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感染症の数理モデル6

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 感染症の数理モデル6

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Daisuke Yoneoka

June 21, 2024
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  1. 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
  2. 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
  3. 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
  4. 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=">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</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
  5. 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=">AAACx3icfVFdb9MwFHUCG6N8dfDIi6FCaumoEsQGLxMTIAEPiILoVqkpkeM6qRU7juybKVWUB/4kEv+FB5w2E/tAXMnW8bnn+voeR7ngBjzvl+Neu761fWPnZufW7Tt373V37x8bVWjKJlQJpacRMUzwjE2Ag2DTXDMiI8FOovRtkz85ZdpwlX2DVc7mkiQZjzklYKmwm4/707CC+rAc4EMcLGJNaBW8J1KSfjosB3VVPjo7DupAsBj6repr6NXNNkzrQPNkCYPv5UbgD89L/qarZ2kddnveyFsHvgr8FvRQG+Nw13kdLBQtJMuACmLMzPdymFdEA6eC1Z2gMCwnNCUJm1mYEcnMvFpbU+MnllngWGm7MsBr9nxFRaQxKxlZpSSwNJdzDbkXyX+lZwXEr+YVz/ICWEY3veJCYFC4sRovuGYUxMoCQjW3z8V0SawvYD/kQqPmblBKGDvNO2an1OyTpT7nTBNQ+mkVEJ1IntV26iTYa9D/hKQ8E1rU6VjL/csGXwXHz0f+wWj/y4ve0ZvW/B30ED1GfeSjl+gIfUBjNEEU/US/nS1n2/3oKvfULTdS12lrHqAL4f74A3HJ3kw=</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
  6. 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