感染症の数理モデル6

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=">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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

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=">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</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=">AAACgHicfVDbahRBEO0dL4nrbWMefWlchF0J64wkUQKLwQv6IkZwc2Fndujprdk06cvQXSMOw3yDX+Orfod/Y89mBZOIBQ2HU6eq+pyskMJhGP7qBNeu37i5tn6re/vO3Xv3exsPDp0pLYcJN9LY44w5kELDBAVKOC4sMJVJOMrOXrf9oy9gnTD6M1YFJIottMgFZ+iptDd8l54M3HD8dupm9UmTjGNXqrSuxmEzi4XOsaLFoBq6WZX2+uEoXBa9CqIV6JNVHaQbnZfx3PBSgUYumXPTKCwwqZlFwSU03bh0UDB+xhYw9VAzBS6pl54a+tgzc5ob659GumT/nqiZcq5SmVcqhqfucq8ltzL1r/a0xPxFUgtdlAian9/KS0nR0DYjOhcWOMrKA8at8N+l/JRZxtEneeFQuxuNkc67eQPepYUPnvpYgGVo7JM6ZnahhG6860W81aL/CdnXP0KPul0feXQ54Kvg8Nko2h3tfNru779ahb9OHpJHZEAi8pzsk/fkgEwIJ9/Id/KD/AyCYBA8DaJzadBZzWySCxXs/QbpoMPi</latexit> GY (s) = E[sY ] = 1 X y=0 p(y)sy

感染症の数理モデル6

感染症の数理モデル6

Daisuke Yoneoka

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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.

Effective distance (Brockmann and Helbing, Science 2013) Effective distanceのレシピ 1.

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