I_{\mathit{PM}}$ in Spain and graphic representation

Using Eqs. (2), (3), (4), (5), the values of the variables ${\mathit{IA}}_{14}$, $u$, $\mathit{PSM}$, and $I_{\mathit{PM}}$, corresponding to the historical series $y\left(x\right)$ of accumulated cases since the start of the pandemic in Spain, have been calculated. The calculations have been made from February 24, 2020, to February 28, 2021, which correspond to the 1st, 2nd and 3rd pandemic waves. The same calculations have also been made for the smoothed series $y^{\ast }\left(x\right)$, obtaining: ${{\mathit{IA}}^{\ast }}_{14}$, $u^{\ast }$, ${\mathit{PSM}}^{\ast }$ and $I_{\mathit{PM}}^{\ast }$. All the data refer to 100,000 inhabitants.

Fig. 2 shows the 1st, 2nd and 3rd waves, whose peaks are the maximum values of $u^{\ast }$(new daily cases) that occurred respectively on March 30, 2020, November 3, 2020, and January 25, 2021. By using the $I_{\mathit{PM}}^{\ast }$ values obtained from the smoothed series $y^{\ast }\left(x\right)$, numerous irregularities due to biases and registration errors are eliminated. However, very marked irregularities persist, especially in March 2021, when, due to error corrections, data were recorded that implied negative values of $u^{\ast }$.

Fig. 2.

Index of pandemic momentum $I_{\mathit{PM}}$ in Spain (blue line) and smoothed $I_{\mathit{PM}}^{\ast }$ (red line). The right axis shows pandemic velocity $u^{\ast }($new daily cases smoothed) (orange bars). The irregularities observed mainly in March 2021 are due to negative values of daily cases, due to corrections by the Health Ministry. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Using graph $u^{\ast }$ versus graph ${\mathit{IA}}_{14}^{\ast }$ the three waves in Spain have been represented in Fig. 3 as cycles. Fig. 4 shows an enlargement of the 1st wave start, a plateau period, and 2nd wave start. This kind of graph provides interesting information on the behavior of the pandemic. In both figures, constant $I_{\mathit{PM}}^{\ast }$ values lines (Eq. (7)) and symmetry line (Eq. (6)) have been plotted. The sequential pairs of values (${\mathit{IA}}_{14}^{\ast },u^{\ast }$) draw a curve that forms a hysteresis cycle (arrows indicate the temporal direction) shaped as an ovoid loop in which the decrease phase ($u^{\ast }$/${\mathit{IA}}_{14}^{\ast }$ < 1/14) travels a different path than that of growth ($u^{\ast }$/${\mathit{IA}}_{14}^{\ast }$ > 1/14), both converging at point (0, 0).

Fig. 3.

The three pandemic waves in Spain. $u^{\ast }$ versus ${\mathit{IA}}_{14}^{\ast }$, with the constant $I_{\mathit{PM}}^{\ast }$ lines (in red) and the symmetry line (dotted blue). Arrows indicate the time direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4.

Plateau at the end of 1st wave and 2nd wave start, with the constant $I_{\mathit{PM}}^{\ast }$ lines (in red) and the symmetry line (dotted blue). Arrows indicate the time direction. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The pattern of lemniscate structures is observed in successive waves in Fig. 3. This lemniscate (Encyclopaedia Britannica, n.d.-b) shaped loop is reproduced at different scales as fractal structures in subsequent pandemic waves. It is also remarkable how the unstable equilibrium of the plateau period breaks when $I_{\mathit{PM}}^{\ast }$ exceeds the value of 1, and the slope ($u^{\ast }$/${\mathit{IA}}_{14}^{\ast }$) is over 1/14 (Fig. 4). Some other observations about the pandemic's behavior in Spain are:

• -NPIs implementation decisions should be made as soon as possible, preferably when $I_{\mathit{PM}}^{\ast }$ < 1. However, the three most significant interventions in Spain were adopted in the 1st wave with $I_{\mathit{PM}}^{\ast }$ = 1.22, in the 2nd wave with $I_{\mathit{PM}}^{\ast }$ = 4.17 and in 3rd wave with $I_{\mathit{PM}}^{\ast }$ = 4.88.
• -Until an appropriate set of NPIs is applied, it is not possible to reverse the direction of the pandemic in the graph $u^{\ast }$ vs. ${\mathit{IA}}_{14}^{\ast }$. However, due to the infection-registration lag time, we cannot expect to observe the change in trend until 14 ± 2 days later.
• -Of the 370 days in the period studied, in 267 days (72 % of the total) the $I_{\mathit{PM}}^{\ast }$ had values >2. Except for the 1st wave, $I_{\mathit{PM}}^{\ast }$ values <2—which would have made it possible to consider the spread of the pandemic under control— were not reached.
• -During the pandemic's decreasing period, a relative control equilibrium was only reached during the “plateau period” (between June 10 and July 6, 2020), with $I_{\mathit{PM}}^{\ast }$ at levels <1 (Fig. 4). In this region of the cycle, the pandemic is relatively under control, but with the possibility of reactivation. This phase has been described by Cacciapaglia et al. (Cacciapaglia et al., 2021) as a “strolling period”, in which the pandemic evolves slowly and without a defined direction. These authors highlight the importance of these periods in attempts to avoid new waves, and argue that limiting the virus spread in these periods is the most effective way to prevent or delay the arrival of successive waves.
• -Once high $I_{\mathit{PM}}^{\ast }$ values are reached (>2), as happened in Spain, it is very difficult to control the virus spread without severe and lasting NPIs. When NPIs begin to relax at values 3, 4 or 5, it is very likely that new waves will start.

4.2. Criteria for interpreting ${\mathit{I}}_{\mathit{PM}}^{\mathbf{\ast }}$

We found that the $I_{\mathit{PM}}^{\ast }$ can be a valid indicator of different levels of risk of disease spread. The different levels of the $I_{\mathit{PM}}^{\ast }$ are established as an alternative to the risk assessment criterion established by the European Union (Council of the European Union. Council Recommendation (EU), 2020) and adopted in Spain (Consejo Interterritorial, 2020) as of October 22, 2020. In both cases, the criteria were based on the values of the cumulative rate of notification of COVID-19 cases in the last fourteen days (${\mathit{IA}}_{14}^{\ast }$) with five risk levels. The criterion based on the $I_{\mathit{PM}}^{\ast }$ proposed in this work has five levels of risk assessment and is based on the analysis of the pandemic waves in Spain. Both criteria are summarized in Table 1 .

Table 1.

Risk criterion based on $I_{\mathit{PM}}^{\ast }$, and Spanish criterion based on ${\mathit{IA}}_{14}$.

	Criterion in this paper	Official criterion in Spain
Risk level	Criterion based on $I_{\mathit{PM}}^{\ast }$	Criterion based on ${\mathit{IA}}_{14}$
New normal		< 25
Low	< 1	25–50
Medium	1–2	50–150
High	2–3	150–250
Very high	3–4	> 250
Extreme	> 4

These risk thresholds of the criterion based on the $I_{\mathit{PM}}^{\ast }$ have been compared with the pandemic development in Spain, represented in Fig. 3 and with the main milestones of each wave summarized in Table 2 , including the date when the most significant restrictions were implemented.

Table 2.

Daily cases $u^{\ast }$, cumulative 14-day case rate ${\mathit{IA}}_{14}^{\ast }$, pandemic momentum index $I_{\mathit{PM}}^{\ast }$, and risk levels based on the $I_{\mathit{PM}}^{\ast }$, at singular dates of the three waves and at the “plateau”.

Pandemic waves	Relevant facts	Date	${\mathit{IA}}_{14}^{\ast }$	${\mathit{u}}^{\ast }$	${\mathit{I}}_{\mathit{PM}}^{\ast }$	Risk level based on ${\mathit{I}}_{\mathit{PM}}^{\ast }$
	Start	24-02-20	0.0015	0.0003	−6.34	Low
1st WAVE 107 days	State of alarm-Lockdown	14-03-20	7.61	2.16	1.22	Medium
	Maximum	30-03-20	137.38	16.52	3.36	Very high
	End	9-06-20	13.01	0.59	0.88	Low
PLATEAU 27 days	Start	10-06-20	12.15	0.62	0.88	Low
	End	6-07-20	10.66	0.86	0.96	Low
	Start	7-07-20	10.71	0.86	0.96	Low
2nd WAVE 156 days	Decree of alarm	25-10-20	391.90	37.37	4.17	Extreme
	Maximum	3-11-20	548.96	44.62	4.39	Extreme
	End	9-12-20	246.59	13.32	3.52	Very high
	Start	10-12-20	238.89	13.86	3.52	Very high
3rd WAVE 81 days	Several restrictionsa	7-01-21	329.21	34.53	4.06	Extreme
	Maximum	25-01-21	963.78	78.16	4.88	Extreme
	End	28-02-21	284.58	15.25	3.64	Very high

^aSeveral different restrictions in Spanish autonomous regions after the Christmas holidays.

In our opinion, the official criterion was illusory and inconsistent. An example of its lack of realism is that, according to this criterion, it would not have made sense to declare confinement in Spain on March 12, 2020 (Ministerio de la Presidencia, Relaciones con las Cortes y Memoria Democrática, n.d.-a) since at that time the ${\mathit{IA}}_{14}^{\ast }$ was 7.61, lower than the “low” risk threshold. Furthermore, during the second wave, strict measures were not taken until October 25, 2020 (Ministerio de la Presidencia, Relaciones con las Cortes y Memoria Democrática, n.d.-b), with the ${\mathit{IA}}_{14}^{\ast }$ value on 391.90, well above the “very high” risk level threshold set at 250. The same situation occurred in early January 2021, when restrictions were reintroduced with the ${\mathit{IA}}_{14}^{\ast }$ = 329.21, also well above the “very high” risk threshold.

The Spanish official criterion (similar to the EU criterion) is more optimistic and less realistic than the one based on $I_{\mathit{PM}}^{\ast }$, especially during the growth stages of the pandemic. For example, the Spanish criterion would consider that ${\mathit{IA}}_{14}$ = 45 corresponds to “low” risk for any value of $u$. However, as Fig. 3 shows, during the growth stage the points (${\mathit{IA}}_{14}$, $u$) are above the line of symmetry defined by Eq. (6) and, if ${\mathit{IA}}_{14}$ = 45, then $u$ > 3.21, and therefore, the corresponding value of the $I_{\mathit{PM}}^{\ast }$ will be >2.16. Therefore, according to our criteria, ${\mathit{IA}}_{14}$ = 45 indicates a “high” risk situation instead of the “low” risk situation diagnosed by the Spanish criterion. This fact is consistent with the idea that, in general, the propagation capacity of SARS-CoV-2 was underestimated in Europe.

Table 2 shows the main dates on which significant restrictions were implemented in each wave. In the 1st wave, the lockdown was established with a medium risk level of $I_{\mathit{PM}}^{\ast }$ = 1.22, although at that time the capacity to carry out tests in Spain was extraordinarily low. So to establish comparisons with the 2nd and 3rd waves, it would be necessary to adjust ${\mathit{IA}}_{14}^{\ast }$ and $u^{\ast }$ indices by a factor between 5.7 and 7.1 (Institute for Health Metrics and Evaluation (IHME), n.d.), which results in a risk level between 1.98 and 2.07, very close to the high-risk level.

In 2nd and 3rd waves, significant restrictions were adopted with an “extreme” level of risk. This fact, together with the infection-registration lag time, is why the pandemic reached an incidence in Spain among the highest in the world (Ritchie et al., n.d.). Once the restrictions were implemented, it took 16, 9 and 18 days until the corresponding pandemic peaks were reached.

4.3. Decision-making scheme

The $I_{\mathit{PM}}^{\ast }$ is not enough to determine the optimal moments to implement or ease the NPIs, which is the second objective of this work. Therefore, we believe that the estimation of the timing of implementing or easing NPIs should be based on three factors: a) the $I_{\mathit{PM}}^{\ast }$ value and its corresponding risk assessment level, b) the stage of the pandemic cycle (growth or decrease phase), which is determined by the location of each point with respect to the axis of symmetry (Eq. (6)) on the plot of $u^{\ast }$ versus $I_{\mathit{PM}}^{\ast }$, and c) the continuous trend of $I_{\mathit{PM}}^{\ast }$ (growing or decreasing), established between 3 and 14 days, depending on the $I_{\mathit{PM}}^{\ast }$ value.

This approach to managing a pandemic is aimed exclusively at stopping its spread without other considerations such as disease severity, economics or politics. This scheme is based on the experience of Spain, but it would be interesting to apply it to individual Spanish regions and other countries to compare and improve it with the experience in other territories.

Table 3 does not indicate which NPIs should be implemented at any given time, but when $I_{\mathit{PM}}^{\ast }$ is still at the low or medium level, it would probably be enough to apply strict border control, quarantine confirmed cases, isolate case contacts, and ban public events. When $I_{\mathit{PM}}^{\ast }$ reaches a high level, it would be necessary to implement restrictions that affect the mobility of a high percentage of the population, such as curfew or lockdown. As to what measures are to be adopted at all times, or what measures can be relaxed during the descent phases, there is literature on the effectiveness of the different measures (Iezadi et al., 2020; Flaxman et al., 2020; ECDC, 2020; Ngonghala et al., 2020; Preiser et al., 2020), but more studies are needed to evaluate the effects of the different interventions.

Table 3.

When to implement or relax NPIs? Decision-making scheme for management based on the $I_{\mathit{PM}}^{\ast }$. “Intervention” means: implementing new NPIs or applying them more strictly. “Easing” means: suppressing or easing current NPIs. Slope = $u^{\ast }/{\mathit{IA}}_{14}^{\ast }$: if slope > 0.0714 the pandemic is growing; if slope < 0.0714 it is decreasing.

Actual ${\mathit{I}}_{\mathit{PM}}^{\ast }$	Risk level	Slope	${\mathit{I}}_{\mathit{PM}}^{\ast }$ trend	Management proposal
$I_{\mathit{PM}}^{\ast }$ < 1	Low	>0.0714	Continuous growth in the last 7 days	Intervention
		<0.0714	Continuous decrease in the last 7 days	Easing
1 < $I_{\mathit{PM}}^{\ast }$ < 2	Medium	>0.0714	Continuous growth in the last 3 days	Intervention
		<0.0714	Continuous decrease in the last 7 days	Easing
2 < $I_{\mathit{PM}}^{\ast }$	High, very high or Extreme	>0.0714	Continuous growth in the last 3 days	Intervention
		<0.0714	Continuous decrease in the last 14 days	Easing

When vaccination against COVID-19 became widespread, the ability of the disease to spread was not altered, but its severity radically decreased and the need to implement severe NPIs was attenuated. This work addresses only the pandemic stages before the vaccines had been widely distributed. In Spain, in August 2021, 70 % of the population already had the complete vaccination schedule against the disease.

4.4. Counterfactual analysis

The suitability of the decision-making scheme is verified by a counterfactual analysis applied retrospectively to the three waves in Spain, roughly estimating the decrease in cases that would have occurred if the NPIs had been implemented in accordance with the decision-making scheme. This means advancing the three dates on which the most significant restrictions were adopted according to the decision-making scheme. In this way, the following counterfactual dates have been considered:

• •1st wave: 2-3-2020, when the pandemic was not yet considered dangerous in Spain (only 150 cases and $I_{\mathit{PM}}^{\ast }$ = −2.239). Twelve days before the actual intervention date of 14-3-2020.
• •2nd wave: 7-7-2020, when NPIs were easing. One hundred ten days before the actual intervention date of 25-10-2020.
• •3rd wave: 15-12-2020, when NPIs were easing. Twenty-three days before the actual intervention date of 7-1-2021.

As explained above, pandemic peaks have been assumed to occur 14 ± 2 days after the respective counterfactual intervention dates (Table 4 ). The counterfactual calculation is performed by imagining that the descent phase of the curve, which represents $u^{\ast }$ versus $x$, is geometrically similar to the downward section of the real curve (the same decreasing proportion between each day and the previous one). In this way, curves geometrically similar to the actual one during the descent period of $u^{\ast }$ are obtained. Given that in each actual wave, only a limited number of descent days from the peak are available (71, 36 and 34, respectively), it is not possible to obtain counterfactual curves longer that that number.

Table 4.

Counterfactual dates of intervention and corresponding peaks, assuming 12, 14 and 16 days between infection and registration and corresponding $I_{\mathit{PM}}^{\ast }$ actual values and risk level at those dates.

	Counterfactual interventions			Counterfactual peaks
	Date	$I_{\mathit{PM}}^{\ast }$	Risk	Dates	$I_{\mathit{PM}}^{\ast }$ range	Risk
1st wave	March 2, 2020	−2.239	Low	March 14, 16 and 18	1.216–1.944	Medium
2nd wave	July 7, 2020	0.965	Medium	July 19, 21 and 23	1.750–2.047	Medium - High
3rd wave	December 15, 2020	3.608	Very high	December 27, 29 and 31	3.695–3.804	Very high

Since we consider the waves as independent events and assume that the development of one wave has not affected the next, the same start dates of each wave have been considered. This is a conservative assumption that forces the counterfactual intervention in the third wave to occur with very high values of $I_{\mathit{PM}}^{\ast }$.

Fig. 5 shows the actual and counterfactual epidemic curves ($u^{\ast }$ versus time) of the 1st wave in Spain (from February 24 to May 25, 2020). The actual total number of cases in this period was 234,128, and if the intervention had been brought forward to March 2 (twelve days before), the number of cases would have been 35,900, 35,782 or 53,696, depending on whether the peak would have occurred 12, 14 or 16 days after the intervention. The mean reduction in cases would have been 83.6 % (sd = 6.0).

Fig. 5.

Counterfactual analysis in Spain from February 24 to May 24, 2020, with intervention date at March 2. Actual daily cases smoothed (blue line) and counterfactual with the peak at 12, 14 and 16 after an intervention. The percentage reduction of cases in the period has been estimated at 88.9, 84.7 and 77.1, respectively (case reduction average = 83.6 %; sd = 6.0). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Similarly, the counterfactual epidemic curves of the 2nd and 3rd waves have been calculated, obtaining reductions in the numbers of cases of 68.1 % (sd = 5.2) and 65.6 % (sd = 3.4). The average reduction of cases obtained in the three periods was 72.4 % (sd = 9.7). These results are summarized in Table 5 .

Table 5.

Decrease in total cases in the three counterfactual periods studied in Spain. The total counterfactual cases were evaluated assuming 12, 14 and 16 days from the intervention date to the pandemic peak.

		COUNTERFACTUAL CASES
WAVES / Calculation periods	ACTUAL CASES	12 days	14 days	16 days	Average	% reduction in cases (sd)
1st / Feb. 24 to May 25, 2020	234,128	25,900	35,782	53,696	38,459	83.6 % (sd = 6.0)
2nd / Jul. 7 to Aug. 24, 2020	141,108	38,313	44,052	52,821	45,062	68.1 % (sd = 5.2)
3rd / Dec. 10, 2020 to Jan. 30, 2021	999,146	309,460	344,156	378,011	343,876	65.6 % (sd = 3.4)
						Average: 72.4 % (sd = 9.7)

Counterfactual calculations could not be performed for the entire period studied (February 24, 2020 to February 28, 2021) since, in the final stages of decline from the pandemic peaks, ratios of similarity were not available to calculate the counterfactual values of $u^{\ast }$. In Fig. 6 , to overcome this difficulty, constant values have been considered equal to the last counterfactual value of each wave in the periods without similarity references. Thus, the reduction in the total number of cases obtained would have been between 80.3 % and 85.4 %, with a mean reduction in cases of 83.0 % (sd = 2.6).

Fig. 6.

Counterfactual analysis in Spain from 24 to 2-20 to 28-2-21, with intervention dates at 2-3-20, 7-7-20 and 15-12-20. Actual daily cases smoothed and counterfactual with a peak at 12, 14 and 16 after interventions. The percentage reduction of cases in the period has been estimated at 85.4, 83.1 and 80.3, respectively (case reduction average = 83.0 %; sd = 2.6).

These results indicate that the impact of the first three pandemic waves in Spain, measured by the number of confirmed cases, was six times greater than it would have been if a decision scheme such as the one presented here had been applied. This reduction would have meant significant reductions in the mortality rate and pressure on the health system.

The technique employed for these counterfactual calculations does not consider the highly probable cumulative effect of the waves, since stopping the 1st wave in time could have meant delaying the start of the second. This effect would have been even more important in the 3rd wave, since it started when the 2nd wave was still at a very high-risk level ($I_{\mathit{PM}}^{\ast }$ = 3.52). Given its complexity, the effect of delaying the relaxation of NPIs according to the proposed decision scheme has also not been evaluated. Taking these factors into account, it is quite possible that earlier implementation of NPIs and an appropriate relaxation of restrictions policy would have resulted in the incidence of cases, hospitalizations, and deaths decreasing by an order of magnitude. The log10 of the accumulated cases would have been around 2.8, a level close to that of countries such as Singapore (3.0) or Japan (2.6).

One example of how early application of NPIs can control the spread of a pandemic is Finland. This country has been one of the least affected by COVID-19 in Europe, with a cumulative incidence of cases 85.0 % lower than in Spain and a mortality rate 90.8 % lower (Ritchie et al., n.d.). Fig. 7 compares the epidemic curves of Spain and Finland, highlighting the importance of an early response.

Fig. 7.

Epidemic curves $u^{\ast }$ in Spain and Finland from 24-02-2020 to 28-2-2021.

In Finland, travellers from risk regions were banned from entering the country in mid-February 2020, and the ban became general in mid-March 2020 (Yarmol-Matusiak et al., 2021). However, in Spain, no control or restriction was implemented on the entry of travellers, even when the 1st state of alarm was in effect. Countries that implemented travel restrictions early in February had fewer deaths from COVID-19 than those that restricted entry to the country later (Koopmans, 2020). When restrictions on entry into the country were implemented in Finland, $I_{\mathit{PM}}^{\ast }$ was around -4 (low risk level), and only two cases had been recorded. On 14-2-2020 lockdown was decreed in Spain with $I_{\mathit{PM}}^{\ast }$ = 1216 (medium risk level). Two days later, a similar lockdown was implemented in Finland, but with $I_{\mathit{PM}}^{\ast }$ = 0.232 (low risk level).

5. Conclusions

This study aimed to find a methodology to quantitatively assess the ability of the pandemic to spread, as a basis for early decision-making on the restriction of social mobility (NPIs). This quantification was based on the analogy observed between the kinematics of the pandemic (evolution of the total number of cases over time) and the kinematics of a moving body with linear acceleration (distance travelled as a function of time).

This analogy suggests the adaptation of the concept of momentum (quantity of movement) from Newtonian mechanics to the dynamics of the pandemic. The pandemic spread momentum is thus defined as the power or capacity of the pandemic to spread, which equates to the effort that will be needed to apply to stop it, as would happen with a snowball rolling down a snowy hill: the greater the momentum, the greater the force needed to stop it. This analogy would explain the paradox found by Kochańczyk et al. (Kochańczyk and Lipniacki, 2021), that the rigor of mobility restrictions has been greater in the most affected countries. It also explains the crucial importance of early response, which has been widely described in the literature (Bertone et al., 2020; Loewenthal et al., 2020; Dergiades et al., 2020; Summers et al., 2020 Nov; Fouda et al., 2020; Park and Chung, 2021; Shakespeare-Finch et al., 2020; Orea and Álvarez, 2020; Steinegger et al., 2021; Pei et al., 2020; Haug et al., 2020). This idea can be summarized with the general rule that the sooner the intervention, the less effort will be required to stop the pandemic and the lower the economic and health burdens suffered.

The risk assessment based on the use of the index of pandemic momentum $I_{\mathit{PM}}^{\ast }$ (defined as the logarithm of the product of the pandemic mass and the pandemic velocity), enables authorities to anticipate and adapt the interventions (NPI) and thus curb the pandemic with the least damage to the economy.

A risk assessment criterion based on the $I_{\mathit{PM}}^{\ast }$ has been proposed as a guide for decision-making. It is stricter than those implemented in the EU and Spain, which only reflected a proxy of the pandemic mass (accumulated cases over 14 days), without considering the velocity at which new cases were presented. Likewise, a decision-making scheme has been proposed that is based on the value of $I_{\mathit{PM}}^{\ast }$, its short-term trend, and the phase (ascending or descending) of the pandemic cycle. According to this scheme, the first measures to restrict social mobility should be adopted during the growth phase ($u^{\ast }$/${\mathit{IA}}_{14}^{\ast }$ > 1/14), $I_{\mathit{PM}}^{\ast }$ < 1 and after a continuous increase in $I_{\mathit{PM}}^{\ast }$ for no more than seven days. This means that, for example, in the case of Spain, travel restrictions (Holmdahl and Buckee, 2020), together with protection, general hygiene and isolation of infected people, should have been implemented in Spain no later than 2-3-2020. Likewise, Loewenthal et al. (Loewenthal et al., 2020) estimated that a 7.49-day delay in starting social distancing would double the number of deaths and that a quick response would have had a prolonged effect on COVID-19-related mortality. We believe that a more precise interpretation of $I_{\mathit{PM}}^{\ast }$ can be reached by studying the particular experiences of the pandemic in each of the Spanish regions and comparisons between different countries.

The decision-making scheme was validated through counterfactual analyses of what would have happened in Spain if the scheme had been applied in the three pandemic waves through early interventions. The result would have been a very significant decrease in the total number of COVID-19 cases: an 83.0 % reduction in average cases (sd = 2.6). These results are consistent with those obtained in other counterfactual analyses (Orea and Álvarez, 2020; Steinegger et al., 2021; Born et al., 2020).

Using the concept of pandemic momentum, a novel way of graphically representing pandemic evolution has been tested by plotting variable $u^{\ast }$ against variable ${\mathit{IA}}_{14}^{\ast }$, whose product is the $\mathit{PSM}$. This graph allows the visualization of the waves in cycles shaped as a lemniscate, with different trajectories in the growing and decreasing phases, and a theoretical axis of symmetry with a slope of 0.0714, which divides each wave into two almost equal phases. The plane is also divided into areas by the intersection of the axis of symmetry and the various fronts of the potential curves that represent constant values of $I_{\mathit{PM}}^{\ast }$. This graphical representation provides a useful visual diagnosis of the pandemic situation.

We believe that the empirical model based on $\mathit{PSM}$ does not only apply to the coronavirus pandemic: it can be applied to other pandemics that evolve in similar ways to the SEIR model. However, the risk assessment criteria and proposals for early response will vary depending on factors such as the transmission routes, incubation period and mortality of the disease.

Author statement

All authors contributed equal to the paper.

Biographies

Enrique Orihuel-Iranzo, Bachelor of Chemical Sciences, University of Valencia, Spain; Ph.D. in Chemistry, Complutense University of Madrid, Spain; Member of the Commission of the UCV-CHRISTEYNS Chair for the Development of a Sustainable Economy, Valencia, Spain; Professor at the UNIR School of Humanities, Logroño, Spain; Main research topics: Emerging foodborne pathogens. Pandemic spread models and forecasting. Sustainable development.

Josep-Miquel Navarro, Graduated in Economics by the University of Valencia, Masters’ Degree in economics by the University of Valencia and Master in Economics and Finance by the Barcelona Graduate School of Economics; Currently working in the Catholic University of Valencia; Areas of interest: Bayesian Econometrics and Monetary policy.

Juan Sapena, BA First Class Degree in Economics and Business (University of Valencia, 1993); Ph.D. in Economics (Cum Laude, University of Valencia, 1999); Head of the Chair “Betelgeux-Christeyns” for a Sustainable Economic Development; Head of Economics Department at the Catholic University of Valencia; Main research topics: international finance, global imbalances, time-series econometrics, time-varying parameters, State-Space modeling, sustainability and economics for the common good.

Ramón Bertó-Navarro, Degree in Biology, University of Valencia, Spain; General Manager de CHRISTEYNS España, Valencia, Spain; Main research topics: Food industry, Food safety and hygiene, marine biology.

Data availability

Data will be made available on request.