Early dynamics of transmission and control of COVID-19: a mathematical modelling study MERC #4 Twitter: @mepbphhond_ 

Table of contents • Self-introduction • Introduction • Method (overview) • Results • Discussion • Method (detail) • My opinion 1 2 3 4 5 6 7 

Introduction Kucharski AJ, Russell TW, Diamond C, et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling study [published correction appears in Lancet Infect Dis. 2020 Mar 25;:]. Lancet Infect Dis. 2020;20(5):553-558. doi:10.1016/S1473- 3099(20)30144-4 

Introduction December, 2019 First reported outbreak of COVID-19 February 13th, 2020 46997 confirmed cases March 11th, 2020 Published in the Lancet 2/7 

Introduction December, 2019 First reported outbreak of COVID-19 February 13th, 2020 46997 confirmed cases March 11th, 2020 Published in the Lancet This study was conducted at the early stage of the outbreak ! 2/7 

Introduction ・Effective reproduction number ・The probability that an outbreak starts with a single imported case What epidemiological characteristics should be explored in the early stage of an outbreak? 2/7 

Introduction Effective reproduction number ・ Insights into the epidemiological situation which are measurable ・ Predictions about potential future growth The probability that an outbreak starts with a single imported case ・ Risk to other countries Guide the design of alternative interventions 2/7 

Introduction Effective reproduction number ・ Insights into the epidemiological situation which are measurable ・ Predictions about potential future growth The probability that an outbreak starts with a single imported case ・ Risk to other countries Guide the design of alternative interventions NEED mathematical modelling!! 2/7 

Introduction Concerns with data sources Real time analysis Time delay between natural history to report Might be incomplete Might capture only certain aspect of dynamics Might be biased 2/7 

Introduction ・Time delay from onset to confirmation Modelling ・The vulnerability of data sources Evidence synthesis approaches Robust estimation 2/7 

Method (overview) 

Data sources ・Daily number of new internationally exported cases (or lack thereof), by date of onset, as of Jan 26. ・Daily number of new cases in Wuhan with no market exposure, by date of onset, between Dec 29, 2019 and Jan 23, 2020. ・Daily number of new cases in China, by date of onset, between Dec 29, 2019 and Jan 23, 2020. ・Proportion of infected passengers on evacuation flights between Dec 29, 2019 and Feb 4, 2020. Four fitted empirical datasets 3/7 

Data sources ・Daily number of new exported cases from Wuhan (or lack thereof) in top 20 most at-risk countries, by date of confirmation, as of Feb 10, 2020. ・Data on new confirmed cases reported in Wuhan between Jan 16, 2020, and Feb 11, 2020. Two datasets for comparison with model outputs 3/7 

Overview of the method ・Estimate the state trajectories using sequential Monte Carlo (SMC), i.e. particle filter to obtain Rt, the symptomatic cases, and prevalence ・States model: Weiner process (Extended SEIR model) ・Observation model: Poisson process (symptomatic cases) ・Observation model: Binomial process (prevalence) 3/7 

Schematic representation of the states model 3/7 

Parameters 3/7 

Overview of the method ・Branching process with a negative binomial offspring distribution to calculate the extinction probability 3/7 

Results 

Results: each states Red line: travel restriction starting on Jan 23, 2020. Blue line: median Light blue shading: 50%CI Dark blue shading: 95%CI Fitted up to Feb 11, 2020 4/7 

Results: each states ・Susceptible on Jan 31, 2020: 94.8 (95%CI 93.1-96.1)% of the Wuhan population ・Around 10 times more symptomatic cases than were reported as confirmed cases. 4/7 

Results: each states ・Confirmed and estimated cases among the 20 countries generally correspond with each other. (USA and Australia as notable outlier) ・100 (51-100)% of cases would eventually have detectable symptoms. 4/7 Most exported cases in late Jan were eventually detectable in theory. 

Results: Rt 4/7 

Results: Rt ・Range (median value): 1.6 to 2.6 between Jan 1, 2020, and Jan 23, 2020. ・Did not predict the slowdown in early February. ・Decline: 2.35 (1.15-4.77) on Jan 16 to 1.05 (0.41- 2.39) on Jan 30. 4/7 You can see the slowdown in observed confirmed cases. 

Results: each states (sensitivity analysis) ・The same result of a decline in Rt from more than 2 to almost 1 in last 2 weeks of January, 2020. 4/7 Assume a large number of initial cases Use different mobility data Assume that pre- symptomatic cases are transmissible 

Results: probability of large outbreak 4/7 : Dispersion parameter 

Results: probability of large outbreak ・Single initial cases: 17-25% ・More than four infection are introduced: over 50% 4/7 Assume SARS-like or MARS-like overdispersion in R0 

Discussion 

Discussion The fluctuations in Rt could be the result of behaviour in the population at risk, i.e. lockdown. Notably more cases exported to France, USA, Australia compared with what the model predicted. increased surveillance and detection 5/7 

Discussion Not necessarily lead to an outbreak when a single case is introduced. Important to rapidly identify and isolate cases and conduct other control measures! 5/7 

Discussion Highlights the value of combining multiple data sources ・Confirmed total cases in some instances apparently doubling every day. (Rt) Once extensive restrictions are introduced, analysing such data can lead underestimation of Rt. 5/7 

Discussion “If COVID-19 transmission is established outside Wuhan, understanding the effectiveness of control measures in different settings will be crucial for understanding the dynamics of the outbreak, and the likelihood that transmission can eventually be contained or effectively mitigated.” 5/7 

Limitation ・The used values of parameters might be refined as more comprehensive data become available. Use multiple datasets to infer model parameters Conduct sensitivity analyses on key area of uncertainty The value of overdispersion is assumed as MERS/SARS-CoV. 5/7 

Method (detail) このsectionはできるだ け日本語でいきます! ある程度の厳密性は犠 牲にしました。 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

Overview of the method (再掲） ・Estimate the state trajectories using sequential Monte Carlo (SMC), i.e. particle filter to obtain Rt, the symptomatic cases, and prevalence ・States model: Weiner process (Extended SEIR model) ・Observation model: Poisson process (symptomatic cases) ・Observation model: Binomial process (prevalence) 6/7 

Method sequential Monte Carlo ってなに？？ states model とか observation model とかわけわからん 6/7 

Method sequential Monte Carlo ってなに？？ states model とか observation model とかわけわからん 6/7 隠れマルコフモデル hidden Markov model の一例の、 状態空間モデル state space model について知る必要あり！ 

A hidden Markov model 6/7 

State space model 6/7 

State space model 6/7 : state model (process model) : observation model (measure model) 

State space model もう少しかみ砕くと・・・ 線形状態空間モデル : states model : observation model 6/7 

State space model もう少しかみ砕くと・・・ 線形状態空間モデル : states model : observation model 6/7 Process noise Measurement noise 

State space model もう少しかみ砕くと・・・ 非線形状態空間モデル : states model : observation model 6/7 Process noise Measurement noise 

State space model 6/7 : state model (process model) : observation model (measure model) 確率微分方程式(SDE)が 基礎理論となります（こ こでその解説はしないで すが・・・） 

State space model -example- 6/7 Process model ; Transmission dynamics 

State space model -example- 6/7 Measure model to take the reporting probability into consideration 

6/7 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

States model 6/7 わかりやすくするた めにWuhan caseと exported caseをま ずは分けて考えてい きます！ 

Wuhan 6/7 

Method ・S: Susceptible ・E: Exposed but not infectious ・I: Infectious ・R: Removed ・Q: The number of symptomatic cases among travellers from Wuhan yet to be reported ・D: The cumulative number of cases among travellers from Wuhan with symptoms ・C: The cumulative number of confirmed cases among travellers from Wuhan 6/7 

Method ・β(t): The transmission rate at time t ・σ: The rate of becoming symptomatic (1/incubation period) ・γ: The rate of isolation (1/delay from onset to hospitalisation) ・κ: The rate of reporting (1/delay from onset to confirmation) ・f: The fraction of cases that travel 6/7 

Method 平均感染性期間持続時間 6/7 t時点でinfectiousであ る確率 (survival functionともいえる) 

Fine points -Wuhan case- 6/7 ① ② ② ③ 

Fine point ① 感染率が時間発展に従う幾何ブラウン運動をするという スムージングを考慮 …気持ち… Empirical time-series dataはnoisyであるためにoverfittingすることを考え て, Gaussian noiseを組み込みsmoothingしよう! 6/7 

Geometric Brown motion 少しだけ確率微分方程式 (stochastic differential equation) の話を. 確率微分方程式とは, 微分方程式にstochastic noiseを乗せたもの. ただし, まるで異なる性質を持つ. その中心が伊藤の公式 Ito’s formula. 6/7 確率過程 stochastic process 微分方程式 differential equation 

Geometric Brown motion 6/7 確率微分方程式 : Brown motion (Weiner process) (平均0, 分散 の正規分布に 従う) のもとでこれを解くと, . なお, これは幾何ブラウン運動の一種であるが, 上記のような の形式の確率過程は「幾何ブラウン運動 geometric Brown motion」という. 

Fine points -Wuhan case- 6/7 ① ② ② ③ 

Fine point ② Erlang distribution を time delay に考慮したことで E も I も2つずつ! The definition of the Erlang distribution mean: , shape parameter: The Erlang SEIR modelは Anderson et al. 1980 にて(おそらく)初めて導入 された. 6/7 

Fine point ② Incubation period: mean 5.2 days, rate=2 Infectious period: mean 2.9 days, rate=2 Erlang distributionにおいてshape parameterが2 Compartmentも2個ずつ 6/7 

Fine points -Wuhan case- 6/7 ① ② ② ③ 

Fine point ③ Confirmation前に回復する確率を考慮 (Exponential distribution) なお, はsurvival function 6/7 Memorylessness 

Exported cases 6/7 Weiner processがあ るため, Euler- Maruyama method を考えます 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

Likelihood Dw: Wuhanにおける有症例 Cw: Wuhanにおけるconfirmed case Dt: Wuhanからのexported caseで有症のもの Ct: Wuhanからのexported caseでconfirmedであるもの ρw: Wuhanにおけるconfirmed caseの割合 ρt: Exported caseにおけるconfirmed caseの割合 δ: Exported caseに対するWuhan内で報告される相対割合 ω: symptomatic caseになる割合 6/7 確定症例はすべて 有症と仮定 Observation state likelihood (1)(2)(3)をそれぞれ尤度としてdataをfitさせる Observation state likelihood (1)を尤度としてdataをfitさせる Positive evacuated caseの割合 Departureまで未発見のWuhanでのInfectious case 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

Sequential Monte Carlo Observation modelにdataをfitさせたとき, そこからstate modelの軌跡を逐 次推定できる. 逐次モンテカルロや粒子フィルタと呼ばれるデータ同化や時 系列分析でよく使われる手法の一つ. なお今回validationとして複数のdata で解析. 細かい内容は時間の関係で省きますけれど, descriptiveですが 以前解説文を書いたので記載しておきます. 6/7 

Sequential Monte Carlo 6/7 さらに詳細は Dropbox内の 参考資料にて. 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

Profile likelihood 95%CIを得るときに利用. あまり本質的ではないので詳細は把握 しないでもいいかと思います… 複数パラメタが存在するとき, それらの尤度平面を考える. 6/7 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

Offspring distribution 6/7 Secondary transmissionの 平均人数は基本再生産数 Offspring distributionは平均R0 のPoisson distributionか！ 

Offspring distribution 6/7 Secondary transmissionの 平均人数は基本再生産数 二次感染させる 人数は人によっ て違うのでは? Offspring distributionは平均R0 のPoisson distributionか！ 本当？ 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

Overdispersion Poisson distributionの平均をgamma distributionの事前分布で 置いたものが, negative binomial distribution 6/7 k: dispersion parameter π: 二次感染させる確率 期待値: 

Overdispersion Poisson distributionの平均をgamma distributionの事前分布で 置いたものが, negative binomial distribution 6/7 (余談) COVID-19の二次感染の過分散推定 の研究結果は, クラスター対策班が 全数把握を優先しなくなった根拠と なっている(Endo et al. 2019) 

Overview of this section ・Offspring distribution ・Overdispersion ・The probability of extinction 6/7 ・State space model ・State model ・Fine point ① ・Fine point ② ・Fine point ③ ・Likelihood ・Sequential Monte Carlo ・Profile likelihood Analysis 1 Analysis 2 

The probability of extinction 6/7 The extinction probabilityとは… １人の感染者から2次感染が生じず自然消滅(絶滅)する確率 p このとき, 1-(1-p)^nがoutbreakが生じる確率!! 

Results: probability of large outbreak 6/7 : Dispersion parameter 

My opinion 

My opinion ・流行初期段階でLockdownの効果や伝播の特徴を検証できたquick studyとしてとても意 義深い論文 ・Multiple data sourcesを用いた解析で一定の妥当性を得る点でkey pointだった! ・Symptomatic caseをtransmission dynamics (SEIR model)に組み込むときいま のままではR0に関してoverestimationでは. (Ejima et al. 2013) ・Overdispersion 推定(Endo et al. 2020)が出るまでは過去のSARS/MARSの overdispersionを参考にする根拠が少し弱かったのでは. ・感染率以外の確率的変動も考慮して本来なら計数過程counting processで記述するべき では. ・Time delayの扱いをconvolutionで示す方が仮定が少なくていいかも. ・感染率をstateとして扱っているが、PMCMCを考えてそのほかのparameterも推定する といいのでは. もしくはSMC^2で逐次推定することも考えていい. 7/7 

Reference • 本論文 Kucharski AJ, Russell TW, Diamond C, et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling study [published correction appears in Lancet Infect Dis. 2020 Mar 25;:]. Lancet Infect Dis. 2020;20(5):553-558. GitHub https://github.com/adamkucharski/2020-ncov • Overdispersion Endo A; Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Abbott S, Kucharski AJ, Funk S. Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. Wellcome Open Res. 2020;5:67. Published 2020 Jul 10. • Erlang SEIR model D. Anderson and R. Watson, On the spread of a disease with gamma distributed latent and 287 infectious periods, Biometrika, 67 (1980), pp. 191–198. A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theoretical Population Biology, 60 (2001), pp. 59–71. H. J. Wearing, P. Rohani, and M. J. Keeling, Appropriate models for the management of infectious diseases, PLoS Medicine, 2 (2005), p. e174. O. Krylova and D. J. D. Earn, Effects of the infectious period distribution on predicted tran326 sitions in childhood disease dynamics, Journal of The Royal Society Interface, 10 (2013), 327 pp. 20130098–20130098. 

Reference • その他発表に際して読んだ中で興味深い論文 Nishiura H, Klinkenberg D, Roberts M, Heesterbeek JA. Early epidemiological assessment of the virulence of emerging infectious diseases: a case study of an influenza pandemic. PLoS One. 2009;4(8):e6852. Published 2009 Aug 31. Ejima K, Aihara K, Nishiura H. The impact of model building on the transmission dynamics under vaccination: observable (symptom-based) versus unobservable (contagiousness-dependent) approaches. PLoS One. 2013;8(4):e62062. Published 2013 Apr 12. doi:10.1371/journal.pone.0062062 Endo A, Leeuwen EV, Baguelin M, 2019. Introduction to particle markov-chain monte carlo for disease dynamics modellers. Epidemics 29, 100363. Yang W, Karspeck A, Shaman J. Comparison of filtering methods for the modeling and retrospective forecasting of influenza epidemics. PLoS Comput Biol. 2014;10(4):e1003583. Published 2014 Apr 24. McDonald SA, Teunis P, van der Maas N, de Greeff S, de Melker H, Kretzschmar ME. An evidence synthesis approach to estimating the incidence of symptomatic pertussis infection in the Netherlands, 2005-2011. BMC Infect Dis. 2015;15:588. Published 2015 Dec 29.