3636institutetext: Center for Data-Driven Discovery in Biomedicine (D3b),
Children’s Hospital of Philadelphia, Philadelphia, PA, USA 22institutetext:
Department of Neurosurgery, University of Pennsylvania, Philadelphia, PA, USA
33institutetext: Center for AI & Data Science for Integrated Diagnostics
(AI2D) and Center for Biomedical Image Computing and Analytics (CBICA),
University of Pennsylvania, Philadelphia, PA, USA 44institutetext: Sheikh
Zayed Institute for Pediatric Surgical Innovation, Children’s National
Hospital, Washington DC, USA 55institutetext: Sage Bionetworks, USA
66institutetext: Medical Artificial Intelligence (MAI) Lab, Crestview
Radiology, Lagos, Nigeria 77institutetext: Montreal Neurological Institute
(MNI), McGill University, Montreal, QC, Canada 88institutetext: Department of
Radiology, Perelman School of Medicine, University of Pennsylvania,
Philadelphia, PA, USA 99institutetext: Department of Pathology and Laboratory
Medicine, Perelman School of Medicine, University of Pennsylvania,
Philadelphia, PA, USA 1010institutetext: Department of Radiology, Duke
University Medical Center, USA 1111institutetext: Mayo Clinic, MN, USA
1212institutetext: Center for Global Health, Perelman School of Medicine,
University of Pennsylvania, Philadelphia, PA, USA 1313institutetext:
Department of Informatics, Technical University Munich, Germany
1414institutetext: TranslaTUM - Central Institute for Translational Cancer
Research, Technical University of Munich, Germany 1515institutetext: Cancer
Imaging Program, National Cancer Institute, National Institutes of Health,
Bethesda, MD, USA 1616institutetext: New Jersey Institute of Technology,
Newark, NJ, USA 1717institutetext: Athinoula A Martinos Center for Biomedical
Imaging, Massachusetts General Hospital, Boston, MA, USA 1818institutetext:
Yale University, New Haven, CT, USA 1919institutetext: PrecisionFDA, U.S. Food
and Drug Administration, Silver Spring, MD, USA 2020institutetext: Helmholtz
AI, Helmholtz Munich, Germany 2121institutetext: Department of Radiation
Oncology, Duke University Medical Center, USA 2222institutetext: Department of
Applied Mathematics and Computer Science, Technical University of Denmark,
Denmark 2323institutetext: Booz Allen Hamilton, McLean, VA, USA
2424institutetext: Biomedical Image Analysis & Machine Learning, Department of
Quantitative Biomedicine, University of Zurich, Switzerland 2525institutetext:
Department of Neuroradiology, Technical University of Munich, Munich, Germany
2626institutetext: Mercy Catholic Medical Center, Darby, PA, USA
2727institutetext: Dana-Farber Brigham Cancer Center & Boston Children’s
Hospital, Boston, MA, USA 2828institutetext: University of California, San
Diego, CA, USA 2929institutetext: Center for Clinical Epidemiology and
Biostatistics, University of Pennsylvania, Philadelphia, USA
3030institutetext: Brain Tumor Institute, Children’s National Hospital,
Washington, DC, USA 3131institutetext: Brain Tumor Center, Cincinnati
Children’s Hospital, Cincinnati, OH, USA 3232institutetext: Pediatric Neuro-
Oncology Program, Nationwide Children’s Hospital, Columbus, OH, US
3333institutetext: Center for Cancer and Blood Disorders, Phoenix Children’s
Hospital, Phoenix, AZ, US 3434institutetext: Center for Cancer and Blood
Disorders, Children’s National Hospital , Washington, DC, USA
3535institutetext: Department of Radiology, The Children’s Hospital of
Philadelphia, PA, USA 3636institutetext: Departments of Radiology and
Pediatrics, George Washington University School of Medicine and Health
Sciences, Washington, DC, USA
† People involved in the organization of the challenge.
‡ People contributing data from their institutions.
§ People involved in annotation process.
* Equal authors.
** Corresponding author: 3636email<EMAIL_ADDRESS>
# The Brain Tumor Segmentation (BraTS) Challenge 2023: _Focus on Pediatrics
(CBTN-CONNECT-DIPGR-ASNR-MICCAI BraTS-PEDs)_

Anahita Fathi Kazerooni 112233††‡‡ Nastaran Khalili 11†† § § * * Xinyang Liu
44††** Debanjan Haldar 11†† § § * * Zhifan Jiang 44†† Syed Muhammed Anwar 44 †
† Jake Albrecht 55 † † Maruf Adewole 66†† Udunna Anazodo 77†† Hannah Anderson
11 § § Sina Bagheri 11 § § Ujjwal Baid 338899†† Timothy Bergquist 55†† Evan
Calabrese 1010†† Verena Chung 55†† Gian-Marco Conte 1111†† Farouk Dako 1212††
James Eddy 55†† Ivan Ezhov 13131414†† Ariana Familiar 11 ‡ ‡ Keyvan Farahani
1515†† Shuvanjan Haldar 1616 § § Juan Eugenio Iglesias 1717†† Anastasia Janas
1818†† Elaine Johansen 1919†† Florian Kofler 2020†† Dominic LaBella 2121††
Koen Van Leemput 2222†† Hongwei Bran Li 1717†† Nazanin Maleki 1818‡‡ Zeke
Meier 2323†† Bjoern Menze 24242525†† Ahmed W Moawad 2626†† Marie Piraud 2020††
Tina Poussaint 2727 ‡ ‡ Zachary Reitman 2121†† Jeffrey D Rudie 2828†† Ibraheem
Salman Shaikh 1616 Russel Taki Shinohara 2929†† Wenxin Tu 11§§ Karthik
Viswanathan 11 Chunhao Wang 2121†† Jeffrey B Ware 88§§ Benedikt Wiestler
2525†† Walter Wiggins 1010†† Anna Zapaishchykova 2727 ‡ ‡ Mariam Aboian 1818††
‡ ‡ Miriam Bornhorst 3030‡‡ Peter de Blank 3131 ‡ ‡ Michelle Deutsch 3232‡‡
Maryam Fouladi 3232‡‡ Lindsey Hoffman 3333‡‡ Benjamin Kann 2727 ‡ ‡ Margot
Lazow 3232‡‡ Leonie Mikael 3232‡‡ Ali Nabavizadeh 1188††‡‡§§ Roger Packer
3030‡‡ Adam Resnick 1122‡‡ Brian Rood 3434‡‡ Arastoo Vossough 113535††§§
Spyridon Bakas 338899†† Marius George Linguraru 443636††‡‡****

###### Abstract

Pediatric tumors of the central nervous system are the most common cause of
cancer-related death in children. The five-year survival rate for high-grade
gliomas in children is less than 20%. Due to their rarity, the diagnosis of
these entities is often delayed, their treatment is mainly based on historic
treatment concepts, and clinical trials require multi-institutional
collaborations. The MICCAI Brain Tumor Segmentation (BraTS) Challenge is a
landmark community benchmark event with a successful history of 12 years of
resource creation for the segmentation and analysis of adult glioma. Here we
present the CBTN-CONNECT-DIPGR-ASNR-MICCAI BraTS-PEDs 2023 challenge, which
represents the first BraTS challenge focused on pediatric brain tumors with
data acquired across multiple international consortia dedicated to pediatric
neuro-oncology and clinical trials. The BraTS-PEDs 2023 challenge focuses on
benchmarking the development of volumentric segmentation algorithms for
pediatric brain glioma through standardized quantitative performance
evaluation metrics utilized across the BraTS 2023 cluster of challenges.
Models gaining knowledge from the BraTS-PEDs multi-parametric structural MRI
(mpMRI) training data will be evaluated on separate validation and unseen test
mpMRI dataof high-grade pediatric glioma. The CBTN-CONNECT-DIPGR-ASNR-MICCAI
BraTS-PEDs 2023 challenge brings together clinicians and AI/imaging scientists
to lead to faster development of automated segmentation techniques that could
benefit clinical trials, and ultimately the care of children with brain
tumors.

###### Keywords:

BraTS, BraTS-PEDs, challenge, pediatric, brain, tumor, segmentation, volume,
deep learning, machine learning, artificial intelligence, AI

##  1 Introduction

Although rare, pediatric tumors of thecentral nervous system are the most
common cause of cancer-related death in children. While pediatric brain tumors
may share certain similarities with adult brain tumors, their imaging and
clinical presentations differ. For example, adult glioblastomas (GBMs),
pediatric diffuse midline gliomas (DMGs, including pediatric diffuse intrinsic
pontine glioma (DIPGs)) are high grade gliomas with short average overall
survival [1, 2]. The incidence of GBMs is 3 in 100,000 people, while DMGs are
about three times more rare. While GBMs are usually found in the frontal
or/and temporal lobes at an average age of 64 years, many DMGs are located in
the pons and often diagnosed between 5 and 10 years of age. Enhancing tumor
regions on post-gadolinium T1-weighted MRI and radiologically-defined necrotic
tissue regions are common imaging findings in GBM, but less common or clear in
DMGs, at least at initial diagnosis. Thus, pediatric brain tumors require
dedicated imaging tools that help in their characterization and facilitate
their diagnosis/prognosis [3, 4, 5].

Tumor segmentation is essential in surgical and treatment planning, as well as
in response assessment and longitudinal monitoring. However, manual
segmentation is time-consuming and has high inter-operator variability [6].
Pediatric brain tumors are variable in their aggressiveness, prognosis, and
heterogeneous histologic subregions, i.e., peritumoral edematous/invaded
tissue, necrotic core, and enhancing tumor core, as reflected in their radio-
phenotypes on multi-parametric magnetic resonance imaging (mpMRI) scans. Some
pediatric brain tumors, such as DIPGs, are heterogeneous and typically located
in surgically inaccessible anatomical locations, leaving them unresectable. As
such, assessment of tumor progression is mainly based on the changes in size
of the tumor measured from longitudinal scans. The current standard, as
recommended by the Response Assessment in Pediatric Neuro-Oncology (RAPNO)
cooperative working group [7, 8], is based on two-dimensional (2D) linear
measurements of the tumor subregions (in the axial slice of the largest tumor
extent), including enhancing tumor, non-enhancing tumor, cystic components,
and peritumoral edematous tissue. Nonetheless, these 2D measurements are
inaccurate surrogates of the volume and they cannot accurately account for
irregularity of the tumor shape and appearance and hence further increase the
inter-operator variability. Studies in adult brain tumors have indicated
superiority of 3D volumetric measurement of tumor extent in the prediction of
clinical endpoints over 2D methods [9]. While the importance of volumetric
tumor measurements, as compared to 2D ones, are being increasingly
acknowledged in response assessment of pediatric brain tumors [10, 2], there
is a lack of available automated tools to facilitate the cumbersome task of
segmentation of tumor subregions on assocaited scans. There are only a handful
of automated tumor segmentation methods explicitly proposed for pediatric
brain tumors [3, 11, 12, 13, 14, 15, 16, 17]. However, the majority of these
methods have been developed only for the segmentation of the T2 fluid
attenuated inversion recovery (FLAIR) abnormal signal [11, 12, 17], also
called whole tumor (WT) in the BraTS terminology. Furthermore, none of the
proposed methods has been evaluated and compared on the same validation data
and through a controlled study design, and therefore there is a gap in
benchmarking the automated segmentation tools in pediatric brain tumors.

The MICCAI brain tumor segmentation (BraTS) challenges have established a
community benchmark dataset and environment for adult glioma over the past 11
years [18, 19, 20, 21]. The focus of this year’s BraTS is expanded to a
Cluster of Challenges spanning across various tumor entities, missing data,
and technical considerations. During BraTS 2022, we organized the first
initiative to include pediatric brain tumors, specifically DMGs, in the test
phase of the BraTS 2022 challenge. The findings of this evaluation study
encouraged us to organize a larger and more diverse initiative in 2023 with
multi-institutional pediatric data. In the BraTS-PEDs 2023 challenge, the
focus is on pediatric brain tumor entities. We have created a retrospective
multi-institutional (multi-consortium) pediatric database, with the data
collected through a few consortia, including Children’s Brain Tumor Network
(CBTN, https://cbtn.org/ ) [22], DIPG Registry (https://www.dipgregistry.org)
and the COllaborative Network for NEuro-oncology Clinical Trials (CONNECT,
https://connectconsortium.org/). Additional data from participating pediatric
institutions have been included in the BraTS-PEDs cohort. The American Society
of Neuroradiology (ASNR, https://www.asnr.org/) collaborated in generating
ground truth annotation for the majority of data in this challenge. These data
will be collectively used to develop and quantitatively evaluate algorithms
developed and models trained by the challenge participants in unseen
validation data until July 2023, when the organizers will stop accepting new
submissions and evaluate submitted containerized algorithms in a completely
unseen pediatric patient population. This manuscript provides details about
the CBTN-CONNECT-ASNR-MICCAI BraTS-PEDs 2023 challenge, as the first
benchmarking initiative on pediatric brain tumor segmentation.

##  2 Materials & Methods

###  2.1 Data

The BraTS-PEDs 2023 dataset includes a retrospective multi-institutional
cohort of conventional/structural magnetic resonance imaging (MRI) sequences,
including pre- and post-gadolinium T1-weighted (labeled as T1 and T1CE),
T2-weighted (T2), and T2-weighted fluid attenuated inversion recovery
(T2-FLAIR) images, from 228 pediatric high-grade glioma. These conventional
multiparametric MRI (mpMRI) sequences are commonly acquired as part of
standard clinical imaging for brain tumors. However, the image acquisition
protocols and MRI equipment differ across different institutions, resulting in
heterogeneity in image quality in the provided cohort. Inclusion criteria
comprised of pediatric subjects with: (1) histologically-approved high-grade
glioma, i.e., high-grade astrocytoma and diffuse midline glioma (DMG),
including radiologically or histologically-proven diffuse intrinsic pontine
glioma (DIPG); (2) availability of all four structural mpMRI sequences on
treatment-naive imaging sessions. Exclusion criteria consisted of: (1) images
assessed to be of low quality or with artifacts that would not allow for
reliable tumor segmentation; and (2) infants younger than one month of age.
Data for n = 228 patients was obtained through CBTN (n = 138), Boston’s
Children Hospital (n = 61), and Yale University (n = 29).

The cohort included in the BraTS-PEDs 2023 challenge is split into training,
validation, and testing datasets. The data shared with the participants
comprise mpMRI scans and ground truth labels for the training cohort, as well
as mpMRI sequences without any associated ground truth for the validation
cohort. Notably, the testing data that will be used for evaluating the
performance of the methods submitted by challenge participants will not be
shared with the participants, but the containerized submissions of the
participants will be evaluated by the synapse.org platform, powered by
MedPerf[23].

Participants are prohibited from training their algorithm on any additional
public and/or private data (from their own institutions) besides the provided
BraTS-PEDs 2023 data, or using models pretrained on any other dataset. This
restriction is imposed to allow for a fair comparison among the participating
methods. However, participants can use additional public and/or private data
only for publication of their scientific papers, if they also provide a report
of their results on the data from BraTS-PEDs 2023 challenge alone, and discuss
potential differences in the obtained results.

####  2.1.1 Imaging Data Description

For all patients, mpMRI scans were pre-processed using a standardized
approach, including conversion of the DICOM files to the NIfTI file format
[24], co-registration to the same anatomical template (SRI24) [25], and
resampling to an isotropic resolution ($1mm^{3}$). The pre-processing pipeline
is publicly available through the Cancer Imaging Phenomics Toolkit (CaPTk)
111https://cbica.github.io/CaPTk/ [26, 27, 28] and Federated Tumor
Segmentation (FeTS) tool 222https://github.com/FETS-AI/Front-End/. De-
identification was performed through removing protected Health Information
(PHI) from DICOM headers in DICOM to NIfTI conversion step [29, 30]. De-facing
was performed via skull-stripping using a formerly proposed pediatric-specific
skull-stripping method [3] to prevent any potential facial
reconstruction/recognition of the patients.

The resulted images were segmented into tumor subregions using an integration
of two pediatric automated deep learning segmentation models [3, 31]. The
result of this step was segmentation of the tumors into four main subregions,
recommended by RAPNO working group for evaluation of the treatment response in
high-grade gliomas and DIPGs. The annotated tumor subregions comprised of
enhancing tumor (ET), nonenhancing tumor (NET), cystic component (CC), and
peritumoral edema (ED) regions. ET is described by areas with enhancement
(brightness) on T1 post-contrast images as compared to T1 pre-contrast. In
case of mild enhancement, checking the signal intensity of normal brain
structure can be helpful. CC typically appears with hyperintense signal (very
bright) on T2 and hypointense signal (dark) on T1CE. The cystic portion should
be within the tumor, either centrally or peripherally (as compared to ED which
is peritumoral). The brightness of CC is here defined as comparable or close
to cerebrospinal fluid (CSF). NET is defined as any other abnormal signal
intensity within the tumorous region that cannot be defined as enhancing or
cystic. For example, the abnormal signal intensity on T1, T2-FLAIR, and T2
that is not enhancing on T1CE should be considered as nonenhancing portion. ED
is defined by the abnormal hyperintense signal (very bright) on FLAIR scans.
ED is finger-like spreading that preserves underlying brain structure and
surrounds the tumor. The automatically-generated four labels (Figure 1) using
the automated segmentation tool were used as preliminary segmentation to be
manually revised by volunteer neuroradiology experts of varying rank and
experience, in accordance with annotation guidelines. The volunteer
neuroradiology expert annotators were provided with four mpMRI sequences (T1,
T1CE, T2, FLAIR) along with the fused automated segmentation volume to
initiate the manual refinements. The ITK-SNAP [32] software was used for
making these refinements. Once the automated segmentations were refined by the
annotators, three attending board-certified neuroradiologists, reviewed the
segmentations. Depending upon correctness, these segmentations were either
approved or returned to the individual annotator for further refinements. This
process was followed iteratively until the approvers found the refined tumor
subregion segmentations acceptable for public release and the challenge
conduction.

While four tumor subregions were annotated, in keeping with the other BraTS
2023 challenges and to allow the participants to incorporate data from adult
glioma challenge for their model training, the participants are provided with
three segmentation labels, including enhancing tumor (ET), nonenhancing
component (NC - a combination of nonenhancing tumor, cystic component, and
necrosis), as well as peritumoral edematous area (ED) (Figure 1).

Figure 1: Illustrative example of tumor subregions in pediatric brain tumors.
Image panels with the annotated tumor subregions along with mpMRI structural
scans (T1, T1CE, T2, and T2-FLAIR). The left-most side image on the bottom
panel with the overlaid annotations showcases the original tumor subregions,
i.e., enhancing tumor (ET - red), non-enhancing tumor (NET - green), cystic
component (CC - yellow), and edema (ED - teal). The image on the right hand
side of the bottom panel with the overlaid annotations on the T2 sequence
demonstrates the tumor subregions provided to the BraTS-PEDs 2023
participants: enhancing tumor (ET), nonenhancing component (NC), and edema
(ED) [3].

####  2.1.2 Common errors of automated segmentations

In our experience with automated segmentation of pediatric brain tumors, some
errors may be noticed:

  1. 1.

Peri-ventricular edema segmented as edema

  2. 2.

Remote areas segmented as tumor (far from actual tumor region)

  3. 3.

Under-segmentation of cysts

  4. 4.

Non-enhancing tumor segmented as cyst, or vice versa

###  2.2 Model Training Using Generally Nuanced Deep Learning Framework
(GaNDLF)

We will offer to the participants a baseline approach implemented in a modular
open-source framework, namely the Generally Nuanced Deep Learning Framework
(GaNDLF) [33], which is maintained by the MLCommons organization
(github.com/mlcommons/gandlf). GaNDLF offers some network architectures, but
also allows the user to leverage the functionality of other libraries, such as
PILLOW and MONAI. The participants can decide to use GaNDLF to develop their
approach or use their own custom source code. For their submission, they need
to package their submitting approach in an MLCube container, instructions for
which will be clearly provided in the Synapse platform. MLCube containers are
automatically generated by GaNDLF and will be used to evaluate all submissions
through the MedPerf platform [23] on each contributing site’s data.

###  2.3 Performance Evaluation

As described earlier and exemplified in Figure 2 (bottom panel), for the
BraTS-PEDs 2023 challenge, the regions to evaluate the performances are: i)
the “enhancing tumor” (ET), ii) the “enhancing tumor/cystic
component/necrosis” (NC) and iii) the entire tumorous region, or the so-called
“whole tumor” (WT).

The participants are required to send the output of their methods to the
evaluation platform for the scoring to occur during the training and the
validation phases. At the end of the validation phase the participants are
asked to identify the method they would like to evaluate in the final
testing/ranking phase. The organizers will then confirm receiving the
containerized method and will evaluate it on withheld testing data. The
participants will be provided guidelines on the form of the container as we
have done in previous years. This will enable confirmation of reproducibility,
comparing these algorithms to the previous BraTS instances and comparison with
results obtained by algorithms of previous years, thereby maximizing solutions
in solving the problem of brain tumor segmentation.

During the training and validation phases, the participants will be able to
test the functionality of their submission through three platforms:

  1. 1.

Cancer Imaging Phenomics Toolkit (CaPTk [5-6], https://github.com/CBICA/CaPTk)

  2. 2.

Federated Tumor Segmentation (FeTS) Tool [7] (https://fets-ai.github.io/Front-
End/)

  3. 3.

Online evaluation platform (Synapse)

###  2.4 Participation Timeline

The challenge is composed of three main stages:

  1. 1.

Training: The four MRI sequences along with the corresponding ground truth
labels will be shared with participants to design and train their methods.

  2. 2.

Validation: The validation data will be released to the participants within
three weeks after the training data. The participants will not be provided
with the ground truth of the validation data, but will be given the
opportunity to submit multiple times to the online evaluation platforms. The
participants can generate preliminary results for their trained models in
unseen data and report them in their submitted short MICCAI LNCS papers , in
addition to their cross-validated results on the training data, to be
published in conjunction with the proceedings of the BrainLes workshop. The
top-ranked participating teams in the validation phase will be invited to
prepare their slides for a short oral presentation of their method during the
BraTS challenge at MICCAI 2023.

  3. 3.

Testing/Ranking: After the participants upload their containerized method in
the evaluation platforms, they will be evaluated and ranked on unseen testing
data, which will not be made available. The final top-ranked participating
teams will be announced at the 2023 MICCAI Annual Meeting.

##  3 Discussion

In this paper, we outlined the design of the first pediatric brain tumor
segmentation (BraTS-PEDs 2023) challenge, to benchmark methods devised for
segmentation of pediatric brain tumors. In 2022, for the first time, the BraTS
(continuous evaluation) challenge included testing the top-ranked methods,
designed for segmentation of adult gliomas, on unseen held out data of
pediatric brain tumors. The results from last year’s BraTS 2022 showed that
while training and optimizing the models on adult brain tumors could result in
relatively high Dice scores in segmentation of the whole tumor region on mpMRI
scans of pediatric brain tumors, they perform poorly on segmentation of tumor
subregions (e.g. in segmentation of enhancing tumor/cystic components as a
counterpart to tumor core in adult brain tumors). These results supported our
hypothesis that the techniques to be applied to pediatric population need to
be trained (at least partly) on curated and annotated cohort of pediatric
brain tumors. However, similar to any other machine and deep learning
problems, training a generalizable model mandates ample dataset processed in a
standardized approach. To achieve this goal, BraTS-PEDs dataset provides the
largest annotated cohort of pediatric high-grade glioma (astrocytoma, and
DMG/DIPGs). We are actively working on increasing the number of subjects in
this cohort to provide the community with a large dataset of these rare
tumors, and to facilitate the future development of tools for computer-aided
treatment planning. Future BraTS-PEDs challenges will include data from more
institutions and will be extended to post-operative or post-treatment scans.

## Acknowledgments

Success of any challenge in the medical domain depends upon the quality of
well annotated multi-institutional datasets. We are grateful to all the data
contributors, annotators, and approvers for their time and efforts. Our
profound thanks go to the Children’s Brain Tumor Network (CBTN), the
Collaborative Network for Neuro-oncology Clinical Trials (CONNECT), the
International DIPG/DMG Registry (DIPGR), the American Society of
Neuroradiology (ASNR), and the Medical Image Computing and Computer Assisted
Intervention (MICCAI) Society for their invaluable support of this challenge.

## Funding

Research reported in this publication was partly supported by the National
Institutes of Health (NIH) under award numbers: NCI/ITCR:U01CA242871 and
NCI:UH3CA236536. The content of this publication is solely the responsibility
of the authors and does not represent the official views of the NIH.

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# Effective Testing Policies for Controlling an Epidemic Outbreak

Muhammad Umar B. Niazi Alain Kibangou Carlos Canudas-de-Wit
Denis Nikitin Liudmila Tumash Pierre-Alexandre Bliman M.U.B. Niazi, C.
Canudas-de-Wit, A. Kibangou, D. Nikitin, and L. Tumash are with Université
Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-Lab, 38000 Grenoble, France.
Emails in the respective order<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Bliman is with
Sorbonne Université, Université Paris-Diderot SPC, Inria, CNRS, Laboratoire
Jacques-Louis Lions, équipe Mamba, 75005 Paris, France. Email: pierre-
<EMAIL_ADDRESS>This work is partially supported by European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (ERCAdG no. 694209, Scale-FreeBack, website:
http://scale-freeback.eu/) and by Inria, France, in the framework of the Inria
Mission Covid-19 (Project HealthyMobility).

###### Abstract

Testing is a crucial control mechanism for an epidemic outbreak because it
enables the health authority to detect and isolate the infected cases, thereby
limiting the disease transmission to susceptible people, when no effective
treatment or vaccine is available. In this paper, an epidemic model that
incorporates the testing rate as a control input is presented. The proposed
model distinguishes between the undetected infected and the detected infected
cases with the latter assumed to be isolated from the disease spreading
process in the population. Two testing policies, effective during the onset of
an epidemic when no treatment or vaccine is available, are devised: (i) best-
effort strategy for testing (BEST) and (ii) constant optimal strategy for
testing (COST). The BEST is a suppression policy that provides a lower bound
on the testing rate to stop the growth of the epidemic. The COST is a
mitigation policy that minimizes the peak of the epidemic by providing a
constant, optimal allocation of tests in a certain time interval when the
total stockpile of tests is limited. Both testing policies are evaluated by
their impact on the number of active intensive care unit (ICU) cases and the
cumulative number of deaths due to COVID-19 in France.

##  I Introduction

On March 16, 2020, five days after the World Health Organization (WHO)
declared COVID-19 to be a pandemic, the Director-General of WHO, Dr. T. A.
Ghebreyesus, ended his media briefing with a simple message to all countries:
“Test, test, test”. This is because testing is well-known to be a crucial
control mechanism for epidemics when no effective treatment or vaccine is
available [1]. It limits the spread of disease to the susceptible population
by enabling the health authority to detect and isolate the infected people.

Recently, several models and techniques have been developed to study testing
policies for epidemic control. Similar to a resource allocation problem in
epidemic control [2], the optimal test allocation problem is posed in [3] as a
well-known sensor selection problem in control theory. A similar problem
studied in [4] considers the specificity and sensitivity of tests and solves
the optimal test allocation problem as a welfare maximization problem.
However, the underlying assumption in these papers is the availability of
information portfolios of all the people in a society, thus limiting their
applicability to larger scales. Another line of research studies testing
policies from an economic perspective, for instance, [5] and [6] aim to find
an optimal trade-off between the economic activity and epidemic mitigation. In
[7], an optimal trade-off between testing effort and lockdown intervention is
computed numerically under the constraint of limited Intensive Care Units
(ICU). However, none of these papers provide a control-theoretic perspective
of testing, namely how testing can be used to change the dynamics of an
epidemic?

In this paper, we study testing policies in terms of testing rate, i.e.,
number of tests performed per day, with the aim to suppress and/or mitigate an
epidemic outbreak. We introduce a new model with the acronym SIDUR—susceptible
(S), undetected infected (I), detected infected (D), unidentified recovered
(U), and identified removed (R)—that has the testing rate as a control input.
The influence of the control input in the SIDUR model is directly linked with
the testing specificity, which determines the probability of detecting an
infected person by a test. Similar to [8, 9, 10], we distinguish the
undetected infected from the detected infected population. Only the undetected
infected people participate in the disease transmission, and the detected
infected are assumed to be either quarantined and/or hospitalized. Moreover,
the unidentified recovered people, who were infected and then recovered
naturally without getting detected, are distinguished from the identified
removed people, who were infected and then recovered or died in a hospital or
quarantine after getting detected.

We propose two testing policies for epidemic control: (1) Best-effort strategy
for testing (BEST) and (2) Constant optimal strategy for testing (COST). The
BEST is a suppression policy for an epidemic that computes the minimum testing
rate to immediately stop the growth of the epidemic when applied from a
certain day onward. The COST is a mitigation policy for an epidemic that
reduces, but not necessarily stops, the growth of the epidemic and is
meaningful only when the total stockpile of tests is limited. It provides the
constant optimal testing rate that must be utilized until the day when the
stockpile of tests finishes. We evaluate both testing policies by predicting
their impact on the number of active intensive care unit (ICU) cases and the
cumulative number of deaths due to COVID-19 in France. Both testing policies
are effective at the beginning of an epidemic outbreak when the treatments or
vaccines against the disease are not available.

This paper focuses solely on the SIDUR model design and testing policies, and
excludes model estimation and validation, which is reported in the detailed
version [11]. The paper is organized as follows: Section II presents the SIDUR
model, identifies the measured outputs in relation to the model states, and
provides a partial solution of the model. Then, in Section III and IV, we
propose the BEST and COST policies, and evaluate them for the case of COVID-19
outbreak in France from February to July 2020. Finally, Section V presents the
concluding remarks.

##  II Model Design

###  II-A Model Equations

We introduce a five-compartment model, SIDUR, with the purpose of devising and
evaluating testing policies. We consider the timescale of days and, at time
$t\in\mathbb{R}_{\geq 0}$, each compartment is characterized by a single state
representing its population, where

  * •

$x_{\mathtt{S}}(t)$: Number of susceptible people who are prone to get
infected by the disease.

  * •

$x_{\mathtt{I}}(t)$: Number of undetected infected people who are infected but
not diagnosed by the health authority.

  * •

$x_{\mathtt{D}}(t)$: Number of detected infected people who are infected and
are diagnosed through a test.

  * •

$x_{\mathtt{U}}(t)$: Number of unidentified recovered people who were infected
and then recovered naturally without getting diagnosed.

  * •

$x_{\mathtt{R}}(t)$: Number of identified removed people who were infected,
diagnosed, and then either recovered or died.

######  Assumption 1.

  1. (i)

The total population $N$ remains constant during the evolution of the epidemic

$x_{\mathtt{S}}(t)+x_{\mathtt{I}}(t)+x_{\mathtt{D}}(t)+x_{\mathtt{U}}(t)+x_{\mathtt{R}}(t)=N,\quad\forall\,t\in\mathbb{R}_{\geq
0}.$

  2. (ii)

Only the undetected infected $x_{\mathtt{I}}(t)$ are responsible for the
disease transmission to the susceptible people $x_{\mathtt{S}}(t)$.

  3. (iii)

All the deaths occurring due to the epidemic are known, reported, and included
in the identified removed $x_{\mathtt{R}}(t)$.

  4. (iv)

After recovery, people acquire permanent immunity and are not infected again.

Let $\beta\geq 0$ be the infection rate defined as the product of the contact
rate by the probability of disease transmission during a contact between a
susceptible and an infected persons. The value of $\beta$ not only depends on
the disease biology but is also influenced by the implementation of non-
pharmaceutical interventions (NPI) such as social distancing, lockdown, travel
restrictions, and preventive measures. Thus, $\beta=\beta(t)$ is a time-
varying parameter that depends on the extent to which the NPIs are
implemented. Depending on the time periods when different NPI policies are
implemented, we assume $\beta$ to be piece-wise constant.

Let $x_{\mathtt{T}}(t)$ be the testable population defined as

$\begin{array}[]{ccl}x_{\mathtt{T}}(t)&:=&x_{\mathtt{I}}(t)+(1-\theta)\left(x_{\mathtt{S}}(t)+x_{\mathtt{U}}(t)\right)\end{array}$

where $\theta\in[0,1]$ is the testing specificity parameter. Note that the
infected population $x_{\mathtt{I}}(t)\leq x_{\mathtt{T}}(t)$, for every $t$,
where the equality holds if $\theta=1$. Given $\theta\in[0,1]$, the
probability of detecting an infected person per test in a homogeneous
population structure is $x_{\mathtt{I}}(t)/x_{\mathtt{T}}(t)$. Thus, if
$\theta=1$, the probability of detection is equal to one, and if $\theta=0$,
the probability of detection is the lowest. Depending on the testing policy,
testing specificity, and the level of contact tracing, we consider
$\theta=\theta(t)$ to be time-varying and piece-wise constant.

Finally, let $\gamma>0$ and $\rho>0$ be the recovery rate and removal rate,
respectively, which are assumed to be constants that depend on the disease
biology. The inverse of the recovery rate $1/\gamma$ is defined to be the
average recovery time after which a typical undetected infected person
naturally recovers, whereas the inverse of the removal rate $1/\rho$ is
defined to be the average removal time after which a typical detected infected
person recovers or dies.

The SIDUR model, depicted in Fig. 1, is given by

$\displaystyle\dot{x}_{\mathtt{S}}(t)$ $\displaystyle=$
$\displaystyle\displaystyle-\beta\;x_{\mathtt{S}}(t)\frac{x_{\mathtt{I}}(t)}{N}$
(1a) $\displaystyle\dot{x}_{\mathtt{I}}(t)$ $\displaystyle=$
$\displaystyle\displaystyle\beta\;x_{\mathtt{S}}(t)\frac{x_{\mathtt{I}}(t)}{N}-u(t)\frac{x_{\mathtt{I}}(t)}{x_{\mathtt{T}}(t)}-\gamma
x_{\mathtt{I}}(t)$ (1b) $\displaystyle\dot{x}_{\mathtt{D}}(t)$
$\displaystyle=$ $\displaystyle
u(t)\frac{x_{\mathtt{I}}(t)}{x_{\mathtt{T}}(t)}-\rho x_{\mathtt{D}}(t)$ (1c)
$\displaystyle\dot{x}_{\mathtt{U}}(t)$ $\displaystyle=$ $\displaystyle\gamma
x_{\mathtt{I}}(t)$ (1d) $\displaystyle\dot{x}_{\mathtt{R}}(t)$
$\displaystyle=$ $\displaystyle\rho x_{\mathtt{D}}(t)$ (1e)

where $u(t)\geq 0$ is the testing rate. Notice that the model (1) fulfils
Assumption 1 for any $u(t)\geq 0$.

Susceptible Infection rate Undetected Infected Testing rate Detected Infected
Recovery rate Unidentified recovered Removal rate Identified Removed Figure 1:
Block diagram of SIDUR model.

###  II-B Control Input and Measured Outputs

The testing rate $u(t)$, which is defined to be the number of tests performed
per day, is the control input in SIDUR model. In order to diagnose the
infected people at time $t$, the tests are allocated to a proportion of the
testable population $x_{\mathtt{T}}(t)$. The testing specificity parameter
$\theta$ allows for the adjustment of the testable population to accommodate
for the detection rate of tests.

The data reported by the health authorities are chosen as the measured outputs
of the SIDUR model. They are five measurements, related directly to the states
as follows:

  * •

Cumulative number of detected people

$y_{1}(t)=x_{\mathtt{D}}(t)+x_{\mathtt{R}}(t).$ (2)

  * •

Cumulative number of removed people

$y_{2}(t)=x_{\mathtt{R}}(t).$ (3)

  * •

Number of positively tested people per day

$y_{3}(t)=u(t)\frac{x_{\mathtt{I}}(t)}{x_{\mathtt{T}}(t)}.$ (4)

Let the number of active ICU patients (or ICU beds occupied) be denoted as
$B(t)$ and the cumulative number of deaths (or extinct cases) be denoted
$E(t)$. Note that $B(t)$ and $E(t)$ are also measured and reported, and are
functions of the number of active infected cases and the cumulative number of
infected cases, respectively.

  * •

Number of active ICU cases

$y_{4}(t):=B(t)=g(A(t-\psi^{-1}))$ (5)

where $A(t)=x_{\mathtt{I}}(t)+x_{\mathtt{D}}(t)$ is the number of active
infected cases, $\psi^{-1}$ is the average time period which a typical
infected ICU case takes from getting infected to being admitted to the ICU,
and the function $g$ is to be chosen such that $y_{4}(t)=B(t)$ fits the data.

  * •

Cumulative number of deaths

$y_{5}(t):=E(t)=h(I(t-\phi^{-1}))$ (6)

where $I(t)=N-x_{\mathtt{S}}(t)$ is the cumulative number of infected cases,
$\phi^{-1}$ is the average time period which a typical non-surviving case
takes from getting infected to death, and the function $h$ is to be chosen
such that $y_{5}(t)=E(t)$ fits the data.

###  II-C Reproduction Number

An important quantity to determine the epidemic potential at the onset of a
disease is the basic reproduction number $R_{0}$. In the later stages,
however, the effective reproduction number $R(t)$ is more suitable [12, 13] to
predict the spread of the disease. The $R(t)$ for the SIDUR model is defined
as the ratio of the number of newly infected people over the number of newly
detected and recovered people at time $t$, i.e.,

$R(t)=\frac{\beta(t)x_{\mathtt{S}}(t)\frac{x_{\mathtt{I}}(t)}{N}}{u(t)\frac{x_{\mathtt{I}}(t)}{x_{\mathtt{T}}(t)}+\gamma
x_{\mathtt{I}}(t)}=\frac{\beta(t)}{\frac{u(t)}{x_{\mathtt{T}}(t)}+\gamma}\frac{x_{\mathtt{S}}(t)}{N}.$

If $R(t)<1$, then more people are being detected and/or recovering than the
people being infected at time $t$, implying that $x_{\mathtt{I}}(t)$ will
decrease in the near future. If $R(t)>1$, then more people are being infected
than the people being detected and recovering at time $t$, implying that
$x_{\mathtt{I}}(t)$ will increase in the near future. The basic reproduction
number $R_{0}=R(0)$ and is given by

$R_{0}\approx\frac{\beta(0)}{\frac{u(0)}{(1-\theta)N}+\gamma}$

where we assume that $x_{\mathtt{S}}(0)\approx N$ and
$x_{\mathtt{T}}(0)\approx(1-\theta)N$. This expression of $R_{0}$ can also be
obtained by following the methodology of [14].

###  II-D Partial Solution of SIDUR Model

Consider the infection time $\xi$ defined as $d\xi=x_{\mathtt{I}}(t)dt$, where

$\xi(t)=\int_{0}^{t}x_{\mathtt{I}}(\eta)d\eta$ (7)

and

$t(\xi)=\int_{0}^{\xi}\frac{1}{x_{\mathtt{I}}(\eta)}d\eta$ (8)

with $\xi=0$ if and only if $x_{\mathtt{I}}\equiv 0$ on $[0,t]$. Moreover,

$\frac{dx_{\mathtt{I}}}{dt}(t)=0\quad\Leftrightarrow\quad\frac{dx_{\mathtt{I}}}{d\xi}(\xi)=0$

because
$dx_{\mathtt{I}}/dt=(dx_{\mathtt{I}}/d\xi)(d\xi/dt)=(dx_{\mathtt{I}}/d\xi)x_{\mathtt{I}}$.
Thus, at the time $t_{p}$ of the infection peak, we have

$\frac{dx_{\mathtt{I}}}{dt}\bigg{|}_{t=t_{p}}=0\quad\Leftrightarrow\quad\frac{dx_{\mathtt{I}}}{d\xi}\bigg{|}_{\xi=\xi_{p}}=0$

where

$\xi_{p}=\int_{0}^{t_{p}}x_{\mathtt{I}}(\eta)d\eta.$

######  Proposition 1.

Consider the infection time $\xi$, then:

$x_{\mathtt{S}}(\xi)=x_{\mathtt{S}}^{0}e^{-\frac{\beta}{N}\xi}$ (9)
$x_{\mathtt{U}}(\xi)=x_{\mathtt{U}}^{0}+\gamma\xi$ (10)

where $x_{\mathtt{S}}^{0}=x_{\mathtt{S}}(0)$ and
$x_{\mathtt{U}}^{0}=x_{\mathtt{U}}(0)$.

###### Proof.

First, from (1a), we obtain

$\frac{dx_{\mathtt{S}}}{d\xi}=-\frac{\beta}{N}x_{\mathtt{S}}(\xi)$

whose solution is given by (9). Then, from (1d), we obtain

$\frac{dx_{\mathtt{U}}}{d\xi}=\gamma$

whose solution is given by (10). ∎

The testing specificity parameter $\theta$ can be close to one under different
circumstances. For instance, if the testing is assisted with efficient contact
tracing, then $\theta\approx 1$. However, if the tests are performed only on
people with apparent symptoms of the disease, then $\theta\approx 1$ also. If
we assume a scenario unlike those mentioned above, then $\theta\ll 1$. In
addition, if we assume that $x_{\mathtt{S}}(t)+x_{\mathtt{U}}(t)\approx N$,
then we can obtain an approximate solution of $x_{\mathtt{I}}(\xi)$ by
assuming $x_{\mathtt{T}}(t)\approx(1-\theta)N$.

######  Proposition 2.

Assume the testable population $x_{\mathtt{T}}(t)\approx(1-\theta)N$. Then, an
approximate solution of the undetected infected population with respect to the
infection time $\xi$ is

$x_{\mathtt{I}}(\xi)=\displaystyle
x_{\mathtt{I}}^{0}+x_{\mathtt{S}}^{0}(1-e^{-\frac{\beta}{N}\xi})-\gamma\xi-\int_{0}^{\xi}\frac{u(\eta)}{(1-\theta)N}d\eta.$
(11)

where $x_{\mathtt{I}}^{0}=x_{\mathtt{I}}(0)$.

###### Proof.

From (1b) and (9), we have

$\frac{dx_{\mathtt{I}}}{d\xi}=\frac{\beta}{N}x_{\mathtt{S}}^{0}e^{-\frac{\beta}{N}\xi}-\gamma-\frac{u}{(1-\theta)N}$

where we assumed $x_{\mathtt{T}}=(1-\theta)N$. Integrating both sides from $0$
to $\xi$ gives (11). ∎

##  III Best-Effort Strategy for Testing

Suppose that the tests can be produced and supplied easily, and there is no
upper bound on the total stockpile of tests. Moreover, we say that, at any
time $t$, the epidemic is spreading if $x_{\mathtt{I}}(t)$ is increasing,
i.e., $R(t)>1$. On the other hand, we say that the epidemic is non-spreading
if $x_{\mathtt{I}}(t)$ is not increasing, i.e., $R(t)\leq 1$. Then, the best
effort strategy for testing (BEST) at any given time $t^{*}$ is the minimum
number of tests to be performed per day from time $t^{*}$ onward such that the
epidemic switches from spreading to non-spreading at $t^{*}$. In other words,
the BEST policy provides the smallest lower bound on the testing rate that is
sufficient to change, at a given time $t^{*}$, the course of the epidemic from
spreading to non-spreading. We remark that the BEST policy is meaningful only
when the epidemic is spreading.

###  III-A Computation of the BEST policy

Define a function

$c^{*}(t)=x_{\mathtt{T}}(t)\left|\frac{\beta(t)}{N}x_{\mathtt{S}}(t)-\gamma\right|_{+}$

where, by definition, for any scalar $z$, $|z|_{+}=z$ if $z>0$ and $|z|_{+}=0$
otherwise.

######  Proposition 3.

Assume that the infection rate $\beta$ is non-increasing while the testing
specificity parameter $\theta$ is non-decreasing on a time interval
$[t^{*},t_{1})$, for some $t^{*}<t_{1}$. Then, the BEST policy at time $t^{*}$
is given by

$u(t)=c^{*}(t^{*})=x_{\mathtt{T}}(t^{*})\left|\frac{\beta(t^{*})}{N}x_{\mathtt{S}}(t^{*})-\gamma\right|_{+},\quad\forall
t\in[t^{*},t_{1}).$

###### Proof.

In order to prove that $c^{*}(t^{*})$ for $t\in[t^{*},t_{1})$ is the BEST at
$t^{*}$, we show the following:

  1. (i)

If $u(t)>c^{*}(t)$ (resp., $u(t)\geq c^{*}(t)$) for any $t\in[t^{*},t_{1})$,
then $x_{\mathtt{I}}$ is decreasing (resp., non-increasing) on
$[t^{*},t_{1})$.

  2. (ii)

If $u(t)>c^{*}(t^{*})$ (resp., $u(t)\geq c^{*}(t^{*})$) for any
$t\in[t^{*},t_{1})$, then $x_{\mathtt{I}}$ is decreasing (resp., non-
increasing) on $[t^{*},t_{1})$.

Assume that $u(t)>c^{*}(t)$ on $[t^{*},t_{1})$. Then,
$\Phi(t):=\beta(t)\frac{x_{\mathtt{S}}(t)}{N}-\frac{u(t)}{x_{\mathtt{T}}(t)}-\gamma<0$
which implies that $x_{\mathtt{I}}$ is decreasing because
$\dot{x}_{\mathtt{I}}(t)=\Phi(t)x_{\mathtt{I}}(t)$ almost everywhere. If only
the weaker assumption $u(t)\geq c^{*}(t)$ on $[t^{*},t_{1})$ is fulfilled,
then, by using the continuity of the solutions of ODE with respect to
perturbations of the right-hand side, one gets that $x_{\mathtt{I}}$ is non-
increasing.

Assume now that $u(t)>c^{*}(t^{*})$ on $[t^{*},t_{1})$, where $c^{*}(t^{*})$
is constant. Then, by continuity, $u(t)>c^{*}(t)$ on a certain interval
$[t^{*},t_{2})$, for some $t_{2}\in(t^{*},t_{1}]$. As a consequence of the
result (i) shown previously, $x_{\mathtt{I}}$ decreases on $[t^{*},t_{2})$.
Moreover, assume that $t_{2}$ is the maximal point in $(t^{*},t_{1}]$ having
this property. In order to show that $t_{2}=t_{1}$, it is sufficient to show
that $u(t_{2})>c^{*}(t_{2})$, otherwise one may consider a larger value for
$t_{2}$ which will lead to a contradiction with the fact that it is maximal.
Since $x_{\mathtt{I}}$ decreases on $[t^{*},t_{2})$ and $\theta$ is non
decreasing, we can conclude that $x_{\mathtt{T}}$ also decreases on this
interval. On the other hand, since $x_{\mathtt{S}}$ is always decreasing and
$\beta$ is non increasing, one can conclude that $c^{*}(t)$ also decreases on
$[t^{*},t_{2})$. This is obtained by upper bounding $c^{*}(t)$. Thus, one has
$c^{*}(t^{*})>c^{*}(t)$, which implies that that $u(t_{2})>c^{*}(t_{2})$.
Therefore, as $t_{2}=t_{1}$, we have established that $x_{\mathtt{I}}$
decreases on the whole interval $[t^{*},t_{1})$. For the case where $u(t)\geq
c^{*}(t^{*})$, we can use the same argument of continuity of the trajectories.

From the previous results, one deduces that the BEST policy is given by
$c^{*}(t^{*})$ and the testing rate $u(t)\geq c^{*}(t^{*})$ for
$t\in[t^{*},t_{1})$. If $u(t)<c^{*}(t^{*})$, for $t\in[t^{*},t_{1})$, then one
can show that the epidemic goes on spreading in the interval $[t^{*},t_{1})$.
Hence, $u(t)=c^{*}(t^{*})$ is the BEST policy at $t^{*}$. ∎

######  Remark 1.

Requiring that $\beta$ must not increase and $\theta$ must not decrease in the
interval $[t^{*},t_{1})$ for some $t_{1}>t^{*}$ is necessary for the BEST
policy. It is thus important to keep the external conditions that determine
the values of $\beta$ and $\theta$ either constant or such that $\beta$
decreases (e.g., through the implementation of lockdown) and/or $\theta$
increases (e.g., through efficient contact tracing). On the other hand, the
case where $\beta$ decreases and/or $\theta$ increases at some time
$t_{1}>t^{*}$ has the effect of speeding up the suppression of the epidemic
under BEST policy. $\lrcorner$

###  III-B BEST Policy for the COVID-19 Case of France

Given the COVID-19 data of France, we first estimate the parameters of the
SIDUR model by fitting the model outputs (2), (3), and (4) with the
corresponding data (see [11] for details). Then, we compute $c^{*}(t^{*})$ for
different values of $t^{*}$ from January 24 to March 15. Figure 2 shows the
testing rate $u(t^{*})$ required by the BEST policy if it is implemented from
day $k=\lfloor t^{*}\rfloor$, where $\lfloor\cdot\rfloor$ denotes the floor
function, and the corresponding value of the peak of undetected infected
$x_{\mathtt{I}}(t)$. It can be seen that the required testing rate increases
exponentially if the BEST policy is not implemented early on in the epidemic
phase. This is because the number of undetected infected $x_{\mathtt{I}}(t)$
becomes very large, which increases the difficulty of controlling the
epidemic.

Figure 2: Testing rate $u(t)$ required by the BEST policy (left y-axis, black)
and the predicted value of the corresponding epidemic peak (right y-axis, red,
in logscale) with respect to time. Figure 3: Predicted number of infected
cases $x_{\mathtt{I}}(t)$: actual testing scenario vs. BEST

Let us now consider a scenario where BEST is implemented on March 01, 2020.
Then, Figure 3 depicts the evolution of undetected infected
$x_{\mathtt{I}}(t)$ when $u(t)$ is given by the data on the number of tests
per day in France and when $u(t)$ is given by BEST policy. To evaluate BEST,
we use $u(t)$ as given by the data until February 29 and then by the BEST
policy from March 01 onward. In the actual COVID-19 case of France, our model
predicted the peak of the undetected infected $x_{\mathtt{I}}(t)$ to be about
$6$ million. However, if the BEST policy was implemented, the peak of
$x_{\mathtt{I}}(t)$ could have been reduced to $363,169$ with the testing rate
$c^{*}\approx 147,000$, which is quite feasible. The impact of the BEST policy
on the number of active ICU cases $B(t)$ and the cumulative number of deaths
$E(t)$ is evaluated after fitting the model outputs (5) and (6) with the data
in the actual testing scenario. The results are illustrated in Figure 4 and 5.
From the figures, we observe that the peak of the number of active ICU
patients could have been reduced by $34.71\%$ and the number of deaths could
have been reduced by $74.45\%$ if the BEST policy was implemented.

Figure 4: The prediction of the number of active ICU cases $B(t)$: actual
scenario vs. BEST policy. Figure 5: The prediction of the cumulative number of
deaths $E(t)$: actual scenario vs. BEST policy.

##  IV Constant Optimal Strategy for Testing

Suppose that the total stockpile of tests is limited and given by
$r_{\max}\in\mathbb{N}$, where $\mathbb{N}$ denotes the set of natural
numbers. Then, the problem is to find the optimal testing rate that minimizes
the epidemic peak, i.e., the peak value of the undetected infected population
$x_{\mathtt{I}}(t)$. We seek an optimal trade-off because if the testing rate
is chosen high, then the stockpile finishes early and leaves us with no tests
at hand, which could result in a second wave of the epidemic more intense than
the first wave. On the other hand, if the testing rate is chosen low, then it
is impossible to limit the epidemic spread (or ‘flatten the curve’), which
could challenge the available medical facilities of the country.

In this section, we determine a constant optimal daily allocation of a limited
stockpile of tests $r_{\max}$ minimizing the epidemic peak. Such an allocation
is called the constant optimal strategy for testing (COST). Given the total
stockpile of tests $r_{\max}$, the constant testing rate is given by

$u(t)=\left\\{\begin{array}[]{ll}C,&\text{if}~{}0\leq t\leq T\\\
0,&\text{if}~{}t>T\end{array}\right.$ (12)

where the time period $T:=r_{\max}/C$ represent the duration of the testing
policy once $C$ is determined. Therefore, the approximate solution (11) of
undetected infected population is given by

$x_{\mathtt{I}}(\xi)=\left\\{\begin{array}[]{ll}\displaystyle
x_{\mathtt{I}}^{0}+x_{\mathtt{S}}^{0}\left(1-e^{-\frac{\beta}{N}\xi}\right)-\frac{C\xi}{(1-\theta)N}-\gamma\xi,&\xi<\Xi\\\\[7.5pt]
\displaystyle
x_{\mathtt{I}}^{0}+x_{\mathtt{S}}^{0}\left(1-e^{-\frac{\beta}{N}\xi}\right)-\frac{C\Xi}{(1-\theta)N}-\gamma\xi,&\xi\geq\Xi\end{array}\right.$
(13)

where $\Xi$ denotes the moment in the infection time coordinate when the
stockpile of tests finishes. This $\Xi$ is given by an implicit formula

$\int\limits_{0}^{\Xi}\frac{d\xi}{x_{\mathtt{I}}^{0}+x_{\mathtt{S}}^{0}\left(1-e^{-\frac{\beta}{N}\xi}\right)-\frac{C\xi}{(1-\theta)N}-\gamma\xi}=T$
(14)

which is derived from (8) as

$\int\limits_{0}^{\Xi}\frac{d\xi}{x_{\mathtt{I}}(\xi)}=T$

where $T:=r_{\max}/C$ and $x_{\mathtt{I}}(\xi)$ is given by (11) and (13).

###  IV-A Computation of the COST policy

The epidemic spread can be controlled through a suitable testing rate
$u(t)=C$, for $t\in[0,T]$. However, when the limited stockpile of tests
finishes, the disease may start spreading again, resulting in a second wave of
the epidemic. Let the testing phase be the time interval when the tests are
being performed, i.e., $t\in[0,T]$, and let the non-testing phase be the time
after the stockpile finishes, i.e., $t>T$. Thus, it is reasonable to suppose
that there can be two epidemic peaks — one that arrives during the testing
phase and the other that arrives during the non-testing phase.

Let $\xi_{p_{1}}$ and $\xi_{p_{2}}$ be the moments in infection time
coordinate when the two epidemic peaks occur. We will prove that
$0<\xi_{p_{1}}\leq\Xi\leq\xi_{p_{2}}$ in the following:. If the two epidemic
peaks exist, then

$\displaystyle\left.\frac{dx_{\mathtt{I}}}{d\xi}\right|_{\xi=\xi_{p_{1}}}$
$\displaystyle=x_{\mathtt{S}}^{0}\frac{\beta}{N}e^{-\frac{\beta}{N}\xi_{p_{1}}}-\frac{C}{(1-\theta)N}-\gamma=0$
$\displaystyle\left.\frac{dx_{\mathtt{I}}}{d\xi}\right|_{\xi=\xi_{p_{2}}}$
$\displaystyle=x_{\mathtt{S}}^{0}\frac{\beta}{N}e^{-\frac{\beta}{N}\xi_{p_{2}}}-\gamma=0$

where the peak times are given as

$\xi_{p_{1}}=\frac{N}{\beta}\ln R_{1},\qquad\xi_{p_{2}}=\frac{N}{\beta}\ln
R_{2}$

with the reproduction numbers

$R_{1}=\frac{x_{\mathtt{S}}^{0}\beta}{\frac{C}{1-\theta}+\gamma N},\quad
R_{2}=\frac{x_{\mathtt{S}}^{0}\beta}{\gamma N},\quad R_{2}>R_{1}>1.$

Therefore, we have $0<\xi_{p_{1}}<\xi_{p_{2}}$. Further, notice that the first
peak occurs only if $\xi_{p_{1}}<\Xi$, otherwise it is ill-defined. Similarly,
$\xi_{p_{2}}>\Xi$.

The values of two epidemic peaks are

$\displaystyle x_{\mathtt{I}}^{p_{1}}$
$\displaystyle=x_{\mathtt{I}}^{0}+x_{\mathtt{S}}^{0}\left(1-\frac{1}{R_{1}}\right)-x_{\mathtt{S}}^{0}R_{1}\ln
R_{1}$ (15) $\displaystyle x_{\mathtt{I}}^{p_{2}}$
$\displaystyle=x_{\mathtt{I}}^{0}+x_{\mathtt{S}}^{0}\left(1-\frac{1}{R_{2}}\right)-x_{\mathtt{S}}^{0}R_{2}\ln
R_{2}-\frac{C\Xi}{(1-\theta)N}$

where $x_{\mathtt{I}}^{p_{1}}=x_{\mathtt{I}}(\xi_{p_{1}})$ and
$x_{\mathtt{I}}^{p_{2}}=x_{\mathtt{I}}(\xi_{p_{2}})$. It is obvious that
$x_{\mathtt{I}}^{p_{1}}$ decreases as $C$ increases, because
$x_{\mathtt{I}}^{p_{1}}$ represents the first epidemic peak during the testing
phase. Further, it is possible to prove that $C\Xi$ decreases as $C$ increases
for every sufficiently large $C$, where $\Xi=\Xi(C)$ is given by the solution
of (14). From (14), we obtain

$\frac{d\Xi}{dC}=-\frac{r_{\max}x_{\mathtt{I}}(\Xi)}{C^{2}}.$

Since $d(C\Xi)/dC=\Xi+Cd\Xi/dC$, therefore

$\frac{d(C\Xi)}{dC}=\Xi-\frac{r_{\max}x_{\mathtt{I}}(\Xi)}{C}=\int\limits_{0}^{T}\left[x_{\mathtt{I}}(\tau)-x_{\mathtt{I}}(T)\right]d\tau$

where $\Xi$ is rewritten according to (7) and $r_{\max}/C$ is represented by
the integral of 1 over the time interval $[0,T]$, where $T=r_{\max}/C$. This
implies that $d(C\Xi)/dC<0$ unless $x_{\mathtt{I}}(T)$ is smaller than the
average number of infected people in the time interval $[0,T]$. Thus, for
every reasonable, sufficiently large $C$ such that $T$ is not too large,
$x_{\mathtt{I}}^{p_{1}}$ decreases and $x_{\mathtt{I}}^{p_{2}}$ increases.
This property can be used to minimize the maximum between two epidemic peaks.
Indeed, the maximum between a decreasing and an increasing function is
minimized when they are equal to each other. Thus, we equate two epidemic
peaks in (15) and solve with respect to $\Xi$ in order to obtain the
optimality condition

$\Xi=\frac{N}{\beta}\left(1+\ln
R_{1}-\frac{R_{1}}{R_{2}-R_{1}}\ln\frac{R_{2}}{R_{1}}\right)$ (16)

which, along with (14), constitutes a complete system to determine the optimal
$C$.

######  Proposition 4.

Consider the testing rate $u(t)$ defined by (12) and assume that
$x_{\mathtt{T}}\approx(1-\theta)N$. Then, the COST policy $C$ is obtained by
solving the system of equations (14) and (16) with $T:=r_{\max}/C$. This
policy is unique provided that $R_{1}>1$.

The proof of this proposition follows from the prior discussion. The solution
$C$ for (14) and (16) can be obtained numerically [11].

###  IV-B COST Policy for the COVID-19 Case of France

Similar to the BEST policy, we evaluate the COST policy through its impact on
the number of active ICU cases $B(t)$ and the cumulative number of deaths
$E(t)$ after fitting the model outputs (5) and (6) with the data in the actual
testing scenario. However, we fit these equations to the data by using
different value of $\theta$ than obtained in the case of BEST policy. For the
COST policy, the value of $\theta$ is obtained through model validation with
the COVID-19 data of France [11] under the assumption that
$x_{\mathtt{T}}(t)\approx(1-\theta)N$.

Figure 6: The comparison between the actual control input $u(t)$ versus the
proposed control input $u(t)$ required by COST.

We consider that the total stockpile of tests $r_{\max}=2,038,037$, which is
the total number of tests performed from January 24 to July 01, 2020, in
France. Then, we obtain the optimal testing rate $C=17,144$ with the terminal
time $T=118$ (i.e., May 20) from Proposition 4. The actual testing rate of
France and the testing rate required by the COST policy are illustrated in
Figure 6. Notice that most of the tests in the actual testing are consumed
after May 13, whereas all the tests in COST are allocated equally in the time
interval from January 24 to May 20. The number of tests performed per day in
the actual testing exceeds $50,000$ per day after May 13, which is about three
times more than what is required by COST, i.e., $17,144$. Thus, COST is
optimal in a sense that it is practical, it decreases the burden on medical
facilities, and it reduces the number of deaths significantly. The impact of
COST policy on the number of active ICU cases $B(t)$ and the cumulative number
of deaths is illustrated in Figure 7 and Figure 8, respectively. From the
figures, we observe that the COST policy reduces the peak of $B(t)$ by
$11.12\%$ and the total number of cumulative deaths $E(t)$ by $37.52\%$.

Figure 7: The comparison between the predicted number of active ICU cases
$B(t)$ in the actual testing scenario and with COST policy. Figure 8: The
comparison between the predicted cumulative number of deaths $E(t)$ in the
actual testing scenario and with COST policy.

##  V Conclusion

We proposed a SIDUR model for demonstrating the impact of testing rate in the
suppression and/or mitigation of an epidemic. The tests enable the health
authority to diagnose and isolate the infected people, which limits the spread
of disease to the susceptible population. Two testing policies have been
proposed: 1) best-effort strategy for testing (BEST) and 2) constant optimal
strategy for testing (COST). The BEST policy is a suppression policy that
provides the minimum testing rate to be implemented from a certain day onward
to stop the growth of undetected infected cases immediately. On the other
hand, the COST policy is a mitigation policy that considers a limited
stockpile of tests and optimally allocates the tests in a time interval so
that the peak of the epidemic is minimized. Although both BEST and COST
policies are easy to compute and implement, they propose constant testing
rates, which may limit the influence of testing if the epidemic parameters
change due to external conditions. In the future, therefore, it will be
interesting to devise a testing policy with a dynamic testing rate that
minimizes the epidemic peak and/or the cumulative number of infected cases.

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