Since its emergence in late 2019, the virus responsible for coronavirus disease 2019 45 (COVID-19) has spread rapidly, causing significant global morbidity and mortality [1, 2] . 46 While COVID-19 has considerable individual health impacts, significant attention has 47 been paid to the strain put on health systems resulting from high rates of hospitalization, 48 critical care, and ventilation among COVID-19 infections [3, 4] . Consequently, a large 49 focus of decision support tools, including epidemiological models, has been to project 50 the impact of COVID-19 on acute care hospitalizations and ICU admissions under 51 different non-pharmaceutical interventions to help guide public health action. 52 Epidemiological models have played a signficant role in shaping the public health 53 response globally. A wide variety of methods have been deployed to project the course 54 of the COVID-19 outbreak, including agent-based models, population-level models with 55 and without age structure, and curve fitting approaches [5] [6] [7] [8] [9] . Despite the variation in 56 methodology, they each estimate the same health outcomes: infections, hospitalizations, 57 ICU hospitalizations, ventilators needed, and deaths. These models played a critical role 58 in helping policy makers design and implement effective interventions to avoid the 59 universal projection that without interventions, the surge of COVID-19 patients at the 60 peak of the outbreak would overwhelm healthcare facilities. While these health 61 outcomes provide necessary projections to address the surge of patients expected during 62 local epidemic peaks, none of these approaches consider what could be described as the 63 "aftershocks" of the surge. That is, a secondary surge due to patients who were 64 discharged from an acute care facility, but still require continued support.
COVID-19 patients may still be infectious, they further increase the already high risk 77 for outbreak at these facilities and, subsequently, the need for even more patients 78 requiring acute hospital care. The staggering number of excess deaths attributed to 79 COVID-19 [13, 14] originating from nursing facilities is a testament to the critical need 80 for surge planning modeling that integrates acute hospital and post-acute care. 81 We consider the downstream effects on post-acute care services of COVID-19 82 patients who previously required hopitalization, ICU care, and/or mechanical 83 ventilation. In this paper, we present a method for projecting post-acute care capacity 84 needs that takes in a time series of discharge estimates from any of the numerous 85 epidemiological models of COVID-19 and extends the results from these models to 86 estimate the flow of patients to three post-acute care services (and direct-to-home), thus 87 providing the same type of surge planning model for post-acute care services that the 88 multitude of high-profile epidemiological models provide for acute care planning. To project patient movement to each of the four post-acute care services, we use the 98 following statistical model, which uses a multinomial distribution to estimate patient 99 flow. We estimate the discharge location for n j patients discharged from ward j, where 100 j is either patients in the hospital for COVID-19 (all wards) or patients in the ICU: 101 (x j,none , x j,hh , x j,snf , x j,hos ) ∼ M ultinomial(n j , p j,none , p j,hh , p j,snf , p j,hos ) (1) where p j,k and x j,k denotes the probability and number, respectively, of patients 102 discharged from ward j to post-acute outcome k (where k ∈ {none, hh, snf, hosp} for 103 our model). Thus, we have that k x j,k = n j . In addition, the set of probability 104 parameters must sum to one, so for our model p j,none + p j,hh + p j,snf + p j,hos = 1.
The multinomial patient allocation function is sufficient to determine flows into 106 post-acute care services. Formally, the model takes a time series of hospital discharges 107 from ward type j, i.e. 108 n j,1 , n j,2 , ...n j,T ,
where T denotes the total length of the time series. At each time step, a draw from 109 the multinomial distribution is used to allocate discharge counts across post-acute care 110 services. The result is a matrix with dimensions T × 4, where each column corresponds 111 to the number of new patients that flow to each post-acute care service at each point in 112 time.
To further operationalize this model, we also estimate the number of patients in each 114 of these services at any time to provide guidance on when existing services may reach or 115 exceed capacity. To determine patient counts in each service, we require baseline 116 estimates of the number of patients in each service and the length of stay for patients in 117 each service. In this paper, we assume the initial patient census is zero, however this 118 assumption can be easily changed. In the next section, we detail how length of stay 119 June 12, 2020 3/15 . CC-BY 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)
The copyright holder for this preprint this version posted June 13, 2020. . https://doi.org/10.1101/2020.06.12.20129551 doi: medRxiv preprint estimates were determined. To calculate the census over time for each service, we use 120 the length-of-stay estimates to create discharge series for each service. The number of 121 discharges each day is equal to the inflow from l k days ago, where l k is the 122 length-of-stay assumption for service k. The cumulative sum of the inpatient flows 123 minus the cumulative sum of discharges gives the census at each point in time.
Model inputs and parameters 125 Hospital and ICU discharges 126 Since our model does not attempt to model the broader COVID-19 outbreak, it requires 127 a time series of hospital and ICU discharges. In this paper, we use model outputs from 128 the Johns Hopkins University Infectious Disease Dynamics (JHU IDD) model, which 129 combines an SEIR model with a statistical model to estimate daily hospitalizations, ICU 130 hospitalizations, and ventilated patients [8] . As a result of the considerable uncertainty 131 surrounding estimates of the proportion of all infections that are hospitalized, the JHU 132 IDD model assumes the hospitalization rate is 10 times the infection fatality rate (IFR) 133 and, due to uncertainty in estimates of the IFR, they explore 3 IFRs: 0.25%, 0.5%, and 134 1%. Here, we explore 0.5% IFR and the corresponding 5% hospitalization rate. 135 We demonstrate the utility of our model on two example runs of the COVID-19 136 outbreak in Utah using the JHU IDD model, simulating the outbreak with R t = 1.2 for 137 "current Utah", based off of the actual transmission rate in Utah on May 2,2020, (Table 1) . We do not calculate a length 147 of stay for patients discharged directly to home since that will not influence facility . CC-BY 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 June 13, 2020. . . CC-BY 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 June 13, 2020. . https://doi.org/10.1101/2020.06.12.20129551 doi: medRxiv preprint Post-acute care probability priors 159 The probabilities that parameterize the multinomial distribution are very uncertain. 160 Even as the COVID-19 pandemic has progressed with significant numbers of 161 hospitalizations and deaths, post-acute care needs lag behind. Considering the 162 generation time of infections coupled with the long duration of hospitalizations and ICU 163 hospitalizations, the number of patients who have been discharged to post-acute care 164 services remains relatively low, leading to uncertainty in parameter estimates. First, we 165 determined preliminary estimates of the fraction of individuals who would end up in 166 each post-acute care service type, for both non-ICU discharges and ICU discharges. 167 Then, we constructed discrete "low," "mean," and "high" estimates to summarize the 168 potential range of outcomes (Table 2) .
The non-ICU discharge destination fractions are based on internal University of 170 Utah Health system discharge data from the previous calendar year. The mean ICU 171 patient discharge fractions for home health and skilled nursing facility care are based on 172 historical estimates for those hospitalized for sepsis, which has a similar inpatient 173 mortality rate [10] . The other ICU estimates are also based on University of Utah 174 discharge data from the previous calendar year. (p j,none , p j,hh , p j,snf , p j,hos ) ∼ Dir(α j,none , α j,hh , α j,snf , α j,hos ).
Thus, for every hospitalization discharge series, we draw a random set of 181 probabilities/parameters for the multinomial distribution from that Dirichlet 182 distribution, where the α values are parameters that will initially be proportional to the 183 mean care service fractions shown in Table 2 . Denoting the vector of these mean values 184 for ward j asp j , we introduce a scaling parameter, φ j , such that our initial α 185 parameters are
The magnitude of φ j will determine how much weight is put on our priors versus the 187 new, actual discharge outcomes we may observe in the future. Smaller α values 188 correspond to wider, flatter distributions around each service type probability and 189 represent less confidence in our priors about these probabilities. If we restrict φ j to be 190 an integer it has a nice interpretation: we are as confident in our priors as if we had 191 observed φ j discharges that were distributed according top j . 192 We used the low and high estimates from Table 2 to calibrate φ j for j = icu and 193 j = non icu discharges. We ran simulations to determine numerically the lowest integer 194 values of φ at which 90% of the individual values drawn from the Dirichlet distribution 195 were within their respective low to high ranges. We do not require that 90% of the 196 draws of each parameter fall within their low-to-high range, only that collectively 90% 197 of all the parameters are within their respective ranges. The former would be too (p j,none , p j,hh , p j,snf , p j,hos )|(x j,none , x j,hh , x j,snf , x j,hos ) ∼ Dir(α j,none + x j,none , α j,hh + x j,hh , α j,snf + x j,snf , α j,hos + x j,hos ) (5)
The expected probabilities in the posterior distributions are 209 E[p j,k ] = x k + α j,k α j,none + x j,none + α j,hh + x j,hh + α j,snf + x j,snf + α j,hos + x j,hos
Updating priors using University of Utah Hospital discharges . CC-BY 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 June 13, 2020. percentage of patients who will not require any services and decreased our expectation 222 of the percentage patients who will require a skilled nursing facility. Table 3 shows the 223 parameters for the posterior Dirichlet distributions. December 31, 2020. We ran the simulations on the hospital discharge time series shown 229 in Figure 1 . . CC-BY 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 June 13, 2020. .
We calculated daily estimates of both the number of patients discharged to each care 231 type and the number of COVID-19 beds needed. Additionally, while we do not consider 232 a baseline occupancy in this paper, one could be used to estimate the total patient 233 census for each facility type at a given spatial scale. 234 We show the model ouput, a time series of the daily number of post-acute care beds 235 needed for each of the three care types considered (home health, hospice, and skilled 236 nursing facilities), for the State of Utah for patients discharged from all hospital wards 237 ( Figure 4 ) and for patients discharged from the ICU ( Figure 5 ). Here, we find the 238 majority of patients in Utah needing additional care are requiring home health care or 239 skilled nursing facilities, with very few needing hospice care. Likewise, we find the 240 expected peak in post-acute care needs occurs about two weeks after the expected peak 241 for acute-care hospitalizations ( Figure 6 ). This is a result of the duration of Further, while little information is available on the types of services that patients 277 with COVID-19 need following discharge from acute care, data on post-discharge 278 probability is also still preliminary. Here, we develop the model and implement 279 probabilities based on preliminary data from Utah, however we also highlight the way in 280 June 12, 2020 9/15 . CC-BY 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 June 13, 2020. . 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 June 13, 2020. . . CC-BY 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 June 13, 2020. . . CC-BY 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 June 13, 2020. This model can also be used and adapted to explore potential alternative strategies 293 for addressing the surge in acute care facilities during future pandemic waves. As 294 suggested by Grabowski et al. [10] , post-acute care services can act as a "pressure 295 release valve" for acute care hospitals, taking in patients who are partially recovered to 296 release beds for more critically ill patients. However, without proper planning, 297 post-acute care facilities can serve as the opposite, acting as a "bottleneck." If patients 298 who are ready to be discharged but require additional care have nowhere to go as a 299 result of post-acute care services being saturated, this can negatively impact hospital 300 capacity. While exploring this relationship is outside the scope of this paper, this 301 modeling framework can be easily adapted to explore these and simliar questions.
Additionally, as treatments for COVID-19 are developed, one unanticipated side 303 effect could be that while treatments successfully prevent death in a large number of 304 patients, they could result in more patients, patients who would have died without 305 treatment, surviving and requiring long-term care. At the current time, any estimates of 306 how treatments may impact acute or post-acute care needs would be entirely 307 speculative, however, as more becomes known about potential successful treatments, the 308 prior estimates for proportion of patients being discharged to the different care services 309 can easily be updated to adapt to this type of new information.
Our modeling approach has several limitations. Our initial assumptions about the 311 fraction of patients requiring each post-acute care service are very uncertain, as they are 312 based on aggregated historical University of Utah Health discharges prior to the 313 appearance of COVID-19. However, we account for this by giving relatively low weight 314 to our priors versus observed COVID-19 discharges. Another limitation is that we do 315 not account for uncertainty about the length-of-stay estimates for post-acute care 316 services, although the model could be modified to incorporate this in future work. The 317 largest limitation is that the model does not directly account for characteristics of the 318 patient population, meaning the posterior allocation probabilities may not generalize 319 well to regions or health systems with dissimilar populations. An extension of this work 320 could condition the probability of requiring a post-acute care service on patient 321 characteristics such as age or comorbidities. Although we highlight several limitations, 322 our approach is well-suited to answer policy questions and provides a unique modeling 323 output that can help guide the post-acute care surge. As most epidemiological models 324 focused on acute care services, we concentrate on post-discharge needs, however, as is 325 the case with standard epidemiological models, it is critical for policy makers to 326 consider the range of possible trajectories and the sensitivity of results to different 327 assumptions. Likewise, since we do not model the community outbreak leading to 328 hospitalizations, it is critical that the inputs to our post-acute care model be carefully 329 considered for their strengths and weaknesses, as any limitations of the input model will 330 flow through our model framework as well. Despite these limitations, this model has 331 already been used to help improve post-acute care services in Utah and we believe its 332 June 12, 2020 13/15 . CC-BY 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)
The copyright holder for this preprint this version posted June 13, 2020. 