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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Phys.</journal-id>
<journal-title>Frontiers in Physics</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Phys.</abbrev-journal-title>
<issn pub-type="epub">2296-424X</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">824369</article-id>
<article-id pub-id-type="doi">10.3389/fphy.2022.824369</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Physics</subject>
<subj-group>
<subject>Perspective</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Local Surveillance of the COVID-19 Outbreak</article-title>
<alt-title alt-title-type="left-running-head">Liu et&#x20;al.</alt-title>
<alt-title alt-title-type="right-running-head">Local Surveillance of COVID-19 Outbreak</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Liu</surname>
<given-names>Caifen</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
<xref ref-type="fn" rid="fn1">
<sup>&#x2020;</sup>
</xref>
<uri xlink:href="https://loop.frontiersin.org/people/1577247/overview"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Xu</surname>
<given-names>Lingfeng</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
<xref ref-type="aff" rid="aff3">
<sup>3</sup>
</xref>
<xref ref-type="fn" rid="fn1">
<sup>&#x2020;</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Bai</surname>
<given-names>Yuan</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Xu</surname>
<given-names>Xiaoke</given-names>
</name>
<xref ref-type="aff" rid="aff3">
<sup>3</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Lau</surname>
<given-names>Eric H. Y.</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
<uri xlink:href="https://loop.frontiersin.org/people/1166382/overview"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Cowling</surname>
<given-names>Benjamin J.</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Du</surname>
<given-names>Zhanwei</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
<uri xlink:href="https://loop.frontiersin.org/people/966342/overview"/>
</contrib>
</contrib-group>
<aff id="aff1">
<sup>1</sup>
<institution>WHO Collaborating Centre for Infectious Disease Epidemiology and Control</institution>, <institution>School of Public Health</institution>, <institution>Li Ka Shing Faculty of Medicine</institution>, <institution>The University of Hong Kong</institution>, <addr-line>Hong Kong</addr-line>, <country>Hong Kong SAR, China</country>
</aff>
<aff id="aff2">
<sup>2</sup>
<institution>Laboratory of Data Discovery for Health</institution>, <institution>Hong Kong Science and Technology Park</institution>, <addr-line>Hong Kong</addr-line>, <country>Hong Kong SAR, China</country>
</aff>
<aff id="aff3">
<sup>3</sup>
<institution>College of Information and Communication Engineering</institution>, <institution>Dalian Minzu University</institution>, <addr-line>Dalian</addr-line>, <country>China</country>
</aff>
<author-notes>
<fn fn-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/797979/overview">Huijia Li</ext-link>, Central University of Finance and Economics, China</p>
</fn>
<fn fn-type="edited-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1531722/overview">Yongxing Li</ext-link>, Beijing University of Technology, China</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1581270/overview">Xueyan Liu</ext-link>, Jilin University, China</p>
</fn>
<corresp id="c001">&#x2a;Correspondence: Zhanwei Du, <email>duzhanwei0@gmail.com</email>&#x200a;</corresp>
<fn fn-type="equal" id="fn1">
<label>
<sup>&#x2020;</sup>
</label>
<p>These authors have contributed equally to this work and share first authorship</p>
</fn>
<fn fn-type="other">
<p>This article was submitted to Social Physics, a section of the journal Frontiers in Physics</p>
</fn>
</author-notes>
<pub-date pub-type="epub">
<day>03</day>
<month>03</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="collection">
<year>2022</year>
</pub-date>
<volume>10</volume>
<elocation-id>824369</elocation-id>
<history>
<date date-type="received">
<day>29</day>
<month>11</month>
<year>2021</year>
</date>
<date date-type="accepted">
<day>13</day>
<month>01</month>
<year>2022</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2022 Liu, Xu, Bai, Xu, Lau, Cowling and Du.</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>Liu, Xu, Bai, Xu, Lau, Cowling and Du</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/">
<p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these&#x20;terms.</p>
</license>
</permissions>
<abstract>
<p>Given the worldwide pandemic of the novel coronavirus disease 2019 (COVID-19) and its continuing threat brought by the emergence of virus variants, there are great demands for accurate surveillance and monitoring of outbreaks. A valuable metric for assessing the current risk posed by an outbreak is the time-varying reproduction number (<inline-formula id="inf1">
<mml:math id="m1">
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<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
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</inline-formula>). Several methods have been proposed to estimate <inline-formula id="inf2">
<mml:math id="m2">
<mml:mrow>
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<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
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</inline-formula> using different types of data. We developed a new tool that integrated two commonly used approaches into a unified and user-friendly platform for the estimation of time-varying reproduction numbers. This tool allows users to perform simulations and yield real-time tracking of local epidemic of COVID-19 with an R package.</p>
</abstract>
<kwd-group>
<kwd>epidemics (covid 19)</kwd>
<kwd>surveillance</kwd>
<kwd>infectious disease</kwd>
<kwd>package</kwd>
<kwd>modeling</kwd>
</kwd-group>
<contract-num rid="cn001">C7123-20G</contract-num>
<contract-sponsor id="cn001">Research Grants Council, University Grants Committee<named-content content-type="fundref-id">10.13039/501100002920</named-content>
</contract-sponsor>
</article-meta>
</front>
<body>
<sec id="s1">
<title>Introduction</title>
<p>The novel coronavirus disease 2019 (COVID-19) pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has led to 257 million confirmed cases and 5.15 million deaths worldwide by November 22, 2021 [<xref ref-type="bibr" rid="B1">1</xref>]. The COVID-19 pandemic continues to pose substantial risks to public health, and the situation is worsened by the emergence of SARS-CoV-2 variants with potentially higher transmissibility&#x20;[<xref ref-type="bibr" rid="B2">2</xref>].</p>
<p>Quantification of the transmissibility during epidemics is fundamental for designing and adjusting public health responses. The time-varying reproduction number <inline-formula id="inf3">
<mml:math id="m3">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, defined as the expected number of secondary cases of disease caused by a single infected individual at time <inline-formula id="inf4">
<mml:math id="m4">
<mml:mi>t</mml:mi>
</mml:math>
</inline-formula>, is a key epidemiological measure of transmissibility, with <inline-formula id="inf5">
<mml:math id="m5">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3c;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> indicating that incidence is in decline because of either successful control measures or population immunity having reached a sufficiently high level to limit further transmission. The real-time monitoring of <inline-formula id="inf6">
<mml:math id="m6">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> provides feedback on the effectiveness of interventions and on the need to intensify control efforts [<xref ref-type="bibr" rid="B3">3</xref>,&#x20;<xref ref-type="bibr" rid="B4">4</xref>].</p>
<p>A large number of methods have been proposed to estimate <inline-formula id="inf7">
<mml:math id="m7">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> from surveillance data [<xref ref-type="bibr" rid="B5">5</xref>&#x2013;<xref ref-type="bibr" rid="B12">12</xref>]. There are generally two categories. One is based on fitting mechanistic transmission models to incidence data, and the other is a statistical approach requiring case incidence data and the distribution of the serial interval (the time between symptom onsets in a primary case and secondary case) [<xref ref-type="bibr" rid="B13">13</xref>]. The mechanistic models are often complicated to deal with because of the potential for biases in the reported incidence data and the context-specific assumptions made. The statistical method proposed by Wallinga and Teunis [<xref ref-type="bibr" rid="B13">13</xref>] is relatively simpler but still has drawbacks. Estimates of <inline-formula id="inf8">
<mml:math id="m8">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> can vary considerably over a short period when the data aggregation time step is small. To overcome these limits, Cori et&#x20;al. [<xref ref-type="bibr" rid="B14">14</xref>] developed a generic tool for estimating <inline-formula id="inf9">
<mml:math id="m9">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> with a ready-to-use R software package <italic>EpiEstim</italic>, which has been frequently used to analyze the recent outbreaks of COVID-19. After searching CRAN package data which retrieve package download information from the RStudio mirror, we found that the most popular packages providing estimation of time-varying reproduction number include <italic>EpiEstim</italic>, <italic>EpiNow2</italic>, <italic>R0</italic>, <italic>epidemia</italic>, and <italic>nbTransmission</italic>. All of these tools use statistical methods to estimate <inline-formula id="inf10">
<mml:math id="m10">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> from surveillance data and are widely adopted to study COVID-19.</p>
<p>Recently, a new method was proposed by Hay et&#x20;al. [<xref ref-type="bibr" rid="B15">15</xref>] using information inherent in cycle threshold (Ct) values from reverse transcription quantitative polymerase chain reaction (RT-qPCR) tests to estimate the time-varying reproduction number from positive samples. Ct values are semiquantitative results provided by RT-qPCR tests. It is common when testing for infectious diseases to use this quantification of sample viral load. Lower Ct values indicate higher viral loads, and a Ct value below 40 gives a positive result. Based on cross-sectional virologic surveys (observed viral loads), this method overcomes the biases in traditional approaches resulting from testing constraint, unrepresentative sampling, and reporting delays. They also developed the R package <italic>virosolver</italic> to infer epidemic dynamics including estimation of&#x20;<inline-formula id="inf11">
<mml:math id="m11">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
<p>In this study, we chose <italic>EpiEstim</italic> and <italic>virosolver</italic> as the representatives of traditional and new methods, respectively. Although the accuracies of the two approaches have been separately demonstrated, there is still a lack of comparison between the two methods to the best of our knowledge. Therefore, we quantify the accuracies of <italic>EpiEstim</italic> and <italic>virosolver</italic> in different transmission scenarios by individual-based simulations and develop a ready-to-use R package for researchers to compare different <inline-formula id="inf12">
<mml:math id="m12">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> methods with the synthetic&#x20;truth.</p>
</sec>
<sec sec-type="methods" id="s2">
<title>Methods</title>
<sec id="s2-1">
<title>SEIR-based simulation</title>
<p>To assess the performance of the methods, we simulate outbreaks in three scenarios with different basic reproduction numbers (<inline-formula id="inf13">
<mml:math id="m13">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#xa0;</mml:mo>
<mml:mn>5</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#xa0;</mml:mo>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, respectively) using SEIR-based simulations as the baselines. The three scenarios could represent the situations of wild type, Delta variants, and potential variants of SARS-CoV-2 with higher transmissibility, according to <inline-formula id="inf14">
<mml:math id="m14">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> estimates given by previous studies [<xref ref-type="bibr" rid="B16">16</xref>, <xref ref-type="bibr" rid="B17">17</xref>]. The model parameters were determined on the basis of existing literature and epidemiological characteristics of COVID-19 in Hong Kong in early 2020 [<xref ref-type="bibr" rid="B14">14</xref>, <xref ref-type="bibr" rid="B15">15</xref>, <xref ref-type="bibr" rid="B18">18</xref>]. In particular, we adopted the prior distributions for the parameters of the SEIR model given by Hay et&#x20;al. [<xref ref-type="bibr" rid="B15">15</xref>]. The SEIR model is a compartmental model which assumes that the growth rate of new infections depends on the current prevalence of infectious and susceptible individuals by modeling the proportion of the population who are susceptible (S), exposed not infectious (E), infectious (I), and recovered (R) with respect to disease over time, as illustrated in <xref ref-type="fig" rid="F1">Figure&#x20;1</xref>. A stochastic SEIR model is implemented, and the R package <italic>odin</italic> is used to solve the model and obtain true infections over time. The true value of <inline-formula id="inf15">
<mml:math id="m15">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is estimated as <inline-formula id="inf16">
<mml:math id="m16">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#xd7;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b2;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mi>&#x3b3;</mml:mi>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf17">
<mml:math id="m17">
<mml:mrow>
<mml:msub>
<mml:mi>S</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the proportion of susceptible population, <inline-formula id="inf18">
<mml:math id="m18">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b2;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the transmission rate at time <inline-formula id="inf19">
<mml:math id="m19">
<mml:mi>t</mml:mi>
</mml:math>
</inline-formula> derived from the compartmental transition equations, and <inline-formula id="inf20">
<mml:math id="m20">
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:mi>&#x3b3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the average infectious period.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>The SEIR structure model used to describe the transmission of infections.</p>
</caption>
<graphic xlink:href="fphy-10-824369-g001.tif"/>
</fig>
<p>
<italic>EpiEstim</italic> and <italic>virosolver</italic> methods were run separately on the same simulations for comparison. For <italic>EpiEstim</italic>, it relies on two inputs: incidence time series and the serial interval distribution. Incidence data by days since the start of outbreak were generated from the simulated SEIR epidemic. We used an empirical serial interval distribution informed by a previous outbreak of COVID-19 in Hong Kong in early 2020 [<xref ref-type="bibr" rid="B18">18</xref>], and we also used the simulated serial interval distribution for comparison, denoted by <italic>EpiEstim</italic> (empirical SI) and <italic>EpiEstim</italic> (simulated SI) in <xref ref-type="fig" rid="F2">Figure&#x20;2</xref>. We assumed that the simulated serial interval distribution has the same standard variation as that of the empirical serial interval distribution and inferred the mean of the simulated serial interval distribution by conducting numerical experiments on a range of means from 1 to 10 with a step of 0.1 and chose the one yielding the least root mean square error (RMSE). For <italic>virosolver</italic>, the input data include population-level Ct values over days since the start of outbreak, and individual-level viral kinetics model over days since the infection. The Ct values were generated for all exposed, infectious, and recovered individuals when they were samples based on the Ct value model proposed by Hay et&#x20;al. [<xref ref-type="bibr" rid="B15">15</xref>], and the viral kinetics parameters were also given in their study. We assumed that the Ct values were observed from randomized samples of the population at selected testing days, and <xref ref-type="fig" rid="F2">Figure&#x20;2</xref> shows the simulated Ct values of the sampled people every 14 days. Each panel presents the distribution of observed Ct values among sampled infected individuals on that testing day. Day 14 and Day 28 had no data because there was no infection among the samples at the early stage of the epidemic.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>A schematic illustrating how our simulation platform generates a comparison of the estimated <inline-formula id="inf21">
<mml:math id="m21">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> from <italic>EpiEstim</italic> and <italic>virosolver</italic>. Incidence data and ground truth were generated from 100 simulations based on the SEIR model (green/gray line and shaded ribbon show mean and the range). Estimates of <inline-formula id="inf22">
<mml:math id="m22">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> were obtained using <italic>EpiEstim</italic> (red line and shaded ribbon show posterior median and 95% CrI using mean incidence data) and <italic>virosolver</italic> (blue line and shaded ribbon show posterior mean and 95% CrI using Ct value model), respectively. <italic>EpiEstim</italic> using the empirical value of serial interval distribution [<xref ref-type="bibr" rid="B18">18</xref>] and the simulated serial interval distribution are denoted by <italic>EpiEstim</italic> (empirical SI) and <italic>EpiEstim</italic> (simulated SI), respectively.</p>
</caption>
<graphic xlink:href="fphy-10-824369-g002.tif"/>
</fig>
</sec>
<sec id="s2-2">
<title>EpiEstim</title>
<p>The framework of <italic>EpiEstim</italic> is based on statistical assumptions and Bayesian estimation. Transmission is modeled by a Poisson process so that the rate, at which individuals infected between infection and symptom onset generate new infections, is equal to <inline-formula id="inf23">
<mml:math id="m23">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi>w</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf24">
<mml:math id="m24">
<mml:mi>s</mml:mi>
</mml:math>
</inline-formula> is the time postinfection; <inline-formula id="inf25">
<mml:math id="m25">
<mml:mi>t</mml:mi>
</mml:math>
</inline-formula> is the time post symptom onset; <inline-formula id="inf26">
<mml:math id="m26">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the time-varying reproduction number at time <inline-formula id="inf27">
<mml:math id="m27">
<mml:mi>t</mml:mi>
</mml:math>
</inline-formula>; and <inline-formula id="inf28">
<mml:math id="m28">
<mml:mrow>
<mml:msub>
<mml:mi>w</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is a probability distribution describing the average infectiousness profile after infection. The incidence at time <inline-formula id="inf29">
<mml:math id="m29">
<mml:mi>t</mml:mi>
</mml:math>
</inline-formula> is assumed to be Poisson distributed with mean <inline-formula id="inf30">
<mml:math id="m30">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:munderover>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>t</mml:mi>
</mml:munderover>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mi>w</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and the likelihood of the incidence <inline-formula id="inf31">
<mml:math id="m31">
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> given the reproduction number <inline-formula id="inf32">
<mml:math id="m32">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is:<disp-formula id="equ1">
<mml:math id="m33">
<mml:mrow>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x7c;</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mi>w</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi mathvariant="normal">&#x39b;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:msup>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:msub>
<mml:mi mathvariant="normal">&#x39b;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>!</mml:mo>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>where <inline-formula id="inf33">
<mml:math id="m34">
<mml:mrow>
<mml:msub>
<mml:mi mathvariant="normal">&#x39b;</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:munderover>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>t</mml:mi>
</mml:munderover>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mi>w</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. <inline-formula id="inf34">
<mml:math id="m35">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is estimated in a time window <inline-formula id="inf35">
<mml:math id="m36">
<mml:mi>&#x3c4;</mml:mi>
</mml:math>
</inline-formula>, under the assumption that the time-varying reproduction number is constant within that time window. Therefore, over time period <inline-formula id="inf36">
<mml:math id="m37">
<mml:mrow>
<mml:mrow>
<mml:mo>[</mml:mo>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>t</mml:mi>
</mml:mrow>
<mml:mo>]</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, the likelihood of the incidence during this time period <inline-formula id="inf37">
<mml:math id="m38">
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> given the reproduction number <inline-formula id="inf38">
<mml:math id="m39">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, conditional on the previous incidences, is as follows:<disp-formula id="equ2">
<mml:math id="m40">
<mml:mrow>
<mml:mi>P</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x7c;</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mi>w</mml:mi>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:munderover>
<mml:mstyle displaystyle="true">
<mml:mo>&#x220f;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>t</mml:mi>
</mml:munderover>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mi mathvariant="normal">&#x39b;</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:msup>
<mml:msup>
<mml:mi>e</mml:mi>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mi>&#x39b;</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
<mml:mo>!</mml:mo>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</disp-formula>
</p>
<p>Using a Bayesian framework with a Gamma distributed prior with parameters of shape <bold>
<italic>a</italic>
</bold> and scale <bold>
<italic>b</italic>
</bold> for <inline-formula id="inf39">
<mml:math id="m41">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, the posterior distribution of <inline-formula id="inf40">
<mml:math id="m42">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is assumed to be a Gamma distribution with parameters <inline-formula id="inf41">
<mml:math id="m43">
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>a</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:munderover>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>t</mml:mi>
</mml:munderover>
<mml:msub>
<mml:mi>I</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mfrac>
<mml:mn>1</mml:mn>
<mml:mi>b</mml:mi>
</mml:mfrac>
<mml:mo>&#x2b;</mml:mo>
<mml:msubsup>
<mml:mstyle displaystyle="true">
<mml:mo>&#x2211;</mml:mo>
</mml:mstyle>
<mml:mrow>
<mml:mi>s</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>t</mml:mi>
</mml:msubsup>
<mml:msub>
<mml:mi mathvariant="normal">&#x39b;</mml:mi>
<mml:mi>s</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>. Hence, inference of <inline-formula id="inf42">
<mml:math id="m44">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is straightforward from the posterior distribution. Note that the choice of the time window size <inline-formula id="inf43">
<mml:math id="m45">
<mml:mi>&#x3c4;</mml:mi>
</mml:math>
</inline-formula> has an impact on the estimates of&#x20;<inline-formula id="inf44">
<mml:math id="m46">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>: small values of <inline-formula id="inf45">
<mml:math id="m47">
<mml:mi>&#x3c4;</mml:mi>
</mml:math>
</inline-formula> lead to a more rapid detection of changes&#x20;in transmission but also more statistical noise; large values lead to more smoothing and reductions in statistical noise.&#x20;By conducting simulation experiments on <inline-formula id="inf46">
<mml:math id="m48">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>7</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#xa0;</mml:mo>
<mml:mn>14</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#xa0;</mml:mo>
<mml:mn>21</mml:mn>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> respectively, we found that <inline-formula id="inf47">
<mml:math id="m49">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>14</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> exhibited the best compromise between high accuracy and easy interpretation, so the window size was set to be 14 in this study. Readers can refer to Gostic et&#x20;al. [<xref ref-type="bibr" rid="B19">19</xref>] for a detailed discussion on the sliding window of the <italic>EpiEstim</italic> method.</p>
</sec>
<sec id="s2-3">
<title>Virosovler</title>
<p>The R package <italic>virosolver</italic> was developed by Hay et&#x20;al. [<xref ref-type="bibr" rid="B15">15</xref>] using virological data and Ct values, to infer epidemic dynamics. Ct values are inversely correlated with <inline-formula id="inf48">
<mml:math id="m50">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>log</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> viral loads, which depend on the time since infection. The distribution of Ct values across positive specimens at a single time point reflects the epidemic trajectory: a growing epidemic will have a high proportion of recently infected individuals with high viral loads, whereas a declining epidemic will have more individuals with older infections and thus lower viral loads. Using a mathematical model for population-level viral load distributions calibrated to known features of the SARS-CoV-2 viral load kinetics, we can use Ct values from a single random cross section of virologic testing to estimate the time-varying reproduction number in a population. For individual <inline-formula id="inf49">
<mml:math id="m51">
<mml:mi>i</mml:mi>
</mml:math>
</inline-formula> sampled on day <inline-formula id="inf50">
<mml:math id="m52">
<mml:mi>u</mml:mi>
</mml:math>
</inline-formula>, the Ct value <inline-formula id="inf51">
<mml:math id="m53">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is assumed to follow the Gumbel distribution as<disp-formula id="equ3">
<mml:math id="m54">
<mml:mrow>
<mml:msub>
<mml:mi>X</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:mi>G</mml:mi>
<mml:mi>u</mml:mi>
<mml:mi>m</mml:mi>
<mml:mi>b</mml:mi>
<mml:mi>e</mml:mi>
<mml:mi>l</mml:mi>
<mml:mrow>
<mml:mo>[</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>d</mml:mi>
<mml:mi>e</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>f</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c3;</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>f</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
<mml:mo>]</mml:mo>
</mml:mrow>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>where <inline-formula id="inf52">
<mml:math id="m55">
<mml:mrow>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>f</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the time of infection, and <inline-formula id="inf53">
<mml:math id="m56">
<mml:mrow>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mi>o</mml:mi>
<mml:mi>d</mml:mi>
<mml:mi>e</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>f</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>&#xa0;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf54">
<mml:math id="m57">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mrow>
<mml:mi>u</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:msub>
<mml:mi>t</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>f</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula> are the location and scale parameters, respectively. The details of the parameterization are found in [<xref ref-type="bibr" rid="B15">15</xref>]. In practice, <italic>virosolver</italic> takes an input data frame of Ct values with associated sample collection dates from RT-qPCR testing and reconstructs the incidence curve that gave rise to those measurements. By capturing this logic in a mathematical model, we can obtain a probabilistic estimate of the underlying incidence curve, thus time-varying reproduction number having observed a set of Ct values at some point in time. Noting that the sampling scheme has an impact on the estimate of incidence, we set the population number to be 8,000 and sampled 1,000 (1/8) of the population to fit the local prevalence data of COVID-19 in Hong Kong in early 2020 as a case study&#x20;[<xref ref-type="bibr" rid="B20">20</xref>].</p>
</sec>
</sec>
<sec sec-type="results" id="s3">
<title>Results</title>
<p>We assessed the performance of <italic>EpiEstim</italic> and <italic>virosovler</italic> in three scenarios where <inline-formula id="inf55">
<mml:math id="m58">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2,5,10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, respectively, in which <inline-formula id="inf56">
<mml:math id="m59">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> can serve as a demonstration of the outbreak of COVID-19 in Hong Kong in early 2020. For each scenario, we generated the incidence data over 100&#x20;days based on the SEIR model from 100 stochastic simulations and estimated the mean incidence. <xref ref-type="fig" rid="F3">Figure&#x20;3</xref> gives the estimated <inline-formula id="inf57">
<mml:math id="m60">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> with the uncertainties (95% credible intervals) across 100 simulations using <italic>EpiEstim</italic> and <italic>virosolver</italic>, respectively, and the ground truths for the <inline-formula id="inf58">
<mml:math id="m61">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> values are presented for comparison. <italic>EpiEstim</italic> with the empirical serial interval distribution [<xref ref-type="bibr" rid="B18">18</xref>] would underestimate <inline-formula id="inf59">
<mml:math id="m62">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. In contrast with <italic>EpiEstim</italic>, <italic>virosovler</italic> provided less biased estimates but exhibited wider intervals of uncertainty. However, both approaches performed well in detecting the timing point when <inline-formula id="inf60">
<mml:math id="m63">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>&#x3c;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>The output of <inline-formula id="inf61">
<mml:math id="m64">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> estimates in three designed scenarios and the corresponding outcomes of accuracy assessment. <bold>(A&#x2013;C)</bold> The graphical interface by setting <inline-formula id="inf62">
<mml:math id="m65">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mtext>&#xa0;</mml:mtext>
<mml:mn>5</mml:mn>
<mml:mo>,</mml:mo>
<mml:mtext>&#xa0;</mml:mtext>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, respectively. We parameterized the serial interval distribution used by <italic>EpiEstim</italic> with the empirical study [<xref ref-type="bibr" rid="B18">18</xref>] and the simulated serial interval distribution, which are denoted by <italic>EpiEstim</italic> (empirical SI) and <italic>EpiEstim</italic> (simulated SI) in figure legends. <bold>(D&#x2013;F)</bold> Results of R squared, Pearson correlation coefficient, and RMSE for both methods in scenarios with <inline-formula id="inf63">
<mml:math id="m66">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mtext>&#xa0;</mml:mtext>
<mml:mn>5</mml:mn>
<mml:mo>,</mml:mo>
<mml:mtext>&#xa0;</mml:mtext>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, respectively.</p>
</caption>
<graphic xlink:href="fphy-10-824369-g003.tif"/>
</fig>
<p>To quantify and compare the accuracies of the methods, we used multiple metrics including coefficient of determination (<inline-formula id="inf64">
<mml:math id="m67">
<mml:mrow>
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>), Pearson correlation coefficient, and root mean square error (RMSE). As <xref ref-type="fig" rid="F3">Figures 3D&#x2013;F</xref> show, in all scenarios, <italic>virosolver</italic> almost had the highest <inline-formula id="inf65">
<mml:math id="m68">
<mml:mrow>
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> and Pearson correlation coefficient with the ground truth of <inline-formula id="inf66">
<mml:math id="m69">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, suggesting that <italic>virosolver</italic> had the highest accuracy and strongest correlation with the synthetic epidemic growth. In terms of RMSE, the performance of <italic>EpiEstim</italic> with the simulated serial interval distribution was the best (lowest RMSE), and <italic>virosolver</italic> had the largest RMSE due to its large estimation uncertainties. We noted that <italic>EpiEstim</italic> with simulated SI always performed better than <italic>EpiEstim</italic> with empirical SI. In conclusion, <italic>virosovler</italic> provided more accurate estimates of <inline-formula id="inf67">
<mml:math id="m70">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and <italic>EpiEstim</italic> relied on the adjustment of serial interval distribution for better performance.</p>
</sec>
<sec sec-type="discussion" id="s4">
<title>Discussion</title>
<p>Quantifying disease transmissibility during outbreaks is crucial for designing effective control measures and assessing their effectiveness once implemented. In the situation where the incidence is still increasing while the time-varying reproduction number is actually dropping, there might be a very different outlook compared to if the incidence and the reproduction number are both increasing. The platform for estimating <inline-formula id="inf68">
<mml:math id="m71">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> provided here can therefore help epidemiologists and policymakers to monitor temporal changes in the transmissibility of COVID-19. The key contributions of our platform are as follows: 1) our software package integrates the most popular method (<italic>EpiEstim</italic>) and the newest approach (<italic>virosolver</italic>) into a unified framework, allowing users to infer real-time viral transmissibility from different perspectives; 2) by setting the value of <inline-formula id="inf69">
<mml:math id="m72">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, users can conduct simulation experiments on our platform to study the epidemic development and compare the performances of two approaches accordingly; 3) this platform is easy enough for nonspecialists to apply by simply inputting the required data and is also flexible for specialists to use by changing the parameters setting if needed.</p>
<p>The estimation tools we used here have several limitations and thus may result in potential bias. For <italic>EpiEstim</italic>, a preexisting estimate of serial interval distribution is required as the input data, which may account for the underestimation of reproduction number in our simulation study (<xref ref-type="fig" rid="F3">Figure&#x20;3</xref>). If data on pairs of infector-infected individuals are available, the serial interval distribution can be estimated jointly, which leads to more precise estimates of transmissibility [<xref ref-type="bibr" rid="B21">21</xref>]. In addition, the inevitable delay between infection and case reporting (the incubation period) could also result in biased estimation of <inline-formula id="inf70">
<mml:math id="m73">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. If data on the incubation period are available, a possible strategy would be to use the incubation period distribution to back-calculate the incidence of infections from the incidence of symptoms and then apply <italic>EpiEstim</italic> to estimate the reproduction number from those inferred&#x20;data.</p>
<p>The virologic data-based method, <italic>virosolver</italic>, as mentioned above, exhibits greater uncertainty of estimated than <italic>EpiEstim</italic>. This is probably caused by insufficient information on Ct value distribution and viral kinetics model. The viral load kinetics model used in <italic>virosolver</italic> was generated on the basis of observed properties of measured viral loads in the literature, and these results were applied to inform priors on key parameters when estimating reproduction numbers. The estimates can therefore be improved by choosing more precise, accurate priors relevant to the observations used during model fitting. For example, the model should be adjusted by specifying different distributions if results come from multiple testing platforms. Results may also be improved if individual-level features such as symptom status, age, antiviral treatment, and vaccination record are available and incorporated into the Ct value&#x20;model.</p>
<p>Apart from the two methods presented in our study, many other approaches are still available, which we will include in this platform in the future to track disease transmissibility by using other data sources (e.g., hospitalization and death). Additionally, genomic data are also of great importance in the inference of transmissibility of COVID-19 considering recent emergence of virus variants [<xref ref-type="bibr" rid="B22">22</xref>]. We only provide incidence and <inline-formula id="inf71">
<mml:math id="m74">
<mml:mrow>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> estimates as the outputs; other epidemiological metrics such as prevalence, hospitalization, admission to ICU, death, and the economic analysis, such as net monetary benefit (NMB), are not included in our platform. Besides, we used the SEIR model for simulation in our package, because Hay et&#x20;al. [<xref ref-type="bibr" rid="B15">15</xref>] had studied four other epidemic models for fitting cross-sectional viral load data, namely, the SEIR model, exponential growth model, SEEIRR model, and Gaussian process model, and they also made a comparison of these models. They found that the SEIR model was the most appropriate as it consistently provided unbiased, constrained estimates of transmissibility during the epidemic growth. We may explore these models and other individual-based models (branching process, for example) in future studies.</p>
<p>Our tool can also be applied to the new variants of SARS-CoV-2 as long as incidence data and Ct values of the infected people are available. Users can obtain a more accurate estimation by adopting the updated parameters of serial interval distribution and viral kinetics model for objective variants informed by recent studies [<xref ref-type="bibr" rid="B23">23</xref>&#x2013;<xref ref-type="bibr" rid="B26">26</xref>]. In conclusion, we have established a platform for simulation and inference of time-varying reproduction numbers by incorporating two commonly used approaches. We would ensure our tool to epidemiologists and public health organizations in a wide range of future outbreak response scenarios.</p>
</sec>
</body>
<back>
<sec id="s5">
<title>Data Availability Statement</title>
<p>The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.</p>
</sec>
<sec id="s6">
<title>Author Contributions</title>
<p>CL, LX, ZD, and BC: conceived the study, designed statistical and modeling methods, conducted analyses, interpreted results, and wrote and revised the manuscript; YB, XX, and EL: interpreted the results and revised the manuscript.</p>
</sec>
<sec sec-type="COI-statement" id="s7">
<title>Conflict of Interest</title>
<p>BC reports honoraria from AstraZeneca, GSK, Moderna, Pfizer, Roche, and Sanofi Pasteur.</p>
<p>CL, LX, YB, EL, BC, and ZD were employed by the company Laboratory of Data Discovery for Health, Hong Kong Science and Technology&#x20;Park.</p>
<p>The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="disclaimer" id="s8">
<title>Publisher&#x2019;s Note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
<ack>
<p>We acknowledge the financial support from the Collaborative Research Fund (Project No. C7123-20G) of the Research Grants Council of the Hong Kong SAR Government.</p>
</ack>
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