Treffer: Optimization of testing protocols to screen for COVID-19: a multi-objective model.

In this paper we develop a new multi-objective simulated annealing (MOSA) algorithm to generate optimal testing protocols for infectious diseases, using the COVID-19 pandemic as our context. A SEIR (susceptible-exposed-infected-recovered) epidemiological model is embedded as the computational platform for our MOSA algorithm to optimize testing protocols for screening across three joint objectives: minimum cost of test materials, minimum total infections over the testing horizon, and minimum number of false negatives over the horizon. We demonstrate the application of this optimization tool to recommend screening protocols for K-12 school districts in the U.S. State of North Carolina. Our approach is scalable by population coverage and can be employed at the level of individual school districts or regional collections of districts, individual schools or collections of schools across a district, business sites, or nursing homes, among other congregate settings where individuals may be screened prior to gaining entry to the site. The algorithm can be solved two ways, generating either independent optimal protocols across individual testing locations, or a common protocol covering all locations in the collection of testing sites. Our findings can be used to inform policy decisions to guide the development of effective testing strategies for controlling the spread of COVID-19 or other pandemic diseases in a wide range of congregate settings across various geographic regions.
(© 2024. The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.)

Declarations. Conflicts of interest: The authors declare that they have no known competing personal relationships or other conflicts of interest that could have appeared to influence the work reported in this paper

AN0181643333;bse01dec.24;2024Dec17.04:08;v2.2.500

Optimization of testing protocols to screen for COVID-19: a multi-objective model: Optimization of testing protocols to screen for COVID-19: a multi-objective model: H. Moheb-Alizadeh et al 

In this paper we develop a new multi-objective simulated annealing (MOSA) algorithm to generate optimal testing protocols for infectious diseases, using the COVID-19 pandemic as our context. A SEIR (susceptible-exposed-infected-recovered) epidemiological model is embedded as the computational platform for our MOSA algorithm to optimize testing protocols for screening across three joint objectives: minimum cost of test materials, minimum total infections over the testing horizon, and minimum number of false negatives over the horizon. We demonstrate the application of this optimization tool to recommend screening protocols for K-12 school districts in the U.S. State of North Carolina. Our approach is scalable by population coverage and can be employed at the level of individual school districts or regional collections of districts, individual schools or collections of schools across a district, business sites, or nursing homes, among other congregate settings where individuals may be screened prior to gaining entry to the site. The algorithm can be solved two ways, generating either independent optimal protocols across individual testing locations, or a common protocol covering all locations in the collection of testing sites. Our findings can be used to inform policy decisions to guide the development of effective testing strategies for controlling the spread of COVID-19 or other pandemic diseases in a wide range of congregate settings across various geographic regions.

Keywords: OR in health services; Multiple objective programming; Simulated annealing; COVID-19; SEIR model

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Highlights

Introduction

Of the many lessons that government healthcare policy-makers learned from the COVID-19 pandemic, two are particularly notable. The first is that a substantial increase in broad viral testing across any population has the potential to mitigate the impact and re-emergence of the pandemic. The second is that any given nation's response to the pandemic will be more effective when coordinated by a small set of centralized organizations providing directives to a wide array of local agencies and public entities. COVID taught us that viral testing efforts by the public health community are best focused on screening — i.e., identifying infected individuals who are pre-symptomatic or asymptomatic including those people who do not have known, suspected or reported exposure to the virus [[1]]. Identifying and quarantining infected individuals by routine testing is a highly effective way to ensure that congregate settings (e.g., schools, work facilities, hospitals) are kept open and operational, as in [[3]] and [[4]]. The most effective tests to deploy in such instances are those that detect viral antigens, as they are inexpensive, easily administered and produce rapid results that are specific and sufficiently sensitive when routinely deployed [[5]]. We note that the approach we take in this paper differs from the one that was employed in the early stages of the COVID-19 pandemic in the U.S., when testing focused only on determining whether symptomatic individuals were infected with the SARS-CoV2 virus. As [[1]] and [[2]], among others, have pointed out, a symptoms-based (i.e., diagnostic) approach to testing is not as effective in containing the spread of a novel virus, as compared to screening all individuals in certain congregate settings, since symptomatic individuals may have already exposed many others to the virus between the point at which they were infected and the point at which they sought diagnostic testing.

By the summer of 2021, a variety of rapid and inexpensive, antigen-based COVID tests were available. One of the biggest problems was the lack of a single, coordinating public/private entity responsible not only for acquiring and distributing these tests, but also for educating the medical community and the public on their use. In addition, no standard methodology emerged for collecting and analyzing the resultant test data to render test distribution more effective. Although test kits were distributed initially in the U.S. by the federal government, it was left up to the state and local departments of health to determine how to allocate the limited supply of testing resources to the most hard-hit communities. These health departments were often at a loss as to how to proceed, given that they had no prior experience with this type of widespread pandemic scenario. This inability to quickly and efficiently test the population obscured the true scope of the pandemic, prevented an effective coordinated response, significantly impacted economic activity, and resulted in tremendous loss of life. This scenario was described in detail by Scott Gottlieb, who served as U.S. Food and Drug Administration Commissioner prior to the start of the COVID pandemic [[6]].

In this paper, we seek to address these problems by developing and demonstrating the use of an optimization tool that captures key tradeoffs related to the deployment of screening tests for COVID-19. Specifically, we present a multi-objective simulated annealing (MOSA) algorithm for protocol optimization, based on three joint objectives: (1) minimizing the cost of procuring test materials, (2) minimizing the total number of infections that ultimately develop over the testing horizon, and (3) minimizing the number of false negatives, given concerns related to test sensitivity for certain kinds of test kits. In addition, our MOSA code is flexible with regard to incorporating various constraints, such as limiting the maximum number of individuals in isolation (i.e., positive test outcomes) across a given testing horizon. Utilizing a SEIR (Susceptible-Exposed-Infected-Recovered) model similarly employed by Paltiel et al. [[7]] as the computational engine embedded in our algorithm, we recommend a selective set of possible screening protocols for a given population in a congregate setting. For instance, a common population of concern — and one that we explore in a computational example — might be all students and staff in K-12 schools in a given U.S. county, along the lines of Kouri et al. [[3]]. Our approach in this context can scale to various levels of population coverage, such that it can be employed at the level of, say, the school population across multiple school districts in an individual county, or for a regional collection of counties that covers all constituent schools in that region. In a multi-county setting, the optimization of the testing protocol can be solved in two ways: generating either independent optimal protocols across the individual counties, or constrained to generate a single optimal protocol covering all counties in the region. We describe the advantages of each solution method in our description of the algorithm and in our example application.

In the sections that follow, we begin by reviewing the relevant literature covering efficient allocation of medical supplies, the use of epidemiology models in establishing medical materials requirements, and the literature on relevant multi-objective optimization tools and methods. We then present our model and illustrate its application in the State of North Carolina. We conclude with a discussion of the results and how this approach may be extended in the future.

Literature review

Optimization techniques have been widely used to solve medical and health care delivery problems and span a number of healthcare policy issues. Recent work in this area includes [[8]], who developed a fuzzy cost function to find the optimal medication dosage for HIV seropositive patient treatment. A fuzzy set approach was also employed by Persi et al. [[9]], who use a decision tree fuzzy optimization method to predict the possible incidence of coronary heart disease. In a setting that involves both diagnosis and treatment of a variety of diseases, Grosan et al. [[10]] presented a multi-criteria programming model to optimize patient outcomes. Similarly, Denton et al. [[11]] developed an optimization model to find the optimal start time for statin therapy of diabetic patients. This type of work also extends to problems in areas supporting medical optimization, such as the routing inventory problem in recycling infectious medical waste investigated by Jingwei and Zujun [[12]] using fuzzy multi-objective approach.

Researchers in the field of epidemiology make wide use of optimization approaches. A welfare maximization problem from the perspective of public health planning was used by [[13]] to explore optimal responses to containing the spread of COVID-19. Measures such as mandatory quarantines were concluded to be optimal for different curvatures of the welfare function. These authors also discovered that testing can be a close substitute for quarantine, which can reduce the number of indiscriminate quarantines. A microsimulation model was used by Hoertel et al. [[14]] to evaluate the potential effect of quarantine duration, quarantine lifting policies, post-quarantine screening and use of an effective COVID-19 treatment. Similarly, a tractable quantitative analysis was performed by [[15]] to measure the effect of lockdowns and detection interventions. Using an extended Susceptible, Infectious, and Recovered compartment model (SIR) for COVID-19, the authors identified specific interventions for successive phases as an optimal lockdown policy.

SEIR is one of the prevalent mathematical algorithms used in epidemiology to describe the transmission of an epidemic disease, compartmentalizing the population of interest into categories of susceptible, exposed, infected, and recovered about the infectious disease of interest. Carcione et al. [[16]] present an excellent example of the use of the SEIR model. A more targeted SEIR model was applied by Kuniya [[17]] to predict the peak of the COVID-19 epidemic in Japan. The authors calculated the peak value using several infection rates and visualized the effect of an intervention on the peak value. However, their proposed model was limited in terms of its application only to specific regions of the country. In a broader study, Dandekar and Barbastathis [[18]] analyzed data from the U.S., China, South Korea, and Italy and applied a SEIR model using neural networks to calculate the effectiveness of quarantine policies. These authors also present a comparison between the original SEIR model and the neural network counterpart. In an extensive, multi-method approach, Rezapour et. al. [[19]] propose a SEIR model for the COVID-19 pandemic and use Caputo fractional derivatives and fractional Euler methods to approximate the steady-state solution for the system. Additionally, these authors propose a simulation model to predict the transmission of COVID-19. Data from United Kingdom was used to investigate the relationship between the mortality and number of vulnerable patients using a modified SEIR model [[20]]. A time-varying SEIR model was used by Huang et al. [[21]] to predict the severeness of COVID-19 pandemic. A new testing rate parameter was proposed by Owusu-Mensah et al. [[22]] for a fractional SEIR model. Their results demonstrated the effectiveness of tracing and testing in reducing the spread of COVID-19.

In other research utilizing SEIR, a modified model was proposed by [[23]] to evaluate the effect of non-pharmaceutical interventions during COVID-19 pandemic. They divided the infected segment of the model into three different sub-segments, namely pre-symptomatic, mild symptoms, and moderate-to-severe symptoms. The authors used a simulation framework to assess the performance of their model. SEIR models are also used to estimate the basic reproduction rate, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> , of a given infectious disease — e.g., Kochańczyk et al. [[24]]. Based on the available data from China, [[25]] developed a forecasting framework to predict the spread of COVID-19 and introduced an improved SEIR model. They then extended the SEIR model to incorporate two additional parameters (deaths and number of cured patients), which significantly improved the prediction accuracy of the model. Using quarantine, isolation and treatment, Tang et al. [[26]] introduced a general SEIR epidemiological model. In other research, Wang et al. [[27]] applied a phase-adjustment to a SEIR model to estimate the number of COVID-19 cases in Wuhan, China. The bifurcations of the periodic SEIR solutions with sinusoidally varying contact rate was investigated by Kuznetsov and Piccardi [[28]]. He and Banerjee [[29]] also studied modified SEIR models by applying a hard limited controller to a model was designed to epidemic spread. Kouri et al. [[3]] present a mathematical tool to evaluate alternative testing cadences for COVID-19, which is applicable to well-contained settings, such as long-term care facilities and public-school systems. In other research, a compartmental epidemic model was adapted by Paltiel et al. [[7]] to capture the essential features of the situation facing university decision-makers related to COVID-19. Grassly et al. [[30]] utilize infectiousness and PCR test sensitivity over time to develop a mathematical model for COVID-19 transmission. The authors used percentage reduction in the value of the transmission rate R as the key metric to evaluate the performance of each testing strategy.

Recently, [[31]], in a special journal issue focused on the role of operations research in future epidemics and pandemics [[32]], proposed a non-linear programming model to address the non-linearity of the disease progression and showed that increased levels of testing resulted in significant reductions of both hospitalizations and deaths. Finally, Zhang et al. [[33]] have shown using the SEIR model that a combination of PCR testing and rapid antigen testing can significantly reduce disease occurrence. While their work shows that the use of masks plus social distancing reduced infection rates alone, those practices in combination with testing suppressed infection rates significantly more.

Graph: Fig. 1 Closed testing network, adapted from [[7]]

Table 1 SEIR system base parameters

<table frame="hsides" rules="groups"><tbody><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>n</mi><mrow><mi mathvariant="italic">cyc</mi></mrow></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq2.gif" /></p></td><td align="left"><p>Number of testing cycles per day</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>n</mi><mrow><mi mathvariant="italic">btw</mi></mrow></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq3.gif" /></p></td><td align="left"><p>Number of cycles between tests</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>t</mi><mrow><mi mathvariant="italic">inc</mi></mrow></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq4.gif" /></p></td><td align="left"><p>Number of days for virus to incubate to infectious state</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>t</mi><mrow><mi mathvariant="italic">rec</mi></mrow></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq5.gif" /></p></td><td align="left"><p>Number of days to recover from infection</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>t</mi><mrow><mi mathvariant="italic">FPr</mi></mrow></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq6.gif" /></p></td><td align="left"><p>Number of days for false positive individual to return to testing pool (<italic>see note)</italic></p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>p</mi><mrow><mi mathvariant="italic">AS</mi></mrow></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq7.gif" /></p></td><td align="left"><p>Fraction of asymptomatic individuals that advance to the symptomatic state</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>p</mi><mrow><mi mathvariant="italic">SD</mi></mrow></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq8.gif" /></p></td><td align="left"><p>Fraction of symptomatic cases resulting in fatality</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>S</mi><mi>e</mi></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq9.gif" /></p></td><td align="left"><p>Test sensitivity</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>S</mi><mi>p</mi></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq10.gif" /></p></td><td align="left"><p>Test specificity</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>R</mi><mn>0</mn></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq11.gif" /></p></td><td align="left"><p>Basic reproductive number of the virus</p></td></tr></tbody></table>

Note: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>t</mi><mrow><mi mathvariant="italic">FPr</mi></mrow></msub><mo>&#8805;</mo><mn>1</mn></mrow></math> based on the structure of the SEIR model computations in [[7]]

There is clearly a broad body of research work that addresses public health optimization modeling, the use of epidemiological modeling like SEIR, and the development and deployment of various interventions to control the spread of epidemic diseases. Examining the literature reveals, however, that none of the previous studies offers a concrete optimization method to efficiently design specific testing protocols for infectious diseases, such as COVID-19. To the best of our knowledge, our work is the first to propose a multi-objective optimization solution for infectious disease testing protocol design.

Testing model

In this section, we describe the elements of our COVID testing model that serves as the basis for defining the screening test protocols that are deployed in congregate setting locations that span a given geographic region. Our screening model is adapted from the SEIR model employed by Paltiel et al. [[7]] in their study of proposed policies for opening college campuses for the Fall 2020 semester during the COVID-19 pandemic in the U.S. The model is shown in Fig. 1. The transitions between the system states that appear in Fig. 1 occur between successive testing cycles. Accordingly, in our modeling we denote the index for testing cycles by c, such that transitions take place as cycles transpire, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>C</mi></mrow></math> .

Table 2 SEIR system computed parameters

<table frame="hsides" rules="groups"><tbody><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>&#964;</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><msub><mi>n</mi><mrow><mi mathvariant="italic">btw</mi></mrow></msub></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq14.gif" /></p></td><td align="left"><p>Fraction of testing pool that is tested in each cycle</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>&#946;</mi><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><mo>&#183;</mo><mrow><mo stretchy="false">(</mo><mi>&#961;</mi><mo>+</mo><mi>&#963;</mi><mo stretchy="false">)</mo></mrow></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq15.gif" /></p></td><td align="left"><p>Fraction of susceptible individuals contacted and infected by infectious individuals in any given cycle</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>&#952;</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mfenced close=")" open="("><msub><mi>t</mi><mrow><mi mathvariant="italic">inc</mi></mrow></msub><mo>&#183;</mo><msub><mi>n</mi><mrow><mi mathvariant="italic">cyc</mi></mrow></msub></mfenced></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq16.gif" /></p></td><td align="left"><p>Fraction of exposed individuals that advance to infectious stage in any given cycle</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>&#961;</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mfenced close=")" open="("><msub><mi>t</mi><mrow><mi mathvariant="italic">rec</mi></mrow></msub><mo>&#183;</mo><msub><mi>n</mi><mrow><mi mathvariant="italic">cyc</mi></mrow></msub></mfenced></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq17.gif" /></p></td><td align="left"><p>Fraction of infected individuals that recover and are removed from testing in any given cycle</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>&#963;</mi><mo>=</mo><mi>&#961;</mi><mo>&#183;</mo><msub><mi>p</mi><mrow><mi mathvariant="italic">AS</mi></mrow></msub><mo stretchy="false">/</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><msub><mi>p</mi><mrow><mi mathvariant="italic">AS</mi></mrow></msub></mfenced></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq18.gif" /></p></td><td align="left"><p>Fraction of asymptomatic individuals that become symptomatic in any given cycle</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>&#956;</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mfenced close=")" open="("><msub><mi>t</mi><mrow><mi mathvariant="italic">FPr</mi></mrow></msub><mo>&#183;</mo><msub><mi>n</mi><mrow><mi mathvariant="italic">cyc</mi></mrow></msub></mfenced></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq19.gif" /></p></td><td align="left"><p>Fraction of false positive individuals who are identified and return to the testing pool in any given cycle</p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>&#948;</mi><mo>=</mo><mi>&#961;</mi><mo>&#183;</mo><msub><mi>p</mi><mrow><mi mathvariant="italic">SD</mi></mrow></msub><mo stretchy="false">/</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><msub><mi>p</mi><mrow><mi mathvariant="italic">SD</mi></mrow></msub></mfenced></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq20.gif" /></p></td><td align="left"><p>Fraction of symptomatic individuals who die from the disease in any given cycle</p></td></tr></tbody></table>

Testing demand model

In Tables 1 and 2, we define all of the parameters needed to compute the flows between the SEIR system states from period to period in Fig. 1. Since the SEIR system transitions take place over testing cycles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mi>C</mi></mrow></math> , we denote the population size of this closed system at any given test cycle c by

1 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>N</mi><mo>=</mo><msub><mi>U</mi><mi>c</mi></msub><mo>+</mo><msub><mi>E</mi><mi>c</mi></msub><mo>+</mo><msub><mi>A</mi><mi>c</mi></msub><mo>+</mo><msub><mi>P</mi><mi>c</mi></msub><mo>+</mo><msub><mrow><mi mathvariant="italic">TP</mi></mrow><mi>c</mi></msub><mo>+</mo><msub><mrow><mi mathvariant="italic">FP</mi></mrow><mi>c</mi></msub><mo>+</mo><msub><mi>V</mi><mi>c</mi></msub><mo>+</mo><msub><mi>D</mi><mi>c</mi></msub><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

which is constant across the testing horizon, i.e., for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>C</mi></mrow></math> . As one can observe from Fig. 1, individuals in this system flow between the system states that are represented by the boxes in the figure and labeled as sets. In the initial state, only sets U and A are populated, starting with values <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>0</mn></msub></math> , such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>N</mi><mo>=</mo><msub><mi>U</mi><mn>0</mn></msub><mo>+</mo><msub><mi>A</mi><mn>0</mn></msub></mrow></math> . From that point forward, transitions between states are governed by the various parameters that appear on the arrows connecting the states in Fig. 1. We also note that, as in Paltiel et al. [[7]], these flows are treated as deterministic and not probabilistic flows — effectively as expected-value transitions — and therefore, many of the flow-related, computed parameters below are defined as "fractions" and not probabilities. Below, we describe a subset of the state transitions represented by the flows across the arrows in Fig. 1

Table 3 Relation of test performance parameters and test results

<table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left"><p>Test negative</p></th><th align="left"><p>Test positive</p></th></tr></thead><tbody><tr><td align="left"><p>Patient negative</p></td><td align="left"><p>True negative</p></td><td align="left"><p>False positive</p></td></tr><tr><td align="left" /><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>Pr</mo><mo>=</mo><msub><mi>S</mi><mi>p</mi></msub></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq26.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>Pr</mo><mo>=</mo><mn>1</mn><mo>-</mo><msub><mi>S</mi><mi>p</mi></msub></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq27.gif" /></p></td></tr><tr><td align="left"><p>Patient positive</p></td><td align="left"><p>False negative</p></td><td align="left"><p>True positive</p></td></tr><tr><td align="left" /><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>Pr</mo><mo>=</mo><mn>1</mn><mo>-</mo><msub><mi>S</mi><mi>e</mi></msub></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq28.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>Pr</mo><mo>=</mo><msub><mi>S</mi><mi>e</mi></msub></mrow></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq29.gif" /></p></td></tr></tbody></table>

The computations detailed in Appendix A incorporate two important parameters used in epidemiology to describe the accuracy of tests for the presence or absence of disease in a patient specimen. Those parameters are test sensitivity and specificity, which we denote by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>e</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>p</mi></msub></math> , respectively. They are defined as follows:

Mathematically, those parameters relate test specimens to test outcomes as described in Table 3.

To aid in understanding the flows along the arrows in the system described in Fig. 1, we walk through the details related to some of the key initial transitions in the system, particularly the transitions of individuals into and out of states U and A, as those are the key starting-point transitions in the model. Thus, we note that, over time, individuals who began in states U and A can flow out of or into these states via several mechanisms:

For details on the computations that govern the entire set of cycle-to-cycle transitions, we refer the reader to Appendix A. We observe here, however, that based on the flows shown in Fig. 1, states V and D are absorbing states. This means that given a sufficiently long period of time, all individuals in the system will end up as either recovered from an infection or deceased as the result of an infection.

Demand for test kits used in screening for the virus is defined as the number of screening tests administered in testing cycle c. This is determined by the sum of the number of individuals in states U, A, and E, as reflected in Fig. 1. In the computational modeling from [[7]], it is assumed that tests are administered at the beginning of each cycle according to the updates to all states in the system from the prior cycle, and therefore,

2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mtext>Screening tests in cycle</mtext><mspace width="0.333333em" /><mi>c</mi><mo>=</mo><msub><mi>U</mi><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>E</mi><mrow><mi>c</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

As discussed above, the number of confirmatory tests to be administered is determined by the the number of individuals in states FP and TP, which is defined as:

3 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mtext>Confirmatory tests in cycle</mtext><mspace width="0.333333em" /><mi>c</mi><mo>=</mo><mi>F</mi><msub><mi>P</mi><mi>c</mi></msub><mo>+</mo><mi>T</mi><msub><mi>P</mi><mi>c</mi></msub><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

Consistent with SEIR models in the field of epidemiology [[26], [27]–[28]], there is no "within-system" testing of symptomatic individuals — i.e., individuals in set P are not included in the testing, as they are assumed to be in isolation. Instead, we assume that symptomatic individuals will behave responsibly, not attempt to enter the congregate setting, and seek a diagnostic test outside the bounds of this testing system. Attempting to capture "bad behavior" would require additional assumptions regarding the fraction of symptomatic individuals that attempt to enter the congregate setting, and we have chosen to avoid such complications in our model. We note, however, that, in Section 5, we conduct a robustness analysis on the exogenous infections parameter in the model, which can be viewed a proxy for various "bad behaviors" that expose the congregate setting to more infections than it would otherwise generate endogenously. Moreover, as of this writing, the guidelines of the U.S. Centers for Disease Control (CDC) [[34]] preventing the spread of respiratory viruses, including SARS-CoV-2, are dramatically relaxed from earlier guidelines, and could result in honest mistakes by parents (e.g., lingering symptoms or an incorrectly executed home test) which lead to similar "leaks" in the system that generate additional infections to be captured in the screening. Finally, our analysis assumes no confirmatory antibody testing is required for recovered individuals (set V), consistent with public health practice throughout the course of the COVID-19 pandemic in the U.S. [[27], [30]]. Finally, this particular model assumes that infected individuals are truly "recovered," and do not require further testing. This was the assumed behavior of COVID-19 in the earlier period of the pandemic and it is the basis of the computations in this paper.

Decision variables

We denote the length of the testing horizon by T, which may be measured in a different time scale than the testing cycles, denoted earlier by c. This simply requires a conversion factor that sets the number of testing cycles, for example, per day or per week. For a given screening horizon length, T, the variables that define a screening testing protocol are as follows:

There are many reasons to split the testing horizon into two portions. Two of the main reasons are as follows. First, per the nature of epidemiological system, infections tend to "bloom" in the early part of the overall testing horizon. It takes some time for the testing and isolation procedure to succeed at identifying infections and isolating people to reduce the rate of spread for the remainder of the horizon (provided testing occurs at a sufficient frequency to achieve this outcome). In that case, testing can ease off after some initial period. Second, a less frequent rate of testing means fewer tests being administered which results in a lower testing cost. Thus, splitting the testing horizon provides flexibility to public officials in charge of testing to create overall testing protocols that are most effective in terms of the trade-off between infections and testing costs.

Objectives and optimization structure

As we discuss above, two important criteria for assessing any screening protocol across a given testing horizon are the number of resulting infections and the cost of testing. We include an additional criterion in our multi-objective approach, namely the number of false negatives. We include this objective in order to address criticism of antigen-based COVID-19 tests (also known as "point-of-care" tests or "at-home" tests in the U.S.) particularly early in the pandemic in the U.S. Antigen tests have lower sensitivity than lab-based testing methods (e.g., PCR tests), but also have the advantages of being not only dramatically less expensive but also, as their name suggests, rapid to administer and generate results within 30 minutes. These issues are discussed in [[1]] and [[2]]. Since lower sensitivity will tend to generate a higher number of false negatives, we include this as the third of our objectives.

Therefore, the objectives for our multi-objective optimization problem are defined as follows:

Computationally, total cost, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><msub><mi>C</mi><mrow><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub></mrow></math> , is the easiest of these to compute, as it is given simply by the product of the number of tests administered and the cost per test. In contrast, computing the other objectives is more complicated, but still straightforward. Specifically, the expression for the number of infections in cycle <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></math> (see Appendix A) is

4 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>I</mi><msub><mi>N</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><munder><munder accentunder="true"><mrow><mi>&#946;</mi><mo>&#183;</mo><mfrac><mrow><msub><mi>A</mi><mi>c</mi></msub><mo>&#183;</mo><msub><mi>U</mi><mi>c</mi></msub></mrow><mrow><msub><mi>U</mi><mi>c</mi></msub><mo>+</mo><msub><mi>A</mi><mi>c</mi></msub><mo>+</mo><msub><mi>E</mi><mi>c</mi></msub></mrow></mfrac></mrow><mo>&#9183;</mo></munder><mrow><mtext>New infections</mtext></mrow></munder><mo>+</mo><munder><munder accentunder="true"><mrow><mi>X</mi><mo>&#183;</mo><msub><mi>I</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo></mrow><mo>&#9183;</mo></munder><mrow><mtext>Exogenous shocks</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

Graph

where the decision triple <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math> on this decision variable is suppressed for brevity, and the expression for false negatives in testing cycle <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></math> is:

5 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>F</mi><msub><mi>N</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfenced close="]" open="["><msub><mi>E</mi><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></msub><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><msub><mi>S</mi><mi>e</mi></msub></mfenced></mfenced><mo>&#183;</mo><mi>&#964;</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

where, as we define in Appendix A, k is the number of cycles required to obtain test results. Note that false negatives include all individuals in set E — i.e., exposed to the virus but currently undetectable due to the relative insensitivity of the test, based on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>e</mi></msub></math> — in addition to a fraction of set A (Asymptomatic), specifically <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>-</mo><msub><mi>S</mi><mi>e</mi></msub><mo stretchy="false">)</mo></mrow></math> , who test negative due to the test sensitivity error level.

Therefore, we can use these defined objectives to formulate the following multi-objective programming model in order to design the most efficient testing protocol that optimizes simultaneously our performance criteria:

6 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><munder><mo movablelimits="true">min</mo><mrow><msub><mi>F</mi><mn>1</mn></msub><mo>&#8712;</mo><msup><mi>S</mi><msub><mi>F</mi><mn>1</mn></msub></msup><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo>&#8712;</mo><msup><mi>S</mi><msub><mi>F</mi><mn>2</mn></msub></msup><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub><mo>&#8712;</mo><msup><mi>S</mi><msub><mi>T</mi><mn>1</mn></msub></msup></mrow></munder><mrow><mo stretchy="false">{</mo><mi>I</mi><msub><mi>N</mi><mrow><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>F</mi><msub><mi>N</mi><mrow><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>T</mi><msub><mi>C</mi><mrow><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">}</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> are the decision variables, as defined above, and where <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>F</mi><mn>1</mn></msub></msup></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>F</mi><mn>2</mn></msub></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>T</mi><mn>1</mn></msub></msup></math> denote the sets of feasible quantities for decision variables <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> , respectively. For example, a possible choice for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>F</mi><mn>1</mn></msub></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>F</mi><mn>2</mn></msub></msup></math> might be <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">{</mo><mtext mathvariant="italic">twice</mtext><mspace width="0.333333em" /><mtext mathvariant="italic">per</mtext><mspace width="0.333333em" /><mtext mathvariant="italic">day</mtext><mo>,</mo><mtext mathvariant="italic">daily</mtext><mo>,</mo><mtext mathvariant="italic">every</mtext><mspace width="0.333333em" /><mtext mathvariant="italic">two</mtext><mspace width="0.333333em" /><mtext mathvariant="italic">days</mtext><mo stretchy="false">}</mo></mrow></math> , whereas <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>T</mi><mn>1</mn></msub></msup></math> might be defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">{</mo><mi>t</mi><mo>&#8712;</mo><mi mathvariant="script">P</mi><mo stretchy="false">|</mo><mn>0</mn><mo>&#8804;</mo><mi>t</mi><mo>&#8804;</mo><mi>T</mi><mo stretchy="false">}</mo></mrow></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">P</mi></math> is the set of non-negative integers and T is the maximum length of time defined in the study. Thus, the optimization defined in expression (6) specifies the optimal values for the initial screening frequency, the secondary screening frequency, and the length of initial screening horizon in such a way that total infections, total false negatives and total cost are jointly minimized. We introduce a multi-objective simulated annealing algorithm in the next section in order to derive the Pareto optimal front for this multi-objective optimization problem.

Optimization algorithm

Simulated annealing, initially proposed by Kirkpatrick et al. [[35]], Johnson et al. [[36]], is a local search algorithm in which non-improving moves are probabilistically accepted in an attempt to avoid becoming trapped in the region of a low-quality, locally-optimal solution and prematurely terminating the algorithm. The algorithm is a heuristic technique drawing on an analogy from physical annealing processes, such as those used in fabricating steel components. It was originally designed for solving single objective optimization problems and was proven to be robust and convergent if the annealing rate that governs the search progress is sufficiently slow [[38]]. The first multi-objective version of simulated annealing was proposed by Serafini [[39]], with a modification of the acceptance criteria of solutions from the original algorithm developed in Serafini [[40]]. Engrand [[41]] proposed maintaining an external archive populated by all non-dominated solutions found at the current point in time in the search process. Subsequently, several Multi-Objective Simulated Annealing (MOSA) methods were developed that incorporated a dynamically updated Pareto set (e.g. Czyzżak and Jaszkiewicz [[42]] and Hapke et al. [[43]]). The solution acceptance criteria of these methods are all derived from the difference between the new and current solutions. However, in the presence of a Pareto set spanning multiple objectives, solely comparing the new solution to the current solution is not necessarily straightforward. That is why there have been a few techniques proposed that use a Pareto domination-based acceptance criterion in MOSA [[44], [46]–[47]], the salient feature of which is that the domination status of a given solution is considered not only with respect to the current solution but also to the archive of non-dominated solutions found up to that point in time. It has been widely demonstrated that simulated annealing algorithms are capable of finding multiple Pareto-optimal solutions in a single run. A detailed survey on single-objective and multi-objective simulated annealing can be found in Suman and Kumar [[48]].

Solution process

As we mention in Section 1, our solution approach is scalable in terms of testing population coverage, and can be employed at the level of a small, focused population, or across several populations that might represent, for example, a collection of school districts or multiple sites of a business firm. In the computational example that we present in Section 5, we provide the details related to identifying the appropriate population values that represent testing in public schools in the U.S. state of North Carolina, in which school districts are organized almost exclusively at the level of counties within the state. Thus, one issue that must be addressed in our solution process is whether a solution that is being applied to a given collection of counties should be solved independently across those counties — searching for the independently optimal solution in each county — or to seek the single, common set of policy variables that provides the best trade-offs of the multiple objectives when implemented across all counties. Given the nature of our policy variables, we refer to a given solution as a testing protocol, which we will, from here forward, refer to simply as a protocol.

Algorithm formulation

In this section, we present a formulation of the MOSA algorithm that allows us to address the two distinct scenarios we describe above — a common protocol across all population sets versus independent protocols across those population sets. The whole formulation is based on an archive-based MOSA initially proposed by Bandyopadhyay et al. [[47]] that uses a dominance measure to compute the probability of acceptance of a new solution. In the first scenario, the original archive-based MOSA [[47]] is adapted in such a way that a common protocol is implemented over all the population centers in the chosen cluster of separate populations. However, the original archive-based MOSA algorithm is independently executed for each population to derive its corresponding Pareto optimal solutions. In other words, while a common protocol is enforced over all populations in the first scenario, each population follows its individually designed protocol in the second scenario. We investigate the difference between the computational outcomes generated by these two scenarios in further detail in Section 5.

We define the following parameters to be used in specifying our archive-based MOSA algorithm:

Before elaborating the steps of the algorithm, we describe and/or formally define some terms common to archive-based MOSA algorithms.

Definition 1

Domination - In the context of a minimization pro-blem, given the set of objective functions as <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">{</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo></mrow><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>f</mi><mi>m</mi></msub><mrow><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo></mrow><mo stretchy="false">}</mo></mrow></math> , a solution X dominates solution <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>X</mi><mo>&#8242;</mo></msup></math> , denoted as <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>&#8827;</mo><msup><mi>X</mi><mo>&#8242;</mo></msup></mrow></math> in this study, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&#8704;</mo><mi>k</mi><mo>&#8712;</mo><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>m</mi><mo stretchy="false">}</mo></mrow><mo>,</mo><msub><mi>f</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo></mrow><mo>&#8804;</mo><msub><mi>f</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><msup><mi mathvariant="normal">x</mi><mo>&#8242;</mo></msup><mo stretchy="false">)</mo></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&#8707;</mo><mi>k</mi><mo>&#8712;</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>m</mi><mo stretchy="false">}</mo></mrow></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>f</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo></mrow><mo>&#60;</mo><msub><mi>f</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><msup><mi mathvariant="normal">x</mi><mo>&#8242;</mo></msup><mo stretchy="false">)</mo></mrow></mrow></math> .

For example, if we have three minimizing objective functions whose values for solutions X and Y are (12.47, 3.18, 0.94) and (12.47, 3.64, 1.07), respectively, then according to the aforementioned definition, solution X dominates solution Y, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>&#8827;</mo><mi>Y</mi></mrow></math> .

A solution that is not dominated by other existing solutions is referred to as a non-dominated solution. Continuing the example above, if we identify another solution, Z, whose solution values are (12.21, 3.43, 0.92), then solutions X and Z are non-dominated, but solution Y remains dominated by the other two solutions. Two non-dominated solutions X and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>X</mi><mo>&#8242;</mo></msup></math> are denoted as <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>&#8776;</mo><msup><mi>X</mi><mo>&#8242;</mo></msup></mrow></math> in this study. The set of non-dominated solutions constitutes the Pareto optimal solution set or Pareto optimal frontier of the problem. The amount of domination between two solutions X and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>X</mi><mo>&#8242;</mo></msup></math> , which will be used in computing the acceptance probability of a new solution, is calculated as follows [[47]]:

7 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi mathvariant="normal">&#916;</mi><mi>d</mi><mi>o</mi><msub><mi>m</mi><mrow><mi>X</mi><mo>,</mo><msup><mi>X</mi><mo>&#8242;</mo></msup></mrow></msub><mo>=</mo><munderover><mo>&#8719;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mfrac><mrow><mrow><mo stretchy="false">|</mo></mrow><msub><mi>f</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo></mrow><mo>-</mo><msub><mi>f</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><msup><mi mathvariant="normal">x</mi><mo>&#8242;</mo></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">|</mo></mrow></mrow><msub><mi>R</mi><mi>i</mi></msub></mfrac></mrow></mtd></mtr></mtable></mrow></math>

Graph

where <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>f</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo></mrow><mo>&#8800;</mo><msub><mi>f</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><msup><mi mathvariant="normal">x</mi><mo>&#8242;</mo></msup><mo stretchy="false">)</mo></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>i</mi></msub></math> is the range of the i-th objective function. As <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>i</mi></msub></math> might be unknown a priori, all the non-dominated solutions in the archive along with the current and new solutions are employed to calculate it. In other words, given <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="double-struck">X</mi></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>X</mi><mo>&#8242;</mo></msup></math> , and X denote the the existing archive, new solution and current solution, respectively, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>R</mi><mi>i</mi></msub><mo>=</mo><mo movablelimits="true">max</mo><mspace width="0.166667em" /><msub><mi>f</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>-</mo><mo movablelimits="true">min</mo><mspace width="0.166667em" /><msub><mi>f</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>&#8712;</mo><mi mathvariant="double-struck">X</mi><mo>&#8746;</mo><mi>X</mi><mo>&#8746;</mo><msup><mi>X</mi><mo>&#8242;</mo></msup></mrow></math> .

Clustering - Clustering is a mechanism that is used to build the MOSA archive in a way that results in a diverse set of non-dominated solutions. Given HL and SL parameters, the size of the archive potentially rises to SL, at which point a clustering procedure is executed to reduce the archive's size to HL. Bandyopadhyay et al. [[47]] recommended using the single linkage algorithm [[49]] as the clustering procedure in their archive MOSA algorithm, where the distance between any two clusters corresponds to the length of the shortest Euclidean distance from any solution in the first cluster to any solution in the second cluster. Given an archive and HL parameter, the clustering operator is denoted by Cluster(archive, HL) throughout this study.

In the following sections, we describe our general MOSA algorithm and its specific implementation with respect to the multiple objective values for our testing protocol optimization problem. With only a slight change in the execution of the algorithm, it can be used to generate a set of Pareto frontiers for either the common-protocol or independent-protocol settings, as we discussed in Section 4.1.

Scenario 1 – Implementing a common protocol

In this section, we step through each of the major components of the MOSA algorithm — initialization, evaluation and perturbation, transition, and temperature updating — in the context of the first of our two solution scenarios, the one for which a common protocol is implemented by all populations in the set being considered. These steps are elaborated more formally in the pseudocode provided in Algorithm 1. Throughout the discussion below, note that we make reference to specific lines that can be found in Algorithm 1.

Graph: Algorithm 1 Multi-objective simulated annealing - scenario 1.

<bold>Initialization</bold> – The algorithm begins with the initialization of a number of non-dominated solutions in line 3. Algorithm 2 elaborates how to initialize an archive for Scenario 1. In this algorithm, a random solution for each decision variable <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> is generated from the corresponding feasible sets <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>F</mi><mn>1</mn></msub></msup></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>F</mi><mn>2</mn></msub></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><msub><mi>T</mi><mn>1</mn></msub></msup></math> , respectively, using the random operator in lines 4 to 6. Since such a solution is implemented across all populations in Scenario 1, the total value of objective functions for all populations is calculated in lines 8 to 10. Based on the total value of objective functions and Definition 1, the dominance status of the current random solution is assessed with respect to all the previous solutions in the archive. If the current random solution is non-dominant compared to all other solutions in the archive, then it is added to the archive. This operation is executed in lines 13 to 20 in Algorithm 2. Finally, the archive's size is reduced to HL in line 25 by means of clustering if it goes beyond SL.

Graph: Algorithm 2 InitScen1 - initialization of an archive in scenario 1.

<bold>Evaluation and perturbation</bold> – After picking a random non-dominated solution from the archive as the current solution (line 4), the values of objective functions associated with the current solution are obtained (line 5) using the ObjEval procedure outlined in Algorithm 3. During this stage, it is crucial to highlight that the entire SEIR procedure, as depicted in Fig. 1, is executed to acquire the objective function values pertaining to a randomly selected non-dominated solution. Given a solution and a set of populations, Algorithm 3 generates the solution for each population, p, sums up the corresponding values of each objective function over all populations, and ultimately returns a tuple of the total values of objective functions, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">{</mo><mi>o</mi><mi>b</mi><msup><mi>j</mi><mrow><mi mathvariant="italic">TC</mi></mrow></msup><mo>,</mo><mi>o</mi><mi>b</mi><msup><mi>j</mi><mrow><mi mathvariant="italic">FN</mi></mrow></msup><mo>,</mo><mi>o</mi><mi>b</mi><msup><mi>j</mi><mrow><mi mathvariant="italic">IN</mi></mrow></msup><mo stretchy="false">}</mo></mrow></math> . Although Algorithm 3 seems relatively straightforward, it is the basis for distinguishing between the Scenario 1 and Scenario 2 solutions, as we describe in Section 4.2.2.

Graph: Algorithm 3 ObjEval - evaluating the objective functions.

Next, a new solution is generated (line 8 of Algorithm 1) by perturbing the current solution. Recall that any solution stored in the archive is a 3-tuple of decision variables that specifies the initial screening frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub></math> , the secondary screening frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub></math> , and the length of the initial screening frequency, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> . Thus, seven perturbation mechanisms may be applied to perturb the current solution: (1-3) perturbing any of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub></math> , or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub></math> individually; (4-6) jointly perturbing any pair of these decision variables, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> , or <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> ; (7) jointly perturbing all three decision variables, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> .

The MOSA algorithm randomly selects one of these mechanisms to be executed, and the variable(s) in question are perturbed within their corresponding feasible domains. Then, the values of objective functions related to the newly generated solution are obtained by applying Algorithm 3 (line 9).

<bold>Transition</bold> – Given current and new solutions with their corresponding objective function values, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>O</mi><mi>b</mi><msub><mi>j</mi><mi>c</mi></msub></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>O</mi><mi>b</mi><msub><mi>j</mi><mi>n</mi></msub></mrow></math> , three different outcomes are possible in terms of the domination status between them:

Graph: Algorithm 4 DomStatus - specifying the domination status of a new solution against an archive.

Before describing the actions of the MOSA algorithm in each of these cases, we first note that, prior to the evaluation of how the current and new solutions compare, Algorithm 1 at line 10 derives K and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">&#175;</mo></mrow></mover></math> by means of Algorithm 4, where K and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">&#175;</mo></mrow></mover></math> are defined as follows:

Now, we can describe the details of the three comparison outcomes:

The internal for statement iterates iter times, such that, at a fixed temperature, iter different perturbed solutions in the neighborhood of the current solution are investigated.

<bold>Updating temperature</bold> – After examining the neighborhood of the current solution, the temperature is reduced in line 59, where a multiplier <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mi>a</mi><mi>n</mi><msub><mi>d</mi><mrow><mi>U</mi><mo stretchy="false">(</mo><msup><mi>&#945;</mi><mrow><mi mathvariant="italic">lower</mi></mrow></msup><mo>,</mo><msup><mi>&#945;</mi><mrow><mi mathvariant="italic">upper</mi></mrow></msup><mo stretchy="false">)</mo></mrow></msub></mrow></math> is used to smoothly reduce the annealing temperature. Our experiments showed that this cooling schedule works sufficiently well when deriving the Pareto optimal front.

At the end of the algorithm, if archive's size is greater than the SL, the clustering procedure reduces its size to HL (line 62).

Scenario 2 – Implementing independent protocols

As mentioned earlier, Scenario 2 describes a situation where each population can have its own protocol designed independently from other counties. In this circumstance, we need only implement the original archive MOSA algorithm on each population to derive its respective Pareto optimal solution. In other words, if the set POP in Algorithms 1 and 3 contains only a single population, then running Algorithm 1 n times, once for each of the n populations in the collection of all populations, yields the independently Pareto optimal solutions for each population.

Computational application and results

To demonstrate our implementation of the SEIR model and its use in developing optimal screening protocols for infectious diseases in congregate settings, we have built a realistic case scenario based on screening for COVID-19 in public schools. Here, we use the State of North Carolina as our case setting. Estimates of the student and staff population of schools in the United States were obtained from the "Global Epidemics" website, maintained by the Brown University School of Public Health [[50]].

Graph: Fig. 2 Prioritized clusters of counties in North Carolina using data from Aug 2020 - Jul 2021 (panel (a)) and Aug 2020 - Feb 2022 (panel (b)) — darker shading represents higher intensity of COVID-19 pandemic in that area

Population data for public school screening in North Carolina

Although the "Global Epidemics" site provides data on school populations by state (actually through estimates provided by the National Education Association—see [[51]]), we chose to narrow the focus to the county level within the state, as policy decisions in schools are made at local levels in the U.S., and in North Carolina most school systems are organized at the county level. This leaves us with a number of modeling and solution decisions to consider, namely:

In order to identify clusters of counties to serve as target solution areas, we employ COVID-19 prediction tools developed by [[52]]. These tools generate a prioritization of counties within a given state in the U.S. via a weighted average of two computational methods for assessing the relative intensity of the COVID-19 pandemic across these counties. One computational method employs machine learning algorithms to predict COVID-19 hospitalizations and deaths based on a large number of county-level statistics regarding various public health measures, demographic factors, and local weather data. The other method utilizes the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), initially introduced by [[53]], to generate an overall priority weighting among multiple measures, in this case, data on COVID-19 hospitalizations and deaths in the weeks leading up to the week for which relative severity of the COVID-19 pandemic is being assessed.

Applying these prioritization methods to all 100 counties in the State of North Carolina, we generated the weighted average priority ranking of all 100 counties. Then, we employed the constrained clustering algorithm [[54]] to create 10 regional clusters of counties, each of which represents a geographical area that is sized in a way that allows us to answer the questions we pose above. Prioritized clusters are shown in Fig. 2 for data covering August 2020 to July 2021 (panel (a)) and August 2020 to February 2022 (panel (b)). The cluster that serves as the basis for our computations is highlighted in panel (a) of the figure, where we also provide the names of the individual counties. We note that this cluster — which includes Wake County, containing the state's capital and second-largest city, Raleigh — was the highest priority cluster in the state (i.e., highest relative intensity of the pandemic) in the period around July 2021, and the second highest priority cluster around February 2022.

In Table 4, we provide estimates of the testing population in the counties shown in Fig. 2. The source of the estimates of the overall school population for the State of North Carolina — which includes both students and staff — come from [[50]]. We allocate the statewide school population across counties by assuming that the school population follows the proportion of the total population in each county relative to the overall population of the state. State and county population statistics were obtained from [[55]]. We assume that each county's school population starts with 5% of students and staff infected with COVID-19 at the outset of the screening. In the context of the SEIR model (Fig. 1), this means that 95% of the population starts the horizon in state U and 5% in state A.

Table 4 School population estimates by county for high-priority cluster in North Carolina

<table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left"><p>Initial</p></th><th align="left"><p>Initial</p></th></tr><tr><th align="left" /><th align="left"><p>County</p></th><th align="left"><p>Proportion of</p></th><th align="left"><p>Estimated</p></th><th align="left"><p>Estimated</p></th><th align="left"><p>Estimated</p></th><th align="left"><p>susceptible</p></th><th align="left"><p>infected</p></th></tr><tr><th align="left"><p>Entity</p></th><th align="left"><p>population</p></th><th align="left"><p>NC population</p></th><th align="left"><p>Grades K-8</p></th><th align="left"><p>Grades 9-12</p></th><th align="left"><p>total</p></th><th align="left"><p>(95% of total)</p></th><th align="left"><p>(5% of total)</p></th></tr></thead><tbody><tr><td align="left"><p>State of North Carolina</p></td><td align="left"><p>&#8211;</p></td><td align="left"><p>&#8211;</p></td><td align="left"><p>1,168,436</p></td><td align="left"><p>362,635</p></td><td align="left"><p>1,531,071</p></td><td align="left"><p>&#8211;</p></td><td align="left"><p>&#8211;</p></td></tr><tr><td align="left"><p>Alamance county</p></td><td align="left"><p>169,509</p></td><td align="left"><p>0.0162</p></td><td align="left"><p>18,884</p></td><td align="left"><p>5,861</p></td><td align="left"><p>24,745</p></td><td align="left"><p>23,508</p></td><td align="left"><p>1,237</p></td></tr><tr><td align="left"><p>Caswell county</p></td><td align="left"><p>22,604</p></td><td align="left"><p>0.0022</p></td><td align="left"><p>2,518</p></td><td align="left"><p>782</p></td><td align="left"><p>3,300</p></td><td align="left"><p>3,135</p></td><td align="left"><p>165</p></td></tr><tr><td align="left"><p>Chatham county</p></td><td align="left"><p>74,470</p></td><td align="left"><p>0.0071</p></td><td align="left"><p>8,296</p></td><td align="left"><p>2,575</p></td><td align="left"><p>10,871</p></td><td align="left"><p>10,327</p></td><td align="left"><p>544</p></td></tr><tr><td align="left"><p>Durham county</p></td><td align="left"><p>321,488</p></td><td align="left"><p>0.0307</p></td><td align="left"><p>35,816</p></td><td align="left"><p>11,116</p></td><td align="left"><p>46,931</p></td><td align="left"><p>44,584</p></td><td align="left"><p>2,347</p></td></tr><tr><td align="left"><p>Granville county</p></td><td align="left"><p>60,443</p></td><td align="left"><p>0.0058</p></td><td align="left"><p>6,734</p></td><td align="left"><p>2,090</p></td><td align="left"><p>8,824</p></td><td align="left"><p>8,383</p></td><td align="left"><p>441</p></td></tr><tr><td align="left"><p>Harnett county</p></td><td align="left"><p>135,976</p></td><td align="left"><p>0.0130</p></td><td align="left"><p>15,149</p></td><td align="left"><p>4,701</p></td><td align="left"><p>19,850</p></td><td align="left"><p>18,858</p></td><td align="left"><p>993</p></td></tr><tr><td align="left"><p>Lee county</p></td><td align="left"><p>61,779</p></td><td align="left"><p>0.0059</p></td><td align="left"><p>6,883</p></td><td align="left"><p>2,136</p></td><td align="left"><p>9,019</p></td><td align="left"><p>8,568</p></td><td align="left"><p>451</p></td></tr><tr><td align="left"><p>Orange county</p></td><td align="left"><p>148,476</p></td><td align="left"><p>0.0142</p></td><td align="left"><p>16,541</p></td><td align="left"><p>5,134</p></td><td align="left"><p>21,675</p></td><td align="left"><p>20,591</p></td><td align="left"><p>1,084</p></td></tr><tr><td align="left"><p>Person county</p></td><td align="left"><p>39,490</p></td><td align="left"><p>0.0038</p></td><td align="left"><p>4,399</p></td><td align="left"><p>1,365</p></td><td align="left"><p>5,765</p></td><td align="left"><p>5,477</p></td><td align="left"><p>288</p></td></tr><tr><td align="left"><p>Wake county</p></td><td align="left"><p>1,111,761</p></td><td align="left"><p>0.1060</p></td><td align="left"><p>123,857</p></td><td align="left"><p>38,440</p></td><td align="left"><p>162,297</p></td><td align="left"><p>154,182</p></td><td align="left"><p>8,115</p></td></tr></tbody></table>

Case parameters for public school screening

Below, we provide details on the case parameters that we explore in our computational experiments. The parameter values chosen are based on a combination knowledge of the tests employed (e.g., sensitivity and specificity of antigen tests and their costs when purchased in bulk), on the nature of the SEIR system described in Fig. 1, or by using virus behavior parameters that match those in [[7]].

<uline>Test parameters</uline>

<uline>Testing environment parameters</uline>

<uline>Virus parameters</uline>

<uline>Decision variable sets</uline>

Graph: Fig. 3 Independent-protocol versus common-protocol solutions for Caswell County, NC

Graph: Fig. 4 Independent-protocol versus common-protocol solutions for Wake County, NC

Finally, we perform robustness tests on parameters that reflect the environment in which the screening policy is being employed. Those key parameters are the number of infections introduced into the congregate setting in question from outside the system and the level of infectiousness of the virus, as measured by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> , which epidemiologists call the "basic reproductive number" of a given virus (see [[56]]) The first robustness parameter that we explore stems from the structure of the underlying SEIR model of virus transmission. This parameter, which is denoted by X in Fig. 1, is the fixed number of new infections that enter the system every q testing cycles. We use the same terminology as [[7]], referring to these as exogenous shocks of new infections. In exploring the sensitivity of our 10-county solutions to the level of exogenous shocks, we base the computation of X on the underlying population of the testing pool. Specifically, we consider the base level to be 1% of the testing pool, and explore two levels less than this, 0.5% and 0.1%, and two larger levels, 2% and 6.25%, the last of which would represent a level of introduced infections that equals the total population (100%) over a 16-week school semester. In addition, we consider three levels of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> . The first, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>R</mi><mn>0</mn></msub><mo>=</mo><mn>2.3</mn></mrow></math> , reflects epidemiologists' estimates of the infectiousness of the SARS-CoV-2 virus in the first year of the COVID-19 pandemic, during which the original, "wild-type" variant of the virus was circulating [[57]]. The lower level, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>R</mi><mn>0</mn></msub><mo>=</mo><mn>1.5</mn></mrow></math> , is intended to represent a level that could be expected in a congregate setting in which most individuals follow good hygiene, social distancing, and masking [[57]]. The higher level, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>R</mi><mn>0</mn></msub><mo>=</mo><mn>5.0</mn></mrow></math> , is intended to represent virus spread at a level that was seen later in the COVID-19 pandemic, as variants of the virus mutated in ways that helped it spread much more rapidly [[58]].

<uline>Robustness parameters</uline> (base case value in bold)

Generating Pareto-optimal protocols for public school screening in North Carolina

We employ the MOSA algorithms, described in Section 4, to search for non-dominated protocols for screening, based on the joint objectives of minimizing testing costs, minimizing infections, and minimizing false negative tests. We employ two solution methods, one that searches for the optimal protocol in each county, independently of the protocols in any other county in the overall target set of counties, and one that constrains the counties to have a common protocol.

The common-protocol approach has practical benefits, both from the standpoint of more easily solving the problem and from the standpoint of implementing the protocol from a public health perspective. On the latter point, we can assume that the public officials in the State of North Carolina would prefer to see more consistent testing protocols employed across counties, as opposed to having protocols that differ across counties, particularly between adjoining counties. Public health officials would clearly prefer to explain a commonly-deployed protocol for screening tests in county schools, as opposed to having to explain why County A's protocol differs from that of adjoining County B.

Table 5 Comparative solutions for common and independent protocols – Caswell County, NC

<table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left"><p>Cost per</p></th></tr><tr><th align="left" /><th align="left"><p>Initial</p></th><th align="left"><p>Weeks of</p></th><th align="left"><p>Secondary</p></th><th align="left" /><th align="left" /><th align="left"><p>Number of</p></th><th align="left"><p>Maximum</p></th><th align="left"><p>Cost per</p></th><th align="left"><p>reduction in</p></th></tr><tr><th align="left" /><th align="left"><p>screening</p></th><th align="left"><p>initial</p></th><th align="left"><p>screening</p></th><th align="left"><p>Testing</p></th><th align="left"><p>Number of</p></th><th align="left"><p>false</p></th><th align="left"><p>number in</p></th><th align="left"><p>case</p></th><th align="left"><p>false</p></th></tr><tr><th align="left"><p>Solution</p></th><th align="left"><p>frequency</p></th><th align="left"><p>screening</p></th><th align="left"><p>frequency</p></th><th align="left"><p>cost</p></th><th align="left"><p>infections</p></th><th align="left"><p>negatives</p></th><th align="left"><p>isolation</p></th><th align="left"><p>averted</p></th><th align="left"><p>negatives</p></th></tr></thead><tbody><tr><td align="left"><p>Common</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>11</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>$423,105</p></td><td align="left"><p>1,802</p></td><td align="left"><p>66</p></td><td align="left"><p>200</p></td><td align="left"><p>$317</p></td><td align="left"><p>$6,899</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Every 3 Weeks</p></td><td align="left"><p>$435,852</p></td><td align="left"><p>1,720</p></td><td align="left"><p>62</p></td><td align="left"><p>200</p></td><td align="left"><p>$308</p></td><td align="left"><p>$6,643</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>12</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$465,751</p></td><td align="left"><p>1,551</p></td><td align="left"><p>59</p></td><td align="left"><p>200</p></td><td align="left"><p>$294</p></td><td align="left"><p>$6,748</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>16</p></td><td align="left"><p>Every 2 Weeks</p></td><td align="left"><p>$498,249</p></td><td align="left"><p>1,460</p></td><td align="left"><p>56</p></td><td align="left"><p>200</p></td><td align="left"><p>$297</p></td><td align="left"><p>$6,950</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$520,607</p></td><td align="left"><p>1,443</p></td><td align="left"><p>56</p></td><td align="left"><p>200</p></td><td align="left"><p>$308</p></td><td align="left"><p>$7,244</p></td></tr><tr><td align="left"><p>Independent</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>2</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>$405,831</p></td><td align="left"><p>1,755</p></td><td align="left"><p>67</p></td><td align="left"><p>287</p></td><td align="left"><p>$294</p></td><td align="left"><p>$6,785</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>$455,370</p></td><td align="left"><p>1,594</p></td><td align="left"><p>59</p></td><td align="left"><p>200</p></td><td align="left"><p>$295</p></td><td align="left"><p>$6,702</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>3</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>$498,249</p></td><td align="left"><p>1,460</p></td><td align="left"><p>56</p></td><td align="left"><p>200</p></td><td align="left"><p>$297</p></td><td align="left"><p>$6,991</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$520,607</p></td><td align="left"><p>1,443</p></td><td align="left"><p>56</p></td><td align="left"><p>200</p></td><td align="left"><p>$308</p></td><td align="left"><p>$7,287</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>14</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$543,923</p></td><td align="left"><p>1,418</p></td><td align="left"><p>55</p></td><td align="left"><p>200</p></td><td align="left"><p>$317</p></td><td align="left"><p>$7,530</p></td></tr></tbody></table>

Table 6 Comparative solutions for common and independent protocols – Wake County, NC

<table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left" /><th align="left"><p>Cost per</p></th></tr><tr><th align="left" /><th align="left"><p>Initial</p></th><th align="left"><p>Weeks of</p></th><th align="left"><p>Secondary</p></th><th align="left" /><th align="left" /><th align="left"><p>Number of</p></th><th align="left"><p>Maximum</p></th><th align="left"><p>Cost per</p></th><th align="left"><p>reduction in</p></th></tr><tr><th align="left" /><th align="left"><p>screening</p></th><th align="left"><p>initial</p></th><th align="left"><p>screening</p></th><th align="left"><p>Testing</p></th><th align="left"><p>Number of</p></th><th align="left"><p>false</p></th><th align="left"><p>number in</p></th><th align="left"><p>case</p></th><th align="left"><p>false</p></th></tr><tr><th align="left"><p>Solution</p></th><th align="left"><p>frequency</p></th><th align="left"><p>screening</p></th><th align="left"><p>frequency</p></th><th align="left"><p>cost</p></th><th align="left"><p>infections</p></th><th align="left"><p>negatives</p></th><th align="left"><p>isolation</p></th><th align="left"><p>averted</p></th><th align="left"><p>negatives</p></th></tr></thead><tbody><tr><td align="left"><p>Common</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>11</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>$20,808,657</p></td><td align="left"><p>88,620</p></td><td align="left"><p>3,263</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$317</p></td><td align="left"><p>$6,899</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Every 3 Weeks</p></td><td align="left"><p>$21,435,561</p></td><td align="left"><p>84,592</p></td><td align="left"><p>3,052</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$308</p></td><td align="left"><p>$6,643</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>12</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$22,905,993</p></td><td align="left"><p>76,273</p></td><td align="left"><p>2,884</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$294</p></td><td align="left"><p>$6,747</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>16</p></td><td align="left"><p>Every 2 Weeks</p></td><td align="left"><p>$24,504,259</p></td><td align="left"><p>71,787</p></td><td align="left"><p>2,753</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$297</p></td><td align="left"><p>$6,950</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$25,603,874</p></td><td align="left"><p>70,979</p></td><td align="left"><p>2,744</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$308</p></td><td align="left"><p>$7,243</p></td></tr><tr><td align="left"><p>Independent</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>7</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$20,279,654</p></td><td align="left"><p>87,146</p></td><td align="left"><p>3,294</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$303</p></td><td align="left"><p>$6,844</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>12</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>$21,624,808</p></td><td align="left"><p>83,013</p></td><td align="left"><p>3,068</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$304</p></td><td align="left"><p>$6,780</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$23,337,088</p></td><td align="left"><p>74,590</p></td><td align="left"><p>2,826</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$293</p></td><td align="left"><p>$6,800</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$24,130,456</p></td><td align="left"><p>72,203</p></td><td align="left"><p>2,756</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$294</p></td><td align="left"><p>$6,891</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>10</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>$24,504,259</p></td><td align="left"><p>71,787</p></td><td align="left"><p>2,753</p></td><td align="left"><p>9,851</p></td><td align="left"><p>$297</p></td><td align="left"><p>$6,991</p></td></tr></tbody></table>

An important question, however, regarding the independent-protocol versus common-protocol solutions is whether they generate outcomes that are consistent in terms of the multi-objective trade-offs. In terms of an "eye test," Figs. 3 and 4 show the cost-versus-infections trade-off in the MOSA solutions for the least populous and most populous counties in our 10-county solution set, Caswell County, NC and Wake County, NC, respectively. One can observe from these figures that the common-protocol trade-off curve and that of the independent protocol solutions are nearly coincident with one another. We also note here that these cost-versus-infections plots are perhaps not as perfectly smooth as one might expect, containing some points that appear to be dominated by other points on the graph. Our MOSA solution, however, is applied to three objectives, so those points that might appear to be dominated on the cost-versus-infections trade-off must exhibit a preferable trade-off on either or both of the infections-versus-false negatives and cost-versus-false negatives trade-offs. We note that the false negative-vs-cost curves are not shown, as they are essentially identical to the infections-vs-cost curves. Thus, while our solutions limit the number of false negatives, we focus only on the outcome of limiting the spread of the viral infections.

Graph: Fig. 5 Sensitivity of infections-vs-cost trade-off to exogenous shock level for 10-county cluster in NC

Rather than depending solely on an "eye test", we offer a more rigorous method is detailed in Appendix B to compare the independent- and common-protocol solutions for each county in the solution set. This approach utilizes two post-hoc statistics that can be computed quickly and analyzed to assess whether there is a statistically-significant difference between the sets of Pareto-optimal solutions from each solution method for each county in our 10-county cluster. The two statistics are what we denote as the cost per case averted (CCA) and the cost per reduction in false negatives (CRFN). The definitions of those statistics may be found in Appendix B. From the formal quantitative comparisons and statistical tests detailed in Appendix B, we conclude that, for our example solution set of 10 counties, the common-protocol and independent-protocol solutions reflect consistent outcomes. Thus, we conclude that a common-protocol solution — which can not only be computed more quickly (approximately 20 minutes, as compared to approximately 35 minutes to generate the set of independent protocols for all 10 counties), but also is arguably easier for public health officials to explain, implement, and coordinate — can be employed for all 10 counties. In Tables 5 and 6 we provide some details on Pareto-frontier solutions from Figs. 3 and 4 that appear in what we call the "elbow" of the infections-cost tradeoff — i.e., the inflection point where the marginal decline in infections with increased testing cost levels off. In Fig. 4, for example, this occurs at around $20M of testing cost. In our view, this is a "balanced" protocol, the point at which greater investment in testing begins to generate less return in terms of reduced levels of infection.

Table 7 Selected solutions across different levels of infection prevalence – NC 10-county cluster

<table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left"><p>Weeks of</p></th><th align="left"><p>Secondary</p></th><th align="left" /><th align="left" /><th align="left" /><th align="left"><p>Maximum</p></th><th align="left"><p>Cost per</p></th><th align="left"><p>Cost per</p></th></tr><tr><th align="left" /><th align="left"><p>Initial screening</p></th><th align="left"><p>initial</p></th><th align="left"><p>screening</p></th><th align="left" /><th align="left"><p>Number of</p></th><th align="left"><p>Number of</p></th><th align="left"><p>number in</p></th><th align="left"><p>case</p></th><th align="left"><p>reduction in</p></th></tr><tr><th align="left"><p>Shock level</p></th><th align="left"><p>frequency</p></th><th align="left"><p>screening</p></th><th align="left"><p>frequency</p></th><th align="left"><p>Testing cost</p></th><th align="left"><p>infections</p></th><th align="left"><p>false negatives</p></th><th align="left"><p>isolation</p></th><th align="left"><p>averted</p></th><th align="left"><p>false negatives</p></th></tr></thead><tbody><tr><td align="left"><p>0.10%</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>11</p></td><td align="left"><p>Every 2 Weeks</p></td><td align="left"><p>$41,433,980</p></td><td align="left"><p>51,360</p></td><td align="left"><p>196</p></td><td align="left"><p>16,951</p></td><td align="left"><p>$168.26</p></td><td align="left"><p>$168</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>12</p></td><td align="left"><p>Every 2 Weeks</p></td><td align="left"><p>$44,062,551</p></td><td align="left"><p>47,317</p></td><td align="left"><p>183</p></td><td align="left"><p>16,951</p></td><td align="left"><p>$176.05</p></td><td align="left"><p>$176</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Every 3 Weeks</p></td><td align="left"><p>$46,023,223</p></td><td align="left"><p>45,192</p></td><td align="left"><p>177</p></td><td align="left"><p>16,951</p></td><td align="left"><p>$182.33</p></td><td align="left"><p>$182</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Every 2 Weeks</p></td><td align="left"><p>$46,679,672</p></td><td align="left"><p>44,674</p></td><td align="left"><p>176</p></td><td align="left"><p>16,951</p></td><td align="left"><p>$184.56</p></td><td align="left"><p>$185</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Every 2 Weeks</p></td><td align="left"><p>$51,886,847</p></td><td align="left"><p>42,193</p></td><td align="left"><p>169</p></td><td align="left"><p>16,951</p></td><td align="left"><p>$203.15</p></td><td align="left"><p>$203</p></td></tr><tr><td align="left"><p>0.50%</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>11</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>$42,662,884</p></td><td align="left"><p>105,247</p></td><td align="left"><p>392</p></td><td align="left"><p>17,567</p></td><td align="left"><p>$221.79</p></td><td align="left"><p>$221</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>10</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$45,443,719</p></td><td align="left"><p>95,211</p></td><td align="left"><p>364</p></td><td align="left"><p>17,567</p></td><td align="left"><p>$224.53</p></td><td align="left"><p>$224</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>12</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$47,479,028</p></td><td align="left"><p>90,670</p></td><td align="left"><p>348</p></td><td align="left"><p>17,567</p></td><td align="left"><p>$229.44</p></td><td align="left"><p>$229</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$50,326,600</p></td><td align="left"><p>86,236</p></td><td align="left"><p>334</p></td><td align="left"><p>17,567</p></td><td align="left"><p>$238.10</p></td><td align="left"><p>$238</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$53,949,202</p></td><td align="left"><p>84,968</p></td><td align="left"><p>333</p></td><td align="left"><p>17,567</p></td><td align="left"><p>$253.72</p></td><td align="left"><p>$253</p></td></tr><tr><td align="left"><p>1.00%</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>8</p></td><td align="left"><p>Every 3 Days</p></td><td align="left"><p>$40,294,719</p></td><td align="left"><p>163,598</p></td><td align="left"><p>618</p></td><td align="left"><p>19,016</p></td><td align="left"><p>$300.70</p></td><td align="left"><p>$299</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>14</p></td><td align="left"><p>Every 3 Weeks</p></td><td align="left"><p>$43,428,779</p></td><td align="left"><p>149,143</p></td><td align="left"><p>550</p></td><td align="left"><p>19,016</p></td><td align="left"><p>$292.53</p></td><td align="left"><p>$291</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Every 3 Weeks</p></td><td align="left"><p>$45,401,054</p></td><td align="left"><p>141,080</p></td><td align="left"><p>533</p></td><td align="left"><p>19,016</p></td><td align="left"><p>$290.06</p></td><td align="left"><p>$289</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>16</p></td><td align="left"><p>Every 4 Weeks</p></td><td align="left"><p>$47,299,600</p></td><td align="left"><p>138,570</p></td><td align="left"><p>531</p></td><td align="left"><p>19,016</p></td><td align="left"><p>$297.42</p></td><td align="left"><p>$296</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$49,422,135</p></td><td align="left"><p>137,010</p></td><td align="left"><p>530</p></td><td align="left"><p>19,016</p></td><td align="left"><p>$307.75</p></td><td align="left"><p>$307</p></td></tr><tr><td align="left"><p>2.00%</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$40,873,986</p></td><td align="left"><p>232,096</p></td><td align="left"><p>890</p></td><td align="left"><p>26,413</p></td><td align="left"><p>$623.96</p></td><td align="left"><p>$611</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>14</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$42,092,565</p></td><td align="left"><p>228,425</p></td><td align="left"><p>878</p></td><td align="left"><p>26,980</p></td><td align="left"><p>$608.47</p></td><td align="left"><p>$597</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$43,560,548</p></td><td align="left"><p>224,340</p></td><td align="left"><p>863</p></td><td align="left"><p>27,504</p></td><td align="left"><p>$594.58</p></td><td align="left"><p>$584</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>12</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$45,304,128</p></td><td align="left"><p>220,025</p></td><td align="left"><p>847</p></td><td align="left"><p>27,992</p></td><td align="left"><p>$583.98</p></td><td align="left"><p>$574</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>11</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$47,336,781</p></td><td align="left"><p>215,552</p></td><td align="left"><p>829</p></td><td align="left"><p>28,446</p></td><td align="left"><p>$576.92</p></td><td align="left"><p>$568</p></td></tr><tr><td align="left"><p>6.25%</p></td><td align="left"><p>Daily</p></td><td align="left"><p>8</p></td><td align="left"><p>Every 4 Weeks</p></td><td align="left"><p>$44,632,459</p></td><td align="left"><p>297,642</p></td><td align="left"><p>1,543</p></td><td align="left"><p>52,005</p></td><td align="left" /><td align="left"><p>$11,696</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>9</p></td><td align="left"><p>Every 4 Weeks</p></td><td align="left"><p>$46,857,795</p></td><td align="left"><p>299,003</p></td><td align="left"><p>1,528</p></td><td align="left"><p>52,005</p></td><td align="left"><p><italic>No cases</italic></p></td><td align="left"><p>$19,081</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>11</p></td><td align="left"><p>Every 4 Weeks</p></td><td align="left"><p>$49,365,290</p></td><td align="left"><p>298,324</p></td><td align="left"><p>1,501</p></td><td align="left"><p>52,005</p></td><td align="left"><p><italic>averted</italic></p></td><td align="left"><p>$15,749</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>12</p></td><td align="left"><p>Every 3 Weeks</p></td><td align="left"><p>$49,904,653</p></td><td align="left"><p>298,306</p></td><td align="left"><p>1,506</p></td><td align="left"><p>52,005</p></td><td align="left" /><td align="left"><p>$15,828</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>12</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>$50,287,597</p></td><td align="left"><p>298,306</p></td><td align="left"><p>1,513</p></td><td align="left"><p>52,005</p></td><td align="left" /><td align="left"><p>$15,949</p></td></tr></tbody></table>

Graph: Fig. 6 Sensitivity of infections-vs-cost trade-off to R0 level for 10-county cluster in NC

Table 8 Selected solutions across different transmission rates – NC 10-county cluster

<table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left" /><th align="left"><p>Weeks of</p></th><th align="left"><p>Secondary</p></th><th align="left" /><th align="left" /><th align="left" /><th align="left"><p>Maximum</p></th><th align="left"><p>Cost per</p></th><th align="left"><p>Cost per</p></th></tr><tr><th align="left" /><th align="left"><p>Initial screening</p></th><th align="left"><p>initial</p></th><th align="left"><p>screening</p></th><th align="left" /><th align="left"><p>Number of</p></th><th align="left"><p>Number of</p></th><th align="left"><p>number in</p></th><th align="left"><p>case</p></th><th align="left"><p>reduction in</p></th></tr><tr><th align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>R</mi><mn>0</mn></msub></math><inline-graphic mime-subtype="GIF" href="10729&#95;2024&#95;9688&#95;Article&#95;IEq236.gif" /></p></th><th align="left"><p>frequency</p></th><th align="left"><p>screening</p></th><th align="left"><p>frequency</p></th><th align="left"><p>Testing cost</p></th><th align="left"><p>infections</p></th><th align="left"><p>false negatives</p></th><th align="left"><p>isolation</p></th><th align="left"><p>averted</p></th><th align="left"><p>false negatives</p></th></tr></thead><tbody><tr><td align="left"><p>1.5</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>6</p></td><td align="left"><p>Every 3 days</p></td><td align="left"><p>$41,729,299</p></td><td align="left"><p>101,863</p></td><td align="left"><p>386</p></td><td align="left"><p>14,997</p></td><td align="left"><p>$213</p></td><td align="left"><p>$213</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Every 3 weeks</p></td><td align="left"><p>$44,686,591</p></td><td align="left"><p>101,189</p></td><td align="left"><p>372</p></td><td align="left"><p>14,997</p></td><td align="left"><p>$228</p></td><td align="left"><p>$227</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Every 2 weeks</p></td><td align="left"><p>$45,256,238</p></td><td align="left"><p>99,690</p></td><td align="left"><p>369</p></td><td align="left"><p>14,997</p></td><td align="left"><p>$229</p></td><td align="left"><p>$229</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>11</p></td><td align="left"><p>Every 3 days</p></td><td align="left"><p>$47,207,803</p></td><td align="left"><p>95,642</p></td><td align="left"><p>363</p></td><td align="left"><p>14,997</p></td><td align="left"><p>$234</p></td><td align="left"><p>$234</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Every 4 weeks</p></td><td align="left"><p>$49,487,645</p></td><td align="left"><p>92,409</p></td><td align="left"><p>349</p></td><td align="left"><p>14,997</p></td><td align="left"><p>$241</p></td><td align="left"><p>$241</p></td></tr><tr><td align="left"><p>2.3</p></td><td align="left"><p>Every 2 Days</p></td><td align="left"><p>13</p></td><td align="left"><p>Weekly</p></td><td align="left"><p>$43,229,090</p></td><td align="left"><p>151,296</p></td><td align="left"><p>563</p></td><td align="left"><p>19,015</p></td><td align="left"><p>$295</p></td><td align="left"><p>$295</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Every 4 weeks</p></td><td align="left"><p>$45,326,331</p></td><td align="left"><p>141,207</p></td><td align="left"><p>533</p></td><td align="left"><p>19,015</p></td><td align="left"><p>$290</p></td><td align="left"><p>$289</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>10</p></td><td align="left"><p>Every 2 days</p></td><td align="left"><p>$47,299,600</p></td><td align="left"><p>138,570</p></td><td align="left"><p>531</p></td><td align="left"><p>19,015</p></td><td align="left"><p>$297</p></td><td align="left"><p>$297</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>15</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$49,422,135</p></td><td align="left"><p>137,010</p></td><td align="left"><p>530</p></td><td align="left"><p>19,015</p></td><td align="left"><p>$308</p></td><td align="left"><p>$307</p></td></tr><tr><td align="left" /><td align="left"><p>Every 2 Days</p></td><td align="left"><p>14</p></td><td align="left"><p>Daily</p></td><td align="left"><p>$51,635,542</p></td><td align="left"><p>134,601</p></td><td align="left"><p>522</p></td><td align="left"><p>19,015</p></td><td align="left"><p>$317</p></td><td align="left"><p>$316</p></td></tr><tr><td align="left"><p>5.0</p></td><td align="left"><p>Daily</p></td><td align="left"><p>5</p></td><td align="left"><p>Every 4 weeks</p></td><td align="left"><p>$34,238,250</p></td><td align="left"><p>297,792</p></td><td align="left"><p>1,230</p></td><td align="left"><p>41,434</p></td><td align="left" /><td align="left"><p>$28,657</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>5</p></td><td align="left"><p>Every 3 weeks</p></td><td align="left"><p>$34,641,803</p></td><td align="left"><p>297,631</p></td><td align="left"><p>1,234</p></td><td align="left"><p>45,002</p></td><td align="left"><p><italic>No cases</italic></p></td><td align="left"><p>$25,550</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>6</p></td><td align="left"><p>Every 4 weeks</p></td><td align="left"><p>$39,455,168</p></td><td align="left"><p>298,166</p></td><td align="left"><p>1,223</p></td><td align="left"><p>38,982</p></td><td align="left"><p><italic>averted</italic></p></td><td align="left"><p>$48,044</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>6</p></td><td align="left"><p>Every 3 weeks</p></td><td align="left"><p>$39,829,388</p></td><td align="left"><p>297,619</p></td><td align="left"><p>1,224</p></td><td align="left"><p>42,339</p></td><td align="left" /><td align="left"><p>$29,128</p></td></tr><tr><td align="left" /><td align="left"><p>Daily</p></td><td align="left"><p>7</p></td><td align="left"><p>Every 4 weeks</p></td><td align="left"><p>$44,346,977</p></td><td align="left"><p>298,609</p></td><td align="left"><p>1,215</p></td><td align="left"><p>36,673</p></td><td align="left" /><td align="left"><p>$117,504</p></td></tr></tbody></table>

Graph: Fig. 7 Wake county cumulative infections with 0% exogenous infections (except where indicated)

The data in Figs. 3 and 4, as well as the data in Tables 5 and 6, suggest an interesting similarity in the optimal testing protocols. For Wake County, the common protocol-generated solutions centered around initial testing of Every 2 Days for 11 to 16 weeks, followed by weekly secondary screening for one to 3 weeks. In one case, the optimal protocol was an initial screening Every 2 Days for 15 weeks followed by daily testing for one week. The results from the independent protocols varied only slightly from these screening cadences, and they were quite similar for both the least populous (Caswell) and the most populous (Wake) counties in our example solution set. Of importance was the fact that these optimal protocols resulted in similar levels of cost per case averted and similar levels of cost per reduction in false negatives between the two counties.

Some discussion of the results in Tables 5 and 6 is warranted here. Regarding the reduction in false negatives with more testing, this is due to the fact that the additional testing serves to capture more true positives, thus removing more individuals from the Asymptomatic (A) stage in the model, as they would otherwise be a source of false negatives. (Specifically, false negatives are computed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">|</mo><mi>E</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo><mo>&#183;</mo><mo stretchy="false">(</mo><mn>1</mn><mo>-</mo><msub><mi>S</mi><mi>e</mi></msub><mo stretchy="false">)</mo></mrow></math> , where |X| is the count of the number of individuals in set X and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>e</mi></msub></math> is the sensitivity of the test.) Regarding the maximum number in isolation (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">|</mo><mi>F</mi><mi>P</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>T</mi><mi>P</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>P</mi><mo stretchy="false">|</mo></mrow></math> ), this will be the same value for testing protocols that use the same initial screening frequency for a sufficiently large number of weeks. This result stems from the fact that more frequent screening will tend to root out infections relatively early in the horizon. Computational details (not shown here) demonstrate that protocols with an initial screening frequency of every 2 days hit their peak number in isolation (e.g., 9,851 for Wake County, as in Table 4) before the end of week 3 of the testing horizon. By contrast, a much less aggressive, weekly screening (again, not shown here) reflects growth in the number in isolation through around 9 weeks into the horizon to a peak well past 9,851 (up to 17,282, to be exact) due to allowing asymptomatic individuals to circulate in the population.

Robustness analysis of Pareto-optimal protocols for public school screening in North Carolina

In this section, we graphically explore the robustness of Pareto optimal front for common-protocol solutions for two key parameters, as discussed in Section 5.2. Those key parameters are the "basic reproductive number" of the virus, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> , and the number of infections introduced into the congregate setting in question from outside the system. Exploring differing levels of outside infections (Fig. 5), we note that if the infection rates outside these school districts remain at 1% or lower, testing protocols can be found that control the spread of the pandemic. However at infection rates approaching 6.5%, no amount of testing will control the infection spread. Table 7 provides detailed comparisons of Pareto-optimal protocols in the "balanced" area of the plot in Fig. 5, around the midpoint of the ranges of both cost and infections. Turning to differing levels of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> , our results (Fig. 6) show that at <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> values of 1.5 to 2.3, testing protocols can be found that control the spread of infection. However, at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>R</mi><mn>0</mn></msub><mo>=</mo><mn>5</mn></mrow></math> , no amount of testing will control the spread of the virus. Table 8 provides a comparison of the details for balanced protocols from Fig. 6.

Finally, a plot of cumulative infections over time for various protocols, applied to the Wake County school population with 0% exogenous infections (Fig. 7), except where noted in the figure, shows that an absence of testing yields almost 100% infectivity within 40 days. The MOSA-generated protocol that resulted in the lowest cost dulled the accumulation of infections somewhat, but still reached almost 100% by 55 days. Both the highest cost protocol and what we described earlier as a "balanced" protocol (i.e., at the inflection point of the infections-vs-cost plot of Pareto-optimal protocols), however, controlled the spread of infection. For either of those MOSA-generated protocols, cumulative numbers of infections decreased by almost 64% by 10 days and almost 84% by 20 days. Even if we allow approximately 1,600 new infections per week (1% exogenous rate), the balanced protocol still slows the number of cumulative infections by almost 72% within 20 days.

Discussion and conclusion

In this paper we used a quantitative methodology to generate optimal testing protocols for infectious diseases, using the COVID-19 pandemic as our context. The protocols were focused on populations that reside in congregate settings such as schools, businesses, hospitals, or nursing homes. For purposes of this paper, we focused on testing of student and staff populations within school districts that are found within counties within the State of North Carolina. The test format chosen was the rapid viral antigen test. These types of tests are inexpensive (approximately $5 in bulk), can be administered quickly ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&#60;</mo><mn>30</mn></mrow></math> minutes), and have reasonably high sensitivity ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&#62;</mo><mn>80</mn><mo>%</mo></mrow></math> ) and specificity ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&#62;</mo><mn>98</mn><mo>%</mo></mrow></math> ) levels.

The basis for our modeling is the SEIR epidemiological framework. Using this model, we compared various test frequencies (e.g. Daily, Every 2 Days, Every 3 Days, Weekly, Every 2 Weeks, Every 3 Weeks, Every 4 Weeks), varying them independently over a primary test period followed by a secondary test period. We developed and deployed a Multi-Objective Simulated Annealing (MOSA) algorithm to find Pareto-optimal sets of solutions that jointly minimize three objectives: 1) the number of infections over the testing horizon, 2) the number of false negative tests observed, and 3) the total cost of testing. Moreover, we employed our MOSA algorithm to solve for protocols that are independently optimal across all geographic regions being considered, and for protocols that are jointly optimal. We believe this comparison of independently solved versus jointly solved collections and our method of demonstrating their equivalence in our solution application are interesting and novel contributions. Moreover, we note that our approach can be applied generally to any public health decision making problem that poses a trade-off between investment and public health outcomes. Thus, we strike a balance in this paper between technical and practical application.

The potential impact of the use of such optimally-generated testing protocols is as follows:

The cost of screening the public school system of North Carolina would be significant. For a 16-week semester, we estimate a testing cost for Wake County, with 192 schools and 160,000 students and staff, of approximately $20 million (see Table 6). Testing costs for the 10-county cluster in our computational example, surrounding Wake County, would be approximately $40 million (see Tables 8 and 7). It is interesting to note, however, that Wake County alone spent over $100 million in public funds to provide COVID testing free-of-charge to the public from June 1, 2020 to Nov 30, 2021 [[59]]. These were almost all PCR-based tests, which were not deployed strategically for screening in school systems or work settings. Thus, while the proposed test protocols can seem expensive, they are likely less expensive and more effective in terms of opening schools and work areas than the previously-used approaches. We also note that our MOSA algorithm could easily be adjusted to address possible constraints on the supply of antigen test kits by either adding a penalty to the objective function for solutions that violate the supply constraint, or by simply excluding constraint-violating solutions. For the sake of brevity, we did not include those results here.

Acquiring and distributing the proper number of test kits to serve specific congregate settings is a significant supply-chain logistics issue. Our experience in advising the State of North Carolina suggests that logistics management might best be directed at the state level. The tools developed in this paper should help decision-makers in determining the number of test kits required for any given geographical area. As discussed in Section 5, we employed machine-learning and TOPSIS modeling to generate a weighted average priority ranking for all 100 counties in NC, and then created prioritized clusters of counties based on the point-in-time state of the pandemic (Fig. 2). We note that the cluster that served as our computational example — which includes Wake County and contains the state's capital and second-largest city, Raleigh — was the highest priority cluster in the state in the period around July 2021, and the second highest priority cluster in the period around February 2022. For this high-priority cluster, we estimate that an average of approximately 500,000 test kits per week would be required for this 10-county cluster over a 16-week school semester. Distribution of these kits to the approximately 400 school districts in these 10 counties is a significant supply-chain challenge, but the tools developed in this paper would serve as significant aids to decision makers in state and local government.

Finally, we note that the availability of COVID-19 vaccines has mitigated, but not eliminated, the need for large scale testing in the U.S. The problems of vaccine hesitancy, the advent of novel viral strains, and the natural waning of immunological competence post vaccine, all suggest an ongoing need for viral screening. Importantly, the issue of long COVID is a huge potential health care burden. Cutler and Summers [[60]] and Cutler [[61]] estimate that the ongoing health care costs could total \$528 billion and the overall economic costs could amount to \$3.7 trillion. Obviously, it is critical to keep the infection rate as low as possible, and tools like the ones developed in this paper should be at the forefront of that effort.

In conclusion, this study provides a holistic and data-based approach for conducting testing of congregate populations, such as schools, hospitals, and public service agencies. Our work is an example of the kind of parametric analysis that policymakers can deploy in future pandemic situations, when there is a limited number of tests available and no pre-determined set of policies on how to allocate them to different regions within a state. This work fills an impotant gap in research on pandemic planning policy, employing well-known optimization methods to recommend allocation of potentially scarce resources based on logical input parameters. In this way, we believe this research provides useful input for current and future practice in healthcare policy-making.

Acknowledgements

This work was supported in part by a grant from the National Institute for Innovation in Manufacturing Biopharmaceuticals (NIIMBL) and the National Institute of Standards and Technology (NIST) under grant number NIIMBL COVID-19-1.03.

Declarations

Conflicts of interest

The authors declare that they have no known competing personal relationships or other conflicts of interest that could have appeared to influence the work reported in this paper

Appendix A. SEIR computations

In this appendix, we specify the computations of the flow quantities between states in the SEIR system (see Fig. 1), following the SEIR model of [[7]]. Extending from their work, we introduce a new parameter, k, which is the number of cycles required to obtain test results. Use of lab-based testing, for example, may require <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>&#8805;</mo><mn>1</mn></mrow></math>

Graph

, meaning that the results of the test will not be known until k cycles in the future. For the case we consider in Section 5, we assume use of rapid-results antigen tests for confirmation, and therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></math>

Graph

.

A.1 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd /><mtd columnalign="left"><mrow><msub><mi>U</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><munder><munder accentunder="true"><mrow><msub><mi>U</mi><mi>c</mi></msub><mo>&#183;</mo><mfenced close="]" open="["><mn>1</mn><mo>-</mo><mi>&#946;</mi><mo>&#183;</mo><mfrac><msub><mi>A</mi><mi>c</mi></msub><mrow><msub><mi>U</mi><mi>c</mi></msub><mo>+</mo><msub><mi>A</mi><mi>c</mi></msub><mo>+</mo><msub><mi>E</mi><mi>c</mi></msub></mrow></mfrac></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>Uninfected minus internal infections</mtext></mrow></munder><mo>-</mo><munder><munder accentunder="true"><mrow><msub><mi>U</mi><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></msub><mo>&#183;</mo><mi>&#964;</mi><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><msub><mi>S</mi><mi>p</mi></msub></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>False positive results</mtext></mrow></munder></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd><mtd columnalign="left"><mrow><mspace width="1em" /><mo>+</mo><munder><munder accentunder="true"><mrow><mi>&#956;</mi><mo>&#183;</mo><msub><mrow><mi mathvariant="italic">FP</mi></mrow><mi>c</mi></msub></mrow><mo>&#9183;</mo></munder><mrow><mtext>Returning false positives</mtext></mrow></munder><mspace width="1em" /><mo>-</mo><munder><munder accentunder="true"><mrow><mi>X</mi><mo>&#183;</mo><msub><mi>I</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo></mrow><mo>&#9183;</mo></munder><mrow><mtext>Exogenous shock infections</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

Graph

where the additional parameters are as follows:

Graph

is the rate at which infectious individuals both contact and infect susceptible individuals, computed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>&#946;</mi><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><mo>&#183;</mo><mrow><mo stretchy="false">(</mo><mi>&#963;</mi><mo>+</mo><mi>&#961;</mi><mo stretchy="false">)</mo></mrow></mrow></math>

Graph

, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math>

Graph

is the viral reproductive number, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#961;</mi></math>

Graph

is the rate at which infected individuals recover from disease, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#963;</mi></math>

Graph

is the rate of symptom onset for infected individuals.

Graph

is the rate at which an individuals in the testing pool are screened for infection, which is driven by the testing frequency (e.g., daily, every 2 days, etc.) and computed as the reciprocal of this frequency (i.e., at an "every 3 days" frequency, 1/3 of the individuals in the testing pool are assumed to be tested each cycle).

Graph

is the specificity of the test.

Graph

is rate at which false positives are identified and returned to the uninfected pool, which is computed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>&#956;</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>n</mi><mrow><mi mathvariant="italic">cyc</mi></mrow></msub><mo>&#183;</mo><msub><mi>t</mi><mrow><mi mathvariant="italic">FPr</mi></mrow></msub><mo stretchy="false">)</mo></mrow></math>

Graph

, where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mi mathvariant="italic">cyc</mi></mrow></msub></math>

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is the number of testing cycles per day and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow><mi mathvariant="italic">FPr</mi></mrow></msub></math>

Graph

is the number of days required for false positives to return to the testing pool, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>t</mi><mrow><mi mathvariant="italic">FPret</mi></mrow></msub><mo>&#8805;</mo><mn>1</mn></mrow></math>

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.

Graph

is an indicator function taking the value 1 if there is a shock of new infections in cycle <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></math>

Graph

and 0 otherwise.

Thus, the "uninfected update" portion of the computation is given by period-starting uninfected ( <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>t</mi></msub></math>

Graph

) minus new infections ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>&#946;</mi><mo>&#183;</mo><msub><mi>U</mi><mi>t</mi></msub><mo>&#183;</mo><msub><mi>A</mi><mi>t</mi></msub><mo stretchy="false">/</mo><mfenced close=")" open="("><msub><mi>U</mi><mi>t</mi></msub><mo>+</mo><msub><mi>A</mi><mi>t</mi></msub><mo>+</mo><msub><mi>E</mi><mi>t</mi></msub></mfenced></mrow></math>

Graph

). From this, false positives (from cycle <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></math>

Graph

) are subtracted, confirmed and returning false positives (from the current cycle, c) are added, and infections contracted by outside "shocks" are subtracted. We note that [[7]] assume that the new infections portion of the "Uninfected update" stems from applying the transmission rate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#946;</mi></math>

Graph

to the uninfected group <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>t</mi></msub></math>

Graph

, but only to those who have incurred probabilistic contact with the asymptomatic and infected group, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>t</mi></msub></math>

Graph

. Thus, it also assumes that the probability that such contact occurs is given by the ratio of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>t</mi></msub></math>

Graph

to the "active testing pool," <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>U</mi><mi>t</mi></msub><mo>+</mo><msub><mi>A</mi><mi>t</mi></msub><mo>+</mo><msub><mi>E</mi><mi>t</mi></msub></mrow></math>

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.

A.2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>E</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><munder><munder accentunder="true"><mrow><msub><mi>E</mi><mi>c</mi></msub><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><mi>&#952;</mi></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>Exposed update</mtext></mrow></munder><mo>+</mo><munder><munder accentunder="true"><mrow><mi>&#946;</mi><mo>&#183;</mo><mfrac><mrow><msub><mi>A</mi><mi>c</mi></msub><mo>&#183;</mo><msub><mi>U</mi><mi>c</mi></msub></mrow><mrow><msub><mi>U</mi><mi>c</mi></msub><mo>+</mo><msub><mi>A</mi><mi>c</mi></msub><mo>+</mo><msub><mi>E</mi><mi>c</mi></msub></mrow></mfrac></mrow><mo>&#9183;</mo></munder><mrow><mtext>New infections</mtext></mrow></munder><mo>+</mo><munder><munder accentunder="true"><mrow><mi>X</mi><mo>&#183;</mo><msub><mi>I</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo></mrow><mo>&#9183;</mo></munder><mrow><mtext>Exogenous shocks</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

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where the additional parameter is:

Graph

, which is the rate at which exposed individuals advance to being infectious, but still asymptomatic.

Thus, the "Exposed update" above reflects the flow of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>&#952;</mi><mo>&#183;</mo><msub><mi>E</mi><mi>c</mi></msub></mrow></math>

Graph

individuals from state E to state A in Fig. 1. The inflows to state E come from infections flowing out of state U, both from internal contact among those individuals and from outside shocks.

A.3 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>A</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><munder><munder accentunder="true"><mrow><msub><mi>A</mi><mi>c</mi></msub><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><mi>&#963;</mi><mo>-</mo><mi>&#961;</mi></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>Asymptomatic update</mtext></mrow></munder><mo>-</mo><munder><munder accentunder="true"><mrow><msub><mi>A</mi><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></msub><mo>&#183;</mo><mi>&#964;</mi><mo>&#183;</mo><msub><mi>S</mi><mi>e</mi></msub></mrow><mo>&#9183;</mo></munder><mrow><mtext>True positive results</mtext></mrow></munder><mo>+</mo><munder><munder accentunder="true"><mrow><mi>&#952;</mi><mo>&#183;</mo><msub><mi>E</mi><mi>c</mi></msub><mo>,</mo></mrow><mo>&#9183;</mo></munder><mrow><mtext>Infection transitions</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

Graph

where the additional parameter is:

Graph

, which is the sensitivity of the test.

Therefore, the inflows to state A come only from infections transitioning from the undetectable state E to detectable state A. The outflows are given by (i) recoveries, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>&#961;</mi><mo>&#183;</mo><msub><mi>A</mi><mi>c</mi></msub></mrow></math>

Graph

, from asymptomatic infections; (ii) transitions, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>&#963;</mi><mo>&#183;</mo><msub><mi>A</mi><mi>c</mi></msub></mrow></math>

Graph

, from asymptomatic to symptomatic state P; and (iii) confirmed true positives results from tests in cycle <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></math>

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, based on the test sensitivity rate, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>e</mi></msub></math>

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.

A.4 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mrow><mi mathvariant="italic">FP</mi></mrow><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><munder><munder accentunder="true"><mrow><msub><mrow><mi mathvariant="italic">FP</mi></mrow><mi>c</mi></msub><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><mi>&#956;</mi></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>False positive update</mtext></mrow></munder><mo>+</mo><munder><munder accentunder="true"><mrow><msub><mi>U</mi><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></msub><mo>&#183;</mo><mi>&#964;</mi><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><msub><mi>S</mi><mi>p</mi></msub></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>False positive results</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

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and

A.5 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mrow><mi mathvariant="italic">TP</mi></mrow><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><munder><munder accentunder="true"><mrow><msub><mrow><mi mathvariant="italic">TP</mi></mrow><mi>c</mi></msub><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><mi>&#963;</mi><mo>-</mo><mi>&#961;</mi></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>True positive update</mtext></mrow></munder><mo>+</mo><munder><munder accentunder="true"><mrow><msub><mi>A</mi><mrow><mi>c</mi><mo>-</mo><mi>k</mi></mrow></msub><mo>&#183;</mo><mi>&#964;</mi><mo>&#183;</mo><msub><mi>S</mi><mi>e</mi></msub><mo>,</mo></mrow><mo>&#9183;</mo></munder><mrow><mtext>True positive results</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

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both of which are based on parameters and expressions that have been defined above.

The cycle transitions in for the symptomatic set, P, and the recovered set, V, are given by

A.6 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>P</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><munder><munder accentunder="true"><mrow><msub><mi>P</mi><mi>c</mi></msub><mo>&#183;</mo><mfenced close=")" open="("><mn>1</mn><mo>-</mo><mi>&#961;</mi><mo>-</mo><mi>&#948;</mi></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>Symptomatic update</mtext></mrow></munder><mo>+</mo><munder><munder accentunder="true"><mrow><mi>&#963;</mi><mo>&#183;</mo><mfenced close=")" open="("><msub><mrow><mi mathvariant="italic">TP</mi></mrow><mi>c</mi></msub><mo>+</mo><msub><mi>A</mi><mi>c</mi></msub></mfenced></mrow><mo>&#9183;</mo></munder><mrow><mtext>Transitions to symptomatic state</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

Graph

and

A.7 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>V</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>V</mi><mi>c</mi></msub><mo>+</mo><munder><munder accentunder="true"><mrow><mi>&#961;</mi><mo>&#183;</mo><mfenced close=")" open="("><msub><mrow><mi mathvariant="italic">TP</mi></mrow><mi>c</mi></msub><mo>+</mo><msub><mi>A</mi><mi>c</mi></msub><mo>+</mo><msub><mi>P</mi><mi>c</mi></msub></mfenced><mo>.</mo></mrow><mo>&#9183;</mo></munder><mrow><mtext>Transitions to recovered state</mtext></mrow></munder></mrow></mtd></mtr></mtable></mrow></math>

Graph

The cycle transitions to the deceased state, D, are given by

A.8 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>D</mi><mrow><mi>c</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>D</mi><mi>c</mi></msub><mo>+</mo><munder><munder accentunder="true"><mrow><mi>&#948;</mi><mo>&#183;</mo><msub><mi>P</mi><mi>c</mi></msub></mrow><mo>&#9183;</mo></munder><mtext>Deaths</mtext></munder><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

where the additional parameter is:

Graph

, which is the rate at which infected, symptomatic individuals die from the disease.

Appendix B. Non-parametric test for independent- versus common-protocol solution consistency

In this appendix, we offer a statistical analysis to assess whether the independent- and common-protocol solutions generate outcomes that are consistent in terms of the multi-objective trade-offs. We base our analysis on two statistics that we denote as the cost per case averted (CCA) and the cost per reduction in false negatives (CRFN), which are computed using the following inputs and outcomes from the set <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">PO</mi></mrow><mi>i</mi></msub></math>

Graph

of solutions <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>&#8712;</mo><msub><mrow><mi mathvariant="italic">PO</mi></mrow><mi>i</mi></msub></mrow></math>

Graph

for county i:

Graph

, which is the number of individuals in the testing population in county i that are initially susceptible to infection (i.e., the number in box U of Fig. 1).

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">FN</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></math>

Graph

, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">TC</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></math>

Graph

, which are the objective values of solution j, reflecting the <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub></mfenced></math>

Graph

protocol values specific to the county i solution.

Then, the CCA and CRFN values for each solution j in county i are computed as follows:

B.1 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>C</mi><mi>C</mi><msub><mi>A</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>T</mi><msub><mi>C</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi mathvariant="italic">IS</mi></mrow><mi>i</mi></msub><mo>-</mo><msub><mrow><mi mathvariant="italic">IN</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math>

Graph

and

B.2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi>C</mi><mi>R</mi><mi>F</mi><msub><mi>N</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>T</mi><msub><mi>C</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><mrow><munder><mo movablelimits="false">max</mo><mrow><mi>k</mi><mo>&#8712;</mo><msub><mrow><mi mathvariant="italic">PO</mi></mrow><mi>i</mi></msub><mo>,</mo><mi>k</mi><mo>&#8800;</mo><mi>j</mi></mrow></munder><mrow><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="italic">FN</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">}</mo></mrow><mo>-</mo><msub><mrow><mi mathvariant="italic">FN</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mfrac><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

Our graphical analysis of these statistics for various solutions that we generated indicated that they did not appear to follow any particular statistical distribution, so we employed non-parametric testing on the CCA and CRFN statistics for each solution method — independent-protocol versus common — for each county in our 10-county solution set, and specifically based the analysis on Mann Whitney U Test (also known as Wilcoxon Rank Sum Test). Results of the non-parametric tests for statistically significant differences in the solutions for each county are shown in Table 9. Since the p-values for each test for every county in the 10-county set all exceed any reasonable confidence threshold, we fail to reject the hypothesis that the cost measures are statistically equal for each county.

Table 9 p-values for non-parametric test of independent versus common solutions for NC 10-county cluster

<table frame="hsides" rules="groups"><thead><tr><th align="left"><p>County</p></th><th align="left"><p>Cost per case averted</p></th><th align="left"><p>Cost per reduction in false negatives</p></th></tr></thead><tbody><tr><td align="left"><p>Alamance county</p></td><td align="left"><p>0.549</p></td><td align="left"><p>0.691</p></td></tr><tr><td align="left"><p>Caswell county</p></td><td align="left"><p>0.956</p></td><td align="left"><p>0.605</p></td></tr><tr><td align="left"><p>Chatham county</p></td><td align="left"><p>0.639</p></td><td align="left"><p>0.651</p></td></tr><tr><td align="left"><p>Durham county</p></td><td align="left"><p>0.307</p></td><td align="left"><p>0.740</p></td></tr><tr><td align="left"><p>Granville county</p></td><td align="left"><p>0.676</p></td><td align="left"><p>0.573</p></td></tr><tr><td align="left"><p>Harnett county</p></td><td align="left"><p>0.652</p></td><td align="left"><p>0.803</p></td></tr><tr><td align="left"><p>Lee county</p></td><td align="left"><p>0.842</p></td><td align="left"><p>0.803</p></td></tr><tr><td align="left"><p>Orange county</p></td><td align="left"><p>0.475</p></td><td align="left"><p>0.963</p></td></tr><tr><td align="left"><p>Person county</p></td><td align="left"><p>0.244</p></td><td align="left"><p>0.631</p></td></tr><tr><td align="left"><p>Wake county</p></td><td align="left"><p>0.933</p></td><td align="left"><p>0.926</p></td></tr></tbody></table>

Table 10 Generalized spread measure for common and independent solutions for NC 10-county cluster

<table frame="hsides" rules="groups"><thead><tr><th align="left" /><th align="left"><p>Common</p></th><th align="left"><p>Independent</p></th></tr></thead><tbody><tr><td align="center"><p>Alamance county</p></td><td align="left"><p>0.992958</p></td><td align="left"><p>0.991463</p></td></tr><tr><td align="center"><p>Caswell county</p></td><td align="left"><p>0.992957</p></td><td align="left"><p>0.991461</p></td></tr><tr><td align="center"><p>Chatham county</p></td><td align="left"><p>0.992952</p></td><td align="left"><p>0.991706</p></td></tr><tr><td align="center"><p>Durham county</p></td><td align="left"><p>0.992956</p></td><td align="left"><p>0.990785</p></td></tr><tr><td align="center"><p>Granville county</p></td><td align="left"><p>0.992960</p></td><td align="left"><p>0.990890</p></td></tr><tr><td align="center"><p>Harnett county</p></td><td align="left"><p>0.992954</p></td><td align="left"><p>0.991274</p></td></tr><tr><td align="center"><p>Lee county</p></td><td align="left"><p>0.992956</p></td><td align="left"><p>0.991688</p></td></tr><tr><td align="center"><p>Orange county</p></td><td align="left"><p>0.992956</p></td><td align="left"><p>0.992925</p></td></tr><tr><td align="center"><p>Person county</p></td><td align="left"><p>0.992962</p></td><td align="left"><p>0.991041</p></td></tr><tr><td align="center"><p>Wake county</p></td><td align="left"><p>0.992957</p></td><td align="left"><p>0.991528</p></td></tr></tbody></table>

One additional comparison we can make between the sets of common and independent solutions is to consider the level of diversity in their collections of Pareto-dominant solutions. The measure we use is from [[62]], specifically the measure they call generalized spread, which accounts for both the coverage of the extreme regions of the objectives and the distribution of solutions across the objective regions generally. Measures closer to 1.0 reflect higher, and therefore more desirable, levels of solution spread. Graphically, a high (close to 1.0) value of generalized spread reflects solutions that are spread throughout the range of objective values forming the Pareto frontier. This is reflected by Figs. 3 and 4 in that there are independent and common solutions spread fairly evenly across a wide range of combinations of pairs of cost and infection levels. Generalized spread allows us to assess this property without explicitly plotting a trade-off plot for every county. Computed values of generalized spread for our common and independent solutions across all 10 counties in our example are shown in Table 10, from which we can see that the generalized spread measures for all the counties our example cluster are quite similar between common and independent solutions.

Thus, we conclude from Table 9 that the common-protocol and independent-protocol solutions have the same statistical centrality measured by median statistic in the Mann Whitney U Test. In addition, they spread in the same fashion in the objective functions space based on the generalized spread measures in Table 10. Therefore, we reliably conclude that common- and independent-protocol solutions generate consistent outcomes.

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By Hadi Moheb-Alizadeh; Donald P. Warsing Jr.; Richard E. Kouri; Sajjad Taghiyeh and Robert B. Handfield

Reported by Author; Author; Author; Author; Author