感染症の数理モデル5

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1. 感染症の数理 セミナー(5) May 24, 2024 @NIID 国⽴感染症研究所 第12室⻑ ⽶岡 ⼤輔
2. 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 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
3. 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
4. デング熱の数理 蚊→ヒト→蚊 のような伝播経路 (蚊→蚊 (経卵巣感染)は今は無視) Notations • 時刻tにおけるヒトの新規感染数: 蚊の新規感染数: • 蚊→ヒトへの伝播の時間間隔のpdf: ヒト→蚊への伝播の時間間隔のpdf: 再⽣産⽅程式（ヒトの場合） 38 <latexit sha1_base64="BJCroYRWVoK86Sy8LVmKPTEt+s8=">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</latexit> jh(t) <latexit sha1_base64="DL2AyjpVLumKplgXHKAl1c3PJ78=">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</latexit> jv(t) <latexit sha1_base64="t1no7A4Aa7V4JWwluEjqNVjpe/w=">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</latexit> fhv(s) <latexit sha1_base64="Y7Sr2/2ibulStOF2Nhes9UfnlHA=">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</latexit> fvh(s) <latexit sha1_base64="lAy4E+fTuuNeiBQxjYpMTBdBYGI=">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</latexit> jh(t) = Rhv Z 1 0 fhv(s)jv(t s)ds <latexit sha1_base64="Zp14HxBDykWcM9B2kbU2OS3Q6w4=">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</latexit> jv(t) = Rvh Z 1 0 fvh(s)jh(t s)ds 再生産方程式 (蚊の場合) 蚊の場合の再生産方程式もvとh を入れ替えるだけ Rhv は一匹の蚊が生み出したヒ トの二次感染者数の平均値 t-sの時点の蚊がうつしている まとめると一本の式で記述可能 <latexit sha1_base64="oXgOJdCMG6PMtZDgsElZyMdB/24=">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</latexit> jh(t) = RhvRvh Z 1 0 Z 1 0 fhv(s)fvh(s)jh(t ⌧ s)d⌧ds ヒトからヒトの⼆次感染者が再⽣産されることを表現
5. デング熱の流⾏初期には 流⾏初期では、「感染数が指数的に増える」 39 <latexit sha1_base64="LMh8BlDi2DM0OyC8jog9M8maJyM=">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</latexit> jh(t) = k exp(rt) と仮定する。 増加率 (⾚線の曲がり具合)がr r(とk)は最⼩⼆乗法とかpure birth processを仮定した最尤法で推定 よくやる仮定 <latexit sha1_base64="lAy4E+fTuuNeiBQxjYpMTBdBYGI=">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</latexit> jh(t) = Rhv Z 1 0 fhv(s)jv(t s)ds <latexit sha1_base64="Zp14HxBDykWcM9B2kbU2OS3Q6w4=">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</latexit> jv(t) = Rvh Z 1 0 fvh(s)jh(t s)ds ただし はfvh ()のモーメント⺟関数 <latexit sha1_base64="h5sRgOAU/PFwOij06oddUfS7xlE=">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</latexit> Mvh( r) = Z 1 0 fvh(⌧) exp( r⌧)d⌧ に代⼊して <latexit sha1_base64="tJjDhw9vCy3NrtvdS2FVMzfdvIg=">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</latexit> jh(t) = kRhvRvh exp(rt)Mvh( r)Mhv( r) ⻩⾊い部分と⻘い部分を⾒⽐べることで を得る <latexit sha1_base64="ftRXc13dl9V7Ouxx26y+OJ0P0EU=">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</latexit> RhvRvh = 1 Mvh( r)Mhv( r) R0 は実はこれの平⽅根（相加相乗平均） <latexit sha1_base64="Ej97xkbn0Dm2O8N7wujg8WkRqDo=">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</latexit> exp r t 1 X i=0 C(i) ! (1 exp( r))C(t) C(0) 増殖率rの推定について 新規感染数がpure birth processに従う (ポワソン過程の⼀般化)と仮定すると 尤度は以下。 ただしC(t)は時刻tにおける累積感染数 これを最⼩化すればOK <latexit sha1_base64="AU7GOABMCyVitbWH29sL19NnI9o=">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</latexit> ) jv(t) = kRvh exp(rt) Z 1 0 fvh(⌧) exp( r⌧)d⌧ = kRvh exp(rt)Mvh( r)
6. （参照：最後の関係式はこの話と似ている）R0 の統計的推測 (1) 今までは、γやβは天下り的に与えてきた→データから推定したいよね 再⽣産⽅程式 時刻 tにおける新規感染者数 I(t) は、a⽇前の過去の新規感染者数を I(t−a)を⽤いて 40 <latexit sha1_base64="KRYtOBHRA80qCdkLhBLyz4qB/bc=">AAACeHicfVBNbxMxEHWWj5bw0bQcuRgiRILaaLeCwgVRWg5wQBSJtJWyIZp1ZlOr/ljZs4holTO/hiv9Lf0rPdWbBom2iJEsP7034/F7WaGkpzg+bUQ3bt66vbR8p3n33v0HK63VtX1vSyewL6yy7jADj0oa7JMkhYeFQ9CZwoPseLfWD76j89KarzQtcKhhYmQuBVCgRq3HHzvU5W94Kg2N4m/hymnK33WgG4QN6I5h1GrHvXhe/DpIFqDNFrU3Wm28TcdWlBoNCQXeD5K4oGEFjqRQOGumpccCxDFMcBCgAY1+WM29zPjTwIx5bl04hvic/XuiAu39VGehUwMd+ataTa5n+l/yoKT89bCSpigJjbjYlZeKk+V1NnwsHQpS0wBAOBm+y8UROBAUEry0qH6brFU+uHmPwaXDT4H6XKADsu55lYKbaGlmwfUkXa/R/xrhx5/GgJrNEHlyNeDrYH+zl2z1Xn550d7eWYS/zB6xJ6zDEvaKbbMPbI/1mWA/2S/2m500ziIePYu6F61RYzHzkF2qaPMcMWi/2A==</latexit> I(t) = Z 1 0 A(a)I(t a)da <latexit sha1_base64="vFZ72HbnVeU8w6LkYvqK9aOFGaY=">AAACb3icfVBdaxNBFJ2sVdv4leqDhYIMBqGVEnZLtX0RW/XBF7GKaQvZuNyd3E2Hzscyc1cMS/pr+lr/jz/Df+BsGsG24oVhDueeO3fOyUslPcXxz1Z0Y+HmrduLS+07d+/df9BZfnjgbeUE9oVV1h3l4FFJg32SpPCodAg6V3iYn7xt+off0HlpzRealDjUMDaykAIoUFln5XMWv0qloSz+Gq6CJnxvDdZHkHW6cS+eFb8OkjnosnntZ8ut1+nIikqjIaHA+0ESlzSswZEUCqfttPJYgjiBMQ4CNKDRD+uZhSl/FpgRL6wLxxCfsX9P1KC9n+g8KDXQsb/aa8iNXP+rPaio2BnW0pQVoREXu4pKcbK8iYSPpENBahIACCfDd7k4BgeCQnCXFjVvk7XKBzfvMLh0+CFQH0t0QNY9r1NwYy3NNLgepxsN+p8Qvv8RBtRuh8iTqwFfBwebveRl78Wnre7um3n4i2yVPWVrLGHbbJe9Z/uszwQ7ZWfsnP1o/YoeR08ifiGNWvOZR+xSReu/AacbvXU=</latexit> R0 = Z 1 0 A(a)da ただし、 <latexit sha1_base64="QvFYNraTbOF3WY09zCRxI8hPGLg=">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</latexit> A(a) R 1 0 A(a)da は世代時間のpdf この解を と⼀旦してみる (rは内的増殖率という) <latexit sha1_base64="g1gRI6Y400KhcDNHvGyOUuELeYQ=">AAAB/XicbVBNS8NAEN34WetXqkcvi0VoQUoiRb0IRS96q2A/oA1ls920SzebsDtRSyn+Eo/qRbz6Szz4b9y2OWjrg4HHezPMzPNjwTU4zre1tLyyurae2chubm3v7Nq5vbqOEkVZjUYiUk2faCa4ZDXgIFgzVoyEvmANf3A18Rv3TGkeyTsYxswLSU/ygFMCRurYuZsCFPEFHrTZY1xQGIodO++UnCnwInFTkkcpqh37q92NaBIyCVQQrVuuE4M3Igo4FWycbSeaxYQOSI+1DJUkZNobTU8f4yOjdHEQKVMS8FT9PTEiodbD0D/2Q9McEujreXsi/ue1EgjOvRGXcQJM0tmuIBEYIjyJAne5YhTE0BBCFTfnYtonilAwgWVNDu7814ukflJyT0vl23K+cpkmkkEH6BAVkIvOUAVdoyqqIYoe0DN6RW/Wk/VivVsfs9YlK53ZR39gff4AmWaTYQ==</latexit> I(t) = k exp(rt) <latexit sha1_base64="YqBVCmEG4Xltxg7eyG3luZYdW48=">AAACG3icbVBNSwMxEM36bf2qevQSLEILWnelqBfBj4vHCrYKbS2zabYNzWaXZFYsxZ/h0V/iUb2IVw/+G9N2D9r6YJjHezMk8/xYCoOu++1MTc/Mzs0vLGaWlldW17LrG1UTJZrxCotkpG99MFwKxSsoUPLbWHMIfclv/O7FwL+559qISF1jL+aNENpKBIIBWqmZ3e/W+UOc1xQL9ITWhcKmS+9sD7BHz/JQSP087kGh0IJmNucW3SHoJPFSkiMpys3sV70VsSTkCpkEY2qeG2OjDxoFk/wxU08Mj4F1oc1rlioIuWn0h4c90h2rtGgQaVsK6VD9vdGH0Jhe6O/6oR0OATtm3B6I/3m1BIPjRl+oOEGu2OitIJEUIzoIiraE5gxlzxJgWtjvUtYBDQxtnBmbgzd+9SSpHhS9w2LpqpQ7PU8TWSBbZJvkiUeOyCm5JGVSIYw8kRfyRt6dZ+fV+XA+R6NTTrqzSf7A+foBwBeeuw==</latexit> k exp(rt) = Z 1 0 A(a)k exp(r(t a))da でないといけない。両辺 で割れば (オイラー=トロカの特性⽅程式) <latexit sha1_base64="g1gRI6Y400KhcDNHvGyOUuELeYQ=">AAAB/XicbVBNS8NAEN34WetXqkcvi0VoQUoiRb0IRS96q2A/oA1ls920SzebsDtRSyn+Eo/qRbz6Szz4b9y2OWjrg4HHezPMzPNjwTU4zre1tLyyurae2chubm3v7Nq5vbqOEkVZjUYiUk2faCa4ZDXgIFgzVoyEvmANf3A18Rv3TGkeyTsYxswLSU/ygFMCRurYuZsCFPEFHrTZY1xQGIodO++UnCnwInFTkkcpqh37q92NaBIyCVQQrVuuE4M3Igo4FWycbSeaxYQOSI+1DJUkZNobTU8f4yOjdHEQKVMS8FT9PTEiodbD0D/2Q9McEujreXsi/ue1EgjOvRGXcQJM0tmuIBEYIjyJAne5YhTE0BBCFTfnYtonilAwgWVNDu7814ukflJyT0vl23K+cpkmkkEH6BAVkIvOUAVdoyqqIYoe0DN6RW/Wk/VivVsfs9YlK53ZR39gff4AmWaTYQ==</latexit> I(t) = k exp(rt) <latexit sha1_base64="J/e5A2LBdnws9TuyTGuk9tcHYis=">AAACDnicbVDLSgNBEJz1GeMr6tHLkCAkoGFXRL0IUS8eI5gHJDH0Tmbj4OzsMtMrhpC7R7/Eo3oRr/6AB//GyeOg0YKmi6puZrr8WAqDrvvlzMzOzS8sppbSyyura+uZjc2qiRLNeIVFMtJ1HwyXQvEKCpS8HmsOoS95zb89H/q1O66NiNQV9mLeCqGrRCAYoJXamaxHT2hTKGy79Nr2AHv0NA+FJr+P83saCh1oZ3Ju0R2B/iXehOTIBOV25rPZiVgScoVMgjENz42x1QeNgkk+SDcTw2Ngt9DlDUsVhNy0+qNbBnTHKh0aRNqWQjpSf270ITSmF/q7fmiHQ8AbM20Pxf+8RoLBcasvVJwgV2z8VpBIihEdZkM7QnOGsmcJMC3sdym7AQ0MbYJpm4M3ffVfUt0veofFg8uDXOlskkiKbJMsyROPHJESuSBlUiGMPJAn8kJenUfn2Xlz3sejM85kZ4v8gvPxDWZHmc8=</latexit> 1 = Z 1 0 A(a) exp( ra)da （上のただしの部分を使うと）よって <latexit sha1_base64="DDEx2FgrbA2KW0FzqNF85JdxCjA=">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</latexit> R0 = 1 Z 1 0 A(a) R 1 0 A(b)db exp( ra)da
7. （参照）R0 の統計的推測 (2) 世代時間のpdfをg(t)と書くと したがって、rとg()の形が決まれば、R0 は推定可能 • 実際の現場的には、g()は過去⽂献から持ってくることが多い • rは右図のようなepi curveに対して あたりをfitting • rの推定は別に最⼩⼆乗法とかでもOK 41 <latexit sha1_base64="DDEx2FgrbA2KW0FzqNF85JdxCjA=">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</latexit> R0 = 1 Z 1 0 A(a) R 1 0 A(b)db exp( ra)da <latexit sha1_base64="XFoBPaOXbyk7gBRw5r4TPBJUrdQ=">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</latexit> R0 = 1 R 1 0 g(s) exp( rs)ds = 1 M( r) M()は統計家ならみんな ⼤好きgのモーメント⺟関数 <latexit sha1_base64="g1gRI6Y400KhcDNHvGyOUuELeYQ=">AAAB/XicbVBNS8NAEN34WetXqkcvi0VoQUoiRb0IRS96q2A/oA1ls920SzebsDtRSyn+Eo/qRbz6Szz4b9y2OWjrg4HHezPMzPNjwTU4zre1tLyyurae2chubm3v7Nq5vbqOEkVZjUYiUk2faCa4ZDXgIFgzVoyEvmANf3A18Rv3TGkeyTsYxswLSU/ygFMCRurYuZsCFPEFHrTZY1xQGIodO++UnCnwInFTkkcpqh37q92NaBIyCVQQrVuuE4M3Igo4FWycbSeaxYQOSI+1DJUkZNobTU8f4yOjdHEQKVMS8FT9PTEiodbD0D/2Q9McEujreXsi/ue1EgjOvRGXcQJM0tmuIBEYIjyJAne5YhTE0BBCFTfnYtonilAwgWVNDu7814ukflJyT0vl23K+cpkmkkEH6BAVkIvOUAVdoyqqIYoe0DN6RW/Wk/VivVsfs9YlK53ZR39gff4AmWaTYQ==</latexit> I(t) = k exp(rt)
8. 次世代⾏列とR0 次世代⾏列Kはこんな感じになりますよね Kの最⼤固有値（スペクトル半径）がR0 になる（っていうかそう定義する）って話を覚えて ますか？ 流⾏初期であれば、こういった関係式を導出しましたね。 したがって、 42 <latexit sha1_base64="AojXVbVI6pzZkY5T6RPJ3uFtm0M=">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</latexit> K = ✓ Rhh Rhv Rvh Rvv ◆ = ✓ 0 Rhv Rvh 0 ◆ 思い出してください： Rhv は一匹の蚊が生み出したヒトの 二次感染者数の平均値 <latexit sha1_base64="/yhbGAoi6Eq4a6j1Z8bg3/4hi9U=">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</latexit> R0 ⌘ eig(K) = p RhvRvh <latexit sha1_base64="ftRXc13dl9V7Ouxx26y+OJ0P0EU=">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</latexit> RhvRvh = 1 Mvh( r)Mhv( r) <latexit sha1_base64="X+DGRqS6s4+vBcQdmPP8vDLfDqc=">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</latexit> R0 = 1 p Mvh( r)Mhv( r) これは、要は 1. 初期の増殖率r 2. 感染の世代時間のpdf がわかればR0もわかるよ、って話
9. 蚊→蚊への伝播も考慮したいよ 経卵巣感染（⺟から⼦へ）も考慮したい：その確率をpとする 次世代⾏列Kは以下に変更 R0 はKの最⼤固有値なので 仮にワクチンにより蚊→ヒトの感染を(1-q)倍にできるとすると R0 <1が⽬標なのでこれをqについて解くと 43 <latexit sha1_base64="wxUizuxYYR1cVKBCY6NNie+p1xk=">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</latexit> K = ✓ Rhh Rhv Rvh Rvv ◆ = ✓ 0 Rhv Rvh p ◆ <latexit sha1_base64="Rs4MmfLz2ifyNaTUTj28ZKEbOak=">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</latexit> R0 ⌘ eig(K) = 1 2 (p + p p2 + 4RhvRvh) 要は、Rvh →(1-q)Rvh <latexit sha1_base64="l8zdNwbAwpY8BSFUgAryGffcctA=">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</latexit> R0 = 1 2 (p + p p2 + 4(1 q)RhvRvh) <latexit sha1_base64="yJFnDlryz+0CWR2r1ylGhA/lDPE=">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</latexit> q > 1 1 p RhvRvh が⽬指すべきワクチン接種率