key: cord-0982931-3fpmatkb
authors: Garbey, M.; Joerger, G.; Furr, S.; Fikfak, V.
title: A Model of Workflow in the Hospital During a Pandemic to Assist Management
date: 2020-05-02
journal: nan
DOI: 10.1101/2020.04.28.20083154
sha: 94328870344e439b57d1172ca6223bbe2586854b
doc_id: 982931
cord_uid: 3fpmatkb

We present a computational model of workflow in the hospital during a pandemic. The objective is to assist management in anticipating the load of each care unit, such as the ICU, or ordering supplies, such as personal protective equipment, but also to retrieve key parameters that measure the performance of the health system facing a new crisis. The model was fitted with good accuracy to France's data set that gives information on hospitalized patients and is provided online by the French government. The goal of this work is both practical in offering hospital management a tool to deal with the present crisis of COVID-19 and offering a conceptual illustration of the benefit of computational science during a pandemic.

care unit (ICU) are or are going to be be severely strained. Healthcare professionals are 5 often times forced to make difficult decisions in patient care and resource allocation. 6 Patient profiles might be out of the ordinary routine of the hospital and workflow must 7 be different. End-to-end on demand visibility with identification of real constraints is 8 needed for the senior management. 9 A manager may have simple but essential questions such as: how many beds do I 10 need on the floor, how many beds are available in the critical care unit, how much 11 supplies should be ordered to take care of our patients and protect our staff from 12 infection, how long will the facility have to work at maximum capacity, is there enough 13 staff to hold this workload long enough, are we doing well with patient outcomes, etc.

people who are going to be symptomatic enough to require hospitalization. This 22 approach has been quickly applied to COVID-19 with success [20, 25] . In the case of the 23 COVID-19 pandemic, it is particularly difficult because large bodies of infected people 24 are asymptomatic. Consequently, the basic reproduction number R 0 factor of 25 COVID-19 is still under active debate. 26 On the hospital workflow side, while there is a large amount of work on this 27 topic [23] , one of the difficulties is to asses the death rate of patients hospitalized at the 28 beginning of the pandemic because the Length of stay (LOS) is rather long and the 29 disease is still not well understood [2, 16, 26] . Every hospital has to adapt to the new 30 crisis as it arrives, so clinical practice may vary greatly from one institution to another. 31 A number of guidelines and great reports have been quickly edited to support the heath 32 community, but it takes time to standardize the healthcare process [3, 5, 13, 24] . 33 Our goal in the paper was to come up with a simple and robust mathematical 34 framework that is easy to use and that supports the management of the patient 35 workflow during a pandemic. Such a model should operate on a relatively limited data 36 set that reports daily on the number of patients admitted for hospitalization, patient acquisition. Much more can be done with the patient electronic records that detail 42 patient comorbidities and chronic conditions, provided that the disease of the pandemic 43 is well understood. 44 We have used a Markov process description of the workflow's graph with probability 45 governing the patient transition from one care unit to another, as well as a simple 46 statistical model of patient LOS at each stage. We will show that with a minimum 47 number of parameters used to fit on the time series listed above for a period of a few 48 weeks, one may start to assemble the information needed to assist the senior 49 management in getting answers and identifying real constraints to reduce speculation or 50 misallocation of resources.

This work is our first iteration to achieve a very ambitious goal: as data becomes 52 available, the quality and level of detail of modeling should keep improving to achieve 53 better results. It is our hope that such an effort, among many others, will once again 54 prove how much digital health can benefit from computational science to improve 55 patient care.

The paper is organized as follows: Section 2 describes our method to construct the 57 model and details the choices we made to work with the data set on hand; Section 3 58 gives the main results and solution to our initial goal in supporting management; 59 Section 4 discusses the benefit and limitation of our method and concludes with further 60 potential development.

Because of the sparsity of data available to construct a predictive model during a 63 pandemic crisis, we are going to use a very simple model that reproduces the workflow 64 of Table 1 . Let's start with a brief description of standard patient workflow -see Figure 65 1 -with respect to disease progression -see Figure 2 .

The patient moves from one care unit to another according to his/her condition.

The first two steps are registration and diagnostics, which in principle should be a 68 relatively quick process. For the patients who stay in the hospital because their health 69 condition justifies a longer stay, they are first put in a ward unit for further assessment 70 and treatment. This step is where a number of medical imaging steps start involving Work Flow Transition Table 1 . Probability of Transition for the Patient in reference to the Workflow of either a chest CT scan in the imaging center or a chest X-ray with a mobile unit.

Meanwhile, significant biological lab work starts to grade the patient's condition more 73 precisely and continues during the patient's stay. These resources, i.e. imaging and lab 74 work, are typically shared by all patients in the hospital and therefore may slow down 75 the process. For simplicity and in the absence of adequate data set for validation, we 76 neglect these constraints. Some of the patients who receive medical attention do well 77 with conservative management only and can be discharged home after a few days. But 78 for others, their health condition may deteriorate and those patients will need to be 79 moved to the IMU for higher level of care and/or to transfer to the ICU for ongoing 80 monitoring and mechanical ventilation. The IMU and ICU require extensive supplies 81 and resources. It is often mentioned that the number of available ventilators is critical 82 to ICU functions. However, it is not the only limiting factor: patients under mechanical 83 ventilation need sedation and might be connected to a number of additional systems to 84 deal with organ failures. Once again for simplicity and because of the lack of input data, 85 our model will not take into account these bottlenecks. There are no technical 86 difficulties required to add those constraints in the mathematical model with our 87 bottom up description of the workflow as in [10, 15] . Additional steps can be recovery 88 for patient being well or unfortunately palliative care when the patient is not responsive 89 to treatment. There are many exceptions and singularities to these standard paths: for 90 example, a patient may go directly from admission to the ICU when their condition is 91 too unstable. In some hospitals, the floor might be shared by patients who are 92 recovering from COVID-19 and palliative care patients.Despite this, we will separate 93 these functional units in our model to clarify the workflow process according to what 94 each patient stage requires in terms of resources and time to deliver adequate care. To 95 summarize, a simple workflow graph is created and the main requirement is to know (i) 96 the probability that a patient goes from one care unit to another and (ii) a statistical 97 estimate of how long the patient should stay in each care unit before moving on. [8, 17] .

Our model follows a Markov process for (i): there is a probability associated with 99 each branch of the graph summarized in Table 1 . With respect to (ii), we use a 100 lognormal distribution that can be reconstructed from the parameters listed in Table 2 101  and Table 3 . This simple framework allows us to construct a generic model that resource allocation, as well as the number of patient outputs, such as the number of 106 patient healed and discharged per day, or the number of death(s). These time series can 107 be fitted to existing data the hospital obtains during a period of a few weeks prior to 108 retrieving the performance parameters of Table 1 . Once the model is calibrated, it can 109 be used to extrapolate the load of each care unit in the next few days and anticipate the 110 need of staff and supplies -see Table 4 . Figure 1 This discrete model is stochastic, so one needs to run many simulations to build a 112 statistical estimate of such quantities. It is appropriate to retrieve the unknown 113 parameters of the model using a form of stochastic optimization method, such as genetic 114 algorithm, since the model workflow process, like the one in the hospital, is discrete, 115 noisy, and nonlinear.

Floor IMU ICU Recovery Palliative Nurse 5 beds 2 beds 2 beds 4 beds 3 beds MD 10 beds 10 beds 6 beds 10 beds 10 beds Table 4 . Number of Staff required at each care unit per beds in reference to the Workflow of Figure 1 Let us describe the data set we are using to construct our model. The French 117 government has kindly decided to release the records of most public hospitals around 118 the country during the COVID19 crisis. From this excel file, we can easily recover the 119 number of patients staying in hospitals, the number of patients in ICU, the number of 120 patients healed and discharged, and the number of patients dying in a medical 121 institution. Those numbers are updated daily and go back to March 18, 2020 [27] . We 122 will extensively use this French Data Set (FDS) to identify the missing parameters of 123 our model.

The number of parameters of our model is relatively large: about one parameter for 125 each branch of the graph minus the number of nodes for (i) and two parameters for the 126 log distribution of (ii) in each care unit. To avoid over-fitting, one should come up with 127 a strategy that lowers the number of unknown based either on literature or hypothesis 128 that can be validated otherwise. We are going to describe thereafter the rationale for 129 our choices to the best of our knowledge and further discuss some of the limitations of 130 our model in Section 4.

First of all, a lognormal distribution of the duration of each step of the process 132 might be justified as follows. Biological process, such as incubation and recovery, are 133 often described as such [18, 19] . First, the patient's condition is indeed dominated by 134 his/her biological time. Second, medical procedures with their associated time lag and 135 delay are also often best described as lognormal processes [10, 22] with a long tail. This 136 is not in contradiction with the fact that patient LOS in the hospital may not ideally be 137 described by a simple exponential distribution or similar. Overall, LOS adds up the 138 time distribution of each step in a Markov process and might be described at the 139 convolution of the probability distribution of each step [14] .

Now let's review the parameters of Table 1 that gives the probability transition from 141 one unit to another, in order to rationalize the construction of our generic model. One 142 can first list the following constraints assuming that all possible paths are exhaustively 143 April 28, 2020 5/18

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The copyright holder for this preprint this version posted May 2, 2020. . https://doi.org/10.1101/2020.04.28.20083154 doi: medRxiv preprint listed in the workflow of Figure 1 , so we have: 144 α 2 + α 3 + α 4 = 1, α 5 + α 6 + α 7 = 1, α 8 + α 9 = 1, α 10 + α 11 = 1, α 12 + α 13 = 1. (1) Overall, the death rate and recovery rate of patients who are staying in the hospital 145 should be within an acceptable limit. Technically, the death rate of hospitalized patients 146 is:

Similarly, the recovery rate of hospitalized patients β h = 1 − β d is:

β d is difficult to asses with a pandemic that just started. As a matter of fact, most 149 infected patients are still in the hospital and their outcome may not be clear. We look 150 thereafter for some lower and upper bounds of β d that limits our search. In France, as of April 17, 2019, the number of deaths in hospitals was 11,842 160 patients and recovered was 35,983 patients [27] . Assuming that the proportion of death 161 versus recovery will be about the same for the patients who are still ill, the death rate of 162 hospitalized patients should be around 25%. Finally, according to [26] , an early estimate 163 of the death rate for hospitalized patients in Wuhan, China based on a case series of 191 164 patients was 54/191 = 28%. 165 We restrict ourselves to the model matching the FDS to a [10%, 40%] death rate 166 interval, that is:

According to several reports including the ICNARC one mentioned above, it is 168 expected that the number of patients dying in ICU is about 50%. [6] provides much 169 further details on the probability of survival of patients with ARDS under mechanical 170 ventilator as a function of the day of the start. It shows that about 25% of the patients 171 in ICU die during the first few days from severe complications. We will introduce an 172 artificial two phases ICU decomposition of the patient stay in the ICU to bypass the 173 limitation of a single lognormal distribution that may not represent an adequate model 174 of LOS in this unit according to [6] clinical studies: a short phase one with mortality 175 driven by α 8 and a longer phase 2 with mortality driven by α 10 .

Consequently, we will assume that: 

There are also few parameters in Table 1 that should have near to no limited effect 178 on statistics when matching our model to FDS. FDS is based on hospitalized patients, 179 so α 1 cannot be recovered from this data set. According to FDS, about 30% of patients 180 who show up at the emergency room (ER) are returning home [27] . We will choose 181 α 1 = 0.3.

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The copyright holder for this preprint this version posted May 2, 2020. . https://doi.org/10.1101/2020.04.28.20083154 doi: medRxiv preprint According to Dr. M. Mueller [29] , 25% of the patients who are not responsive to 183 treatment may leave palliative care alive and are discharged home. This may vary 184 depending on each country or hospital policy. Because patients with COVID-19 in 185 palliative care are still very contagious, we will assume they stay in the hospital until 186 the end. We will choose α 12 = 0., for all our calculations.

To sum up, our model essentially needs the calibration of 6 parameters, namely A = (α 2 , α 3 , α 5 , α 6 , α 8 , α 10 ) under the set of constraints (1), (2), (3).

Let us denote F admission (jd) the number of patients admitted per day jd ∈ 1..N in 189 the hospital who have a positive diagnosis and must stay in the hospital. We will use We find A as the solution of the minimization problem of the weighted norm:

where F s is the mean of a large number of runs of the model. This number of runs is 198 set large enough to let the solution of the optimization problem be independent of it.

As mentioned above, we will use a genetic algorithm to solve that minimization problem. 200 The weight factor (γ 1 , γ 2 , γ 3 ) in (4) can be set equal or unequal to favor the quality of 201 the fitting for one of the variables, such as the number of patients in the ICU that is 202 critical to management. Table 2 and 3 give the time window we used for each transient stage. We construct a 204 lognormal distribution of duration for the patient stay in such a way that about 90% of 205 the patients' stay will be within a coarse approximation [P, Q] listed in these tables.

The choice of the parameters in Table 2 and Table 3 might be easier to come up with. 207 One of the most remarkable features is that patients with COVID-19 who stay in the 208 ICU can be longer than usual [1] . The LOS in palliative care was set according to Dr. 209 M. Mueller's data [29] . We have used extensively [2, 26] , as well as the feedback from 210 clinicians in the field to estimate the interval of variation for the parameters [P, Q] the 211 best we could. We used a fairly large interval since it can be observed that the standard 212 deviation for LOS in each care unit is large as described in this report from the Imperial 213 College London COVID-19 Response Team [30] . One may fine tune the interval value 214 [P, Q] if needed in the fitting process of the model to the data set of time series available. 215 To distinguish those unknown parameters that are important from those who are 216 less significant, we run linear sensitivity analysis for each of our results. This method is 217 used to confirm that the time window parameters of Table 2 and Table 3 have a 218 secondary effect on the quality of the model fitting.

Finally, we derive from our model some predictions on staffing and supplies for the 220 next week or so, as well as the load foreseen for each care unit. The nature of the 221 stochastic simulation automatically gives an uncertainty estimate on these predictions 222 that increases as time grows. To compute supplies such as personal protection kits, we 223 can use some adaptation of the reference of the CDC web site [31] that was constructed 224 for Ebola. Our software can then be used to feed the stock management scheme 225 implemented by CDC for COVID-19 [32] . In this paper, we use a growth estimate of 226 two personal protective equipment (PPE) per shift and per staff member for simplicity. 227

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The copyright holder for this preprint this version posted May 2, 2020. . Table 4 a gross approximation of the number of nurses 228 and staff per bed site in each unit. Those figures are depending on the crisis situation 229 and might differ depending on the country [5] . 230 In order to take into account the fact that staff and supplies are limited and require 231 hard management choices during a pandemic crisis, we tested the model further against 232 the scenario of a shortage on nurses who are essential in intensive care units. To 233 introduce a risk factor due to the shortage of nurses, we have extrapolated from [9] To get a continuous approximation, we assume that the shortage of nurses has a linear 239 effect, and use linear interpolation for shortages from 0% to 40% maximum. This is 240 certainly a gross approximation, but we felt that it was important to bring awareness to 241 those effects with a simulation tool. We will present in the next section our results.

Let us first report on the model fitting with the FDS. We sum up the number of 244 admissions, patients in ICU, number of recoveries and deaths for the whole country of 245 France in order to get a robust data set that averages the noise of the data. We 246 calibrated the model to this largest data set that covers the period 3/18/20 to 4/11/20 247 and found a death rate of about 25%. This result is in agreement with the estimate we 248 did in Section 2, as of April 17, 2019 [27] . Table 1 to 3. All 250 numbers have been scaled by a factor to represent an average hospital size. The 251 sensitivity analysis on the alpha unknown vector A is reported in Figure 3 .

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The copyright holder for this preprint this version posted May 2, 2020. . We observed that the number of patient admissions is not a smooth curve. Typically, 253 Sunday's have less activity with less patients discharged than weekdays. However, the 254 model fitting seems adequate and robust to a small variation of parameters. The logic 255 on the influence of parameters is simple, α 2 being the one who is most important for all 256 output. Each of the six parameters seems to have some significant influence for at least 257 April 28, 2020 9/18 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 2, 2020. In order to compare the results obtained with designated subset of the FDS that 273 corresponds to the hospital in Paris and the hospitals in Alsace, we used the simulation 274 with the exact same set of parameters found for the data set with the whole country.

Alsace has been the busiest cluster at the beginning of the pandemic, followed later on 276 by Paris and Ile de France. The results for Alsace are reported in Figure 6 and Figure 7 . 277 We observed a fairly large difference of the model's prediction on the number of patients 278 under mechanical ventilation. It seems that at the peak of the pandemic in Alsace, the 279 number of patients under mechanical ventilation was less in reality than in the model. 280 One possible factor would be the shortage of available beds in the ICU. On the other 281 hand, the number of deaths did not go higher than significantly expected. A better 282 explanation might be the fact that a fairly large number of patients in critical condition 283 were transferred to hospitals in different parts of the country or neighboring countries: 284 according to local newspaper more than 110 patients from Alsace have been 285 transferred [33] . This seems coherent with our results: the scaling factor for the Alsace 286 data set to get a maximum hospitalization rate of about 50 patients per day is 6; the 287 overshoot on the ICU prediction is about 20 in Figure 6 ; The total maximum overshoot 288 is therefore about 120; considering that the average LOS in ICU (see Table 1 ) is roughly 289 12 days, our model still seems to give an adequate approximation. But unfortunately, 290 we do not have enough information to add this new patient path in the workflow of 291 Figure 1 .

This phenomena is less present in the results for the data set with Paris but are still 293 there -see Figure 8 and Figure 9 . One can indeed refine the parameter fitting to be 294 specific for Alsace and Paris in order to reflect that the clinical decision process in the 295 workflow, i.e parameters of Table 1 to 3, might be sensitive to how much the local 296 system is under stress, but we should then take into account those number of 297 transferred patients that are not negligible.

Next, let us describe the use of our model to assist daily management in the hospital 299 during the pandemic. One key factor is to anticipate the load of each care unit and 300 required resources, either to match the increase in number of patients or to reallocate 301 resources to other patients who have seen their surgery postponed. 302 We choose a hypothetical scenario that might occur if confinement conditions to 303 contain the pandemic are lifted too early. We assume that the hospital has a nominal 304 low flux of patients from week 1 to 7, and a recurrence with a daily 20% increase of new 305 patients coming in occurs in week 8. Figure 10 shows the dynamic of the load of each 306 care unit, in particular the large delay in the number of patients in the ICU that 307 becomes saturated the latest. The black curves are a simulation of the previous week's 308 load (week 7), while red curves are the prediction for the following week (week 8). The 309

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The copyright holder for this preprint this version posted May 2, 2020. . As an illustration of the capability of the model, Figure 10 and Figure 11 provide an 314 estimate of the growth of resources needed to face the new patient wave. A number of 315 decisions should be made in regards to patient care. Figure 12 compares the patient 316 April 28, 2020 11/18

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The copyright holder for this preprint this version posted May 2, 2020. . output with or without shortage of nurses. Those results are speculative since it is 317 difficult to quantify the risk for patients beyond the nice publication results of [9] 318 and [8] . It is our hope that data accumulated during crises such as the present episode 319 of COVID-19 will give the mathematical modeling the base to do this estimate 320 rigorously in future work. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 2, 2020. . mathematical models available, even for the present crisis, see [25] and [26] . It might be 325 difficult to assess the basic reproduction number R 0 factor, which is under active debate. 326 It is probably even more difficult to assess the exact impact of global confinement or 327 targeted confinement on those parameters that characterized the pandemic model. 328 We should however be able to use our model to test if the effect on the most critical 329 resource, such as ICU beds and delay in care, are linearly or nonlinearly related to those 330 parameters.

Let us use the most simplistic ordinary differential equation epidemiology model: The function I(t) is used as the input of our workflow model, and represents the 336 number of patients admitted to the hospital. We test the influence of the transmission 337 rate on the number of ICU beds over a 16-week period. Figure 13 shows that the 338 maximum number of ICU beds required during the epidemic is significantly higher when 339 April 28, 2020 13/18

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The copyright holder for this preprint this version posted May 2, 2020. admission goes up to 50 patients per day. This is a significant load for any hospital 347 system because all patients suffer from the same disease and cannot be triaged using the 348 existing departmental structure. The hospital system needs to recruit resources quickly 349 enough to deliver quality patient care while keeping the staff safe from infection.

There are many ways of developing such a mathematical model. We chose a Markov 351 process that can augment a workflow graph provided by the clinicians and used a simple 352 statistical model for the LOS of the patient at each stage corresponding to a graph node. 353 A number of variations in the model construction are available: for example, changing 354 the probability distribution of LOS for specific stages with a more sophisticated model 355 than lognormal or decomposing the graph nodes into subgraphs of the workflow with 356 more details. In particular, the ICU supports different paths of medical care depending 357 on patient conditions. Because of the sparsity of data on hand, we kept the model as 358 simple as possible and we were able to fit the French Data Set with good accuracy.

Using this approach, we could:

• recover important parameters that are characteristics of the workflow such as the 361 probability for a patient to transition from one unit to another, and important 362 patient outcomes such as healing rate or death rate.

April 28, 2020 14/18 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

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• on a pragmatic side, we use the model to assist the senior manager in answering 364 his/her questions as listed in our introduction: how many beds do I need on the 365 floor, how is this affecting patient outcomes, do we need to transfer patients to a 366 different facility, etc.?

There are a number of limitations to our approach. The smaller the hospital, the less 368 predictable the outcome will be. With time, the characteristics of the population of 369 patients who show up to the ER may change and the pandemic management by the 370 governing organizations would evolve. One can think, for example, that systematic 371 testing would provide early diagnostics and impact the performance of the health 372 system as shown by the statistics of countries who were early adopters of that strategy. 373 Due to the heterogeneity of the patient population and disease patterns that depend 374 heavily on patient characteristics, our next step in improving this model would be to 375 include patients' medical history listed in the electronic medical record.

Above all, any model of workflow especially during a pandemic should be aware of 377 the Human Factor. Staff can get sick or burnout during a pandemic and there should 378 be a number of strategies to compute that risk and enter this into the constraints 379 imposed on the health care system [4, 11, 12, 21] . Further, human behavior and decision 380 process changes under stress: it can be for economical or psychological reasons. The 

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We would like to thank Patrick Doolan for sharing his view with us on management and 385 risk evaluation from his great experience acquired from the energy sector.

Declarations of interest: none 387