key: cord-0826348-fpf5n0kn authors: Kozyreff, Gregory title: Hospitalization dynamics during the first COVID-19 pandemic wave: SIR modelling compared to Belgium, France, Italy, Switzerland and New York City data date: 2021-02-03 journal: Infect Dis Model DOI: 10.1016/j.idm.2021.01.006 sha: 04da4755a188aee7af7e5e75e27fc78a34524d5e doc_id: 826348 cord_uid: fpf5n0kn Using the classical Susceptible-Infected-Recovered epidemiological model, an analytical formula is derived for the number of beds occupied by Covid-19 patients. The analytical curve is fitted to data in Belgium, France, New York City and Switzerland, with a correlation coefficient exceeding 98.8%, suggesting that finer models are unnecessary with such macroscopic data. The fitting is used to extract estimates of the doubling time in the ascending phase of the epidemic, the mean recovery time and, for those who require medical intervention, the mean hospitalization time. Large variations can be observed among different outbreaks. as many information as possible on the nature of virus and its ability to spread. At the epidemiological level, some parameters are of course particularly desirable to ascertain such as the rate of infection, β, within a given population and the average time of spontaneous recovery, t R . But perhaps even more important from a crisis management point of view is to be able to predict how many beds in hospital are needed and how long they will be occupied. Some estimates are of course already available [22, 12, 6, 5, 21, 16] , however they usually rest on the analysis of rather small cohorts. This paper is an attempt to extract such an information from the data made available in several similar but different contexts: Belgium, France, Italy, Switzerland, and New York City (NYC). Note that NYC contains a well-delimited population whose size is comparable to those of Switzerland or Belgium and, hence, can be treated on an equal footing as a small country from an epidemiological point of view. It will be shown that an analytical curve derived from the simplest possible SIR compartment model can be made to fit remarkably well with the data, with a correlation coefficient ranging from 0.988 in New York City to beyond 0.997 elsewhere. With such good fits, the obtained curves can safely be extrapolated to forecast bed occupation several weeks in advance. This calls into question the necessity to resort to more complex modelling, involving more parameters than the SIR model, to confront macroscopic data in the absence of more detailed information [10, 17, 9] . Note that SIR modelling has been applied to several outbreaks of COVID-19 in various parts of the world [4, 1, 20, 2] . However, at the time of their publication, the local outbreak was still in the ascending phase and the wave was therefore 2 J o u r n a l P r e -p r o o f not complete, which limited the accuracy of data fitting. The study also shows that a great disparity of epidemiological parameters can exist between different countries, despite their similarities. This reflects both the policies put in places to mitigate local epidemics and also, perhaps, underlines differences between health systems. Besides the managerial motivation invoked above, the focus in this study is on the hospitalization dynamics for two reasons. Firstly, since the beginning of the pandemic, a large uncertainty has been surrounding the number of cases, as the ability and protocols to test patients varies from one country to the next. Estimates of the number of infected people, as well as when the epidemics started in a given region appear poorly reliable. By contrast, bed occupation numbers are easier to monitor. Next, the processes leading to the number of occupied beds are multiple and additive, leading to a much smoother data than, for instance, the daily numbers of admitted and discharged patients. Hence, curve fitting is expected to yield more reliable information with hospitalization data. Except for NYC, where the data is lacking, we will both exploit general bed occupancy and the number of patients in Intensive Care Units (ICU). The simplest of all epidemiological models is the SIR model, which separates a given population into a set of susceptible (S), infected (I), and recovered (R) 3 J o u r n a l P r e -p r o o f individuals. These populations evolve in time according to where S(t) + I(t) + R(t) = N , the size of the population, β is the infection rate and t R is the spontaneous recovery time. In the majority of countries where confinement measures have been taken, the growing exponential phase of the local outbreak was stopped well before a sizeable fraction of the population was infected. Hence, thanks to public intervention, I(t), R(t) N at all time, even if they could reach considerable values. Therefore, one has S(t) ≈ N and the equation for I(t) becomes, with good approximation, The effect of confinement and social distancing is to reduce the coefficient β, so that this parameter is a function of time. For simplicity, we assume that there is a well defined date at which β switches from a large value β 0 to a smaller one, β 1 . This, of course, is an approximation, but it appears acceptable since there has been, in most of the setting considered in this study, a well defined date where the local authority has declared some form of lockdown [8] . Taking, for each outbreak, t = 0 as the time when lockdown started, we thus have and are, respectively, the initial growth rate and the late-time decay rate. Equivalent to c, and more convenient to discuss, is the doubling time t d = ln(2)/c during the initial phase of the local outbreak. In Eq. (5), I 0 is the value of I(t) at t = 0, a number difficult to determine with accuracy. Note that all we can learn from the data of I(t) is c and Γ, which is not quite enough to know β 0 , β 1 and t R . Hopefully, β 1 is close to zero, but in all probability it isn't. Hence, t R < Γ −1 . However, one may hypothesize that the populations involved with the pandemic in Belgium, France, Switzerland, Italy and New York City all have similar response to the virus, so that they share the same value t R . Hence, the smallest of the values of Γ −1 extracted from these five epidemic events may count as the best estimation of the upper bound on t R . Knowing I(t), the evolution of the number of hospitalized patients P (t) is straightforward to model. It obeys the equation which expresses, simply, that the number of hospitalisations increases at a rate proportional to the number of infected people and that, once admitted into hospital, the mean time of stay is t H . Above, α is the probability, if infected, to be hospitalised. In this last Eq., I(t) appears with a delay τ . This delay accounts, for the most part, for the average time elapsed between being infected 5 J o u r n a l P r e -p r o o f and requiring to be hospitalized; additionally, one may conjecture that social event such as mass gatherings may have further delayed the response to the measures, leading to a larger value of τ . Combining Eqs. (5) and (7), one derives where t 0 is the time of the first hospital admission and where p = αI 0 e cτ Γt H /(1+ ct H ). Finding p, it would be particularly interesting to deduce α. Unfortunately, this requires the knowledge of I 0 , which we don't have. In the same way as for P (t), one may derive an evolution model for the number of occupied beds in Intensive Care Unit (ICU), P ICU (t). The simplest way is to write The above equation neglects intermediate stages between being infected and integrating the ICU. Accordingly, the evolution of P ICU (t) is given by the same expressions as in Eqs. (8) and (9) but with the substitutions t H → t ICU , τ → τ , and p → p ICU . One may argue that Eqs. (5) to (10) are oversimplified in that the model neglects an intermediate population E(t) of exposed, not-yet contagious individuals, and that the population P ICU should rather be coupled to the larger 6 J o u r n a l P r e -p r o o f set P (t) rather than I(t) as in [6] . In the same vein, we have not separated the population in age categories, even though this would be highly relevant [14] . However, the attitude in the present paper is to invoke the simplest possible model in order to exploit simple explicit formulas like Eqs. (8) and (9). As we will see, this yields excellent fit to the data. Hospitalization data was gathered for 1. Belgium, during the period extending from 2020-03-15 to 2020-05-28 [19] . Official lockdown was imposed on 2020-03-18. The first patient was hospitalized on 2020-02-04. 2. France, during the period extending from 2020-03-18 to 2020-05-28 [18] . Official lockdown was imposed on 2020-03-17. The first patient was hospitalized on 2020-01-24. 3. Italy, during the period extending from 2020-02-24 to 2020-05-28 [7] . Official lockdown was imposed on 2020-03-9. Estimation of the first hospitalization is 2020-02-07. 4 . Switzerland, during the period extending from 2020-02-25 to 2020-05-28 [15] . Containment measures were taken as of 2020-03-20. First hospitalization was on 2020-02-26. 5. New York City, during the period extending from 2020-02-29 to 2020-05-23 [13] . Stay-at-home order was enforced on 2020-03-22. Estimation of the first hospitalization is 2020-02-29. Curve fitting was done separately with ICU data. Given the set of data points (t i , y i ) for a given outbreak, with mean valueȳ, the correlation coefficient was computed as The comparisons of the analytical curves with the official data is shown in Figs. 1 and 2 . In all cases, a close fit is obtained with the analytical formula, with C almost equal to 1. A close inspection of the curves shows that the growth phase is not purely exponential, meaning that β is not simply a constant β 0 during that phase. This was anticipated. The ranges of values for the various parameters are summarized in Table 1 . Notable similarities, but also differences, can be seen from one country/city to another. Below, we highlight some of them. Belgium and NYC have been exposed to the most rapidly growing outbreaks been reported [11, 3] . In all cases where ICU data were available, t ICU is significantly less than t H : Italy, Switzerland, and Belgium display similar figures for t ICU : slightly more than 16 days in Italy, slightly less in Belgium, and between 13 and 17.25 for Switzerland. In France, again, t ICU appears to be significantly longer. In this paper, we have shown how the simplest of all epidemiological models suffices to match macroscopic data with almost perfection. Having not let the pandemic evolve freely, political decision makers have curbed the outbreaks in a way that can be modelled by simple analytical formulas. These provide mathematical models for bed occupation numbers as a function of time that can be fitted very closely to the data supplied by health agencies. The fitting procedure yields estimates of some important epidemiological parameters of COVID-19. ological models (see, e.g. [6] ). 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