Labor and employment dynamics during the coronavirus crisis: an update

Confirmations, clarifications and a mystery

Bernard Gazier

Among the data and statistical analyses published between April 30 and May 15, 2020, three deserve particular attention either because they confirm and clarify trends that are already visible or because the questions that they have brought to the fore are far from having a satisfactory explanation of the adjustment processes already underway concerning work and employment.

Confirmation 1 : Unemployment insurance claims as a major adjustment mechanism in the United States

A confirmation enriched with new details has just come from the United States. We noted that the dominant adjustment in this country comes from unemployment insurance registrations, which suddenly increased starting at the end of March. The latest news release from the US Department of Labor (DOL)[1] provides the following figures for the week of April 27 to May 2: in gross terms (direct observation), the unemployment benefit rate stands at 14.5% after reaching 14.9% the previous week, corresponding to 21 million beneficiaries. In seasonally adjusted terms, the corresponding figures are 15.7% and 22.8 million. The week-to-week evolution profiles now illustrate a sharp drop in initial claims from the beginning of April, down from 6 million to 3 million per week. The evolution of the workforce compensated in gross terms shows a stabilization during the last week observed here. We can conclude that this compensation channel has largely been refueled, playing its role as a shock absorber until the deconfinement and recovery measures come into play. An additional element is provided by the brief news release from the DOL: the US state that has the highest number of registrations for unemployment insurance claims is California (with an unemployment rate of 27.7%), followed by Michigan (23.1%). The unemployment rate for the state of New York is 18.8%.

Confirmation 2 : “Partial unemployment” as an important adjustment mechanism in Europe

We know that for many European countries, the main channel for adjustment has been through a  “partial unemployment”[2] scheme. The first European comparative figures became available at the beginning of May, coming from an analysis published by the European Research Institute ETUI (European Trade-Unions Institute) and carried out by T. Müller and T. Schulten.[3] Data including 22 European countries (including Switzerland and the United Kingdom) indicate a massive use of this instrument, judging by the proportion of workers affected by the claims made through this channel at the end of April. The average of the sample provided here is 26.8% of the workers affected, more than a quarter of the total workforce.

Figure 1: Proportion of workers (actual or applied for) participating in STW and similar schemes (End of April/beginning of May 2020, in per cent of workers*)

 Note: Figures for all workers taken from the Eurostat Labour Force Survey (annual figures for 2019); figure for Luxembourg includes cross-border commuters.

Source : Müller et Schulten (2020), p. 3

As the graph above illustrates (ibid. p. 3), the countries with the highest coverage are ... Switzerland with 48.1%, France with 47.8% and Italy with 46.6%. Next comes a series of countries with rates of coverage between 35% and 30%: Slovenia, Croatia, Austria, Belgium and Ireland. The next block, between 27% and 23%, includes Germany, Spain, the United Kingdom and the Netherlands. The Nordic countries (Sweden, Denmark, Finland) are at the bottom of the distribution, with Portugal and most of the Eastern European countries (Romania, Czechia, Slovakia, Bulgaria and Poland).

These figures, however, represent only the requests for possible authorization for compensation and not the compensation actually paid. We know that in the French case, in March, these figures were half less than the demands: 5 million workers rather than 10 million. They provide, nonetheless, two pieces of information: on the one hand, almost all European countries are using this instrument; and on the other hand, they are doing it in a very differentiated way, some massively and others more modestly.

3. The mystery of unemployment registrations in France

Finally, the latest French indicators, published on May 12,|4] far from dissipating questions concerning previous unemployment registrations (which have remained very low since the start of the crisis), have strengthened them. Indeed, we discover on p. 8 of the report that "on average, between April 5 and May 2, weekly requests for registration at Pôle Emploi [the French employment agency], decreased compared to similar weeks in 2019". The increase in unemployment, which has remained modest (see our previous update), is therefore currently due to the slowdown in hiring, not to job losses. Perhaps these trends, as we’ve seen elsewhere, have led to some people to temporarily retreat into inactivity, such as workers at the end of their careers renouncing to apply for employment or women who return to the home. These adjustments, which are very important in the US and undoubtedly happening now, as they did during the global financial crisis that broke out in 2007, have not been observed at all in France. Perhaps the "shutdown" of the French economy resulting from the two months of confinement has generated a temporary retreat into inactivity? The future will tell.

Notes

[1] US Department of Labor (2020), “Unemployment Insurance Weekly Claims”, News Release, 14 May.

[2] Let me remind the reader here that chomâge partiel ("partial unemployment") is a poorly named mechanism in French, because it has nothing to do with unemployment: the salaried employment contract is preserved and the non-activity compensated; furthermore, in the case of the coronavirus crisis, the compensation is not partial, but most often total. The English generic term "short-time work" (STW) helps avoid the first error, but not the second. A specificity of the crisis is that the benefits of this instrument have been extended to workers temporarily experiencing a total interruption in their work lives.

[3] Müller, T., and T. Schulten (2020), “Ensuring fair short-time work. A European Overview”, ETUI Policy Brief, n°7.

[4] DARES, Délégation Générale à l’Emploi et à la Formation Professionnelle et Pôle emploi 2020,
« Situation du marché du travail durant la crise sanitaire - 12 mai 2020 » Tableau de bord hebdomadaire.

Uncertainty quantification of a SEIR-type model applied to the Covid-19 outbreak

Josselin Garnier (Ecole polytechnique)

March 27, 2020

We consider the model proposed in P. Magal and G. Webb [2020], Predicting the number of reported and unreported cases for the COVID-19 epidemic in South Korea, Italy, France and Germany to quantify the uncertainty of the predictions. We apply a general methodology that propagates the uncertainties of the a posteriori distribution of the most important parameters of the model in a Bayesian framework. The methodology could be extended to other models.

1 Model

The model proposed in [1] has the form

S'(t) = −τ(t)S(t)[I(t) + U(t)],                                                                                             (1)

I'(t) = τ(t)S(t)[I(t) + U(t)] − νI(t),                                                                                     (2)

R'(t) = v₁I(t) − ηR(t),                                                                                                         (3)
U'(t) = v₂I(t) − ηU(t),                                                                                                        (4)

where S(t) is the number of individuals susceptible to infection at time t, I(t) the number of asymptomatic infectious individuals at time t, R(t) the number of reported symptomatic infectious individuals at time t, and U(t) the number of unreported symptomatic infectious individuals at time t.

The fraction f of asymptomatic infectious individuals becomes reported as symptomatic infectious individuals, and the fraction 1 − f becomes reported as unreported symptomatic infectious individuals. The asymptomatic infectious rate becomes reported as symptomatic individuals: v₁ = fν. The asymptomatic infectious rate becomes reported as unreported symptomatic individuals: v₂ = (1 − f)ν.

Reported symptomatic individuals are infectious for an average period of 1/η days, as are unreported symptomatic individuals. The daily number of reported cases from the model can be obtained by computing the solution of the following equation:

DR'(t) = v₁I(t) − DR(t).                                                                                                     (5)

The transmission rate τ(t) at time t is parameterized as

τ(t) = τ0 exp − [−µ(t − N)+].                                                                                               (6)

[This means: τ is constant and equal to τ0 till time N; it then decays exponentially with the rate µ.]

We will also consider the parameterized model
τ(t) = τ0 [1 − µ(t − N)+]₊.                                                                                                  (7)

[This means: τ is constant and equal to τ0 till time N; it then decays linearly with the rate µ, until it reaches 0 at time N + 1/µ, and finally it stays at 0.]

2 Strategy

We fix some of the parameters as prescribed by [1]: ν = 1/7, η = 1/7, f = 0.1, and we consider the exponential form (6) for the transmission rate. We will study later the impact of f and the form of the transmission rate.


1) We calibrate the parameters of the model by least-squares on the daily number of reported cases (with Poisson noise). See Figure 1 for the fit between the observed data and the model data with the estimated parameters.

2) We can predict (I(t), R(t), U(t)) with these estimated parameters. See Figure 2. So far, this looks like [1], with a few more days in the data set.

3) We find that µ and N are the important parameters. We fix all other parameters to their estimated values, and we perform a Bayesian estimation (with uniform prior) of the pair (µ, N). See Figure 3.

4) We propagate the uncertainty of the a posteriori distribution of (µ, N) onto the predictions of (I(t), R(t), U(t)). See Figures 4-5.

3 Discussion

- The a posteriori distributions of µ and N are anticorrelated, while the overall result on the maximal value of I is very sensitive to µ. This may explain why the result in [1] looks so bad for Germany, because the estimated µ is two times smaller than the one for France, while, in fact, the estimation is not robust.

Figure 1: Observed daily number of reported cases in France from February 25 to March 26 (dots) and predicted daily number of reported cases with the calibrated model (blue solid line).

- There is a lot of uncertainty in the predictions! Of course, this is not surprising for such models (with exponential growth with estimated growth rates).

- The parameter f (the ratio of the umber of reported symptomatic infectious cases over the total number of symptomatic infectious cases) is rather large (the results on I are essentially inversely proportional to it). Unfortunately, it cannot be calibrated from the data. If it is included in the Bayesian analysis; its a posteriori distribution is its prior distribution. If we consider the two models with f = 0.1 and f = 0.4, the Bayes factor is close to 1.1, in favor of the first one, which is not significant. In Figures 6-10, we give the results obtained with f = 0.4. We need other types of data to estimate this parameter (for instance, a survey from a representative sample of the general population).

- The time-dependent form of the transmission rate is important. If we impose the linear form as in (7), then the results are very different from the ones obtained with the exponential form (6). In Figures 11-15, we give the results obtained with f = 0.1 and the linear form (7). The Bayes factor is, however, 2.2, in favor of the exponential model: the data set favors the exponential rather than the linear model for the transmission rate.

Figure 2: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t).

Figure 3: A posteriori distribution of (µ, N). The dot is the maximum a posteriori.

Figure 4: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). The solid lines are the median values; the dashed lines are the mean values. The median values are close to the maximum a posteriori values.

Figure 5: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). The solid lines are the median values; the dashed lines are the 10% and 90% quantiles.

Figure 6: Observed daily number of reported cases in France from February 25 to March 26 (dots) and predicted daily number of reported cases with the calibrated model (blue solid line). Here f = 0.4.

Figure 7: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). Here f = 0.4.

Figure 8: A posteriori distribution of (µ, N). The dot is the maximum a posteriori. Here f = 0.4.

Figure 9: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). The solid lines are the median values; the dashed lines are the mean values. The median values are close to the maximum a posteriori values. Here f = 0.4.

Figure 10: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). The solid lines are the median values; the dashed lines are the 10% and 90% quantiles. Here f = 0.4.

Figure 11: Observed daily number of reported cases in France from February 25 to March 26 (dots) and predicted daily number of reported cases with the calibrated model (blue solid line). Here f = 0.1 and τ(t) have the linear form (7).

Figure 12: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). Here f = 0.1 and τ(t) have the linear form (7).

Figure 13: A posteriori distribution of (µ, N). The dot is the maximum a posteriori. Here f = 0.1 and τ(t) have the linear form (7).

Figure 14: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). The solid lines are the median values; the dashed lines are the mean values. The median values are close to the maximum a posteriori values. Here f = 0.1 and τ(t) has the linear form (7).

Figure 15: Predictions of the calibrated model: the blue line is the number of asymptomatic infectious individuals I(t), the red line the number of reported symptomatic infectious individuals R(t), the green line the number of unreported symptomatic infectious individuals U(t). The solid lines are the median values; the dashed lines are the 10% and 90% quantiles. Here f = 0.1 and τ(t) have the linear form (7).



References

[1] P. Magal and G. Webb, Predicting the number of reported and unreported cases for the COVID-19 epidemic in South Korea, Italy, France and Germany, https://www.medrxiv.org/content/10.1101/2020.03.21.20040154v1