Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
感染症の数理モデル6
Search
Sponsored
·
Your Podcast. Everywhere. Effortlessly.
Share. Educate. Inspire. Entertain. You do you. We'll handle the rest.
→
Daisuke Yoneoka
June 21, 2024
Research
170
0
Share
Embed
Copy iframe code
Copy JS code
Copy link
Start on current slide
感染症の数理モデル6
Daisuke Yoneoka
June 21, 2024
More Decks by Daisuke Yoneoka
See All by Daisuke Yoneoka
感染症の数理モデル15
kingqwert
0
91
感染症の数理モデル14
kingqwert
0
160
感染症の数理モデル13
kingqwert
0
73
感染症の数理モデル12
kingqwert
0
140
感染症の数理モデル11
kingqwert
0
140
感染症の数理セミナー_10_.pdf
kingqwert
0
170
感染症の数理モデル9
kingqwert
0
130
感染症の数理モデル8
kingqwert
0
140
感染症の数理モデル7
kingqwert
0
140
Other Decks in Research
See All in Research
非試合日の野球場を楽しむためのARホームランボールキャッチ体験システムの開発 / EC79-miyazaki
yumulab
0
230
2026-01-30-MandSL-textbook-jp-cos-lod
yegusa
1
1.4k
機械学習で作った ポケモン対戦bot で 遊ぼう!
fufufukakaka
0
320
【Zozo Research 技術共有会】三次元領域の現在と展望
mickey_0226
3
420
衛星×エッジAI勉強会 衛星上におけるAI処理制約とそ取組について
satai
4
560
AIで最適化を解けるか?
mickey_kubo
0
120
PGDM: Physically Guided Diffusion Model for L Downscaling
satai
2
290
Ghost in the 7‑Zip: The Shadow of Residential Proxies Creeping into Your Life
nttcom
0
1.2k
typst の使い方:言語学を研究する学生のために
gitomochang
0
460
2026 東京科学大 情報通信系 研究室紹介 (すずかけ台)
icttitech
0
3.9k
論文紹介 "ReSim: Reliable World Simulation for Autonomous Driving"
kogo
0
640
The Landscape of Agentic Reinforcement Learning for LLMs: A Survey
shunk031
4
1.1k
Featured
See All Featured
The AI Search Optimization Roadmap by Aleyda Solis
aleyda
1
5.9k
Agile Actions for Facilitating Distributed Teams - ADO2019
mkilby
0
210
Understanding Cognitive Biases in Performance Measurement
bluesmoon
32
2.9k
How to Create Impact in a Changing Tech Landscape [PerfNow 2023]
tammyeverts
55
3.4k
How to make the Groovebox
asonas
2
2.2k
Visual Storytelling: How to be a Superhuman Communicator
reverentgeek
2
560
Refactoring Trust on Your Teams (GOTO; Chicago 2020)
rmw
35
3.5k
A Modern Web Designer's Workflow
chriscoyier
698
190k
Evolving SEO for Evolving Search Engines
ryanjones
0
220
Principles of Awesome APIs and How to Build Them.
keavy
128
18k
Dominate Local Search Results - an insider guide to GBP, reviews, and Local SEO
greggifford
PRO
0
200
10 Git Anti Patterns You Should be Aware of
lemiorhan
PRO
659
62k
Transcript
感染症の数理 セミナー(6) June 21, 2024 @NIID 国⽴感染症研究所 第12室⻑ ⽶岡 ⼤輔
⽬次 1. 感染症のコンパートメントモデル 2. 基本再⽣産数 3. 最終流⾏規模 4. R実装 5.
⼈⼝の異質性とSIR 6. 再⽣産⽅程式とエボラ vs インフル 7. R0の推定⽅法(流⾏初期) 8. 内的増殖率の検定 9. Effective distance 10. 分岐過程 (Branching process) 本書の内容をカバーします。 具体的なコードなどは右の本 詳細なプログラムなどは https://github.com/objornstad/epimdr/tree/ master/rcode (結構間違ってる。。。) 2/48
はじめに 本セミナーシリーズは数理重めです。 簡単な微分/積分、線形代数が出てきます。 なるべく平易に解説しますが、完全に数学アレルギーの⽅はここ で終わられることをおすすめします。 セミナー終了時にはある程度次のパンデミックに向けて、 (ある程度) 数理モデリングができるようになることを⽬標としてます。 ⾃由参加なので、もし無理そうならお気軽に休んでください。 3/20
Effective distance (Brockmann and Helbing, Science 2013) Effective distanceのレシピ 1.
まずは隣接⾏列を⽤意しましょう (ある国j → 国iに⾏く旅客数をmij とすると) 2. 渡航確率を定義: 3. Effective distanceは以下 ( はj→iに⾏く全ての経路) 44 <latexit sha1_base64="WVw7D9rsHenA5hX8CxnWYUqV6xY=">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</latexit> Pij = mij P a maj <latexit sha1_base64="iqBytTQvdwlHfqjJfZhVIJxOIco=">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</latexit> 0 B @ 0 m12 m13 . . . m1N m21 0 m23 . . . m2N . . . . . . · · · ... 0 1 C A <latexit sha1_base64="e26i39QtTSV6Woh1x9x58yxVr6Q=">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</latexit> min ij { ij X (k,l)2 ij log Pkl } 経路の数 <latexit sha1_base64="Nd7CElvhdhj4HSLPTdkG2ZI4WH8=">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</latexit> ij
実データとの⾼い説明⼒ • ネットワーク上のSIRとかごちゃごちゃ考えないでいいんだよ • 横にED,縦にArrival timeをplot • めっちゃ相関⾼い!のか。。。? 45
分岐過程 • Galton-Watson processの定式化が⼀番有名 を⾮負の整数値をとるiidな確率変数: (ただし ) 46 <latexit sha1_base64="GL0Z3QrIIO4C2/WhI0xbfQk3Ms4=">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</latexit>
Zt+1 = Zt X i=1 Xt,i <latexit sha1_base64="dnBFeKrJ/oDLZmTXmiBMLy/SMRU=">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</latexit> Xt,i <latexit sha1_base64="itvO7QQ8osiOx0s2uq1sUfmGpdI=">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</latexit> P(Xt,i = m) = pm Zt は世代tの感染者数 Xt,i は世代tの個体iが感染させた数 N世代後に感染者数はどうなるか?が⽬的 • その定義よりZはマルコフ連鎖(もっと⾔うと普通は状態0を吸 収状態とする吸収的マルコフ連鎖を仮定) • 世代tの感染者数Zt の期待値は、ある個体がうつす 感染数の期待値から <latexit sha1_base64="/QJrU3OHsX386wyBoDoLP9nW310=">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</latexit> p0 > 0, p0 + p1 < 1 <latexit sha1_base64="6p2Tnm+ggx9c59MScxarcZT1nDM=">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</latexit> E[Zt] = E[Xt,i]t
MERSの伝播リスク ⼀⼈が平均R0 ⼈にうつすとすると感染者数は 第t-1世代時の総感染者数は R0 <1なる感染症において、総感染者数の情報を得ることが感染性を知 るうえでとても⼤事 でも、Spatially heterogeneousなpopulationで これやると推定値にバイアスが⼊る
(Birello et al. 2024) 47 <latexit sha1_base64="DnbtaWbc3n81nGFLZ0YCTE/7GZM=">AAACbnicfVBdSxtBFJ2s9StajRb0oRQXQ0FKCLt+vynqQ19KVYwKSRruTm7i4HwsM3elYcmv6Wv7g/ov+hM6GyP4hRdmOJx77tw5J0mlcBRFf0vBxLvJqemZ2fLc/PuFxcrS8qUzmeXY4EYae52AQyk0NkiQxOvUIqhE4lVye1z0r+7QOmH0BQ1SbCvoa9ETHMhTncpqXDvvRMX5sTm6t2qtriHXqVSjejSq8CWIx6DKxnXaWSod+EGeKdTEJTjXjKOU2jlYElzisNzKHKbAb6GPTQ81KHTtfORgGH72TDfsGeuPpnDEPp7IQTk3UIlXKqAb97xXkLVEvdZuZtTbb+dCpxmh5ve7epkMyYRFImFXWOQkBx4At8J/N+Q3YIGTz+3JouJtMkY67+YEvUuL3zz1PUULZOyXvAW2r4Qeetf9Vq1Abwnh54PQo3LZRx4/D/gluNysx7v1nbPt6uHROPwZ9pGtsw0Wsz12yL6yU9ZgnA3ZL/ab/Sn9C1aCT8HavTQojWc+sCcVbPwHzl28Fg==</latexit> 1, R0, R2 0 , R3 0 , . . . <latexit sha1_base64="I/0wJq8xSxF3i5ymZMMtAvaZW3k=">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</latexit> In ⌘ n 1 X t=0 Rt 0 = 1 Rn 0 1 R0 <latexit sha1_base64="qrynYPzU6iu7RXDTR3fx8npLUqI=">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</latexit> R0 ⇡ 1 1 In 今、MERSではR0 は⼩さ いので、R0 nはめっちゃ ⼩さいはず
MERSの伝播リスク (Cont.) 今、1感染者だけが輸⼊されたとする 最終規模は ⼀⼈の⼈が何⼈にうつすかの確率分布 (offspring dist)を負の⼆項分布 メリット:⼤きい分散(つまりスーパスプレッダー)が表現可能 最終規模の分布は以下 (Nishiura
et al. (2012)) 48 <latexit sha1_base64="tAi4/IL43ps4yVMV3NtDSadnb3s=">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</latexit> Y = 1 X t=0 Xt <latexit sha1_base64="gZOlhTZMa1R0IqYcI1EFRd6fsLM=">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</latexit> P(Xt = x) = (k + x) x! (k) ✓ R0 R0 + k ◆x ✓ 1 + R0 k ◆ k <latexit sha1_base64="JEkbeFjhXzWt1IV2qX5g+K5O9c0=">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</latexit> P(Y = y) = Qy 2 j=0 j k + y y! ✓ k R0 + k ◆ky ✓ R0k R0 + k ◆y 1
使い⽅ データが揃い、負の⼆項分布のパラメータが(最尤)推定できると Q1. 例えば、1例輸⼊されたときに、⼆次感染の発⽣確率は? Q2. ⼆次感染で終わる(絶滅)確率は? ( と書く) Q3. 総感染者数が8⼈以上になる確率は?
49 <latexit sha1_base64="gZOlhTZMa1R0IqYcI1EFRd6fsLM=">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</latexit> P(Xt = x) = (k + x) x! (k) ✓ R0 R0 + k ◆x ✓ 1 + R0 k ◆ k <latexit sha1_base64="tmBDkU20ZGE8b/aXy0bJusbL1/c=">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</latexit> P(X = m) = pm 統計家向け: ここがXの確率⺟関数(PGF) の形になっていることに気づこう <latexit sha1_base64="RU6tyaAFnVMR2XcVHTpV/QAsGXM=">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</latexit> P(X2 = 0|X0 = 1) = 1 X k=1 P(X2 = 0|X1 = k, X0 = 1)P(X1 = k|X0 = 1) = 1 X k=1 {P(X = 0)}k pk = 1 X k=1 pk 0 pk <latexit sha1_base64="rDyff7u0uOXh/EqJKEUo9PykeLo=">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</latexit> P(X1 > 0) = 1 P(X1 = 0) = 1 ✓ 1 + R0 k ◆ k ⇡ 22.7% (19.3 25.1%) <latexit sha1_base64="XfPKZ5rwTjdk6J9nBvRCTZC8B7c=">AAADHnicfVHLbhMxFPUMrxIeTWHJxhAVJbSJZiqgZRFRAQs2iPBIW1QnI4/jmbjjGQ+2BzWy/C98DawQW/gbnEclkiCuZOnonHN9fY/jkjOlg+C351+6fOXqtY3rtRs3b93erG/dOVKikoT2ieBCnsRYUc4K2tdMc3pSSorzmNPjOHs51Y+/UKmYKD7qSUkHOU4LljCCtaOi+vde8xNK6Wd40IIPuzBsI1XlkeHd0A73oRO7vAURgjUnrqholEhMDCqlGEXmrBvYoeHtPYs4TXQTzcQzazK7w5Fk6Vi3rOH3L+R5c2bN+yjYyezCMTQZX3Y4ed3E26FFuHSTz2EYdJ6h7ajeCDrBrOA6CBegARbVi7a852gkSJXTQhOOlToNg1IPDJaaEU5tDVWKlphkOKWnDhY4p2pgZolbuO2YEUyEdKfQcMb+3WFwrtQkj50zx3qsVrUpuRvn/5JPK50cDAwrykrTgsxnJRWHWsDpD8IRk5RoPnEAE8nccyEZY5eVdv+8NGh6txaCK7fNK+q2lPSNo96WVGIt5CODsExzVli3dYp2p+h/Rnx+YXSoVnORh6sBr4OjvU74tPPk3ePG4YtF+BvgHngAmiAE++AQvAY90AfEa3sfPOQN/K/+N/+H/3Nu9b1Fz12wVP6vPyB+/Mw=</latexit> P(Y 8) = 1 7 X l=1 P(Y = l) = 1 7 X l=1 Ql 2 j=0 j k + l l! ✓ k R0 + k ◆kl ✓ R0k R0 + k ◆l 1 ⇡ 10.9% <latexit sha1_base64="zdM3ub26NHPLf88BSULhbww9BNg=">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</latexit> GY (s) = E[sY ] = 1 X y=0 p(y)sy