The dynamics of the Swiss epidemic[1]

Covid dynamics: the calm before the storm After the first wave of spring, the evolution of Covid gave rise to questions. First, the number of hospitalizations and deaths until end of September remained low in proportion to the increase in cases. Then the dynamics accelerated. A simple statistical model accounts for these aspects, and indicates that we are heading for some very difficult weeks. The link between cases and deaths: a two-step dynamic During the first wave, the curves of hospitalizations and deaths closely followed those of new cases. A divergence has emerged since the summer, as the sharp increase in cases was not followed by the other curves until recently. A simple statistical analysis shows that the link between cases on the one hand and hospitalizations and deaths on the other has changed significantly since the beginning of the summer. This reflects an increase in testing. While it implies that a given number of cases is less worrying than in the spring, the sharp increase in the recent number of cases nonetheless leads to a strong number of hospital stays and deaths. We can analyze the dynamics between the different variables in a simple way based on the figures on https://www.corona-data.ch/ aggregated by week. The table below shows the results of an econometric analysis for new deaths, new hospitalizations (inflow-outflow in hospital), new cases in intensive care, and new cases under respiratory assistance. The first column shows that new deaths during a week (t) reflect new cases during the previous week (t-1), with one new case leading to 0.055 extra deaths. This link is statistically significant, as indicated by the Student statistic whose value of 23.6 implies that there is only a small probability that the link is simply the result of chance. Of course this link would imply that the recent increase in cases should result in an increase in deaths, which turned out not to be the case. This reflects the increased testing effort that now identifies mild cases, whereas the spring testing was focused on severe cases. In order to take this into account, the statistical model allows the relationship between cases and deaths to take on a different value since the beginning of June. The first column of the table indicates that the link has indeed been weaker since then, and significantly so. Since early June, one new case leads to 0.007 extra deaths (=0.055 - 0.048). Of course this weaker link can still lead to a substantial number of deaths if the number of cases is sufficiently high. 1

New cases (t) New deaths New persons in New persons New persons (t) hospital (t) in intensive under respiratory New cases (t-1) 0.162 0.055 28.1 care (t) assistance(t) Persons in hospital (t-1) 23.6 0.0% 0.015 0.013 0.0% 15.4 11.0 Persons in intensive care (t-1) -0.376 0.0% 0.0% -0.048 -23.1 0.018 0.012 Persons under respiratory -17.9 0.0% 12.1 6.1 assistance (t-1) 0.0% 0.0% 0.0% Post-May * New cases (t) -0.130 0.053 0.94 -22.9 3.9 -0.566 Post- May * New cases (t-1) 0.0% 0.1% -18.4 -0.787 0.0% Post- May * Persons in 0.97 -10.6 -0.014 intensive care ( t-1) 0.0% -9.9 R2 0.0% -0.012 -0.008 -9.7 -3.2 0.0% 0.3% -0.020 -8.8 0.97 0.0% 0.343 4.6 0.0% 0.99 t and t-1 indicate the specific week. Each cell of the table shows the estimated coefficient the value of the Student t- statistic, and the likelihood that the coefficient differs from zero simply by chance (p-value). It is commonly considered that a p-value of 5% or less indicates that the estimated relation is robust. Although the statistical model is simple, it captures the evolution of the number of deaths well. The figure below shows the weekly number of deaths since the end of February. The blue curve shows the actual number, while the red curve shows the number estimated by applying our model to the number of observed cases. We can see that the two curves are very close. The model captures well the first wave, the limited number of deaths from May to September, and the increase in recent weeks. 2

New deaths per week 450 400 350 300 250 200 150 100 50 0 2020-02-25 2020-03-22 2020-04-19 2020-05-17 2020-06-14 2020-07-12 2020-08-09 2020-09-06 2020-10-04 Actual data Estimated values Impact of cases on hospitalizations We can also estimate the impact of new cases on the number of people in hospital. The second column of the table shows that the net hospital admissions (admissions - discharges) during a week reflects the number of new cases during that week (t), as well as the number of people who were already hospitalized at the end of the previous week (t-1). The negative effect of the latter variable simply shows that the number of people leaving the hospital is simply proportional to the number of people that were already in hospital. Here again we see that a given number of new cases results in a reduced number of new hospitalizations since June, with the coefficient falling from 0.162 to 0.032 (=0.162-0.130). We can also analyze the impact on the number of people in intensive care (third column) and under respiratory assistance (last column). We observe, not surprisingly, that these numbers increase with the flow of new cases during the week (t) and the previous week (t-1), and decrease with the number of people already in intensive care (or under assistance) at the end of the previous week, as for hospitalizations. Again, the impact of the number of new cases has been decreasing since June. A comparison of the curves of new hospitalizations (admissions - discharges), and changes in the number of people in intensive care, and under respiratory assistance shows that our statistical model fits the data well. The three following figures show the evolution of the actual figures (blue lines) as well as the estimates generated by our model (red lines). The figures clearly show that the model captures the different stages of the epidemic. 3

New hospital cases per week (inflow - outflow) 1 000 2020-02-25 800 2020-03-22 600 2020-04-19 400 2020-05-17 200 2020-06-14 0 2020-07-12 -200 2020-08-09 -400 2020-09-06 -600 2020-10-04 Actual data Estimated values New intensive care units cases per week (inflow - outflow) 200 150 100 50 0 2020-02-25 2020-03-22 -50 2020-04-19 2020-05-17 2020-06-14 2020-07-12 2020-08-09 2020-09-06 2020-10-04 -100 Actual data Estimated values 4

New respiratory assistance cases per week (inflow - outflow) 120 100 80 60 40 20 0 -20 2020-02-25 2020-03-22 -40 2020-04-19 2020-05-17 -60 2020-06-14 2020-07-12 2020-08-09 2020-09-06 2020-10-04 -80 Actual data Estimated values Of course our model may evolve in the coming weeks, and it should be recalculated on a regular basis to identify possible changes in the statistical links. Scenario for the coming weeks As our model gives the evolution of the numbers of deaths and hospitalizations conditional on the trajectory of new cases, we can carry out a scenario analysis for the immediate future. For this we consider the trajectory of new cases presented in the figure below by the dotted curve. It should be emphasized that the figure shows a scenario, not a forecast of the flow of new cases based on a statistical model. The number of new cases during the week ending 25 October amounts to just over 35,000. We consider that it doubles during the following week, which is in line with the specialists' estimates. Thereafter, we take a relatively optimistic scenario where the sanitary measures lead to a stabilization of the number of new cases and then a regular decrease. This is only one scenario among many, and our model easily allows us to estimate the consequences of alternative scenarios. 5

New cases per week 90 000 2020-02-25 80 000 2020-03-22 70 000 2020-04-19 60 000 2020-05-17 50 000 2020-06-14 40 000 2020-07-12 30 000 2020-08-09 20 000 2020-09-06 10 000 2020-10-04 0 2020-11-1 2020-11-29 Once we have the curve of new cases, we can estimate the number of new deaths, as well as the change in the number of people hospitalized each week, and therefore the number of people in hospital at the end of the week. The following figures show the evolution of these variables, the estimated future values in the scenario being indicated by the dotted lines. The outlook is worrying, even though our scenario assumes a fairly rapid decrease in the number of new cases. The situation is expected to deteriorate rapidly and exceed the spring peaks. Estimates indicate an increase in the number of deaths to almost 500 per week (25% higher than at the peak of the first wave). The number of people hospitalized is estimated to be double that of the spring, with a peak of almost 6,000, with a similar evolution in the number of people in intensive care and under respiratory assistance. Admittedly, this is only one scenario, but it is based on a relatively simple (and therefore robust) statistical relationship and a plausible evolution of the number of cases. The scenario indicates that hospital capacities will come under greater pressure than in the first wave, and that it is therefore important to act quickly. 6

500 450 400 350 300 250 200 150 100 50 0 6 000 5 000 4 000 3 000 2 000 1 000 0 2020-02-25 7 2020-02-25 New deaths per week2020-03-22 2020-03-22 Number of persons in hospital2020-04-19 2020-04-19 2020-05-17 2020-05-17 2020-06-14 2020-06-14 2020-07-12 2020-07-12 2020-08-09 2020-08-09 2020-09-06 2020-09-06 2020-10-04 2020-10-04 2020-11-1 2020-11-1 2020-11-29 2020-11-29

2020-02-25 Number of persons in intensive care2020-02-25 2020-03-22 2020-03-22 2020-04-19 800 2020-04-19 2020-05-17 700 2020-05-17 2020-06-14 600 2020-06-14 2020-07-12 500 2020-07-12 2020-08-09 400 2020-08-09 2020-09-06 300 2020-09-06 2020-10-04 200 2020-10-04 100 2020-11-1 2020-11-1 2020-11-29 0 2020-11-29 Number of persons under respiratory assistance 450 400 350 300 250 200 150 100 50 0 8