Hospitalization dynamics during the first COVID-19 pandemic wave: SIR modelling compared to Belgium, France, Italy, Switzerland and New York City data
HHospitalization dynamics during the first COVID-19 pandemic wave: SIR modellingcompared to Belgium, France, Italy, Switzerland and New York City data
Gregory Kozyreff
Optique Nonlin´eaire Th´eorique, Universit´e libre de Bruxelles (U.L.B.), CP 231, Belgium (Dated: July 6, 2020)Using the classical Susceptible-Infected-Recovered epidemiological model, an analytical formulais derived for the number of beds occupied by Covid-19 patients. The analytical curve is fitted todata in Belgium, France, New York City and Switzerland, with a correlation coefficient exceeding98.8%, suggesting that finer models are unnecessary with such macroscopic data. The fitting isused to extract estimates of the doubling time in the ascending phase of the epidemic, the meanrecovery time and, for those who require medical intervention, the mean hospitalization time. Largevariations can be observed among different outbreaks.
I. INTRODUCTION
As the COVID-19 pandemic continues to develop in various parts of the world, the scientific community at large isproducing a massive effort to gather 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 ofinfection, β , within a given population and the average time of spontaneous recovery, t R . But perhaps even moreimportant from a crisis management point of view is to be able to predict how many beds in hospital are needed andhow long they will be occupied. Some estimates are of course already available [1–6], however they usually rest onthe analysis of rather small cohorts. This paper is an attempt to extract such an information from the data madeavailable in several similar but different contexts: Belgium, France, Italy, Switzerland, and New York City (NYC).It will be shown that an analytical curve derived from the simplest possible SIR compartment model can be madeto fit remarkably well with the data, with a correlation coefficient ranging from 0.988 in New York City to beyond0.997 elsewhere. With such good fits, the obtained curves can safely be extrapolated to forecast bed occupationseveral weeks in advance. This calls into question the necessity to resort to more complex modelling, involving moreparameters than the SIR model, to confront macroscopic data in the absence of more detailed information [7]. Thestudy also shows that a great disparity of epidemiological parameters can exist between different countries, despitetheir similarities. This reflects both the policies put in places to mitigate local epidemics and also, perhaps, underlinesdifferences between health systems.Besides the managerial motivation invoked above, the focus in this study is on the hospitalization dynamics for tworeasons. 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 infectedpeople, as well as when the epidemics started in a given region appear poorly reliable. By contrast, bed occupationnumbers 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 datais lacking, we will both exploit general bed occupancy and the number of patients in Intensive Care Units (ICU). II. MATHEMATICAL MODELLING
The simplest of all epidemiological models is the SIR model, which separates a given population into a set ofsusceptible (S), infected (I), and recovered (R) individuals. These populations evolve in time according tod S d t = − βSI/N, (1)d I d t = βSI/N − I/t R , (2)d R d t = I/t R , (3)where S ( t ) + I ( t ) + R ( t ) = N , the size of the population, β is the infection rate and t R is the spontaneous recoverytime. In the majority of countries where confinement measures have been taken, the growing exponential phase ofthe local outbreak was stopped well before a sizeable fraction of the population was infected. Hence, thanks to public a r X i v : . [ q - b i o . P E ] J u l intervention, I ( t ) , R ( t ) (cid:28) 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,d I d t = ( β − /t R ) I. (4)The effect of confinement and social distancing is to reduce the coefficient β , so that this parameter is a functionof time. For simplicity, we assume that there is a well defined date at which β switches from a large value β to asmaller one, β . This, of course, is an approximation, but it appears acceptable since there has been, in most of thesetting 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 I ( t ) = I × (cid:26) e ct , t < ,e − γt , t > . (5)where c and γ are given by c = β − /t R , γ = 1 /t R − β (6)and are, respectively, the initial growth rate and the late-time decay rate. Equivalent to c , and more convenient todiscuss, is the doubling time t d = ln(2) /c during the initial phase of the local outbreak.In Eq. (5), I is the value of I ( t ) at t = 0, a number difficult to determine with accuracy. Note that all we can learnfrom the data of I ( t ) is c and γ , which is not quite enough to know β , β and t R . Hopefully, β is close to zero, butin all probability it isn’t. Hence, t R < γ − . However, one may hypothesize that the populations involved with thepandemic in Belgium, France, Switzerland, Italy and New York City all have similar response to the virus, so thatthey share the same value t R . Hence, the smallest of the values of γ − extracted from these five epidemic events maycount 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 theequation d P ( t )d t = αγI ( t − τ ) − P ( t ) /t H , (7)which expresses, simply, that the number of hospitalisations increases at a rate proportional to the number of infectedpeople and that, once admitted into hospital, the mean time of stay is t H . Above, α is the probability, if infected, tobe hospitalised. In this last Eq., I ( t ) appears with a delay τ . This delay accounts, for the most part, for the averagetime elapsed between being infected and requiring to be hospitalized; additionally, one may conjecture that socialevent 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 P ( t ) = p e c ( t − τ ) (cid:16) − e − ( ct H +1)( t − t ) /t H (cid:17) , t < τ, (8)= p (cid:20)(cid:16) − e − ( ct H +1)( τ − t ) /t H (cid:17) e − ( t − τ ) /t H + ct H + 1 γt H − (cid:16) e − ( t − τ ) /t H − e − γ ( t − τ ) (cid:17)(cid:21) , t > τ, (9)where t is the time of the first hospital admission and where p = αI e cτ γt H / (1 + ct H ). Finding p , it would beparticularly interesting to deduce α . Unfortunately, this requires the knowledge of I , 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 CareUnit (ICU), P ICU ( t ). The simplest way is to writed P ICU ( t )d t = α ICU γI ( t − τ (cid:48) ) − P ICU ( t ) /t ICU . (10)The above equation neglects intermediate stages between being infected and integrating the ICU. Accordingly, theevolution 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 (cid:48) , 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 set P ( t ) rather than I ( t ) as in [3]. In the same vein, we have not separated the population in age categories, even thoughthis would be highly relevant [9]. However, the attitude in the present paper is to invoke the simplest possible modelin order to exploit simple explicit formulas like Eqs. (8) and (9). As we will see, this yields excellent fit to the data. III. DATA AND FIT
Hospitalization data was gathered for1. Belgium, during the period extending from 2020-03-15 to 2020-05-28 [10]. Official lockdown was imposed on2020-03-18. The first patient was hospitalized on 2020-02-04.2. France, during the period extending from 2020-03-18 to 2020-05-28 [11]. Official lockdown was imposed on2020-03-17. The first patient was hospitalized on 2020-01-24.3. Italy, during the period extending from 2020-02-24 to 2020-05-28 [12]. 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 [13]. Containment measures were takenas 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 [14]. Stay-at-home order was enforcedon 2020-03-22. Estimation of the first hospitalization is 2020-02-29.The data sets were analyzed with Mathematica 8.0 using the functions ‘FindFit’ and ‘NonlinearModelFit’. Bothcommands allow to obtain parameter set by least-square regression and the latter yields 95% confidence intervals(CI). One notes that the parameters γ − and t H (or t ICU ) appear in separate but very similar mathematical terms inEq. (9), making these two parameters strongly correlated and rendering their determination ambiguous. For example,with the french data, the pairs ( γ − , t H ) ≈ (12 ,
42) and ( γ − , t H ) ≈ (42 ,
12) can be made to yield almost equallygood fit. In order to remove the ambiguity and narrow down 95% CI, one should fix one of these two parameters.To this end a first round of parameter fitting was carried out for each geographical region in which t d , the doublingtime in the initial phase, was varied between 3 and 7. It turned out that there was little ambiguity with Belgian data,where both γ − and t H were close to 16 days. For Italy γ − was found to lie between 15 and 21 days, whereas NYCand Switzerland gave lower values, around 9 days. From this initial investigation, one makes the following informedguesses: γ − (Belgium) = 16, γ − (Italy) = 20, γ − (NYC, Switzerland) = 9. Finally, for France, one assumes the samevalue as in Belgium: γ − (France) = 16. What is output as 95% IC in the following should of course be regarded, atbest, as conditional probabilities. They are mere indicators of uncertainty.Curve fitting was done separately with ICU data.Given the set of data points ( t i , y i ) with mean value ¯ y for a given outbreak, the correlation coefficient was computedas C = (cid:80) i ( P ( t i ) − ¯ y ) ( y i − ¯ y ) (cid:113)(cid:80) i ( P ( t i ) − ¯ y ) (cid:113)(cid:80) i ( y i − ¯ y ) . (11) IV. RESULTS AND DISCUSSION
The comparisons of the analytical curves with the official data is shown in Fig. 1. In all cases, a close fit is obtainedwith the analytical formula, with C almost equal to 1. A close inspection of the curves shows that the growth phaseis not purely exponential, meaning that β is not simply a constant β during that phase. This was anticipated. Theranges of values for the various parameters are summarized in Table I. Notable similarities, but also differences, canbe 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 with doubling time under 4 days;this is consistent with their high population densities. t d was between 4 and 5 days in France and Switzerland, andbetween 5 and 6 days in Italy.Italy, France, and Belgium imposed very similar lockdown measures with very similar restrictions. Their rangesof values for τ are also similar in their lower bound. However, one clearly sees that the response to the lockdownmeasures was delayed by a longer time in Italy and in France than in Belgium. The larger value of τ in Italy thanin France may be due to the fact that lockdown was imposed in two steps: on the 8th of March in the northernregion and the next day to the rest of the country. In the case of France, one recalls that the first round of localelections took place two days before the lockdown all over the country. Despite precautions, this may have given alast-minute boost to the outbreak, accounting for the larger values of τ . Ranges of values obtained with ICU data aresystematically shifted towards longer times, indicating an extra delay between the development of severe symptomsand the further degradation of health condition necessitating ICU care.
10 20 30 40 50 60 7010002000300040005000 number of days since lockdown nu m b e r o f o cc up i e db e d s τ (cid:0)(cid:0)(cid:0)(cid:9) Belgium
10 20 30 40 50 60 7020040060080010001200 number of days since lockdown nu m b e r o f o cc up i e db e d s i n I C U τ (cid:0)(cid:0)(cid:0)(cid:9) Belgium
20 40 60 8050001000015000200002500030000 number of days since lockdown nu m b e r o f o cc up i e db e d s τ (cid:0)(cid:0)(cid:0)(cid:9) France
20 40 60 801000200030004000500060007000 number of days since lockdown nu m b e r o f o cc up i e db e d s i n I C U τ (cid:0)(cid:0)(cid:0)(cid:9) France
20 40 60 8050001000015000200002500030000 number of days since lockdown nu m b e r o f o cc up i e db e d s τ (cid:0)(cid:0)(cid:0)(cid:9) Italy
20 40 60 801000200030004000 number of days since lockdown nu m b e r o f o cc up i e db e d s i n I C U τ (cid:0)(cid:0)(cid:0)(cid:9) Italy -
20 20 40 60 8010002000300040005000 number of days since lockdown nu m b e r o f o cc up i e db e d s τ (cid:0)(cid:0)(cid:0)(cid:9) Switzerland -
20 20 40 60 80200400600800 number of days since lockdown nu m b e r o f o cc up i e db e d s i n I C U τ (cid:0)(cid:0)(cid:0)(cid:9) Switzerland -
20 20 40 6050010001500 number of days since lockdown nu m b e r o f o cc up i e db e d s τ (cid:0)(cid:0)(cid:0)(cid:9) NYC
FIG. 1. Number of occupied beds as a function of time in general hospitalization and in ICU. Dots: data points communicatedby official agencies. Thick curves: 95% confidence band using the analytical model. Gray lines indicate the value of τ . TABLE I. Fitting values (95%CI) of doubling times, effective lock-down times, hospitalization times and coefficient p . Thecharacteristic time γ − of exponential decrease of the infected was set to a fixed value for each country or city. Times areexpressed in days. t d p, p ICU τ, τ (cid:48) t H , t ICU γ − Belgium 3.5 - 4.1 2920 - 3210 7.5 - 8.3 16.1 - 17.0 16Belgium (ICU) 3.5 - 4.1 671 - 739 8.0 - 8.8 14.8 - 15.7 16France 4.0 - 4.5 13970 - 15250 9.1 - 9.8 The Belgian CI for τ suggests a time from exposition to severe symptoms of 7 . ± . et. al report a time from onset of symptoms to hospital of 4 (2-7) days [4], while anincubation time of 5.1 days (4.5-5.8) has been reported [15, 16].In all cases where ICU data were available, t ICU is significantly less than t H : by approximately 1 day in Belgium,20 days in France, 8 days in Italy and 5 days in Switzerland. Combining Belgium, Italy and Switzerland, t H lies inthe range 16-22 days. The study in Shanghai (China) reports a similar number for discharge time: 16 days (12-20) [4].With an hospitalization time of 35 ± . t H (NYC) = 12 . ±
1. This is much less than in the other data sets. However, Richardson et al. conducted a study of 5700 patients hospitalized in NYC area and found even lower values: the overall length of staywas only 4.1 days (2.3-6.8) [6]. Possible causes of discrepancy are (i) over-simplification of the present approach or(ii) the limited interval of the study by Richardson et al. , between March 1 and April 4, 2020, shortly before the peakof the outbreak.Italy, Switzerland, and Belgium display similar figures for t ICU : slightly more than 16 days in Italy, slightly less inBelgium, and between 13 and 17.25 for Switzerland. In France, again, t ICU appears to be significantly longer.
V. CONCLUSION
In this paper, we have shown how the simplest of all epidemiological models suffices to match macroscopic data withalmost perfection. Having not let the pandemic evolve freely, political decision makers have curbed the outbreaks ina way that can be modelled by simple analytical formulas. These provide mathematical models for bed occupationnumbers as a function of time that can be fitted very closely to the data supplied by health agencies. The fittingprocedure yields estimates of some important epidemiological parameters of COVID-19. Assuming values of the decayrate γ of the outbreak, one derives estimates of the time from contamination to hospital, the time in hospital, and thetime in ICU. Numbers obtained are consistent with previously published values. Ranges of confidence are given, butthey are conditioned by the value γ . Still, interesting trends are observed, notably the much shorter hospitalizationtime inferred for NYC compared to the other geographical areas. Overall, a great disparity of values is observeddepending on geographical location. Local circumstances, in the form of numbers of available beds, massive publicgathering, peculiarities in the lockdown measure, and also public awareness, certainly have impacted the parametersof the local outbreaks. This should be taken into account in epidemiological models. If only macroscopic data, suchas those analyzed here, are available, it appears unnecessary to resort to more complicated models than SIR or closevariants thereof. Acknowledgement
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