key: cord-0961761-h9mbg8fv
authors: Calabrese, Justin M.; Demers, Jeffery
title: How optimal allocation of limited testing capacity changes epidemic dynamics
date: 2022-01-24
journal: J Theor Biol
DOI: 10.1016/j.jtbi.2022.111017
sha: 321fb08a2e4d53f52fb2d6167eb5aa6a5b833bb3
doc_id: 961761
cord_uid: h9mbg8fv

Insufficient testing capacity has been a critical bottleneck in the worldwide fight against COVID-19. Optimizing the deployment of limited testing resources has therefore emerged as a keystone problem in pandemic response planning. Here, we use a modified SEIR model to optimize testing strategies under a constraint of limited testing capacity. We define pre-symptomatic, asymptomatic, and symptomatic infected classes, and assume that positively tested individuals are immediately moved into quarantine. We further define two types of testing. Clinical testing focuses only on the symptomatic class. Non-clinical testing detects pre- and asymptomatic individuals from the general population, and a concentration parameter governs the degree to which such testing can be focused on high infection risk individuals. We then solve for the optimal mix of clinical and non-clinical testing as a function of both testing capacity and the concentration parameter. We find that purely clinical testing is optimal at very low testing capacities, supporting early guidance to ration tests for the sickest patients. Additionally, we find that a mix of clinical and non-clinical testing becomes optimal as testing capacity increases. At high but empirically observed testing capacities, a mix of clinical testing and non-clinical testing, even if extremely unfocused, becomes optimal. We further highlight the advantages of early implementation of testing programs, and of combining optimized testing with contact reduction interventions such as lockdowns, social distancing, and masking.

The COVID-19 pandemic caught the world off-guard and continues to result in devastating conse-Model development 84 We develop a modified SEIR model to determine how limits on the number of tests administered per 85 day influence disease controllability, and to determine how limited resources can be best distributed 86 among compartments in the modeled population. Our study was motivated by the COVID-19 87 crisis, both in terms of model structure, and in terms of the pressing need to make the most of assume two separate infectious categories based on observable symptoms. One, the "symptomatic 91 class," collects moderate to severely symptomatic cases for which one would typically seek clinical 92 treatment, and the other, the "asymptomatic class," collects all remaining cases, which may be 93 either properly asymptomatic, or may simply be mild enough that the infected individual does 94 not consider themselves sick or seek clinical treatment. We first develop a baseline disease model 95 without interventions, and then incorporate testing and quarantine control strategies. 96 Baseline SEIR model 97 We assume a homogeneously mixed population divided into S susceptible, E exposed, A asymp-98 tomatic and infectious, Y symptomatic and infectious, and R recovered classes. Newly infected 99 individuals first enter the exposed class where they are unable to transmit the disease, and after 100 a latent period, will enter the symptomatic or asymptomatic infectious class. For clarity, we take 101 "asymptomatic" to include individuals who will show only mild to no symptoms over the course 102 of the disease. The portion of individuals in the exposed class who eventually transition to the 103 symptomatic class are considered "pre-symptomatic". Although some evidence suggests that pre- and that recovered hosts obtain permanent immunity. The model equations are as follows:

Here and throughout this paper, over dots denote derivatives with respect to time, and we mea- . Upon infection, susceptible individuals S move into the exposed class E where they are neither symptomatic or infectious. A fraction f A of exposed individuals transition to the asymptomatic infectious class A at rate ε, and a fraction f Y transition to the symptomatic infectious class Y at rate ε. Infectious individuals transition to the recovered class R at rate r. 133 Testing model 134 To analyze testing and quarantine control strategies operating with testing capacity constraints, 135 we construct a simple model that scales smoothly between extremes of abundant and severely 136 constrained testing resources. This model is governed by the testing capacity, C, and the testing 137 time, τ . The testing capacity C denotes the maximum achievable per capita testing rate assuming a 138 fixed level of resources. This maximum testing rate represents the limitations of a finite health care 139 infrastructure and finite testing supplies, and we take "increased resources" to mean an increased 140 value of C. The testing time represents the average amount of time required for an individual 141 be tested and obtain results, absent any backlogs or waiting times due to other patients. Time-

consuming factors independent of the number of people needing to be tested determine the value 143 of τ , for example, procrastination, travel time, and test processing times.

Suppose that some sub-population P (t) of the total population Z is eligible to be tested at 145 time t, and letṪ (t) denote the rate at which tests are administered and processed for results. 146 We demand that our model forṪ (t) attain two limiting expressions representing "resource-limited" 147 7 and "testing time-limited" testing regimes as follows:

(2) 

The above expression limits to the testing time-limited regime for small P (t), monotonically in-158 creases with P (t), and saturates to the resource-limited regime as P (t) approaches ∞. We justify 159 this testing model based on the fact that it exhibits the correct limiting behavior, and that it 160 incorporates the reasonable assumption that the average waiting time required to administer and 161 process a single test increases linearly with the patient load P (t) (see Appendix A Eq. (10)).

It is important to note that despite its frequent use in the literature, a simple linear testing Eq. (2), which represents a resource-rich rather than resource-limited scenario.

170 populations subject to distinct testing capacity constraints. Specifically, suppose two distinct sub-172 populations P 1 (t) and P 2 (t) are subject to two distinct testing policies with distinct resource pools 173 limited by the capacities C 1 and C 2 , respectively. In this scenario, the total rate at which tests are 174 administered to the two populations is given by the following:

(4) applies resources to the exposed class E(t) and the asymptomatic class A(t), as well as to some When an infected individual is tested in our model, they will instantly transition to the quaran-195 tine class Q(t), and will subsequently recover from the disease and transition to the recovered class. 196 We also introduce the "unknown status" class U (t), which is the subset of recovered hosts who did 9 not receive any testing or quarantine, and are therefore unaware that they previously had 19. We assume that recovered individuals who have previously been tested and quarantined will 199 exclude themselves from non-clinical testing due to assumed immunity, and therefore assume that 

Suppose that a fraction ρ of the testing capacity C is allocated to non-clinical testing, with the 240 remainder devoted to clinical testing. The parameter ρ denotes the "strategy parameter," and its 241 value represents a government's policy for balancing health care resources between reservation for 242 more critical symptomatic cases and for use in contact tracing, testing centers, and surveillance 243 programs. Our modified SEIR model including testing, quarantine, and resource allocation is as 244 follows:

In Appendix A, we analyze a closed-form expression for R 0 under our full SEIR + testing and 248 quarantine model, and we provide expressions in Eqs. (11) and (12) for average testing waiting 249 times for non-clinical and clinical patients, respectively.

A summary of all control related parameters is given in Table 2 for reference, and a schematic 251 summarizing the flow of infected individuals through our control model is given in Fig. 1 In accordance with the goal of "flattening the curve" typically communicated by government and 262 health agencies (World Health Organization, 2020a), we simulate our model dynamics to determine 263 Figure 2 : Diagram indicating the flow of infectives in our SEIR model with testing or quarantine control. Blue arrows represent natural disease transitions, and red arrows represent transitions due to testing and quarantine interventions. Exposed E and asymptomatic infectious A individuals enter the quarantined class Q via non-clinical testing, while symptomatic infectious individuals Y enter quarantine Q via clinical testing. Quarantined individuals are prevented from generating new infections, and enter the recovered class R at the natural recover rate r. Infectious individuals who do not enter the quarantined class also recover at rate r, and subsequently enter the subset U of recovered individuals with unknown infection histories, signifying that they are unaware that they were ever infected with COVID-19.

if and to what extent appropriately allocated resources can reduce the peak number of infections.

First, we calculate optimal resource allocation strategies ρ for reducing the epidemic peak (defined 265 as the maximum value of the sum of the E, A, and Y classes), assuming parameter values in Table   266 1 and an initial outbreak of one exposed individual as our baseline case. Optimization is executed 

Optimal resource allocation strategies 288 We find that, even under extremely limited testing capacities, the epidemic peak can be reduced 289 to the initial outbreak size of 1 infected individual, provided that resources are optimally allocated 290 and that non-clinical resources are sufficiently concentrated on the infected population (Fig. 3a) .

Reducing the epidemic peak to the initial outbreak size signifies that disease spread has been 292 effectively suppressed. For a given η at low testing capacities, the optimal strategy is to devote all 293 resources to clinical testing, and a minimum threshold capacity C th (η) exists, above which optimal 294 strategies call for a mix of clinical and non-clinical testing (Fig. 3b) . As testing capacity increases 295 above C th (η) optimal strategies require an increasing share of resources to be devoted to non-296 clinical testing until a second threshold capacity C * (η) is reached. The threshold C * (η) represents 297 the smallest testing capacity for which the outbreak can be suppressed to its initial size with a for by the optimal strategy at C = C * (η). As a result, optimal strategies are not unique when 304 C > C * (η). To see this non-uniqueness explicitly, consider a concentration level η, and let ρ * (η) 305 denote the optimal strategy parameter at the critical capacity C * (η). At this critical capacity, the 306 optimal action is to devote ρ * (η)C * (η) total resources to non-clinical testing and 1 − ρ * (η) C * (η) 307 total resources to clinical testing, the result of which reduces the epidemic to the smallest possible 308 value 1. If the testing capacity C exceeds the critical level C * (η), one can always allocate at least 309 ρ * (η)C * (η) and 1 − ρ * (η) C * (η) total resources to non-clinical and clinical testing, respectively, 310 thereby guaranteeing the epidemic peak to be reduced to 1. The allocation of the remaining 311 C − C * (η) resources will therefore be irrelevant, as adding resources to either strategy can not 312 further decrease the peak size beyond the initial infection size. In other words, for a given η, if 313 C > C * (η), the epidemic peak will be reduced to 1 whenever ρ is selected such that ρC ≥ ρ * (η)C * (η)

. These inequalities imply that any ρ drawn from the interval

will reduce that epidemic peak to the minimum possible 316 value, thus showing that the optimal strategy is not unique for C > C * (η).

For a given capacity C, there exists a critical non-clinical concentration value η crit (C), below 318 which the optimal strategy is clinical testing only, and above which the optimal strategy is mixed 319 (Fig. 3) . From the definition of C th (η) as the minimal capacity below which the optimal strategy 320 is clinical testing only for a given η, we have the relation C th (η 0 ) = C 0 if and only if η crit (C 0 ) = η 0 , 321 and numerical values for η crit (C) at specific C values can therefore be inferred from Table 3 . should be considered. In Fig. 4 , we plot η crit as a function of testing capacity C. Here, the curve 327 defined by η crit (C) divides the (C, η) plane into two regimes, one where the optimal strategy calls 328 for clinical testing only, one where optimal strategies are a mix of clinical and non-clinical testing.

In particular, we find that for C > 8.0 tests per thousand per day, η crit (C) = 0. Thus, for testing 330 capacities above 8.0, it is always optimal to devote at least some resources to non-clinical testing, 331 15 even if the non-clinical testing is a simple randomized population sampling program lacking the 332 efficacy of targeted contact tracing efforts.

Numerical Values (tests per thousand per day) Optimal resource allocation strategies ρ for reducing the epidemic peak as a function of testing capacity. An optimal ρ curve which terminates at a testing capacity C * below than the maximally considered value 25.0 tests per thousand per day indicates a non-clinical concentration level for which the optimal strategy is not unique at capacities above C * , and for which the optimal epidemic peak size can be reduced to the initial value of one infected at capacities above C * . Note that for the idealized omniscient limit η = 1, the optimal testing strategy is not unique down to the smallest non-zero testing capacity considered 0.01 tests per thousand per day. Note also that for η = 0.85, 0.90, 0.95, and 0.97, the optimal ρ values at C = C * appear to be close to 1, but are not actually equal to 1. Figure 4 : Optimal resource allocation strategy regimes for reducing the epidemic peak as a function of testing capacity C and non-clinical testing concentration level η. For (C, η) values within the shaded region, optimal strategies call for sharing resources between a mix of clinical and non-clinical testing. Within the non-shaded region, optimal strategies call for all resources to be focused to clinical testing only. The black curve indicates a critical concentration level threshold which for a given testing capacity, determines whether the optimal strategy will be mixed or clinical testing only.

Social distancing and delays in testing program implementation 334 Unsurprisingly, delaying the implementation of a testing program by 30 days has negative impact 335 on optimal peak reduction, with the delay being most detrimental at the lowest testing capacities 336 (cf . Figs 5a and 5b) . Specifically, a delay of this magnitude makes it impossible to reduce the epi-337 demic peak to its initial value, regardless of the non-clinical concentration level, within the range of 338 testing capacities [0, 1.2] tests per thousand per day (Fig. 5a) . This is not the case for immediate 339 testing program implementation, where the peak can be reduced to its initial value at any non-zero 340 testing capacity given a sufficient concentration level (Fig. 3) . Reducing the peak to its initial value 

Halving the contact rate, which simulates the influence of social distancing, has a strong effect 347 on optimal policies and peak sizes (Figs. 5c and 5d ). At zero testing capacity (which corresponds 348 to the disease dynamics without testing and quarantine), the epidemic peak reaches a proportion of 349 0.23, which is approximately half of the no testing peak proportion without social distancing. This for which the peak can not be suppressed to its initial size for η = 1, and therefore no range of 357 testing capacities over which the optimal ρ is unique (Figs. 5d and 3b) .

Combining the two modulating factors shows that the beneficial effects of social distancing at per thousand per day with 50% contact reduction social distancing (Fig. 5e) , as compared to [0, 1.2] 363 without social distancing (Fig. 5a) . For all delays between 1 day and the time of the uncontrolled 364 epidemic peak, 62 days, larger degrees of contact reduction from social distancing yield larger re-365 ductions in the range of testing capacities for which the peak can not be reduced to its initial size 366 in the idealized omniscient limit η = 1 (Fig. 6) . Note that after day 62, the infection curve turns 367 downward in the uncontrolled model, so for delays greater than 62 days in the controlled model, 368 the epidemic peak value will always be equal to the initial value regardless of testing capacity, and 369 peak reduction is not a useful control goal. Also note that in Fig. 6 Figure 5 : Effects of social distancing and control delays on optimal testing strategies for reducing the epidemic peak. See Fig. 3 for a comparison to our baseline case and an explanation of the meaning of each plot. Figure 6 : Combined effects of social distancing and delays in testing implementation on epidemic controllability. Threshold testing capacities are plotted as a function of implementation delay, where different curves represent different social distancing strengths as percent reduction in the contact rate. For a given implementation delay time, if testing capacity falls below the value indicated by a curve in the figure, the epidemic will not be forced into a downturn upon control implementation despite perfectly omniscient non-clinical testing, assuming the indicated level of social distancing. Plotted curves terminate at a 62 day delay because the uncontrolled epidemic curve peaks and begins to decrease on its own after day 62. Plotted curves begin to decrease after about a 50 day delay because the slope of the uncontrolled epidemic curve begins to decrease after about 50 days.

The COVID-19 pandemic has exposed a critical lack of capacity for diagnostic testing in an emerg-377 ing pandemic. Using a modified SEIR model, we explored how distributing a limited amount of 378 testing effort can affect the course of an epidemic when testing is directly coupled to quarantine.

The model is tailored to the epidemiology of SARS-CoV-2, and divides infected individuals into 380 symptomatic and non-symptomatic classes, with the latter class including individuals who have 381 been exposed but are not yet infectious as well as those who are infectious but not strongly symp- testing is never the optimal strategy. In other words, non-clinical testing plus a small but finite 402 amount of clinical testing will always be better than a purely non-clinical strategy for epidemic 403 peak reduction.

Since the onset of the pandemic, testing capacity has steadily increased throughout much of the and meaningful control goal, we also explored the common approach of minimizing R 0 (see Ap-416 pendix A). A mathematical advantage of R 0 minimization is that it leads to closed-form expressions 417 for key threshold parameter values that delimit the conditions under which different testing strate-418 gies are optimal. However, we found that for our model, results between these two control goals 419 often differed markedly. Specifically, we identified conditions under which testing policies resulting 420 in R 0 < 1 still yielded large outbreaks, which suggests limited utility of R 0 as a control target. 421 We hypothesize that this phenomenon results from the combination of a finite system size and a 422 finitely small initial condition (see Appendix A). We further note that the choice of control goal 423 can also lead to qualitatively different conclusions about optimal strategies. For example, purely 424 clinical testing strategies are never optimal under R 0 minimization, which contrasts sharply with 425 low testing capacity results for peak minimization.

Our results suggest that testing early is critically important to control efforts. Specifically, the 427 range of testing rates that allows full epidemic control is broadest when testing is implemented im- 

Taken together, our results suggest that focusing exclusively or mostly on clinical testing at very 437 low testing capacities is often optimal or close to optimal. As testing capacities increase, which can 438 typically be expected to happen with time since epidemic onset, the options for optimally distribut- In this appendix, we provide an analytic expression for our model's basic reproduction number, R 0 , 461 and we demonstrate that R 0 reduction is not a reliable metric of control efficacy for epidemic peak 462 reduction. The basic reproduction number is a threshold quantity which determines the stability of 463 a disease-free population with no natural or acquired immunity: small numbers of initial cases will 464 produce large epidemic outbreaks when R 0 > 1, and will result in rapid disease die-out when R 0 < 1 465 (Diekmann et al., 1990) . Intuitively, R 0 quantifies the number of secondary cases produced by a 466 typical initial case when interacting with the disease-free state. Because we are able to obtain an 467 analytic expression for R 0 , the question of its suitability as a metric for control efficacy is especially 468 prescient; the ability to analytically minimize R 0 rather than numerically minimize the peak itself 469 would provide exact expressions and deep mechanistic insight into optimal control strategies if R 0 470 were indeed found to be a reliable metric for control efficacy.

Analytic expression for R 0

The analytic expression for our model's basic reproduction number is found utilizing the next- 

26

The case C = 0 corresponds to the uncontrolled model in Eq. (1), and R 0 is a discontinuous 482 function of C at C = 0 except for the special case ρ = 1, η = 1. Under uncontrolled conditions, the 483 parameters in Table 1 give an R 0 = 5.0, with 3.0 originating from the asymptomatic contribution, 

Extending this definition to our disease model with testing and quarantine in Eq. (5), we find two 505 effective testing times for non-clinical and clinical testing, denoted τ N ef f and τ C ef f , respectively:

These effective testing times represent the average delays for asymptomatic and symptomatic in- ing times when the patient load is negligibly small), so for larger patent loads, a fixed number of 512 resources will move individuals to quarantine at a slower effective per-capita rate. In this sense, 513 lower patient loads allow a given number of resources to be leveraged more efficiently. 514 We hypothesize that the large outbreaks observed when R 0 < 1 arise due to a finite system of initial infectives will produce slightly longer than minimal effective testing times, and that this 524 small increase can become exaggerated when ρ is very close but not equal to 1 or 0. Thus, initial 525 conditions can yield testing efficacies much smaller than those assumed by R 0 , sometimes to a 526 degree which allows epidemics to grow even when R 0 < 1. In support of our hypothesis, we have 527 found that reducing the initial condition size by a factor of 10 (which corresponds to less than one 528 infected individual) eliminates the effect of large peaks when R 0 < 1 for all cases pictured in Fig. 7 . In this appendix, we provide a definition for the concentration parameter η in terms of test-positive 532 and prevalence rates, and use the resulting expression to estimate plausible values from data. To 533 begin, consider the case η = 0 representing a monitoring program conducted via random population 534 sampling. LetṪ + 0 (t) andṪ − 0 (t) denote the rates at which positive and negative tests, respectively, 535 are processed and administered under non-clinical testing for η = 0:

Let f + 0 (t) and f − 0 (t) denote the corresponding respective test-positive and negative rates, defined 537 as the fractions of tests returning positive and negative results:

The above expression show that for η = 0, test positive and negative rates are equivalent to 539 the overall disease prevalence and non-prevalence, respectively, within

population. This result agrees with the notion that η = 0 represents a random population sampling, 541 as the test positive rate from a random sample should be an unbiased estimate for disease prevalence.

Consider now the case of η > 0, and letṪ + (t) andṪ − (t) denote the rates at which positive and 543 30 negative tests, respectively, are processed and administered under non-clinical testing:

The corresponding test-positive rate f + (t) and test-negative rate f − (t) are given by the following:

Combining the above expressions with Eqs. (15) and (16), we find the following expression for η:

Equation (21) shows that η is a measure of the efficacy of a non-clinical testing program's ten-547 dency to focus tests on infected individuals relative to overall prevalence levels. When non-clinical 548 testing performs little to no better than a random sampling program, the test-positive to negative 549 ratio will nearly equal the positive to negative prevalence ratio, so the fraction term in Eq. (21) will 550 be close to one, and η will be close to zero. As the ratio of test-positive to negative rates increases 551 beyond the level of positive to negative prevalence, the fraction term decreases in magnitude, and 552 η grows larger. When the test-positive to negative ratio becomes much larger than the ratio of 553 positive to negative prevalence, the fraction term in Eq. (21) will be small, and η will be close 554 to one. Interestingly, because η is constant, Eq. (21) shows that, as a consequence of our model 555 structure, the time-dependencies of the test-positive to negative ratio and the positive to negative 556 prevalence ratio cancel one another.

Substituting the identities f − 0 (t) = 1 − f + 0 (t) and f − (t) = 1 − f + (t), Eq. (21) gives a math-558 31 ematical relationship between η, the test-positive rate, and the prevalence rate. In Fig. 8a , we 559 plot the test-positive rate as a function of the prevalence rate for a number of η values. In our 560 disease+testing model, as the epidemic grows, the point (f + 0 (t), f + (t)) will travel to the rightwards 561 along one of the corresponding η curves in Fig. 8a , stop and reverse direction once the epidemic 562 peaks, and eventually approach the origin as the disease dies out.

To properly estimate η for a real system, one must acquire test-positive rates and prevalence 564 rates which exclude data from moderate to severely symptomatic cases in clinical settings. To the 565 best of our knowledge, such data are not readily available. As a substitute, we use test-positive where we see that for η less than 0.50, test-positive rates are only slightly above prevalence rates, 584 and this would not be reasonable for a testing program which does not randomly sample both 585 infected and non-infected individuals. Figure  8a exemplifies the degree to which non-zero η values increase the test-positivity rate beyond the level of prevalence that would be measured by random population sampling at η = 0. Figure 8b zooms into the ranges of prevalence and test-positivity rates for the entire clinical plus non-clinical population estimated over the first year of the pandemic 2020 in the United States. Test-positive rates are taken from (Ritchie et al., 2020) , and estimated prevalence rates are taken from (Noh and Danuser, 2021) . Specific values pairs of test-positivity and prevalence values on specific dates in 2020 are indicated by the marks in Fig. 8b Appendix C: Limited non-clinical testing access 587 In this appendix, we consider the effects of limiting the overall population accessible to non-clinical 588 testing. Such limitations may be especially relevant for large η values representing extremely 589 efficacious contact tracing programs, as the time and effort required to run such programs may 590 limit the number of individuals able to be reached, and many individuals may not be amenable to The above expressions show that limited non-clinical testing access effectively increases the non-596 clinical testing time to τ /γ. Importantly, we see that limited testing access does not change the 597 interpretation of η in terms of test-positivity rates and prevalence rates outlined in Appendix B.

In Fig. 9 , we plot optimal infected population proportions at the epidemic peak and correspond-599 ing allocation strategies for the same η values as in Fig. 3 , assuming only a fraction γ = 0.20 of 600 E(t) + A(t) + S(t) + U (t) class can be reached by non-clinical testing. Generally, we find that when 601 non-clinical testing has limited access to the population, a larger testing capacity is required to 602 achieve a given level of controllability compared to the full testing access case. Interestingly, we 603 find that the critical threshold testing capacities at which optimal actions become a mix of clini-604 cal and non-clinical testing are equivalent to the full testing access case. This occurs because the 605 critical thresholds C th indicate the points at which the optimal fraction ρ of resources devoted to 606 non-clinical testing switches from 0 to an infinitesimal but non-zero amount, and so the associated This assumption is equivalent to increasing the non-clinical testing time equal to τ /γ = 5τ . Comparing Fig. 9a to Fig. 3a shows that limited testing access generally requires larger testing capacity to achieve a given level of peak reduction. Comparing Fig. 9b to Fig. 3b shows that for a given η value, the threshold testing capacity at which optimal strategies become a mix of clinical and non-clinical testing are equivalent under limited and full testing access.

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Appendix B: The concentration parameter η