Age-structured estimation of COVID-19 ICU demand from low quality data
AAge-structured estimation of COVID-19 ICU demand fromlow quality data
R. V
EIGA , R. M URTA and R. V ICENTE
Instituto de F´ısica - Universidade de S˜ao Paulo, 05508-090, S˜ao Paulo-SP, Brazil Looqbox - 04547-130, So Paulo-SP, Brazil Experian DataLab LatAm - 04547-130, S˜ao Paulo-SP, Brazil Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo - 05508-090, S˜ao Paulo-SP, Brazil
PACS – Diseases
PACS – Dynamics of social systems
Abstract –We sample aggravated cases following age-structured probabilities from con-firmed cases and use ICU occupation data to find a subnotification factor. A logistic fit isthen employed to project the progression of the COVID-19 epidemic with plateau scenar-ios taken from locations that have reached this stage. Finally, the logistic curve found iscorrected by the subnotification factor and sampled to project the future demand for ICUbeds.
Introduction. –
The COVID-19 pandemic is rav-aging the world and requiring every research energyavailable to help local public administrators dealingwith the crisis. Brazil, unfortunately, is an emblematiccase of a public health emergency mismanagement.Despite many voluntary initiatives [1–9], the countrylacks publicly available data that is complete, consis-tent and timely to monitor the pace of the epidemic.A main concern in many locations like Brazil is howto use incomplete data of low quality to anticipate thedemand for crucial and limited Intensive Care Units(ICU).Compartment based epidemiological models [10,11], like SIR, SEIR or many other more realistic vari-ants, require the estimation of a number of param-eters. Any future scenarios derived from the equa-tions defining these models are critically dependenton these parameters that, in their turn, depend on thequality of the data available. Having data of admit-tedly low quality, makes the task of fitting realisticmodels questionable, at best.The situation is further complicated by Sars-CoV-2being a new virus with uncertain epidemiological pa-rameters and by the complexities of severely unequalsocieties. To overtake these limitations, here we usethe fact that the first wave of epidemics has alreadybeen resolved in several locations to inform the propo-sition of sensible and simple scenarios based only onvery generic, yet robust, dynamical features. We thenuse clinical data from those same locations, data on ICU utilization and demographic data to estimate ICUdemand by Monte Carlo simulation [12].
Building scenarios. –
We start by considering theevolution in time of the number of confirmed cases. Avery general feature, captured by compartment mod-els, is that there is an initial exponential growth fol-lowed by a plateau, eventually reached when thenumber of susceptible declines. The simplest struc-ture like that is provided by a logistic function. Wethus model the evolution of confirmed cases ˆ n ( t ) for t > n ( t ) = ˆ n ∗ + e − α ( t − t ) , (1)where α is the rate of the early exponential growth andˆ n ∗ is the number of cases attained when the epidemichits the plateau for α ( t − t ) (cid:29)
1. The time shift t marks the inflection point n ( t ) = ˆ n ∗ /2.Figure 1(a) represents Brazilian official records forconfirmed cases and deaths as they were presented atMay 17, 2020. It can be verified that, at this date, theepidemic hadn’t plateaued yet.We focus our analysis at the state of S˜ao Paulo, themost populous state ( ≈
21% of Brazilian population)and also the state housing the megalopolis that is theepidemic epicenter: the city of S˜ao Paulo ( ≈
27% ofS˜ao Paulo state’s population). Figure 1(b) depicts con-firmed cases for the state and the city of S˜ao Paulo un-til the same May 17, 2020. We do that because for S˜aoPaulo we have daily ICU occupation reports.p-1 a r X i v : . [ phy s i c s . s o c - ph ] J un . Veiga et al. Date Confirmed casesConfirmed deaths (a) Confirmed COVID-19 cases and deaths in Brazil [3].
Date São Paulo StateSão Paulo City (b) Confirmed COVID-19 cases in S˜ao Paulo [3].
Fig. 1: Epidemic dynamics: confirmed cases.
In order to find plausible values for ˆ n ∗ we look atthe epidemic dynamics across all countries affectedand calculate relative contagion velocities β ( t ) / ˆ n ( t ) ,with β ( t ) = a ∆ k ˆ n ( t ) + ( − a ) β ( t − ) , (2)for k-lagged first differences ∆ k ˆ n ( t ) = ˆ n ( t + k ) − ˆ n ( k ) k , (3)where we have used a = k =
5. We thenadopt to neglect countries where the relative conta-gion speed is greater than 0.5 % as a criteria to iden-tify locations that have reached an epidemic plateau.We further restrict our search to countries with totalpopulation from five to tens of million inhabitants .For each country L that have already reached the epi-demic plateau we considered the number of cases perinhabitant as providing a different scenario, to say,ˆ n ∗ L = ˆ n L ( t c ) N L N , (4)where t c time when β ≤ β c , N and N L are, respec-tively, the populations we want to model and the pop-ulation of the country that provides the scenario.With this simple approach we have selectedSwitzerland to provide an “optimistic” scenario andSpain to provide a “pessimistic” scenario. Table 1 liststhe number of cases expected in the plateau for theeach scenario. L ˆ n L ( t c ) N L Spain 229,047 46,795,540Switzerland 163,071 8,513,227
Table 1: Scenarios for ˆ n ∗ . Total population provided by [14]. We estimate α and t by linear regression oflog ( ˆ n ∗ / ˆ n ( t ) − ) , discarding the first 45 days since Code available at https://github.com/rodsveiga/ICU demand case one. The state of S˜ao Paulo has a populationof N =
46, 289, 333 [15], while the city has N =
12, 252, 023 inhabitants [16] . The expected epidemicdevelopment for the two scenarios can be viewed inFigure 2.Figure 2(b) makes explicit the grave situation of thecity of S˜ao Paulo. Subnotification of cases and noti-fication delays are not taken into account. Followingcurrent trends, we expect that both scenarios will soonbecome obsolete.Studies report that COVID-19 agravation are age-dependent [17–19], making the age-pyramid centralto the task of estimating demand for ICU beds. Wesuppose that confirmed cases, both for the state andfor the city of S˜ao Paulo, follow the age-pyramid [15]as shown in Figure 2(c).After age-structured sampling of cases, we sampleover age-dependent probabilities of ICU admissionfollowing. For that we use data by age group reportedfor United States from February 12 to March 16 [20](see Figure 2(d)). These data might be preliminary,however ICU admission probability for individualsunder 60 is clearly non-negligible.To build our estimates we assume that a individualremains in ICU for τ =
14 days before being removed.Given an specific day t j , the total ICU beds demand onthis day is given byˆ N ICU ( t j ) = ˆ n ICU t j + ˆ n ICU t j − + · · · + ˆ n ICU t j − τ , (5)where ˆ n ICU t j is the Monte Carlo estimation obtainedfrom sampling ˆ n ( t j ) , Eq.(1), from age-structure prob-abilities, followed by sampling from ICU admissionrates.We also introduce a multiplicative constant S ,which accounts for subnotification of cases. This con-stant is found by fitting public ICU occupation datato the median of the proposed scenarios and it isassumed to hold for all simulations throughout thiswork. Unfortunately, ICU bed occupation data is alsonot widely available in Brazil [21].p-2stimation of COVID-19 ICU demand Date T o t a l C a s e s ×10 SpainSwitzerlandData (a) Expected epidemic development. State of S˜ao Paulo. Pa-rameters: α Spain = t =
95 days; α Swiss = t =
87 days.
Date T o t a l C a s e s ×10 SpainSwitzerlandData (b) Expected epidemic development. City of S˜ao Paulo. Pa-rameters: α Spain = t =
68 days; α Swiss = t =
60 days. -
04 05 -
09 10 -
14 15 -
19 20 -
24 25 -
29 30 -
34 35 -
39 40 -
44 45 -
49 50 -
54 55 -
59 60 -
64 65 -
69 70 -
74 75 -
79 80 -
84 85 -
89 90 + Age interval P o p u l a t i o n f r a c t i o n ( % ) (c) Age-structured population fraction in S˜ao Paulo State [15]. -
04 05 -
09 10 -
14 15 -
19 20 -
24 25 -
29 30 -
34 35 -
39 40 -
44 45 -
49 50 -
54 55 -
59 60 -
64 65 -
69 70 -
74 75 -
79 80 -
84 85 -
89 90 + Age interval I C U a d m i ss i o n ( % ) (d) ICU admission by age group in United States from Febru-ary 12 to March 16. Fig. 2: Data (until May 17, 2020) and model projections for both scenarios from Table 1 for the state of S˜ao Paulo and thecity of S˜ao Paulo [3]. Age-pyramid and age-dependent ICU admission.
Results for the State of S˜ao Paulo. –
ICU occupa-tion for the state of S˜ao Paulo is reported occasionallyby the government on social networks [22–31]. Thesescarce data points are represented by the blue circlesin Figure 3(a) together with both scenarios. Modelsare fitted to data up to May 17, 2020 and results untilthis date are depicted in gray.Public data indicates that the state of S˜ao Paulo has5934 COVID-19 ICU beds (availability on 05/22; cal-culated from [32]). We can thus find the time intervalfor system collapse, that is shown in Table 2. The tablealso shows demand peaks for each scenario and thesubnotification factor S .Curves like the ones in Figure 3(a) could be con-structed considering only the demand in the publichealth system (SUS), since the SUS dependent popu-lation is known for each and every Brazilian state [33].However, we lack reliable data to estimate the subno-tification factor S and the number of SUS ICU bedsavailable for COVID-19 is not clear.At the time of writing, we were able to extract ICUbed occupation from social networks for eight days after May 17 [32, 34–40]. In Table 3 we use these datapoints to verify the quality of our predictions. We ob-serve that data points are mostly compatible with theintervals suggested by the scenarios. Results for the City of S˜ao Paulo. –
Our samplingestimation method is quite simple and can be appliedto any population once ICU occupation data is avail-able.The city of S˜ao Paulo is country’s pandemic epicen-ter. Validation will be possible only when reliable datais available. Unfortunately only occupation percent-ages are reported, but it is not clear how many ICUbeds in total are available. Thus we are unable to findthe subnotification factor S . Taking S = et al. Collapse (68% CI) Collapse (95% CI) Max date Max value (68% CI) Max value (95% CI) S Spain 05/21 to 05/31 05/18 to 07/06 06/07 7154 ±
987 7154 ± ±
937 6180 ± ±
861 5389 ± Date N u m b e r o f I C U b e d s DataSpainSwitzerlandCapacity (a) State of S˜ao Paulo. Shaded areas denote Monte Carlo errorbars.Detailed figures are depicted in table 2.
Date N u m b e r o f I C U b e d s SpainSwitzerlandCapacity (b) City of S˜ao Paulo. Shaded areas denote Monte Carlo errorbars.Detailed figures are depicted in table 4.
Fig. 3: Scenarios for the state and city of S˜ao Paulo.68% CISpain Median Switzerland Data05/19 4976 4801 4627 3659 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± seem to indicate that the city of S˜ao Paulo would havealready reached epidemic peak (see Table 4). How-ever, data on daily new cases seems to indicate other-wise, pointing towards a situation worsening beyondthe worst scenario employed. Concluding remarks . –
The number of confirmedcases is modelled by a logistic function, describing aninitial exponential increase followed by a plateau. The
Max date 68% CI 95% CISpain 05/18 2166 ±
504 2166 ± ±
474 2033 ± ±
431 2023 ± exponential increase rate is estimated from availabledata. Epidemic plateau is estimated from scenariosbased on the dynamics observed on other countries.We employ limited ICU occupation data to estimate asubnotification factor and then use age-structured es-timates to project scenarios for the progression of ICUdemand.Information is critical to deal with a sanitary crisisas we are facing. It should be government responsi-bility to use its resources to protect taxpaying citizens.Fortunately, we still have science as a guide. Acknowledgement. –
R. Veiga is financially sup-ported by CNPq under process 162857/2017. Thiswork is also supported by the Covid Radar initiative.
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