covid-19 healthcare demand projections: arizona · 5/13/2020 · covid-19 healthcare demand...

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COVID-19 Healthcare Demand Projections: Arizona

Esma S. Gel, Megan Jehn, Timothy Lant,Anna R. Muldoon, Trisalyn Nelson, Heather M. Ross

Modeling Emerging Threats for Arizona (METAz) WorkgroupArizona State University, Tempe, AZ 85281

Last Update: May 7, 2020

Abstract

Beginning in March 2020, the United States emerged as the global epicenter for COVID-19cases. In the ensuing weeks, American jurisdictions attempted to manage disease spread on aregional basis using non-pharmaceutical interventions (i.e. social distancing), as uneven diseaseburden across the expansive geography of the United States exerted different implications forpolicy management in different regions. While Arizona policymakers relied initially on state-by-state national modeling projections from different groups outside of the state, we soughtto create a state-specific model using a mathematical framework that ties disease surveillancewith the future burden on Arizona’s healthcare system. Our framework uses a compartmentalsystem dynamics model using an SEIRD framework that accounts for multiple types of diseasemanifestations for the COVID-19 infection, as well as the observed time delay in epidemiologicalfindings following public policy enactments. We use a bin initialization logic coupled with afitting technique to construct projections for key metrics to guide public health policy.

1 Introduction

Since its documented onset in December 2019 and formal identification in January 2020 in Wuhan,China, COVID-19 (SARS-CoV-2) has spread around the globe, infecting more than 3.5 millionpeople globally by early May 2020 [4]. In an atmosphere of intense uncertainty around many ofthe epidemiological parameters for modeling including true case counts as a result of low testingavailability, the Modeling Emerging Threats for Arizona (METAz) Workgroup of Arizona StateUniversity has developed and refined models for predicting the burden of disease in order to informpolicy related to nonpharmaceutical interventions (i.e., social distancing). Because the burden ofdisease and transmission dynamics differ by location due to a variety of factors including geography,population, and environmental conditions, METAz chose to focus on state-level modeling to informpublic health response efforts with greater precision. The modeling approaches we describe canbe applied to any region or state where region-specific data are available. Here, we focus on thestate of Arizona in the American Southwest (population of around 7.3 million, 113,990 sq. miles,majority population concentrated in centrally-located Maricopa County) as a proof of concept.

At the time of the last update to the report, the following case counts have been reportedin Arizona since March 4, 2020. As of May 5, Arizona’s healthcare system has not experiencedan overwhelming surge of COVID-19 cases exceeding systemwide capacity to care for critically illpatients. Since Arizona has not yet exceeded existing hospital capacity for COVID-19 cases, aplateau in daily hospital admissions is a hopeful sign that social distancing measures may have

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helped Arizona to avoid overwhelming hospital systems as other states have experienced and allowtime to prepare options for future management of the disease in Arizona.

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© METAz, 2020; Not for release

Figure 1: Cumulative confirmed COVID-19 casesin Arizona, between March 4 to May 5, 2020

However, Arizona is still experiencing widecommunity spread of SARS-CoV-2. Due to arelatively low rate of testing statewide, there isongoing debate and uncertainty about whetherArizona’s case prevalence data provides an ac-curate portrait of the true public health riskburden and whether we have passed an (ini-tial) peak of infections and hospitalizationsstatewide and in individual counties. Currentprojections from a variety of modeling groups(i.e., IHME, UA, ASU) indicate that the peaknumber of cases will be reached in Arizona inmid-April - mid-May. However, it is importantto note that modeling projections are inher-ently uncertain, and accurate assessment of casepeaks will be possible only once the peak haspassed. In light of the transmission dynamicsand laboratory reporting delays for the SARS-

CoV-2 outbreak, peak determination will be possible approximately two to four weeks followingpeak occurrence. It is also important to note that there is still significant uncertainty about thetransmission dynamics of the virus, including the degree of asymptomatic infection and transmis-sion and the results do not capture the full range of uncertaintly. We demonstrate this observationthrough our modeling below.

On April 16, the United States Government released Guidelines for Opening Up America Again,proposing a phased approach to re-opening the country. In order to progress into and through threesequential phases of opening businesses and other public and private services, states are expectedto meet a set of gating criteria outcome metrics along with a set of capacity responsibilities forcarrying out core public health and management functions. In order to move into Phase 1 withlimited reopening of businesses and other services, states must demonstrate flattening the caserates, and in order to move into Phase 2 with expanded reopening of businesses and services statesmust demonstrate no rebound in case counts from the limited reopening in Phase 1.

As a comparison, California has identified six controls required for “safe” re-opening: (i) expandtesting to trace and track individual outbreaks and ensure the isolation and quarantine of thoseaffected, (ii) protect vulnerable populations from infection and spread, most notably, seniors, peoplewho are immunocompromised, and unhoused individuals, (iii) the ability of hospital and healthcaresystems to meet the need of anticipated surges as stay at home orders are loosened, (iv) Work withacademia and the pharmaceutical industry as they develop treatments and a vaccine for the virus,as well as protocols to distribute them, (v) develop strategies for schools and businesses to adhereto safe physical distancing guidelines, (vi) a plan to toggle back and forth between stricter andlooser social distancing requirements depending on how the situation changes. Arizona has notyet developed a comprehensive plan that incorporates the full testing capabilities within the state(both molecular and serological) with a program linked to non-pharmaceutical interventions (NPI)including stay-at-home and other social distancing and infection mitigation policies and procedures.In order to re-open Arizona safely, a phased approach needs to be data-driven and focused onavoiding a rapid surge in cases through appropriate and effective policy for non-pharmaceuticalinterventions.

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This paper proposes a mathematical framework that ties disease surveillance with future burdenon Arizona’s hospital system and hospital resources. The mathematical model links together policyinterventions with estimated outcomes for infections, hospitalizations, and deaths in an epidemi-ological analysis. One of the key features of such a model is the time-delay of new infections onconfirmed case counts and the impact on the healthcare system. We propose methods to evaluatethe likely outcomes for a range of policy decisions intended to keep Arizona safe while re-openingin a responsible and defensible sequence.

2 Methodology

2.1 Structure of the Model

We make use of a compartmental system dynamics model using an SEIRD framework that includesmultiple compartments for infected individuals. This model structure allows us to estimate thenumber of patients in the hospital and assess model fit with respect to two sources of data: dailynew cases and cumulative reported deaths over time. In essence, the population of interest, in thiscase, the population of the State of Arizona (assumed to be 7,278,717 in this study) is dividedinto states of Susceptible (S), Exposed but not yet infectious (E), Asymptomatic infected (Ia),infectious and presymptomatic (Ip), Symptomatic with a mild infection (Is), symptomatic with asevere infection and hospitalized (H), symptomatic with a critical infection and in the ICU (C),undergoing additional recovery in ICU (B), Recovered and immune (R) and Dead (D), as shownin Figure 2.

40%

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Figure 2: Depiction of the compartmentalized system dynamics model used to represent transmis-sion and disease progression for State of Arizona projections

Our model defines separate bins for asymptomatic and presymptomatic individuals to explicitlyaccount for transmissions by infected individuals who do not exhibit symptoms. Several results inthe literature point to the fact that the viral shedding by these individuals are also different. Incomparison, individuals who are exposed to the virus generally go through an incubation period(modeled by a rate of ζ) during which they are exposed but not yet infectious. This duration ismodeled as 3 days in our study, to support an ensemble estimate of 5 days for time from exposureto symptoms and an estimate of 6 days serial time obtained from the literature [22, 9, 12, 1, 13,5]. After the preinfectious period, individuals become infectious, either as an asymptomatic or apresymptomatic patient. The presymptomatic duration (modeled with rate δ) is assumed to be 2days [19, 23]. Asymptomatic patients recover at a rate of γ = 1/6, corresponding to an average

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recovery duration of 6 days after the preinfectious period. We modeled a number of variations onhow symptomatic individuals experience COVID-19. After the presymptomatic period of 2 days,a large fraction, 81% (denoted by ρ in the model) estimated by [21, 24], of individuals go througha 6-day period of relatively mild symptoms and recover similar to the asymptomatic patients [20].The remaining 19% of symptomatic patients develop a severe or critical infection and seek care ata hospital.

A large portion of the patients admitted to the hospital only have a severe infection and re-cover after an average duration of 7 days on a regular hospital bed. The clinicians we have beenpartnering with in Phoenix metropolitan area and Tucson hospitals have indicated that an averageof 20% of these patients, however, progress to a critical infection, requiring ICU care and possi-bly intubation. In addition to patients that progress to the ICU from a regular hospital bed, ourpartnering clinicians informed us of a small fraction of patients that report to the hospital witha critical infection with breathing problems and are directly admitted to the ICU. Hence, in ourmodel, there are two modes of admission to the ICU; one directly from the emergency room andthe other one from a regular ward, after the patient’s infection progresses to a critical condition.The parameters for these splits are set to ensure that (i) fraction of symptomatic patients withmild infection is 81% [21, 24], (ii) the total fraction of symptomatic patients that develop a criticalinfection that requires ICU care is 5% [21] and (ii) 20% of patients in a regular bed progress to acritical infection [15].

Studies in the literature cite a diverse range of outcomes for patients in the ICU, but mostagree that the ICU duration for patients that eventually recover is generally longer. For example,[18] cites point estimates for the duration of onset-of-symptoms to death to be 17.8 days and fromonset-of-symptoms to hospital discharge to be 22.6 days. This additional time is also due to varioussteps that caregivers have to take to arrange for care after the ICU period since generally patientsthat underwent intubation and other invasive procedures require subsequent care in other post-acute facilities. This additional “recovery” time is represented as another bin, with a duration of4 days (modeled with rate α). The reported average ICU stays in the literature are generally verydiverse; we adopted a conservative point estimate of 8 days to align with the symptom onset torecovery/death estimates [18] as well as other more detailed studies that tracked patients’ progressthrough the hospital [24].

One of the important parameters in the model is ω, which represents the fraction of asymp-tomatic patients. Several studies point to the importance of modeling transmissions by asymp-tomatic individuals, who may never be aware that they were transmitting the virus during a periodof infectiousness. However, point estimates on the fraction of individuals that experience asymp-tomatic infection vary greatly from context to context. In our models we adopted a relativelyhigh asymptomatic rate of 40% based on point estimates observed by (author?) [16] because thisassumption allows us to obtain worst-case estimates on the prevalence of infections in the generalpopulation given that in the absence of widespread testing of asymptomatic individuals, the asymp-tomatic patients are generally undetected. In our modeling and analysis, we explicitly consider thepossibility that only a small fraction of the true incidence of infections are detected as COVID-19cases and reflected in the reported case counts and deaths. One such example that points to a largeundetected fraction of cases is [10], indicating that 86% of the early infections in China were undoc-umented, implying that the “actual” cases in a population may be more than 7 times the detectedcases. The same study also offers a rate of transmissions by asymptomatic individuals at 55% of thetransmission rate by symptomatic individuals, which we reflect in the force of infection, λ(t) shownin Figure 2. Subsequently several other papers have offered additional understanding on the roleof asymptomatic infections in transmission and its prevalence in different contexts [14, 8, 16, 7, 2].We use these papers along with the actual data on new cases and deaths in Arizona to obtain point

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estimates for model parameters. We also devise an initialization algorithm to identify initial valuesof the compartments in the model.

The ordinary differential equations (ODE) that define the system dynamics are given by Equa-tions (1) thru (10).

S ′(t) = −βt λ(t)S(t) (1)

E ′(t) = βt λ(t)S(t)− ζ E(t) (2)

I ′a(t) = ζ ω E(t)− γ Ia(t) (3)

I ′p(t) = ζ (1− ω) E(t)− δ Ip(t) (4)

I ′s(t) = δ Ip(t)− γ Is(t) (5)

H′(t) = γ ψ Is(t)− µH(t) (6)

C′(t) = γ (1− ρ− ψ) Is(t)− ν C(t) + µφH(t) (7)

B′(t) = ν (1− υ) C(t)− αB(t) (8)

R′(t) = γ Ia(t) + γ ρ Is(t) + µ (1− φ)H(t) + αB(t) (9)

D′(t) = ν υ C(t) (10)

The time dependent force of infection, λ(t) is modeled as

λ(t) =0.55Ia(t) + Ip(t) + Is(t) + 0.05[H(t) + C(t) +B(t)]

N −D(t). (11)

This expression is motivated by the fact that asymptomatic individuals transmit the disease at areduced rate as discussed above, and patients under care in the hospital are relatively well isolatedvia institutional infection control measures so they only transmit at a rate that is equal to 5%of the presymptomatic or symptomatic patients. Studies that point to the high-infectiousness ofpresymptomatic patients [3] implies that infections are mostly driven by patients in these compart-ments. We model a time dependent transmission rate, βt, denoted by the subscript t to representthe time dependency. Together with the force of infection and the current pool of susceptible in-dividuals, the transmission rate βt yields the rate at which susceptible individuals get exposed tothe infection. The force of infection term can be thought of as the probability that an arbitraryindividual is infectious at a rate equivalent to that of a symptomatic patient.

The transmission rate, βt represents the average rate of contact between susceptible and (symptomatic-equivalent) infectious people multiplied by the probability of transmission given contact. Hence,given the above force of infection and the number of individuals in the susceptible compartment,the rate at which individuals become exposed to the virus at time t is strongly driven by the term,βt. A good way of thinking about the impact of non-pharmaceutical interventions such as socialdistancing, stay-at-home orders, school closures, wearing masks, etc. is through the term βt, andhow the different interventions impact either (i) the average number of infectious individuals thatsusceptible individuals contact, or (ii) the probability of transmission given contact.

Note that an increase in either of these two values would lead to an increase in the effectivetransmission rate at a given time, which will then increase the rate at which susceptible individualsget exposed to the virus. We find Figure 3 to be informative to understand the effect of these twovariables to understand the impact of social distancing and other NPI interventions. Staying on thesame β curve of 0.20 while increasing the average number of contacts for a susceptible individualfrom 10 to, say, 20 requires that the probability of transmission given contact be reduced from0.02 to 0.013 through measures that reduce the probability of transmission given contact with aninfectious individual. Such measures may involve hand washing practices, wearing masks, keeping

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Description Parameter Value Sources

Time to infectiousness ζ−1 3 days [22, 9, 12, 1, 6, 17, 11]Presymptomatic duration δ−1 2 days [23, 19]Asymptomatic infectious period γ−1 6 days [21, 20]Mild infection recovery time γ−1 6 days [21]Severe infection recovery time µ−1 7 days [21, 5]Critical infection to death ν−1 8 days [18, 24]Additional days to recover after ICU α−1 4 days [18]

Fraction of asymptomatic infections ω 40% [16]Fraction of mild symptomatic infections ρ 81% [21]Fraction hospitalized on regular bed ψ 17.5% [21]Fraction of hospitalized progressing to ICU φ 20% [15]Mortality among ICU patients υ 50-60% data fit

Table 1: Point estimates used for model parameters and sources

6+ ft apart, etc. As the interactions between individuals are expected to increase after the stay-at-home orders are lifted, the importance of such measures should be more rigorously emphasized. Itis also useful to note that a modest 15% increase in both values would result in a 32.25% increasein β, which would have dire consequences for the transmission dynamics.

β=0.15

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Figure 3: Transmission rate, β repre-sented as a product of the average num-ber of contacts and the probability oftransmission given contact with an in-fectious individual

The model parameters and point estimates for themobtained from the literature are given in Table 1. We nextexplain our approach of initializing the compartments andfitting the transmission rate β for the month of April (dur-ing which Arizona has enacted a stay-at-home order) andmortality rate at the ICU, υ, using publicly available dataon case counts and COVID-19 related deaths in Arizona.

2.2 Initialization of Compartments

We first present a methodology to initialize the model ina manner that is independent of the transmission rate, β.In particular, we consider the data on cumulative num-ber of confirmed cases in Arizona, where the first reportedcases were on March 4, 2020, as shown in Figure 1. Weuse these data to obtain the number of new cases on eachday. The average reporting delay on COVID-19 tests isabout 6 days in Arizona. Given that our model indi-cates an incubation period of 5 days and average timeto seek testing (when it is available) is about 3 days aftersymptom-onset, we obtain “presumed” exposure dates forthe reported new cases on each day. A visual that showsthis logic is shown with the blue bars (reported new casesover time) and the orange bars (numbers eventually de-tected, shown on the presumed exposure dates) in Fig-ure 4. Note that the orange bars show the number of individuals exposed to the virus on the givenday, who are then eventually detected by testing.

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Initialization Scheme

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Figure 4: Reported new cases and presumed exposure dates

As discussed above, a large portion of the individuals exposed to the virus on a given dayare never detected due to the fact that (i) a significant portion of these individuals never developsymptoms; and (ii) some symptomatic individuals are never tested, their infections are attributedto another influenza-like illness, or their case is missed due to false negative results in COVID-19tests. To account for the large rate of undetected infections, we have devised a novel approach usingan “X-factor” initialization scheme where we multiply the number of eventually detected-exposedindividuals by the X-factor to obtain the “underlying overall exposures” on a given presumedexposure day. In Figure 4, the grey points depict exposures in an “X-factor of 4 scenario.”

The X-factored exposures on presumed exposure days are then fed into our SEIRD model,keeping the transmission rate to zero. We obtain an approximate continuous time function byinterpolating over these presumed exposures, called W (s). Figure 5a shows the approximated rateof exposures over time in X-factor of 4 scenario between 3/4 and 3/29 in Arizona; the black dotsare the daily presumed exposures also shown in Figure 4.

We then numerically evaluate the convolution

E[Ni(t)|W (s), 0 ≤ s ≤ t] =

∫ t

0W (s)fi(t− s)ds (12)

to obtain the expected number of individuals in compartment i, i ∈ {Ia, Ip, Is,H, C,B,R,D} at thetime t, where fi(τ) denotes the probability that an individual would be in Bin i τ time units afterexposure to the virus. The fi(·) functions for each bin in the model can be obtained by simulatingthe above stated model with one exposed individual and transmission rate of zero. As an example,Figure 5b shows the fraction at the hospital, fH+C+B(·) versus time.

The solution to the ODEs is unique given a set of initial values for the number in each bin attime “zero.” Using the above initialization logic, we calculate the number that we expect to see ineach bin on a chosen presumed exposure day, using all of the data on new cases reported on thepresumed exposure days prior to this point, and using the number of presumed exposures on thatday to initialize the E compartment. We are then -almost- ready to simulate the model startingfrom that day and observe the number in each compartment to obtain projections.

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5 10 15 20 25 30 35

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1 Exposed at time 0, Fraction in the hospital t days after


(b) fH+C+B(t) under assumed parameters

Figure 5: Building blocks for backcasting procedure used for initialization

2.3 Fitting Transmission Rate and Mortality

In our study, we initialize the bins on 3/30 (i.e., this calendar day is our t = 0) and use the actualdata on presumed exposures (under any assumed X-factor scenario) starting from 3/31 until 4/15to fit the transmission rate, βt. We tried a number of different initialization dates, but the resultson the β fit were comparable, and furthermore, since the stay-at-home order was issued in Arizonaon March 31, we expect that the transmission rate starting from 3/31 to stay relatively consistentduring the period 3/31 to 4/15 with respect to the exposures that happened during this time period.This is why we selected to initialize on this date. We use a least-squares method to identify thebest value of β to describe the transmission rate under the stay-at-home order in Arizona. As anexample, Figure 6a shows the model fit along with the 95% prediction intervals for the number ofexposures under a 4X scenario. The grey points in the graph show the raw values (without the 4Xcorrection) of the presumed exposures (or, 1X scenario).

In addition to fitting the transmission rate, β, we use the cumulative number of COVID-19related deaths in Arizona to fit the mortality rate among the ICU patients. Note that in our model,we assume that all patients who die will do so in the ICU, which ignores the deaths that occuroutside the hospital. At the time of the writing of this manuscript, Arizona’s healthcare capacity,beds, ICU have been sufficient to care for COVID-19 patients. Therefore, to our knowledge, Arizonahas not experienced significant reported deaths outside the hospitals due to an inability for patientsto access critical care services. There is, however, ongoing debate about whether COVID-19 relateddeaths are under-reported in Arizona and nationwide.

Given our assumption for this modeling exercise that deaths are primarily occurring in the ICU,for this analysis we assumed that the information on the reported deaths is relatively accurate.Figure 6b shows the cumulative number of deaths that the model predicts under a 4X loadingscenario, with death rate υ = 0.5408 and β fixed at the fitted value of 0.2226. We see that themodel produces relatively tight prediction intervals and generally does a good job of predicting thedeaths between 3/31 and 5/5 when we compare it to actually reported data. Note that under the1X loading scenario, the ICU death rate produced by the model fit procedure was on the order of2.5; that is, the assumption of 100% detection rate was not aligned with the point estimates weused in the model to predict the reported death rates. Given that there is widespread belief thatCOVID-19 deaths are underreported, we understand this finding to be in support of the idea thatonly a small fraction of the infections are detected that thus reported in the official case counts. In

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the next section, we present projections for 1X (as an overly optimistic case), 4X, and 6X loadingscenarios to provide a range of scenarios.

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Presumed Exposures for 4X Scenario, β=0.2226


(a) Model predicted 4X exposures between 3/31and 5/5 with 95% prediction intervals; best fit forβ = 0.2226. Red dots are 4X presumed exposuresobtained from real data, gray dots are 1X presumedexposures, obtained through backcasting of dailynew cases

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(b) Model predicted cumulative number of deaths be-tween 3/31 and 5/5 with 95% prediction intervals;best fit for υ = 0.5408 under 4X loading scenario.Red dots are the reported COVID-19 deaths for theperiod in Arizona.

Figure 6: Fitting values for β and υ using case counts and deaths

3 Model Projections

We provide projections on the number of deaths, number hospitalized and total infections for anumber of cases that differ in X-factor and the transmission rate over time. We first start with thebenchmark cases of 1X, 4X and 6X loading scenarios simulated under the assumption of constanttransmission rates obtained by fitting the model to data during the stay-at-home period.

While the transmission rate that is indicated by the model fitting exercise outlined above isrelatively low, indicating a Reffective value of approximately 1, it is useful to observe the dynamicsin a relatively long horizon of 500 days, as given in Figure 7. Day 0 in this simulation is 3/31, wherewe initialize our model and run it with the beta and υ values that we fitted using the new remainingdata on new cases and deaths after this point (i.e., 16 points of backcasted presumed exposuresand 30 days of data on deaths). Given the large initial susceptible population that we use for themodel (i.e., 7,278,717) it takes around 200 days to reach herd immunity as long as the transmissionrate is maintained at this current constant level. The number of hospitalizations reaches 26,650 onday 175, which corresponds to mid-September 2020. This figure demonstrates that policies thatwere initially put in place to limit contacts have been effective in reducing transmission, but alsopreserve a high pool of susceptibles that can be returned into mixing with the general public andcan ignite future outbreaks.

While visualization of the epi curves is useful to gain insights into the long-term behavior andother concerns such as peaks and herd immunity, it is more informative to focus on shorter-termpredictions since it is hard to imagine a real-life scenario under which β stays constant over a verylong period of time due to measures taken by individuals and public health officials. The “baseline”plots given in Figure 8 show the total infected deaths and hospitalizations as well as exposures anddeaths under the 1X, 4X and 6X loading scenarios with fitted transmission and mortality rates atthe ICU. The best fit β values were 0.2221, 0.2226 and 0.2229 for 1X, 4X and 6X scenarios. The

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Figure 7: Baseline case of 4X loading with constant β = 0.2157

υ values used for 1X, 4X and 6X scenarios were, 0.95, 0.5408 and 0.3386. To fit the mortalityrate in the 6X scenario, we kept the data on reported deaths in tact and fitted the value of υ tothe data in a 6X loaded model. An alternative approach would have been to inflate the deathnumbers to account for the observation that COVID-19 related deaths may be underreported. Ourapproach yields a mortality a rate of about 34% which we decided to adopt for our projectionsunder 6X loading scenario. Assuming a 30% underreporting of deaths, for example, results in afitted mortality rate of 46% in the ICU but we do not use that number in our projections to keepto a evidence-based conservative set of estimates on the death toll of the epidemic.

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Figure 8: Fitted model predictions with 1X, 4X and 6X loading scenarios, with transmission rateβ of 0.2221, 0.2226, 0.2229 and ICU death rate υ of 0.95, 0.54 and 0.34, respectively. The red dotsin plots correspond to data used for fitting 4X scenario parameters.

We see that both 4X and 6X provide plausible fits to the data on cumulative deaths (by adjusting

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the mortality rate υ). While we don’t directly use hospitalization numbers in our analysis, thecurrent hospitalization numbers appear to be close to the number predicted by the 4X loading(1065 patients). In these graphs, the light orange shading indicates the two-week period after 5/5(last data point used).

The projections show that it is reasonable to expect a slowly increasing number of patients in thehospital due to the above stated fact that there is still a large number of susceptible individuals inthe population. Better data on the numbers in the hospital through testing and rigorous reportingpractices would be invaluable to track the progress and fit improved models.

3.1 Summer Effect

There is currently debate on the impact of higher summer temperatures of Arizona on the transmis-sion rate of COVID-19, both with regard to a potentially suppressive effect on virus survivabilityin elevated temperatures, as well as the behavioral effect of extremely hot temperatures changingpatterns of indoor and outdoor activity in Arizona’s desert environment. While we are not certainabout a potential summer effect, particularly on virus survivability, we have found it to be a usefulcontext to demonstrate the sensitivity of outbreak dynamics to the transmission rate β.

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Figure 9: Fitted model predictions with 4X loading with transmission rate β of 0.2226 and ICUdeath rate υ of 0.54 under five possible summer effect cases

In Figure 9, we plotted the total infected and the hospitalizations under four different scenarioswith respect to the summer effect. The plot shows the tradeoff between an early summer effectversus a later but more significant summer effect. The figure also demonstrates the impact of a25% to 50% reduction in the transmission rate as well as the impact of the timing of the summereffect in flattening the curve further.

3.2 Opening Scenarios

A more likely scenario awaiting Arizona is an increase in the transmission rate due to the liftingof the Governor’s stay-at-home orders and re-opening of businesses that were deemed to be non-essential. While it is not possible to know how β will change as a result, it is useful to understandthe way in which increases in transmission rates will influence the overall dynamics of the epidemic.

We start by demonstrating the impact of a very modest, mere 30% increase in transmissionrate from its current estimated value. We are not arguing that there will be a 30% increase inβ since we do not have the data to support such an argument, but a 30% increase in β may, forexample, be a result of 15% increase in the average number of contacts and a 15% increase in theprobability of transmission given contact as demonstrated in Figure 3. One can easily imagine such

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increases in the average number of contacts as well as probability of transmission given contactwith an infectious individual after the NPIs are relaxed. Hence, these scenarios are presented asrelatively moderate levels of increase; the increase in transmission rate β can certainly be muchhigher.

Figure 10 shows the hospitalizations over a period of about five months for three different waysthat this increase may happen: an “early” 30% increase on 5/15 upon the lifting of the currentstay-at-home orders in place on 5/1, a “late” 30% increase on 6/15 and a gradual 10% increase on5/15, 6/1 and 6/15, as well as keeping the transmission rate the same at β = 0.2226 for the 4Xloading scenario. The qualitative insights will remain the same for the other scenarios.

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Figure 10: Total infections and hospitalizations for the early, late and gradual increase in trans-mission rate

Even for a relatively modest estimate of the opening on the transmission rate, we see that thetransmission dynamics changing drastically, even for very gradual increases of 10%. The hospital-izations predicted as a result of these transmission rate changes show the impact on the healthcareresources.

Even though the results in the above figure are alarming, it is hard to imagine that we will not bereacting to such increases in hospitalizations, etc. so it represents a worst-case in how the epidemicwill unfold. The more realistic scenario is that we will be tracking the behavior of the epidemicand will be triggering NPI actions to manage the impact on healthcare resources and public health.Hence, it is useful to view the short-term behavior of the curve to gain some insight into the signalsthat one can look for in the data as we relax the stay-at-home orders. Figure 11 plots the presumedexposures inferred from data (for 4X loading) as well as the reported COVID-19 deaths in Arizona(the blue curve in these plots is the same as the orange curves in Figure 8). The figure clearly showsthat we will be observing an increase in transmission rate, but unfortunately with a significant timelag. The increase in presumed exposures starting from 5/15 shown in Figure 11a will be observedin the case counts data starting from 6/1. This fact is also clearly shown in Figure 11b; we expectto start seeing increases in death counts starting from the beginning of June. If the transmissionrate increases more gradually, then it will be even harder to conclusively say that the transmissionrate has increased. Depending on the case, the detection of a rate increase and its likely magnitudecan take a lag of about a month, or may be more, depending on the quality of data.

Suppose that we detect an increase in the transmission rate in the very optimistic case of twoweeks or relatively pessimistic case of four weeks after the actual increase in β, and react with astay-at-home order that returns the transmission rate to its baseline value. To keep the discussionsimpler, let’s consider the case of gradual increase in β under which the transmission rate was setto increase in increments of 10% every two weeks starting from 5/15. Let’s assume that after the

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Figure 11: Presumed exposures and deaths for 4X loading under different β increase scenarios;gradual increases of 10% on 5/15, 20% on 6/1, 30% on 6/15 and slow increases 5% on 5/15, 15%on 6/1 and 30% on 6/15 (each from baseline β)

detection of rate increase, the rate goes back to the low baseline value after the reactive shutdown.Figure 12c and 12d clearly demonstrate the impact of the reactive shutdown in comparison to

the curve with constant β = 0.2226 for 4X loading. While there is a difference in the height of thepeaks between the curves, the figure also indicates that reactive shutdowns can be very effective tomitigate the consequences of increases in transmission rate, particularly if the detection time lag iskept to the minimum of two weeks.

As the detection time lag increases, however, the effectiveness of the reactive shutdown tomitigate the impact of the peak decreases. It is also worthwhile to note that in the scenariopresented, the reactive shutdown is issued on 5/29 and 6/15 in the 2-week lag case and 4-weeklag case, respectively. Even after the reactive shutdowns, we expect to continue seeing increasednumbers of total infections and hospitalizations, which will likely raise public questions about therationale for the shutdowns if they failed to mitigate infections and deaths, making the public policymanagement aspects of the intervention more challenging.

In our framework, it is possible to calculate the expected excess number of exposures or deaths(or any other epidemiological measure) as a function of increased β values over time. Figure 13depicts the excess observations that we would “expect” to see in the data, over time. Note thatboth plots start from the date of 5/15, which is the date for which Arizona is going to be liftingthe stay-at-home orders, as of 5/1. We can observe from the plots that the excess new cases anddeaths start getting reported about 2 weeks after the first 10% increase in β. We assumed that theassumption on detecting 1/4 of the actual cases (i.e., 4X scenario) and plotted the excess that youwould expect to be detected in this scenario in Figure 13a. Note that these are expected increasesin the case counts and deaths assuming a homogeneous increase in the transmission rate across theentire susceptible population in the State. Hence, typically the numbers we expect to pick up fromthe data will be smaller. This also points to the importance of rigorous testing and monitoring ofnew cases since this framework, supported with good quality data, can yield an ability to developreactive shut-down strategies like the one depicted in Figure 12.

Figure 13 omits the prediction intervals to keep the demonstration of excess new cases anddeaths simple but the prediction intervals on the number of new cases and deaths, allows usto construct “control limits” that would signal an “out-of-control” situation with respect to thetransmission rate, β. Our current work involves development of these trigger control limits underexplicit consideration of the various sources of model uncertainty. The success of such approaches,however, depend on the quality and consistency of the data that we get on the case counts and

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Figure 12

deaths, which in turn depends on the testing policies and rigorous reporting of COVID-19 relateddeaths.

4 Discussion

In future work, we look forward to refining the model, applying it to other areas, creating additionaltools. We note that this model iteration was constructed based on the stated Arizona stay-at-homemodel remaining in place until 5/15. After this model was constructed based on the 5/15 reopeningexpectation, the Arizona governor announced on 5/4 that businesses, including salons and dine-inrestaurants, would begin reopening between 5/8 and 5/11. We anticipate that our projections willshift in future iterations based on these policy changes.

In future work, we further look forward to testing this model against data from other statesbeyond Arizona in an effort to validate this approach for other public health policy making juris-dictions.

Acknowledgements: We would like to acknowledge the contributions of the Arizona Departmentof Health Services Modeling Working Group: Amber Asburry, Steven Robert Bailey, TimothyFlood, Joe Gerald, Heidi Gracie, Katherine Hiller, Ken Komatsu, Josh Labaer, Mark Manfredo,Anita Murcko, Vern Pilling, Timothy Richards, George Runger, Marguerite Sagna, Lisa Villarroel,Patrick Wightman, and Neal Woodbury.

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Figure 13: Expected excess new cases and excess COVID-19 deaths (in comparison to the baselineβ = 0.2226 case, assuming a 10% gradual increase on 5/15, 6/1, 6/15 in baseline β)

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