感染症の数理 セミナー(2) Mar 1, 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 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

SIRモデル Kermack and McKendrick (1927) 4/20 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 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) 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 I = 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 N = S + I + R ↑ここ、前回サクッと⾶ばし ちゃってけどなんで？

平均感染期間Dについて 感染期間T：ある個体が感染してから２次感染を起こすまでの 期間(generation timeとも呼ぶ) 治癒率γ: 時間不変 (時変とすると⾊々おもしろい) 【証明】(ほぼほぼ⽣存時間分析のところに出てくるのと同じ) γはある時間tまで治らず、その次の瞬間に治る確率 ここで とおくと、 よって、 このときTの期待値Dは 5/20 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 dt = P(t < T < t + dt|T > t) 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 q(t) = P(T > t) 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 dt = P(t < T < t + dt|T > t) = q(t) q(t + dt) q(t) = q0(t) q(t) dt 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 q0(t) = q(t) , q(t) = P(T > t) = exp( t) 指数分布になってる！: pdfが 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 f(t) = exp( t) 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 D = Z 1 0 t exp( t)dt = 1

⼈⼝の異質性 (heterogeneity) これまでのSIRは、どの⼈ともランダムに接触 でも、⼦ども同⼠、⼤⼈同⼠、男同⼠、野球部同⼠接触するでしょ？ 12/20 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 K = ✓ RCC RCA RAC RAA ◆ 次世代⾏列 (NGM) Aは⼤⼈ Cは⼦供 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 Rji i →j の接触パターンで 「⼀⼈の接触者が⽣み出す感染者の平均値」 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 ✓ IC,t+1 IA,t+1 ◆ = ✓ RCC RCA RAC RAA ◆ ✓ IC,t IA,t ◆ 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 ✓ IC,t IA,t ◆ = ✓ RCC RCA RAC RAA ◆t ✓ IC,0 IA,0 ◆ t+1世代とt世代の関係 t世代と0世代の関係 (重要な)定理 1. : t+1世代とt世代の総感染者の割合はある値に収束 2. : 総感染者に対するある⼤⼈と⼩児の感染者数の⽐はある値に収束 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 = IC,t+1 + IA,t+1 IC,t + IA,t ! c 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 IC,t IC,t + IA,t ! c1, IA,t IC,t + IA,t ! c2 ↓これが実は基本再⽣産数R0 と同じ！

次世代⾏列の例 例: ⼀⼈の接触者が⽣み出す感染者の平均値 ex. ⼤⼈→⼤⼈ = 1.4、⼦供→⼤⼈ = 0.4 このときλ=1.53に収束する 13/20 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 K = ✓ 1.4 0.4 0.4 0.3 ◆ ↑R0 と同じじゃん！ 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 K ✓ IC,t IA,t ◆ = ✓ RCC RCA RAC RAA ◆ ✓ IC,t IA,t ◆ = ✓ IC,t+1 IA,t+1 ◆ = ✓ IC,t IA,t ◆ = R0 ✓ IC,t IA,t ◆ ここがポイント t+1世代の感染者数に対するt世代の感染者の 割合がλに収束するということは、こう書け るということ。 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 K ✓ IC,t IA,t ◆ = R0 ✓ IC,t IA,t ◆ ここがポイント R0 がまさしくKの固有値、固有ベクトルを求める問題になっている！ ここがポイント Kの最⼤固有値(スペクトル半径)がR0 となっている。

⼈⼝の異質性を加味したSIRモデル あるサブグループaの⼈⼝Na において とおく 14/20 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 dsa dt = sa X b abib dia dt = sa X b abib ia dra dt = ia 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 sa = Sa Na , ia = Ia Na , ra = Ra Na ⾏列表記すると 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 d dt 0 B B B @ i1 i2 . . . in 1 C C C A = 0 B B B B @ 11 12 . . . 1n 21 22 . . . 2n . . . . . . . . . . . . n1 n2 ... nn 1 C C C C A 0 B B B @ i1 i2 . . . in 1 C C C A 初期ではsa =1 AAAD6XicjVJdb9MwFHUbPkb42sYjErKoGBOMKokY8IKYgAdeEEOs26Smqm5cN7MaO5HtTKui/Ii9IV75VfwTHrnJmildEcKSpeNz7zn3XttRlghjPe9Xp+tcu37j5tot9/adu/fur29sHpo014wPWJqk+jgCwxOh+MAKm/DjTHOQUcKPotmHKn50yrURqTqw84yPJMRKTAUDi9R4o3O+9ZaGkSwOSvq8BuE3EUsow9CtIzwWqsgkWC3OShfPFsaF75d0izaHoD5MUmtapCppGNJGELQFAQpWFUGjOG34S7BMtGxV21Y1tsu+qvHlanI5SD3r8mgvwhikBBR6rZpera2YJr4S/K+GvTq93V7jt9LbeL3n9b160VXgL0CPLNY+PuI7rMhyyZVlCRgz9L3MjgrQVrCE46vlhmfAZhDzIUIFkptRUX+ekj5BZkKnqcatLK3ZtqIAacxcRpiJDZ6Yq7GK3Ink38LD3E7fjAqhstxyxS5qTfOE2pRWn5FOhObMJnMEwLTAdik7AQ3M4pddKlR52zRNDE7zkeOUmn9G6kvGNdhUPytC0LEUqsSp43CnQv9KhLMmEZHr4pX7Vy94FRwGff9Vf/fry97e+8Xlr5GH5DHZJj55TfbIJ7JPBoR1fncfdZ92t52Zc+58d35cpHY7C80DsrScn38ATKM2DA== = T + ⌃ = 0 B B B B @ 11 12 . . . 1n 21 22 . . . 2n . . . . . . . . . . . . n1 n2 ... nn 1 C C C C A + 0 B B B B @ 0 . . . 0 0 . . . 0 . . . . . . . . . . . . 0 0 ... 1 C C C C A NGM with Large domain 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 KL = T ⌃ 1 これの最⼤固有値がR0 AAACkHicfVDbahRBEO0db3G9ZBMffWlchBjMMiOa5CFivDyIIEZxk8DOutT01kya9GXsrgkuw/yMX+Orvvk39mw2YBKxoJvDOVVdfU5WKukpjn93oitXr12/sXSze+v2nbvLvZXVfW8rJ3AorLLuMAOPShockiSFh6VD0JnCg+z4dasfnKDz0prPNCtxrKEwMpcCKFCT3s6nScxT/FrJE55OcweiTjMkaOq0AK2h4c/5xpmgbLGWbLgv64+aOtzNpNePB/G8+GWQLECfLWpvstLZSadWVBoNCQXej5K4pHENjqRQ2HTTymMJ4hgKHAVoQKMf13ObDX8YmCnPrQvHEJ+zf0/UoL2f6Sx0aqAjf1FryX9po4ry7XEtTVkRGnG6KK8UJ8vbzPhUOhSkZgGAcDL8lYsjCJFQSPbclvZtslb5YOUNBosO3wfqQ4kOyLr1OgVXaGmaYLlIH7fof43w7awxoG435J1cTPcy2H8ySDYHzz4+7e++WiS/xO6zB2yNJWyL7bK3bI8NmWDf2Q/2k/2KVqPt6EX08rQ16ixm7rFzFb37A4hZykI= R0 ⌘ = log(1 r⇤) r⇤ Recall: : これにKL は相当している (Tの要素がβでΣの要素がγ(のinverse)なので) 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 za = 1 exp X b Rabzb ! 最終規模⽅程式は前と同じ計算で以下となる

こっから更にadvancedになります Endemic SIRモデルと解の安定性 (線形微分⽅程式、ヤコビアン、リアプノフ安 定の計算) 15/20 この本の第⼀章 がわかるくらい

エンデミックSIR 単位時間あたりb⼈誕⽣ 感受性者も感染者も回復者も、単位時間当たり⼀定の割合μ(⼈/単位時間) で (その感染症でない理由によって) 死亡（感染症が理由でないことを表すため「⾃然死」とも呼ぶ） 16 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 dS(t) dt = b µS(t) S(t)I(t) dI(t) dt = S(t)I(t) ( + µ)I(t) dR(t) dt = I(t) µR(t) 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 dN(t) dt = b µ(S(t) + I(t) + R(t)) = 0 ) N(t) = b µ N(t) = S(t)+I(t)+R(t)はb/μを安定な平衡値としてもつ ので、S, I, Rの２つのうち⼀つが決まれば後は決まる ので、上２つで⼗分じゃん ⾏列表記しておこう 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 x(t) = ✓ S(t) I(t) ◆ 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 f(S, I) = ✓ b µS SI SI (µ + )I ◆ とおくと、上２つの式は 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 dx(t) dt = f(x(t)) よく使うので、ヤコビアンも定義しとく 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 f0(S, I) = ✓ µ I S I S (µ + ) ◆

定常解の安定性 定常解: dS/dt = 0, dI/dt=0を満たす点を探すと 当然、E1 とE2 の安定性が気になる 17/20 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 dS(t) dt = b µS(t) S(t)I(t) dI(t) dt = S(t)I(t) ( + µ)I(t) dR(t) dt = I(t) µR(t) 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 E1 = ✓ b µ , 0 ◆ , E2 = ✓ µ + , µ + µ ◆ 感染者0⼈なんで⾃明 ⾮⾃明: R0 >1のときだけ現れる 感染症が常にある=エンデミックな状態の定常状態 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 dx dt = Ax 線形微分⽅程式の安定性の基礎 定理 Aの固有値に関して 1. 固有値の実部がすべて⾮正ならば、リアプノフ安定 2. 固有値の実部がすべて負ならば、漸近安定 3. ある固有値の実部が正ならば、不安定 https://math-fun.net/20180720/789/

18/20 https://www.cis.twcu.ac.jp/~asakawa/ MathBio2010/lesson07/index.html But, 今考えている式はこうは表せない 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 dx(t) dt = f(x(t)) 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 dx dt = Ax 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 dS(t) dt = b µS(t) S(t)I(t) dI(t) dt = S(t)I(t) ( + µ)I(t) dR(t) dt = I(t) µR(t) 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 x(t) = ✓ S(t) I(t) ◆ 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 f(S, I) = ✓ b µS SI SI (µ + )I ◆ いまはこう ⾏列表記だと ただし ヤコビアン (上で計算したよね) 線形化⽅程式

知りたいのは⾮線形⽅程式の安定性 これは線形化⽅程式の安定性と どのような関係にあるの？ 定理 • 平衡点において、線形化⽅程式 の⾏列の固有値を考える。すべ ての固有値が負の実部を持つな らば、もとの⾮線形⽅程式は漸 近安定（証明はリアプノフ関数を使⽤) • ある固有値が正の実部を持ち、 すべての固有値の実部が0でない （双曲型不動点） ならば、もとの⾮線形 ⽅程式は不安定≠ 解の発散（ハート マン・グロブマンの定理） 19/20 このヤコビアンの 固有値を考える つまり、線形化した場合は、固有値が全て負の場合以外 は、安定性はよくわかんない→リアプノフ関数 https://math-fun.net/20180720/844/

E1 の安定性 E1 におけるfのヤコビアンは この固有値は 20/20 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 dx(t) dt = f(x(t)) 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 x(t) = ✓ S(t) I(t) ◆ 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 f(S, I) = ✓ b µS SI SI (µ + )I ◆ ただし 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 f0(E1) = 0 B @ µ b µ 0 b µ (µ + ) 1 C A 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 1 = µ < 0 2 = b µ (µ + ) = b µ (1 1 R0 ) 8 > < > : > 0(R0 > 1) = 0(R0 = 1) < 0(R0 < 1) 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 R0 = b µ(µ + ) ただし 定理 Aの固有値に関して 1. 固有値の実部がすべて負ならば、漸近安定 2. ある固有値の実部が正ならば、不安定

E2 の安定性 E2 におけるfのヤコビアンは この固有値は 21/20 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 f0(E2) = ✓ µR0 (µ + ) µ + µR0 0 ◆ 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 = 1 2 ⇣ µR0 ± p (µR0)2 4µ(µ + )(R0 1) ⌘ • R0 <1のとき より正と負の両⽅があ るから不安定 • R0 = 1のとき 。0があるのでリアプノフ安定？(実はすぐには判定 不可) • R0 >1のとき より固有値の実部は常 に負なので安定 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 p (µR0)2 4µ(µ + )(R0 1) > µR0 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 p (µR0)2 4µ(µ + )(R0 1) < µR0 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 = 0, µR0 実はリアプノフ関数を書くと、E2はラ ・サールの不変性原理より境界上を除 いて⼤域的にも安定