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感染症の数理モデル1

Daisuke Yoneoka
February 15, 2024

 感染症の数理モデル1

Daisuke Yoneoka

February 15, 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 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. SIRモデル Kermack and McKendrick (1927) 4/20 <latexit sha1_base64="Aw425Lop8YfQ4vugYIF3fQA859A=">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</latexit> dS dt

    = SI dI dt = SI I dR dt = I 感受性⼈⼝ 感染性⼈⼝ 治癒⼈⼝ 仮定 1. Sは病原体に暴露次第Iへ移⾏ 2. ⼆次感染はIのみ 3. Iは⼀定期間で⾃動的にRへ以降 4. 治癒後は免疫がつき⼆度⽬の感染はない (=RからSやIに移⾏はない) 意味: SとIの⼈数に⽐例した⼈数だけIに⾏く は感染⼒(⼀⼈のSが⼆次感染する率) 意味: (第⼀項)SとIの⼈数に⽐例した⼈数だけIに来た (第⼆項)γの割合だけ治ってIへいく 意味: γの割合だけ治ってIからきた β: 伝達率 1⼈のIが1⼈のSと接触する率 γ:治癒率 平均感染期間D の逆数(γ=1/D) (逆に、D=1/γなので、γがわかるとDがわかる) <latexit sha1_base64="KiuUxCGDoouemn4CwdEFHjafzu0=">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</latexit> I = <latexit sha1_base64="P3u458EFEPOF+uskm1we2BhddFU=">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</latexit> N = S + I + R
  3. 感染時刻からの経過時間も考慮しよう Iの⼈って、感染したときからずっと同じ感染⼒もたないよね? →伝達率βや治癒率γは感染齢a(感染時刻からの経過時間)に依存 5/20 <latexit sha1_base64="gKvdUdHUfZkI494qnQB0eyJKqIY=">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</latexit> dS(t) dt = S(t)

    Z 1 0 (a)I(t, a)da I(t, 0) = S(t) Z 1 0 (a)I(t, a)da ✓ @ @t + @ @a ◆ I(t, a) = (a)I(t, a) dR(t) dt = Z 1 0 (a)I(t, a)da 意味: 感染齢a=0の時の感染⼈⼝ = 新規感染者数 意味: 感染齢aの時の感染⼒は <latexit sha1_base64="gKvdUdHUfZkI494qnQB0eyJKqIY=">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</latexit> dS(t) dt = S(t) Z 1 0 (a)I(t, a)da I(t, 0) = S(t) Z 1 0 (a)I(t, a)da ✓ @ @t + @ @a ◆ I(t, a) = (a)I(t, a) dR(t) dt = Z 1 0 (a)I(t, a)da 意味: 時刻が1進むときの治癒⼈数 時刻tが1進むとaも1進むことに注意(だから偏微分 の合計でOK) 意味:感染齢aのときの治癒⼈数は <latexit sha1_base64="gKvdUdHUfZkI494qnQB0eyJKqIY=">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</latexit> dS(t) dt = S(t) Z 1 0 (a)I(t, a)d I(t, 0) = S(t) Z 1 0 (a)I(t, a)da ✓ @ @t + @ @a ◆ I(t, a) = (a)I(t dR(t) dt = Z 1 0 (a)I(t, a)da 感染者がどう変化するかの式が2つに別れた!
  4. 基本再⽣産数 R0 知りたいのは、流⾏(epidemic)が起こるのか? 6/20 <latexit sha1_base64="LgBXHs3a4r/G6q4Gk1pB0As+k2Q=">AAACz3icfVFbb9MwFHbCZSNc1sIjLxYVaCBaJYgNHhiagAf6gNgY3SY1VXXinGTW7DiyHaCKgnjlJ/LIP8FtU7QL4kiWPn2XOOdzUgpubBj+8vwrV69dX1u/Edy8dfvORqd799CoSjMcMSWUPk7AoOAFjiy3Ao9LjSATgUfJ6du5fvQFteGq+GxnJU4k5AXPOAPrqGmnfhSnmQZWpwdNndqG7tB+nKAFekCHNI6Dv/pwpdNWH9I+jXOQEpxzh262dH/JPR6+Ds/GP63iq0gQTDu9cBAuhl4GUQt6pJ29add7FaeKVRILywQYM47C0k5q0JYzgU0QVwZLYKeQ49jBAiSaSb1oqaEPHZPSTGl3CksX7NlEDdKYmUycU4I9MRe1OfkvbVzZ7OWk5kVZWSzY8qKsEtQqOq+cplwjs2LmADDN3b9SdgKuFese5twt829bpYRxq7xDt6LGD476WKIGq/STOgadS140buU8fjpH/zPCt5XRoUXf0cV2L4PDZ4Noe7C1/7y3+6Ztfp3cJw/IJonIC7JL3pM9MiKM/PbWvI7X9ff9r/53/8fS6ntt5h45N/7PPzcV268=</latexit> dS dt = SI

    dI dt = SI I = ( S )I > 0 dR dt = I • I>0は当たり前なの で、実質ここが正 • t=0(初期)では S=N(IとRがいない) <latexit sha1_base64="5UR3Q1gCVQ+AfJmy8gWT2mIaFjo=">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</latexit> R0 ⌘ N > 1 基本再⽣産数 ⼀⼈の感染者が起こす⼆次感染の総数 <latexit sha1_base64="HB7u1JxdbR6H6NBAErm2Ru0rf/I=">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</latexit> Z = R0 + R2 0 + R3 0 + . . . = T X i=1 Ri 0 T !1 ! 8 < : 1 1 R0 (R0 < 1) 1 (R0 1) 重要な関係式: ⼀⼈⽬はR0 に⼆次感染、その次の⼈もR0 ⼈に⼆次感染、その次の⼈も、、、となっていくと最終的には何⼈感染するの? • ⼀⼈が必ずR0 に⼆次感染させるなら、 総感染者数Zは収束 or 発散 • Zがどれくらいになるか⾒積もれる!
  5. 最終流⾏規模 (Final epidemic size) もうちょっと現実的に⼈⼝あたりで考える SIRには2つの均衡点 (s, i, r) =

    (1, 0 , 0)と(s*, 0, r*) 7/20 t→∞としたときに感染者は0⼈になり、 s* (= s(∞)): 何%が感染を逃れたか r* (= r(∞)): 何%が感染したか <latexit sha1_base64="GZ7GT/t1BrHuSYH75Oq0B+oFT9Y=">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</latexit> ds dt = si di dt = si i dr dt = i <latexit sha1_base64="TWw88aSbJzklo45MPjvq1kp19nw=">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</latexit> s = S N , i = I N , r = R N <latexit sha1_base64="Aw425Lop8YfQ4vugYIF3fQA859A=">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</latexit> dS dt = SI dI dt = SI I dR dt = I <latexit sha1_base64="xH6Y+u0OyWqjHpz+UFtZmTKuzBw=">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</latexit> r⇤ = 1 exp( r⇤R0) <latexit sha1_base64="y4Drxc9cmIJphOIsqsikfGkQF1o=">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</latexit> R0 ⌘ = log(1 r⇤) r⇤ 新しい基本再⽣産数 r*にてつい て解く (解析的には解けないので)数値計算 <latexit sha1_base64="lxjKR111eVZZbnFb7RGt1wYQNmw=">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</latexit> r⇤ ⇡ 1 exp( R0) R0 が⼤きいとき はr*≒1なので R0 だけから、最終的な流⾏規模が⾒積もれる! 求め⽅:上の3番⽬を1番⽬に代⼊し て積分するだけ
  6. ここまでのまとめ 簡単なSIRからでもいろんなことがわかるよ! 1.伝達率β(1⼈のIが1⼈のSと接触する率)と治癒率γが分かればR0 がわかる 2. 平均感染期間D=1/γ なので、γ=1/Dでγがわかる 3.β=R0 γなので、R0 とγが分かればβがわかる

    4. R0 が分かれば、最終規模 r*(t→∞で何%が感染したか) がわかる 8/20 <latexit sha1_base64="y4Drxc9cmIJphOIsqsikfGkQF1o=">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</latexit> R0 ⌘ = log(1 r⇤) r⇤ <latexit sha1_base64="xH6Y+u0OyWqjHpz+UFtZmTKuzBw=">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</latexit> r⇤ = 1 exp( r⇤R0)
  7. 実装: Final epidemic size 10/20 • Final epidemic sizeは左図の緑⾊の左の⽅の収束先の値 <latexit

    sha1_base64="y4Drxc9cmIJphOIsqsikfGkQF1o=">AAACkHicfVDbahRBEO0db3G9ZBMffWlchBjMMiOa5CFivDyIIEZxk8DOutT01kya9GXsrgkuw/yMX+Orvvk39mw2YBKxoJvDOVVdfU5WKukpjn93oitXr12/sXSze+v2nbvLvZXVfW8rJ3AorLLuMAOPShockiSFh6VD0JnCg+z4dasfnKDz0prPNCtxrKEwMpcCKFCT3s6nScxT/FrJE55OcweiTjMkaOq0AK2h4c/5xpmgbLGWbLgv64+aOtzNpNePB/G8+GWQLECfLWpvstLZSadWVBoNCQXej5K4pHENjqRQ2HTTymMJ4hgKHAVoQKMf13ObDX8YmCnPrQvHEJ+zf0/UoL2f6Sx0aqAjf1FryX9po4ry7XEtTVkRGnG6KK8UJ8vbzPhUOhSkZgGAcDL8lYsjCJFQSPbclvZtslb5YOUNBosO3wfqQ4kOyLr1OgVXaGmaYLlIH7fof43w7awxoG435J1cTPcy2H8ySDYHzz4+7e++WiS/xO6zB2yNJWyL7bK3bI8NmWDf2Q/2k/2KVqPt6EX08rQ16ixm7rFzFb37A4hZykI=</latexit> R0 ⌘ = log(1 r⇤) r⇤ より、今R0 =4.2くらい 98.4%くらいの⼈がこの感染症に最終的にはかかる