COVID-19に関する理論疫学的見解

 COVID-19に関する理論疫学的見解

色々教えていただけるとありがたいです.

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Hiro (葉色)

May 07, 2020
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  1. COVID-19ʹؔ͢Δཧ࿦Ӹֶతݟղ ༿৭1 2020 ೥ 5 ݄ 8 ೔ 1twitter: @mepbphhond

  2. ໨ ࣍ ୈ 1 ষ ͸͡Ίʹ 3 ୈ 2 ষ

    ং࿦ -Basic/Effective reproduction number- 4 ୈ 3 ষ Time Interval ͷਪఆ 6 3.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1 Doubly interval-censoring . . . . . . . . . . . . . . . . 6 3.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1 Estimation of the time interval distribution using dou- bly interval-censored likelihood . . . . . . . . . . . . . 7 3.2.2 Estimation of the time interval distribution using Bayesian framework . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.3 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ୈ 4 ষ ྲྀߦॳظஈ֊Ͱͷ cCFR ͱ R0 ͷਪఆ 11 4.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.1.1 CFR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.1.2 The factor of underestimation . . . . . . . . . . . . . . 12 4.1.3 R0 of an infection from the growth rate of an outbreak (epidemic) . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . 14 4.2.1 Estimation of the delay distribution . . . . . . . . . . 14 4.2.2 Statistical inference . . . . . . . . . . . . . . . . . . . 15 4.2.3 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ୈ 5 ষ Lockdown ͷఆྔతධՁ 18 5.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.1.1 Counterfactual model . . . . . . . . . . . . . . . . . . 18 5.1.2 Hazard function . . . . . . . . . . . . . . . . . . . . . 19 5.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . 19 5.2.1 Reduced number of exported cases . . . . . . . . . . . 19 5.2.2 Reduced probability of major epidemic overseas . . . . 20 5.2.3 Time delay to major epidemic gained from the reduc- tion in travel volume . . . . . . . . . . . . . . . . . . . 21 1
  3. 5.2.4 ·ͱΊ . . . . . . . .

    . . . . . . . . . . . . . . . . . . . 22 ୈ 6 ষ Heterogenerous population Ͱͷਪఆ 24 6.1 Heterogeneity of population . . . . . . . . . . . . . . . . . . . 24 6.2 Next generation matrix . . . . . . . . . . . . . . . . . . . . . 24 ୈ 7 ষ ײછऀ਺ͷਪఆ 26 7.1 Back-calculation . . . . . . . . . . . . . . . . . . . . . . . . . 26 ୈ 8 ষ ʮ8 ׂݮʯͷҙຯ 27 ୈ 9 ষ ͓ΘΓʹ 28 2
  4. ୈ1ষ ͸͡Ίʹ ຊߘ͸, ݱࡏྲྀߦ͍ͯ͠Δ৽ܕίϩφ΢Πϧε COVID-19 ͷ఻೻ಈଶ΍ײ છ֦େ๷ࢭͷ؍఺͔ΒཱͯΒΕͨཧ࿦Ӹֶ (Theoretical epidemiology) ʹΑ

    Δ਺ཧϞσϧʹ͍ͭͯഭΔͱͱ΋ʹ, ݱࡏऔΒΕ͍ͯΔ੓ࡦ (͍ΘΏΔʮׂ̔ ݮʯͱ͸ͳʹ͔ͳͲ) ͷ਺ཧతഎܠͷཧղʹزڐ͔໾ʹཱͭͱ͍͏໨తͷ΋ ͱॻ͔Εͨ΋ͷͰ͋Δ. Ϋϥελʔରࡦ൝͕ͲͷΑ͏ͳ෼ੳΛߦ͍ͬͯΔ͔ Θ͔Βͳ͍ͱ͍͏੠͕ଟ͍ݱঢ়Λ἞ΜͰগ͠Ͱ΋ͦ͏͍͏੠ΛݮΒ͢ͱ͍͏ ಈػ΋͋Δ. ଞʹ΋ LaTeX ͷײ֮ΛऔΓ໭͢͜ͱ΍ࣗ෼ͷࢥߟͷ੔ཧ, ݱঢ় ͷࣗ෼ͷཧղ౓ͷ೺Ѳ΋ಈػʹ͋Δ. ͳ͓, ࣮ࡍʹ੢Ӝത1ઌੜ͕ൃද͞Εͨ ಺༰ (࿦จ΍ͦͷଞެࣜൃදͳͲ) Λ΋ͱʹิ଍ͳͲΛೖΕͳ͕Βड़΂͍ͯ͘ ͕, ඞͣ͠΋੢ӜઌੜΒͷҙΛ͢΂ͯ἞ΈऔΕ͍ͯΔΘ͚Ͱ͸ͳ͍͜ͱΛ༧ Ί஫ҙ͓ͯ͘͠. ༻͍ΒΕΔख๏2ͱͯ͠͸, ౷ܭֶʢBayes ౷ܭ΍࣌ܥྻ෼ ੳɾۭؒ౷ܭɾҼՌਪ࿦ɾଟมྔղੳɾੜଘ࣌ؒղੳͳͲ΋ؚΉʣ ɾ֬཰աఔɾ ඍ෼ํఔࣜ (ৗඍ෼ํఔࣜ, ภඍ෼ํఔࣜ, ֬཰ඍ෼ํఔࣜ)ɾྗֶܥɾػցֶ शͳͲଞʹ΋ଟذʹΘͨΓ਺ཧਓޱֶͱͷ༥߹΋Α͘ߦΘΕΔ. ຊߘͰ͸ओ ʹઢܗ୅਺3ͷॳา΍౷ܭֶ͸͋Δఔ౓ཧղ͕͋Δ͜ͱΛલఏʹͨ͠.4 1๺ւಓେֶେֶӃҩֶݚڀӃ ࣾձҩֶ෼໺ Ӵੜֶڭࣨ 2༻͍Δϓϩάϥϛϯάݴޠͱͯ͠͸ R ΍ Python, Julia ͳͲ͕͋Γ MCMC ༻ʹ Stan ΍ Julia ͳͲ͕༻͍ΒΕΔ. 3۩ମతʹ͸, ݻ༗஋΍ϠίϏΞϯ͕͋Δఔ౓ཧղͰ͖͍ͯΔ͜ͱΛ૝ఆͨ͠ 4͔ۭؒ͠͠౷ܭֶ͸ݱঢ়ࣗ෼͸ษڧͨ͜͠ͱ͕ͳ͘શ͘Կ΋Θ͔Βͳ͍ͷͰѻΘͳ͍ (=ѻ ͑ͳ͍). 3
  5. ୈ2ষ ং࿦ -Basic/Effective reproduction number- ͜ͷষͰ͸ཧ࿦Ӹֶʹ͓͚Δ࠷΋جຊతͱ͞ΕΔϞσϧΛ༻͍ͯجຊ࠶ ੜ࢈਺ Basic reproduction number

    ΍࣮ޮ࠶ੜ࢈਺ Effective reproduction number ͷಋग़Λߦ͏. ͯ͞ײછ঱ͷ਺ཧϞσϧͱ͸, SIR model ͱ͍͏έϧϚοΫ-ϚοέϯυϦο Ϋ͕ఏҊͨ͠ҎԼͷΑ͏ͳৗඍ෼ํఔࣜܥΛجૅͱ͠, ϞσϧΛ༻͍ͯ෼ੳ Λߦ͏ཧ࿦Ӹֶͷओཁͳख๏Ͱ͋Δ.                dS dt = −βSI dI dt = βSI − γI dR dt = γI (2.1) S I R β γ ͜͜Ͱ, β ͸ײછ཰, γ ͸ִ཭཰1Λද͠, S : Susceptible I : Infected R : Removed Ͱ͋Γ, S(t) + I(t) + R(t) = N0 Ͱ͋Γ, S(0) = N0 ͱ͢Δ. ͯ͞, ײ છ͕֦େ͢Δ͔Ͳ͏͔͸ͲͷΑ͏ʹ൑அ͢Ε͹͍͍ͩΖ͏͔. ͦΕ͸ୈ̎ࣜ ʹ஫໨͢Ε͹ྑ͍. dI dt = βSI − γI = 0 (∗) ͜ͷͱ͖, ɹײછ঱ͷ఻೻͸ఆৗঢ়ଶͱͳΔ. (*) Λ੔ཧ͢Δͱ࣍ͷ௨Γ. 1ճ෮ͱִ཭͞ΕΔׂ߹Λද͢. ೔ຊޠ͚ͩݟΔͱগࠞ͠ཚ͢Δ͔΋͠Ε·ͤΜͶ. R ͕ Recover Ͱ͸ͳ͘ Removed ͳ͜ͱͱ߹ΘͤΔͱཧղ͕ਐΉ͔΋͠Εͳ͍. 4
  6. βSI − γI = I(βS − γ) = 0 ⇄

    βS γ − 1 = 0 (∵ I ̸= 0) ⇄ βS γ = 1 (2.2) ͜ͷ βS γ ʹ͍ͭͯ, t=0 ͷͱ͖ βN0 γ ͱදͤ͜ΕΒΛͦΕͧΕ࣍ͷΑ͏ʹஔ͘. R0 = βN0 γ = βND (D = 1 γ ) Re = βS γ (2.3) ͜ͷͱ͖, R0 1Λجຊ࠶ੜ࢈਺ (Basic Reproduction Number) ͱ͍͍,t=0, ͭ·Γ໔ӸΛ࣋ͨͳ͍ूஂʹ͓͍ͯײછऀ਺͕ 0 ਓͷͱ͖ʹײછऀ͕ 1 ਓੜ ͨ͡ͱ͖ͦͷਓʹΑͬͯೋ࣍ײછ͢Δฏۉͷਓ਺Λද͢. ैͬͯ, R0 < 1 ͷ ͱ͖ײછ͸֦େͤͣ R0 > 1 ͷͱ͖͸ײછ͕֦େ͢Δͱߟ͑Δ͜ͱ͕Ͱ͖Δ. ·ͨ Re Λ࣮ޮ࠶ੜ࢈਺ (Effective Reproduction Number) ͱ͍͍, ೚ҙͷ̓ ʹ͓͍ͯ, ͭ·Γ͋Δ࣌఺Ͱ̍ਓͷײછऀʹΑͬͯೋ࣍ײછ͢Δฏۉͷਓ਺Λ ද͢. ैͬͯ, Re < 1 ͷͱ͖ײછ͸֦େͤͣ Re > 1 ͷͱ͖͸ײછ͕֦େ͢Δ ͱߟ͑Δ͜ͱ͕Ͱ͖Δ. R0 ͸͋ΔϞσϧʹ͓͍ͯॳظ৚݅ʹґଘ͢ΔͨΊਓ ޱີ౓΍஍ҬੑʹΑΓมಈ͢Δ. Re ͸ײडੑਓޱʹ͋Δਓʑͷߦಈม༰ (ख ચ͍͏͕͍΍ϚεΫΛ͢Δ͜ͱ, ਓͱͷ઀৮ΛݮΒ͢ͳͲ) ΍໔ӸΛ࣋ͭ͜ͱ ʹΑͬͯ௿Լͤ͞Δ͜ͱ͕Ͱ͖Δ. ࣮ޮ࠶ੜ࢈਺ͷ஋ʹΑͬͯײછ঱Ӹֶʹ ͓͚Δײછ఻೻ͷఔ౓Λ೺Ѳ͢Δ. ͳ͓, D ͸ฏۉײછظؒΛද͢. Ҏ্Ͱ঺հͨ͠ϞσϧҎ֎ʹ΋ SEIR model ΍ SIS model, SIRS model ͳ ͲίϯύʔτϝϯτϞσϧ͸༷ʑଘࡏ͠, ߟ͍͑ͨײછ঱ͷੑ࣭ʹ߹Θͤͯ ϞσϧΛߏங͢Δ. ࠓճྫࣔͨ͠ SIR model ͸ Deterministic model Ͱ͋Δ ͕, ΄͔ʹ΋ Stochastic model ΍ Statistical model, ·ͨ Network theory Λ ར༻ͨ͠ϞσϧͳͲ͍Ζ͍Ζͳ֯౓͔Βߟ͍͑ͯ͘.23 1R θϩͰ͸ͳ͘, R naught ͱಡΉ. 2ͳ͓, ຊߘ͸ײછ঱਺ཧϞσϧͷೖ໳ͱͯ͠ॻ͔Εͨ΋ͷͰ͸ͳ͍ͨΊ͜ΕҎ্ͷෆཁͳ঺ հ͸ճආ͢Δ. 3SIR model ΋·ͩ·ͩࢁఔࢥߟ͢Δ༨஍͸͋Δ͠, ਖ਼௚΋ͬͱ৭ʑݴ͍͍͚ͨͲ࢒೦... 5
  7. ୈ3ষ Time Intervalͷਪఆ ໨త   ͜ͷষͰ͸, ੢ӜઌੜͷνʔϜ͕ Jounal of

    Clinical Medicine ʹग़ͨ͠ ࿦จ Incubation Period and Other Epidemiological Characteristics of 2019 Novel Coronavirus Infections with Right Truncation: A Statistical Analysis of Publicly Available Case Data(J. Clin. Med. 2020, 9(2), 538; https://doi.org/10.3390/jcm9020538) ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔ΛഎܠΛ֬ೝͨ͠ޙʹղઆ ͨ͠.   3.1 ४උ 3.1.1 Doubly interval-censoring ײછ೔͕ෆ໌ͳ৔߹, ௥੻ௐࠪͱ͍ͬͨϑΟʔϧυݚڀͰ͸જ෬ظؒΛਪ ఆ͢Δ͜ͱ͕Ͱ͖ͳ͍. ͜ͷ৔߹ʹ, interval-censoring Λߦ͏. ͯ͞, ͜͜Ͱ͸͋Δײછऀ͕ະײછऀʹ͏ͭ͢ͱ͖Λߟ͑Δ. ͜ͷͱ͖, ײ છ೔ times of exposure ΋ൃ঱೔ illness onset ΋ෆ໌Ͱ͋Δ৔߹Λߟ͑ͯΈΑ ͏.1 感染⽇ 発症⽇ 治癒 潜伏期間τ EL ER SL SR e s 1࣮ࡍ, ෆݦੑײછऀ΍௥੻ͷݶքͳͲʹΑΓײછ೔͔ൃ঱೔ͷͲͪΒ͔, ͋Δ͍͸ͲͪΒ΋ ͕ෆ໌Ͱ͋Δ͜ͱ΋Α͋͘Δ. 6
  8. ͜͜Ͱ, EL ͸ະײછऀͷൃ঱೔ͱͯ͠ߟ͑ΒΕΔதͰ࠷΋ૣ͍೔, ER ͸ ࠷΋஗͍೔ͱ͢Δ. SL ͸ײછ೔ͱͯ͠ߟ͑ΒΕΔதͰ࠷΋ૣ͍೔, SR ͸࠷΋

    ஗͍೔ͱ͢Δ. ·ͨજ෬ظؒʹؔͯ͠͸ τ = s − e ͱදͤΔ. ͜͜Ͱ, ҎԼͷؔ਺Λఆٛ͢Δ. f(τ) : જ෬ظ͕ؒτͱͳΔ֬཰ີ౓ؔ਺ g(t) : t ࣌఺ʹ͓͍ͯײછ͢Δ֬཰ (ײછऀ͕ະײછऀʹײછͤ͞Δ֬཰) h(t) : ̓࣌఺ʹ͓͍ͯൃ঱͢Δ֬཰ ͕ͨͬͯ͠, p(e, s) = p(e)p(s|e) = g(e)h(s|e) = g(e)f(s − e) Ͱ͋ΔͷͰ໬ ౓ likelihoodʢύϥϝλ͸ Θf , Θg , Θh ) ͸࣍ͷ௨Γ.   L(Θf , Θg , Θh |D) = ∏ i ∫ ER,i EL,i ∫ SR,i SL,i g(e)f(s − e)dsde   ͜ΕΑΓ, ࠷໬ਪఆ maximum likelihood estimation(MLE) ʹΑͬͯײછ ೔ͱൃ঱೔ͷ࠷໬ਪఆ஋ estimated maximum likelihood ͕ಘΒΕΔͷͰજ ෬ظ͕ؒΘ͔Δ. ͜ͷख๏Λ doubly interval-censoring ͱ͍͏. 3.2 ࣮ࡍʹߦΘΕͨղੳ ຊߘͷझࢫʹΑΓ, ݁࿦ʹ͍ͭͯΑΓ݁࿦ͷಋ͖ํ (Methods) Λத৺ʹղઆ ͢Δ. ͜ͷ࿦จͰ͸, જ෬ظؒ΍ൃ঱͔ΒೖӃ·Ͱͷظؒ, ൃ঱͔Βࢮ๢·Ͱ ͷظؒ, ೖӃ͔Βࢮ๢·Ͱͷظؒͱ͍͏ time interval ͷਪఆʹ͍ͭͯ doubly interval-censored likelihood Λ࢖ͬͨख๏ͱ Bayesian framework Λ༻͍ͨख ๏ͷ྆ํͰߦΘΕͨ. 3.2.1 Estimation of the time interval distribution using doubly interval-censored likelihood doubly interval-censored likelihood ͸ҎԼͷ௨Γ. L(Θ|D) = ∏ i ∫ ER,i EL,i ∫ SR,i SL,i g(e)f(s − e)dsde 7
  9. ͜͜Ͱ, f(t) Λͦͷ··࠾༻͢Δͱબ୒όΠΞε1͕͔͔ΔͨΊ, ӈଆ੾அ right truncation Λߟྀͯ͠ҎԼͷࣜ2Λ༻͍ͨ. f′(s − e,

    e) = f(s − e) ∫ T −e 0 re−ru 1 − e−ru F(T − e − u)du ͜͜Ͱ, r ͸ࢦ਺ؔ਺త૿Ճ཰Ͱ F(·) ͸ f(·) ͷྦྷੵ෼෍ؔ਺. T ͸࠷৽ͷ ؍ଌ࣌ࠁ3Λࣔ͢. ͯ͞, ैͬͯ໬౓͸ҎԼͷΑ͏ʹͳΔ. ·ͨ͜ͷͱ͖ΠϯςάϥϧҎԼΛ ιi ͱͯ࣍͠ͷΑ͏ʹද͢. L′(Θ|D) = ∏ i ∫ ER,i EL,i ∫ SR,i SL,i g(e)f′(s − e, e)dsde ιi = ∫ ER,i EL,i ∫ SR,i SL,i g(e)f′(s − e, e)dsde (3.1) SL,i > ER,i ͷͱ͖, s′ = s − e ͱஔ׵ͯ࣍͠ͷΑ͏ʹมܗͰ͖Δ. ιi = ∫ ER,i EL,i de g(e) ∫ SR,i−e SL,i−e f′(s′, e)ds′ = ∫ ER,i EL,i g(e) {F′(SR,i − e, e) − F′(SL,i − e, e)} de (3.2) ER,i > SL,i > EL,i Ͱ͋Δͱ͖, ҎԼͷΑ͏ʹͳΔ. ιi = ∫ Sl,i EL,i g(e){F′(SR,i − e, e) − F′(SL,i − e, e)}de + ∫ ER,i SL,i g(e)F′(SR,i − e, e)de (3.3) ࠷ޙʹ, EL,i > SL,i Ͱ͋Δͱ͖, ҎԼͷΑ͏ʹͳΔ. ιi = ∫ ER,i EL,i g(e)F′(SR,i − e, e)de (3.4) ͜ͷΑ͏ʹ৔߹෼͚Λͯ͠ time interval ΛͦΕͧΕٻΊͨ.4 f(s − e) ͸Ψ ϯϚ෼෍, ର਺ਖ਼ن෼෍, ϫΠϒϧ෼෍Λߟ͑ͨ. 1ੜଘ࣌ؒղੳͷ࿩Ͱ͢. ଧͪ੾Γ censoring ͱ੾அ truncation ͕ߟྀ͞ΕΔ΂͖ิਖ਼Ͱ ͢. ࠓճͷ right truncation ͸જ෬ظ͕ؒ௕͍৔߹ʹΤϯυϙΠϯτͷઃఆʹΑΓ؍ଌ͞Εͣ, population of interest ͱ sampled population ͕ҟͳͬͯ͠·͍ selection bias ͕ൃੜ͢Δঢ় ଶΛ͍͏. Inverse probability weighting methods for Cox regression with rightʖtruncated data (https://doi.org/10.1111/biom.13162). 2Selection bias ͷิਖ਼ͷ࢓ํ͸Α͘Θ͔Βͳ͍Ͱ͢... 3͜ͷͱ͖͸ 2020 ೥ 1 ݄ 31 ೔. 4࣮ࡍͷσʔλΛ༻͍ͨܭࢉ͸ R Λ࢖༻ͨ͠. 8
  10. 3.2.2 Estimation of the time interval distribution using Bayesian framework

    Bayes ਪఆʹΑΓ time interval distribution ΛٻΊΔ. ࣄલ෼෍͸ର਺ਖ਼ن ෼෍, ΨϯϚ෼෍, ϫΠϒϧ෼෍Λߟ͑Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏ (MCMC)1 ʹΑͬͯϞϯςΧϧϩੵ෼͔Βඇ੾அର਺໬౓ʹର͢Δࣄޙ༧ଌ෼෍ (time interval distribution) Λਪఆ͢Δ. ͳ͓, ର਺ਖ਼ن෼෍ͱϫΠϒϧ෼෍͸Ͳͪ Β΋ඪ४ਖ਼ن෼෍ʹै͏. ΨϯϚ෼෍͸ shape parameter ͸ฏۉ 3, ඪ४ภ ࠩ 5 ͷਖ਼ن෼෍ʹै͍, inverse scale parameter ͸, location parameter Λ 0, scale parameter Λ 5.0 ʹͱΔίʔγʔ෼෍ͱ͠֊૚ϕΠζΛߟ͑ͨ.2ͳ͓, ࣄ લ෼෍ͷબ୒͸ Stan developer community ʹΑΔਪ঑3ʹैͬͨ. ·ͨ, times of exposure ͱ illness onset ͷࣄલ෼෍͸ҎԼͷΑ͏ʹఆࣜԽ ͨ͠. ei = EL,i + (ER,i − EL,i )˜ ei si = ˆ SL,i + (SR,i − ˆ SL,i )˜ si (3.5) ͜ͷͱ͖, ei > SL,i ͳΒ͹ ˆ SL,i = ei , ei = SL,i ͳΒ͹ ˆ SL,i = SL,i Ͱ͋Δ. ei < SL,i ͷͱ͖͸ ˜ ei ͱ˜ si ͸࣍ͷ෼෍ʹै͏. ˜ ei ∼ nomal(mean = 0.5, SD = 0.5), ˜ si ∼ nomal(mean = 0.5, SD = 0.5) ͜ͷͱ͖ Stan ʹΑΔඇ੾அର਺໬౓Λ࣍ͷΑ͏ʹఆٛͨ͠. logL(Θ) = ∑ i distribution lpdf(si − ei |Θ) (3.6) ͜͜Ͱ, lpdf ͸֬཰ີ౓ؔ਺ͷର਺Ͱ͋Γ, lognomal lpdf ͔ gamma lpdf ͔ weibull lpdf ͷ͍ͣΕ͔Λද͢. ·ͨ, જ෬ظؒʹؔͯ͠͸ଧͪ੾Γ໬౓Λ ߟ͑ͨ. MCMC ͸, the simulation phase ͱͯ͠ 5000 ճΛ burn in ͯ͠ the tuning phase Ͱ 10000 ճΛ࠾༻͠, ͦΕΛ̐ܥྻ෼ߦͬͨ. Ҏ্͔Βࣄޙ༧ଌ෼෍Λਪఆ֤͠ϞσϧΛ WAIC ͰධՁ͠࠷దͳϞσϧΛ બ୒ͨ͠. ͳ͓, WAIC ͸ҎԼͷ௨Γ. WAIC = −2(lppd − pW AIC ) (3.7) 1No U-Turn Sampler(NUTS) Λ࣮૷ͨ͠ Stan ʹΑΓཚ਺ੜ੒͸ߦΘΕͨ. 2Ұൠ࿦ͱͯ͠֊૚ϕΠζ͸࠷໬๏͕೉͘͠ AIC ͰධՁͣ͠Β͍ͷͰ, MCMC Λ༻͍ͯ WAIC Λߟ͑Δࣄ͕ଟ͍. 3Stan developer team. Prior choice recommendations. https://github.com/stan- dev/stan/wiki/Prior-Choice-Recommendations. Assessed 6 February 2020. 9
  11. ͜͜Ͱ, lppd = ∑ i log Pr(yi ), pW AIC

    = ∑ i V (yi ) (3.8) Ͱ͋Δ. ͳ͓, Pr(·) ͓Αͼ V (·) ͸ର਺໬౓ͷฏۉͱ෼ࢄΛͦΕͧΕද͢. 3.2.3 ·ͱΊ Ҏ্ͷ̎௨Γͷਪఆ๏͔Β time interval ͸ਪఆ͞Εͨ. ࣮ࡍʹ࿦จ͕ൃ ද͞Εͨͱ͖ʹಘΒΕͨৄ͍݁͠Ռ౳ͷߟ࡯ʹ͍ͭͯ͸, ࿦จͷ Results ΍ Discussion ΛಡΜͰ΋Β͍͍ͨ. ҎԼʹ࿦จͷ Abstract ͷΈࡌ͓ͤͯ͘.1 Abstract   The geographic spread of 2019 novel coronavirus (COVID-19) infections from the epicenter of Wuhan, China, has provided an opportunity to study the natural history of the recently emerged virus. Using publicly available event-date data from the ongoing epidemic, the present study investigated the incubation period and other time intervals that gov- ern the epidemiological dynamics of COVID-19 infections. Our results show that the incubation period falls within the range of 2–14 days with 95% confidence and has a mean of around 5 days when approxi- mated using the best-fit lognormal distribution. The mean time from illness onset to hospital admission (for treatment and/or isolation) was estimated at 3–4 days without truncation and at 5–9 days when right truncated. Based on the 95th percentile estimate of the incubation pe- riod, we recommend that the length of quarantine should be at least 14 days. The median time delay of 13 days from illness onset to death (17 days with right truncation) should be considered when estimating the COVID-19 case fatality risk.   1Bayes ਪఆʹؔͯ͠͸͋Δఔ౓ཧղͷ͋Δ͜ͱΛ૝ఆͯ͠ॻ͖·ͨ͠... Ͱͳ͍ͱ 3 ষ͚ͩ ͰͱΜͰ΋ͳ͍ྔॻ͔ͳ͍ͱ͍͚ͳ͍ͷͰ... ͢Έ·ͤΜ... 10
  12. ୈ4ষ ྲྀߦॳظஈ֊ͰͷcCFRͱ R0 ͷਪఆ ໨త   ͜ͷষͰ͸, ੢ӜઌੜͷνʔϜ͕ Jounal

    of Clinical Medicine ʹग़ͨ͠ ࿦จ Real-Time Estimation of the Risk of Death from Novel Coronavirus (COVID-19) Infection: Inference Using Exported Cases (J. Clin. Med. 2020, 9(2), 523; https://doi.org/10.3390/jcm9020523) ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆ ͨ͠.   4.1 ४උ 4.1.1 ͱ 4.1.2 Ͱ͸, ͜ͷষͰղઆ͢Δ࿦จͷ४උͱͯ࣍͠ͷઌߦ࿦จͷ಺ ༰ʹ͍ͭͯҰ෦ղઆ͢Δ.   Nishiura, H.; Klinkenberg, D.; Roberts, M.; Heesterbeek, J.A. Early epidemiological assessment of the virulence of emerging infectious dis- eases: A case study of an influenza pandemic. PLoS ONE 2009, 4, e6852.   4.1.1 CFR ެऺӴੜ্ֶͷॏཁͳࢦඪͷ 1 ͭͱͯ͠, ࣬පͷॏ঱౓Λද͢ CFR ͕͋ Δ. CFR ͱ͸͋Δ࣬පʹΑΔࢮ๢਺Λ͋Δ࣬පͷጶױ਺Ͱׂͬͨ஋Ͱக໋ ཰ case-fatality rate ͷུͰ͋Δ. े෼ʹ௕͍؍࡯ظؒΛऔͬͨ৔߹͸ࢮ๢ ཰ death rate1Λጶױ཰ morbility2Ͱׂͬͨ஋ͱͳΔ. ཧ૝͸෼฼͕ײછऀͷ 1ጶױ਺Λةݥ๫࿐ਓޱ population at risk Ͱׂͬͨ஋. follow-up bias Λආ͚ΔͨΊʹ෼ ฼ͷ population at risk ͸ਓ-೥๏ person-year Λ༻͍Δ. 2ࢮ๢਺Λ population at risk Ͱׂͬͨ஋. 11
  13. ૯਺Ͱ͋Δ΂͖͕ͩ͢΂ͯͷײછऀΛ਺্͑͛Δ͜ͱ͸ෆՄೳͳͷͰ਍அ͞ Εͨ঱ྫͷΈΛࢦ͢͜ͱ͕ଟ͍. Outbreak ͷॳظஈ֊1Ͱ͸৘ใ͕֬ఆ਍அ ʹݶఆ͞ΕΔ͜ͱ͕ଟ͍. ͕ͨͬͯ͠, ࿦จ಺Ͱ͸֬ఆ਍அͷΈΛର৅ͱ͠ confirmed CFR(cCFR) ͱݺͿ.

    cCFR ͸฼ूஂͷײછ͕े෼ʹ֬ೝ͞Εͯ ͍ͳ͍ͨΊ, CFR ΛաେධՁ͢Δ܏޲ʹ͋Δ. ͔͠͠ COVID-19 ͷΑ͏ͳ ෆ࣮֬ੑͷߴ͍ঢ়گԼͰ͸ symptomatic CFR (sCFR) ͷ্ݶΛࣔ͢ࢦඪͱ ͯ͠ॏཁࢹ͞ΕΔ. ޿͘༻͍ΒΕ͍ͯΔ crude estimate of the cCFR ͸ t ࣌ ఺ʹ͓͚Δྦྷੵࢮ๢਺ͱͷൺͰٻΊΒΕΔ͕, ຆͲ͕աখධՁ͞Εͯภͬͨ cCFR Λ΋ͨΒ͢܏޲ʹ͋Δ.2 Ҏޙ, bias ʹݴٴ͢Δͱ͖ʹ͸ biased cCFR ͱ unbiased cCFR Λ༻͍Δ. CFR ͸̏ͭͷΞϓϩʔν͕ଘࡏ͢Δ. (1) ͋Δ࣌఺Ͱͷྦྷੵ঱ྫ਺ʹର͢Δྦྷੵࢮ๢਺ͷׂ߹ (2) clinical outcomeʢࢮ๢͔ճ෮) ͕Θ͔͍ͬͯΔױऀͷྦྷੵ਺ʹର͢Δྦྷ ੵࢮ๢਺ͷׂ߹ (3) onset ͔Β death ·Ͱͷ time delay Λௐ੔ͨ͠ྦྷੵ঱ྫ਺ʹର͢Δྦྷੵ ࢮ๢਺ͷׂ߹ 4.1.2 The factor of underestimation cCFR Λද͢౷ܭతࢦඪ͸ (1) bt : t ࣌఺ʹࢉग़͞Εͨ biased cCFR, (2) π: ٻΊ͍ͨ unbiased cCFR (3) pt : π ͷਪఆ஋Λ΋ͨΒ͢ϥϯμϜม਺Ͱ, ͋Δ ಛఆͷ outbreak Ͱͷ࣮ݱ஋ͱΈͳ͞ΕΔ ͷ̏ͭͰ͋Δ. t ࣌఺Ͱͷྦྷੵࢮ๢ ਺ͱྦྷੵ֬ఆ঱ྫ਺ΛͦΕͧΕ Dt ͱ Ct ͱ͢Δ. ͜ͷͱ͖, bt ͸࣍ͷ௨Γ. bt = Dt Ct (4.1) ͜Ε͕ unbiased cCFR ʹൺ΂ͯաখධՁͰ͋Δ͜ͱΛࣔ͢. t ೔໨ͷ৽ن֬ ఆ঱ྫ਺Λ ct , ࢮ๢͢Δͱ͖ͷ onset ͔Β death ·Ͱͷ࣌ؒʹ͍ͭͯͷ৚݅ ෇͖֬཰ີ౓ؔ਺ Λ fs ͱ͢Δͱ, Ct ͱ Dt ͸ҎԼͷΑ͏ʹදͤΔ. Ct = t ∑ i=0 ci Dt = pi t ∑ i=0 ∞ ∑ j=0 ci−j fj (4.2) ͜͜Ͱ, pt ͸ײછ঱Ͱࢮ๢͢Δ֬ఆ঱ྫͷ࣮ݱׂ߹͔ͭϥϯμϜม਺ͳͷ Ͱ͜Εͷਪఆ஋ʹภΓ͸ͳ͍. (4.1), (4.2) Λ΋ͱʹ bt ͸࣍ͷΑ͏ʹදͤΔ. 1͜ͷষͰղઆ͢Δ࿦จ͸ύϯσϛοΫલͰ͋Γ outbreak ॳظஈ֊Ͱ͋ͬͨ. 2ੜଘ࣌ؒղੳ্ͷ໰୊Ͱ͋Δ. ৄ͘͠͸࣍ͷ࿦จΛࢀর͞Ε͍ͨ. Methods for estimating the case fatality ratio for a novel, emerging infectious disease. (DOI: 10.1093/aje/kwi230) 12
  14. bt = pt t ∑ i=0 ∞ ∑ j=0 ci−j

    fj t ∑ i=0 ci (4.3) ͔͜͜ΒΘ͔ΔΑ͏ʹ, time delay Λද̎͢ॏ࿨Λ༻͍ͨ෼ࢠͷ஋͕෼฼Α Γখ͍ͨ͞Ί, bt ͸ pt ΑΓখ͘͞ͳΔ.1 ͜ΕΛ࣍ͷΑ͏ʹมܗ͢Δ. pt = bt t ∑ i=0 ci t ∑ i=0 ∞ ∑ j=0 ci−j fj (4.4) pt ͸ϥϯμϜม਺Ͱ͋Δ͕, (4.4) ΑΓྦྷੵࢮ๢਺ Dt , ৽ن֬ఆ঱ྫ਺ ct , onset ͔Β death ·Ͱͷ࣌ؒ෼෍ fs ͱ͍͏̏ͭͷ৘ใ͔Βਪఆ͞ΕΔͱ͍͑Δ. ut = t ∑ i=0 ∞ ∑ j=0 ci−j fj t ∑ i=0 ci (4.5) ͱ͓͘ͱ, ut ͸ the factor of underestimation2ͱͳΔ. ͕ͨͬͯ͠, pt = bt ut Ͱ͋Δ. ͜͜·Ͱ͸, ཭ࢄ࣌ؒʹ͍ͭͯٞ࿦͖͕ͯͨ͠࿈ଓ࣌ؒʹ͍ͭͯߟ͑ͯΈ Α͏. ྲྀߦͷॳظஈ֊Ͱ growth rate ͕ r ͷࢦ਺ؔ਺తͳ૿ՃΛೝΊΒΕΔͱ ͖৽ن֬ఆ঱ྫ਺ͷظ଴஋͸ҎԼͷ௨Γ. E(ct ) = c0 ert (4.6) ͳ͓, r ͸಺ࡏతͳೋ࣍ײછ͚ͩͰͳ͘ impoted ͳײછ΋ؚ·ΕΔ. (4.6) Λ ༻͍ͯ, the factor of underestimation ͸࣍ͷΑ͏ʹͳΔ. u = ∫ t 0 erτ ∫ ∞ 0 e−rsf(s)dsdτ ∫ t 0 erτ dτ = ∫ ∞ 0 e−rsf(s)ds (4.7) 1right censoring ͷ࿩. ঱ྫ֬ೝ͞Ε t ࣌఺Ҏ߱ʹࢮ๢͢ΔՄೳੑ͕͋Δ. 2աখධՁ܎਺ (?) ͷΑ͏ͳ༁ޠ͕͍͍͔΋. ༁ޠ͕͋Δͷ͔͸ෆ໌. 13
  15. 4.1.3 R0 of an infection from the growth rate of

    an outbreak (epidemic) 4.1.2 Ͱ, ʮྲྀߦͷॳظஈ֊ʹ͓͍ͯ growth rate ͕ r ͷࢦ਺ؔ਺తͳ૿Ճʯ ΛԾఆͯٞ͠࿦ΛਐΊ͕ͨ͜Εͷ৴ጪੑΛ͔֬ΊͯΈΑ͏. ̎ষͰઆ໌ͨ͠ ྲྀೖྲྀग़ͷͳ͍࠷΋جຊతͳ SIR model ʹ͍ͭͯߟ͑Δ. ྲྀߦॳظʹ͓͍ͯ ͸ S(t) ≈ N0 ͱΈͳ͢͜ͱ͕Ͱ͖ΔͷͰҎԼͷ͕ࣜߟ͑ΒΕΔ. dI dt ≈ βN0 I − γI ≈ (βN0 − γ)I (4.8) ͕ͨͬͯ͠, (βN0 − γ) ͸ఆ਺ͳͷͰྲྀߦॳظʹ͓͍ͯ͸ࢦ਺ؔ਺૿Ճ͢Δͱ ͯ͠Α͍. ͯ͜͞ͷͱ͖, βN0 − γ = Λ ͱ͢Δͱ, R0 = βN0 γ = 1 + Λ γ = 1 + ΛD (4.9) ໌֬ʹ pre-infectious and/or infectious periods ͕Θ͔Βͳ͍ͱ͖, લऀΛ D, ޙऀΛ D′ ͱ͢Δ1ͱ serial interval Ts = D + D′ ʹࢦ਺ؔ਺૿ՃΛ͢Δͱߟ ͑ R0 ͸࣍ͷΑ͏ʹදͤΔ. R0 = 1 + ΛTs (4.10) 4.2 ࣮ࡍʹߦΘΕͨղੳ cCFR ͷਪఆʹ͍ͭͯ͸, ࣌ؒ஗ΕΛߟྀͨ͠ real-time Ͱͷਪఆ͕ߦΘ Εͨ. 4.2.1 Estimation of the delay distribution t ࣌఺Ͱͷጶױ਺2Λ i(t) = i0 ert ͱ͢Δͱ, ྦྷੵጶױ਺͸ I(t) = ∫ t 0 i(s) ds = i0 (ert − 1) r (4.11) ͱද͞ΕΔ. ൃ঱ onset ͔Βࢮ๢ death ͷ࣌ؒ஗Ԇ෼෍͸ 3 ষͷ݁ՌΑΓର ਺ਖ਼ن෼෍Λબ୒ͨ͠.3 f(t; θ) Λ parameter θd = {ad , bd } ͷର਺ਖ਼ن෼෍ͱ ͢Δͱ, the factor of underestimation ͸ҎԼͷ௨Γ. u(r, θd ) = ∫ ∞ 0 e−rsf(s; θd )ds (4.12) 1SEIR model ʹ͓͚Δ I ͷظؒΛ D, E ͷظؒΛ D′. 24.1.2 ʹ͓͚Δ E(c(ct)) ͱಉ͡. 3ର਺ਖ਼ن෼෍, ΨϯϚ෼෍, ϫΠϒϧ෼෍Λ̎छྨͷํ๏Ͱݕ౼͠·ͨ͠ΑͶ. 14
  16. ࣌ؒ஗ΕΛௐ੔͞Εͨྦྷੵጶױ਺͸ u(r, θd )I(t) ʹै͏. t ೔·Ͱʹใࠂ͞Ε ͨྦྷੵࢮ๢਺ D(t) ͸ҎԼͷೋ߲෼෍ʹै͏.

    D(t) ∼ binom(size = u(r, θd )I(t), prob = CFR(t)) (4.13) ae ͱ be ΛͦΕͧΕ, exported cases ͷ ill onset ͔Βใࠂ·Ͱͷ time interval distribution Λ modeling ͨ͠ΨϯϚ෼෍ͷܗঢ়ͱ inverse scale1ͱ͢Δ. ͜ ͷͱ͖, the factor of underestimatoin Λ ˜ u(r, θe = {ae , be }) ͱ͢Δͱ the number of expoted cases E(t) ͸ҎԼͷ௨Γ. E(t) ∼ binom(size = ˜ u(r, θe )I(t), prob = p) (4.14) ͜͜Ͱ p ͸΢Πϧεͷݕग़࣌ؒ the detection time window of virus (i.e. incubation + infectious periods) T = 12.5 ೔ؒ Ͱதࠃ͔Βͷཱྀߦऀͷத͔ Β෢׽͔ΒͷཱྀߦऀΛݟ͚ͭΔ֬཰Ͱ͋Δ. M = 5.56 million ͸தࠃ͔Βͷ ೥ཱྀؒߦऀ਺, ϕ = 0.021% ͸෢׽͔Βͷཱྀߦऀͷׂ߹, n = 11million ͸෢׽ ͷਓޱͱ͢Δͱ p ͸ҎԼͷΑ͏ʹܭࢉͰ͖Δ. p = M × T 365 × ϕ n = ϕMT 365n ≈ 0.0036 (4.15) 4.2.2 Statistical inference Expoted cases ʹ͓͚Δ ill onset ͔Βใࠂ·Ͱͷ the delay distribution Λ ∆Te,a ͱ͢Δͱର਺໬౓͸ҎԼͷ௨Γ. logLe (θe |∆Te,a ) = ∑ k log(gamma(∆Te,k |shape = ae , scale = be )) (4.16) ࠷໬๏͔Β best-fit parameters {ae , be } ΛಘΔ. The number of expoted cases ͱྦྷੵࢮ๢਺ͷର਺໬౓͸ͦΕͧΕҎԼͷ ௨Γ. logLE ({r, i0 }|{E(t), te ≤ t ≤ T}) = T ∑ t=te log ( binom(E(t)|size = ˜ u(r, ˜ θe I(t), prob = p) ) (4.17) logLE ({r, i0 , CFR(t)}|{D(t), te ≤ t ≤ T}) = T ∑ t=td log ( binom(D(t)|size = u(r, ˜ θe I(t), prob = CFR(t)) ) (4.18) 1scale Λ θ ͱ͢Δͱ inverse scale λ = 1 θ Λ༻͍ͯΨϯϚ෼෍͸ f(x) = λk Γ(k) xk−1e−λ. 15
  17. ͨͩ͠, te ͱ td ͸ͦΕͧΕ࠷ॳʹ expoted case ͕ى͖Δ೔ͱ death ͕ى͖Δ

    ೔Λද͢. (4.14), (4.15) Λ͋Θͤͨର਺໬౓͸࣍ͷ௨Γ. logLΣ (θΣ = {r, i0 , CFR(t)}|D(t), E(t)) = logLE ({r, i0 }|E(t)) + logLD ({r, i0 , CFR(t)}|D(t)) (4.19) ͜ΕΑΓ࠷໬๏͔Β best-fit parameters {r, i0 , CFR(t)} ΛಘΔ. ͜͜Ͱ, 2 ͭͷγφϦΦΛߟ͑ͨ. ·ͣ̍ͭ໨ʹ͍ͭͯ. ॳΊͯ COVID-19 ͷൃ঱Λ֬ೝͨ͠ 12 ݄ 8 ೔ʹ i0 Λ̍ʹݻఆͯ͠ྦྷੵጶױ཰ͷى఺Λͦͷ೔ ʹ͓͍ͯߟ͑Δ৔߹, ى఺ͱͨ͠ 12 ݄ 8 ೔͕ॳΊͯͷ঱ྫ͔Ͳ͏͔ʹؔͯ͠ ͸ෆ࣮֬ੑ͕࢒Δ. ͕ͨͬͯ͠ى఺Λ 12 ݄̍೔͔Β 12 ݄ 10 ೔·ͰมԽ͞ ͤΔײ౓෼ੳ sensitivity analysis Λߦͬͨ. ࣍ʹ 2 ͭ໨ͷγφϦΦʹ͍ͭͯ. ॳΊͯ expoted case ͕֬ೝ͞Εͨ 2020 ೥ 1 ݄ 13 ೔Λى఺ͱͯ͠ܭࢉΛ͠ ͨ. ྆γφϦΦͱ΋, cut-off times, the detection window of virus T, ෢׽ࠃ ࡍۭߓͷ catchment population n ͷେ͖͞Λม͑ͯ growth rate, ྦྷੵൃੜ ਺, cCFR Λਪఆ͢Δ sensitivity analysis Λߦͬͨ. ͜ͷ݁ՌΛ༻͍ͯ R0 ΛٻΊΔ. R0 ͸ҎԼͷ௨Γ. ͨͩ͠ S ͸ฏۉͷ serial interval Λද͢. R0 = 1 + rS (4.20) ͜ͷݚڀͰ͸, MCMC ͕༻͍ΒΕͨ.1 Tuning phase ͱͯ͠ 2500 ճ෼Λ burn in ͯ͠ 4250 ճ෼Λ࠾༻ͨ͠. ͜ΕΛ 8 ܥྻಘͨ. ͳ͓, ˜ u(r, θe )I(t) · ͨ͸ u(r, θd )I(t) Λ཭ࢄతͳೋ߲෼෍͔Β࿈ଓۙࣅΛಘͯ modeling ͨ͠.2 · ͨ, sequential fitting Λߦͬͨ. ͜Ε͸·ͣ໬౓ Le ͚ͩΛߟ͑, ࣍ʹͦͷͱ ͖ಘΒΕͨਪఆύϥϝʔλͷฏۉΛ༻͍ͯ໬౓ LΣ Λ fitting ͤͨ͞. ࠷ޙ ʹ, ϓϩϑΝΠϧ໬౓Λ༻͍ͯಋग़͞Εͨ৴པ۠ؒ3ͱ߹Θͤͯ, ࠷໬ਪఆΑ Γ pointwise estimates Λܭࢉ͢Δ͜ͱͰ, fitting ͨ͠ਪఆ஋Λݕূͨ͠. ͦ ͷ݁Ռ, ݁Ռ͕Ұக͍ͯ͠Δ͜ͱ͕Θ͔ͬͨ. 4.2.3 ·ͱΊ ࠷໬๏ʹΑͬͯ, ྲྀߦॳظஈ֊Ͱ cCFR ͳͲͷύϥϝλΛਪఆͦ͠ͷ݁Ռ Λ༻͍ͯ R0 Λਪఆͨ͠. ࣮ࡍʹ࿦จ͕ग़͞Εͨͱ͖ʹಘΒΕͨৄ͍݁͠Ռ ౳ͷߟ࡯ʹ͍ͭͯ͸, ࿦จͷ Results ΍ Discussion ΛಡΜͰ΋Β͍͍ͨ. ࣍ ͷϖʔδʹ࿦จͷ Abstract ͷΈࡌ͓ͤͯ͘. 1Python PyMC3 package ͷ No-U-Turn Λ࢖༻. 2ΨϯϚ෼෍Λར༻ͯ̎ͭ͠ͷϞʔϝϯτΛҰகͤͯ͞࿈ଓԽͨ͠. 3Delta ๏ͱϓϩϑΝΠϧ໬౓Λ༻͍Δํ๏͕໬౓͔Β৴པ۠ؒΛಘΔํ๏. ࠓճ͸ৄࡉ͸શ ͯল͍ͨ. ͳ͓, ৴པ۠ؒͷਪఆ͸ bootstrap ๏ͳͲ΋༻͍ΒΕΔ. 16
  18. Abstract   The exported cases of 2019 novel coronavirus

    (COVID-19) infection that were confirmed outside of China provide an opportunity to esti- mate the cumulative incidence and confirmed case fatality risk (cCFR) in mainland China. Knowledge of the cCFR is critical to character- ize the severity and understand the pandemic potential of COVID-19 in the early stage of the epidemic. Using the exponential growth rate of the incidence, the present study statistically estimated the cCFR and the basic reproduction numberʕthe average number of secondary cases generated by a single primary case in a naive population. We modeled epidemic growth either from a single index case with illness onset on 8 December, 2019 (Scenario 1), or using the growth rate fitted along with the other parameters (Scenario 2) based on data from 20 exported cases reported by 24 January, 2020. The cumulative incidence in China by 24 January was estimated at 6924 cases (95% CI: 4885, 9211) and 19,289 cases (95% CI: 10,901, 30,158), respectively. The latest estimated values of the cCFR were 5.3% (95% CI: 3.5%, 7.5%) for Scenario 1 and 8.4% (95% CI: 5.3%, 12.3%) for Scenario 2. The basic reproduction number was estimated to be 2.1 (95% CI: 2.0, 2.2) and 3.2 (95% CI: 2.7, 3.7) for Scenarios 1 and 2, respectively. Based on these results, we argued that the current COVID-19 epidemic has a substantial potential for causing a pandemic. The proposed approach provides insights in early risk assessment using publicly available data.   17
  19. ୈ5ষ LockdownͷఆྔతධՁ ໨త   ͜ͷষͰ͸, ੢ӜઌੜͷνʔϜ͕ Jounal of Clinical

    Medicine ʹग़ͨ͠ ࿦จ Assessing the Impact of Reduced Travel on Exportation Dynamics of Novel Coronavirus Infection (COVID-19) (J. Clin. Med. 2020, 9(2), 601; https://doi.org/10.3390/jcm9020601) ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆ ͨ͠.   தࠃࠃ಺ʹ͓͍݄͔ͯ̍Β্݄̎०·Ͱͷ travel volume ͷݮগ͕ COVID- 19 ͷࠃ֎఻೻ಈଶʹ༩͑ΔӨڹΛݕ౼ͨ͠. ۩ମతʹ͸ (1) the number of expoted cases, (2) the probability of a major epidemic, (3) the time delay to a major epidemic ΛΞδΞͰதࠃ͔Βͷཱྀߦ٬͕࠷΋ଟ͍೔ຊʹ͍ͭͯ Statistical model Λ༻͍ͯఆྔԽͨ͠. 5.1 ४උ 5.1.1 Counterfactual model ҼՌਪ࿦ Causal Inference1ʹ͓͚Δ൓ࣄ࣮Ϟσϧ counterfactual model ͱ ͸ͳʹ͔ʹ͍ͭͯ؆୯ʹ৮ΕΔ. ߏ଄తҼՌϞσϧ͸࣮ࡍʹ࣮ߦ͞Ε͍ͯͳ͍ࣄ৅ͷޮՌΛਪఆͰ͖Δ͕, ͜ ͷͱ͖࣮ࡍʹ࣮ߦͯ͠ͳ͍ࣄ৅͸൓ࣄ࣮ͱݺͿ. ൓ࣄ࣮ϞσϧʹΑͬͯҼՌ ޮՌ causal effect ͕Θ͔Δ͕, ൓ࣄ࣮Λ̍ඪຊ͔ΒಘΔ͜ͱ͸Ͱ͖ͣඪຊ਺ Λଟͯ͘͠ूஂϨϕϧͰͷ൓ࣄ࣮Λݕ౼͢Δ. ͜ͷͱ͖ಘΒΕΔͷ͸ causal effect Ͱ΋ಛʹฏۉҼՌޮՌ average causal effect Ͱ͋Δ. ࣮ݧݚڀͷ৔߹͸ causal effect ΛಘΔʹ͸ަབྷόΠΞε cofounding bias2 Λճආ͢ΔͨΊʹཧ૝తͳ randomized experiments Λߦ͏͜ͱ͕େ੾͕ͩ, 1Pearl ͷ” Causal Inference In Statistics: A Primer”ͱ͔”Causal Inference: What If” ͸ҼՌਪ࿦ͷษڧʹ͓͢͢ΊͰ͢. Pearl ͷํ͸΋͏ऴΘͬͯ·͚͢Ͳ What if ͸·ͩษڧதͳ ͷͰͦΕҎ্ͷ͜ͱ͸ݴ͑ͳ͍Ͱ͢... 2DAG ͷ࿩Ͱग़ͯ͘Δ΍ͭͰ͢. 18
  20. ؍࡯ݚڀͷ৔߹͸ͦΕͰ͸આಘྗʹ͚ܽΔ. ؍࡯ݚڀʹ͓͍ͯ average causal effect ΛಘΔʹ͸ (1) Exchangeability, (2) Consistency

    (3)Positivity ͷ̏ཁ ૉΛຬͨ͢͜ͱ͕ඞཁͰ͋Δ.1 झࢫ͔Β֎ΕΔͷͰඞཁ࠷௿ݶ͜͜·ͰͰࢭ Ί͓ͯ͘. 5.1.2 Hazard function ੜଘ࣌ؒղੳʹ͓͚Δϋβʔυؔ਺ hazard function ʹ͍ͭͯղઆ͢Δ. Hazard function λ(t) ͸ t ࣌఺ʹ͸ੜଘ͍͕ͯͨ͠ δ ͚ͩܦͬͨͱ͖ʹ͸ ࢮ๢͢Δةݥ౓Λද͢. ·ͨ, ͜ͷྦྷੵ෼෍ؔ਺ F(t) ͸ੜଘؔ਺ survival functionS(t), ͭ·Γ࣌ࠁ t Ͱੜଘ͍ͯ͠Δ֬཰Λ༻͍ͯҎԼͷΑ͏ʹදͤΔ. F(t) = 1 − S(t) (5.1) ࣍ʹҎԼͷؔ܎ࣜΛߟ͑Δ. λ(t) = 1 S(t) limδ→0 F(t + δ) − F(t) δ = F′(t) S(t) = − S′(t) S(t) (5.2) Αͬͯ, λ(t) ͱ S(t) ͷؔ܎͸࣍ͷ௨Γ. λ(t) = − S′(t) S(T) ↔ S(t) = exp ( − ∫ t 0 λ(s)ds ) (5.3) ͜ΕΑΓ, ྦྷੵ෼෍ؔ਺ F(t) ͸ hazard function λ(t) Λ༻͍ͯ࣍ͷΑ͏ʹද ͤΔ. F(t) = 1 − exp ( − ∫ t 0 λ(s)ds ) (5.4) 5.2 ࣮ࡍʹߦΘΕͨղੳ 5.2.1 Reduced number of exported cases தࠃࠃ֎Ͱ਍அ͞Εͨ࠷ॳͷ঱ྫ͸ 2020 ೥ 1 ݄ 13 ೔ʹ͓͍ͯ֬ೝ͞Ε ͨ. ྲྀߦ։࢝೔Λ 2019 ೥ 12 ݄ 1 ೔ (0 ೔໨) ͱ͢Δͱ, ෢׽ࢢ͸ 53 ೔໨͔Β lockdown ʹೖͬͨ. COVID-19 ͷજ෬ظؒ͸໿ 5 ೔2ͳͷͰ travel volume ͷ ݮগͷӨڹ͸ 58 ೔໨͔ΒղऍՄೳͱͳΔ. λΠͰ֬ೝ͞Εֳͨ֎ॳͷ঱ྫ͸ 43 ೔໨ʹ౰ͨΔ. 1ҼՌਪ࿦ͷղઆ΍·ͱΊ͕ຊߘͷ໨తͰ͸ͳ͍ͷͰ౿Έࠐ·ͳ͔͕ͬͨ, ଞʹ΋ measurement error/misclassification, model specification, selection bias ͷΑ͏ͳԾఆ͕ஔ͔ΕΔ. 2̏ষͷ࿦จͷ݁ՌͰ͋Γ, ຊߘʹ΋ͦΕ͕ࡌͬͨ Abstract ͸̏ষͰҾ༻ͨ͠. 19
  21. Lockdown ʹΑΔ reduced number of expoted cases ͷਪఆͷͨΊʹ coun- terfactual

    model Λߟ͑Δ. Lockdown ͕ͳ͍ͱ͖͸ t ೔໨ͷࠃ֎Ͱͷൃ঱ൃ ੜ཰ c(t) Λஔ͘ͱ, ϙΞιϯճؼʹΑͬͯҎԼͷ model ͕ਪఆ͞ΕΔ. ͜Ε ͸ 57 ೔໨·Ͱͷσʔλ͔Β fitting ͢Δ. E (c(t)) = c0 ert (5.5) ͜ͷͱ͖, c0 ͸ॳظ஋Ͱ r ͸தࠃࠃ֎Ͱͷࢦ਺ؔ਺త૿Ճ཰. h(t) Λ t ೔໨Ͱͷ؍ଌ஋ͱ͢ΔͱҎԼͷࣜʹΑΓ reduced number of ex- ported cases ͕ಘΒΕΔ. V = 67 ∑ 58 (h(t) − E(c(t))) (5.6) ͜ͷͱ͖, ύϥϝʔλͷਪఆ஋ͱͦͷڞ෼ࢄߦྻ͔ΒϞσϧΛཱͯͨ. ڞ෼ࢄ ߦྻΛ༻͍ͯ 95%CI ͕ಘΒΕΔ. 5.2.2 Reduced probability of major epidemic overseas ͋ΔҰ࣍ײછ঱ྫ͔Βੜͨ͡ೋ࣍ײછ঱ྫ਺ͷ෼෍͕ R0 ͱෛͷೋ߲෼෍ʹ ै͍ the dispersion parameter Λ k ͱԾఆ͢Δ. 1 ͜ͷͱ͖, the probability of extinction (ײછ঱͕͋Δ 1 ਓ͔Β 2 ࣍ײછ͕ى͖ͣࣗવফ໓ (ઈ໓) ͢Δ ֬཰) Λ π ͱ͢Δͱෛͷೋ߲෼෍ʹ͓͚Δظ଴஋ µ ʹ͍ͭͯײછ͕ੜͨ͡ͱ ͖ͷ R0 ਓͱߟ͑ΒΕΔͨΊ µ = R0 (1 − π) ͱ͠ π ͸ҎԼͷΑ͏ʹදͤΔ. π = Γ(0 + k) 0!Γ(k) ( 1 1 + µ k )k ( 1 1 + µ k )0 = 1 ( 1 + R0 k (1 − π) )k (5.7) ͜͜Ͱޓ͍ʹಠཱͳ untraced cases Λ n ݅ͱ͢Δͱ, major epidemic ͕ى͖ Δ֬཰ p ͸ҎԼͷ௨Γ. p = 1 − πn (5.8) ͜͜Ͱ counterfactual ͳγφϦΦͱݱ࣮ͷγφϦΦʹ͍ͭͯݕ౼͢Δ. ۩ମ తʹ͸, ҎԼͷ m ͱm Ͱ, travel volume ͷݮগ͕ى͖ͳ͔ͬͨ৔߹͕ coun- 1ͳͥϙΞιϯ෼෍Ͱ͸ͳ͘ෛͷೋ߲෼෍Λબ୒ͨ͠ͷͩΖ͏͔. ͜Ε͸ҎԼͷΑ͏ͳཧ༝ʹ ΑΔ. ೔ຊʹ͓͍ͯ COVID-19 ͸ 8 ׂͷਓ͕ 2 ࣍ײછΛىͣ͜͞࢒Γ 2 ׂ͕ײછ֦େʹد༩ ͍ͯ͠Δ͜ͱ͕Θ͔͍ͬͯΔ. ͜ͷΑ͏ͳͱ͖͸ฏۉ λ ͱͯ͠ϙΞιϯ෼෍ΛԾఆ͢Δʹ͸෼ ࢄ͕େ͖͘ͳͬͯ͠·͏. ͜ΕΛ overdispersion ͱ͍͏. dispersion parameter ͕̍ͷͱ͖ෛ ͷೋ߲෼෍͸ϙΞιϯ෼෍ʹҰக͢Δ. ෛͷೋ߲෼෍ΛϙΞιϯ෼෍ͷظ଴஋ λ ͷࣄલ෼෍͕ ΨϯϚ෼෍Ͱ͋Δͱ͢Δߟ͑ํͰѻͬͨ. 20
  22. terfactual ͳ৔߹Ͱ͋Δ. m = 67 ∑ t=58 h(t) m =

    67 ∑ t=58 E(c(t)) (5.9) ͕ͨͬͯ͠, ྲྀߦൃੜ֬཰ͷݮগ͸࣍ͷ௨Γ. ͳ͓, 10%, 30%, 50%͕ traced Ͱ͋Δ৔߹͸ͦΕͧΕॱʹ 0.9, 0.7, 0.5 Λ m ʹֻ͚ͨ. ϵ = πm − πm (5.10) 5.2.3 Time delay to major epidemic gained from the reduction in travel volume ϋβʔυؔ਺ hazard function λ(t) Λ༻͍ͯ travel volume ͷݮগʹΑͬ ͯಘͨ࣌ؒ஗ΕΛٻΊΔ. travel volume ͷݮগ͕ͳ͍৔߹ʹ t ࣌఺·Ͱʹ major epidemic Λى֬͜͢཰͸ H0 (t) = 1 − exp ( − ∫ t 0 λ(s)ds ) (5.11) Ͱ༩͑ΒΕΔ. ɹ࣍ʹ travel volume ͷݮগΛߟྀͨ͠৔߹Λߟ͑Δ. ͦͷͨ Ίʹ travel volume ʹΑΔྲྀߦൃੜϦεΫͷ૬ରతݮগ ϵr Λ࣍Ͱ༩͑Δ. ϵr = 1 − 1 − πm 1 − πm (5.12) ͜ͷͱ͖, ૬ରةݥ౓ (1 − ϵr ) Λ༻͍ͯ travel volume ͷݮগ͕͋Δ৔߹ʹ t ࣌఺·Ͱʹ major epidemic Λى֬͜͢཰͸ҎԼͷΑ͏ʹදͤΔ. H1 (t) = 1 − exp ( − 1 − πm 1 − πm ∫ t 0 λ(s)ds ) (5.13) ͜͜Ͱ, hazard function ͷੵ෼஋Λ Λ(t) ͱ͢Δͱײછऀ૿৩౓ r Λ༻͍ͯ Λ(t) = C(exp (rt) − 1) Ͱ͋Δ (C ͸ఆ਺). (5.11) ͱ (5.13) ͷྲྀߦ·Ͱͷ ࣌ؒͷதԝ஋ΛͦΕͧΕ σ0 ͱσ1 ͱ͢Δͱ, Hi (σi ) = 0.5 ͕੒Γཱͪ1, Αͬ ͯ Λ(σ0 ) = ln 2 ͱ Λ(σ1 ) = ln 2 1 − ϵr ͕Θ͔Δ. ͳ͓, r ͸ COVID-19 ͷ৔߹ ͸ 0.14 ͱਪఆ͞ΕΔ.2 ͜ͷΑ͏ʹੈ୅͕࣌ؒࢦ਺ؔ਺ʹै͏ͱ͖3ഒՃ࣌ؒ 1͜ͷੑ࣭Λར༻͢ΔͨΊʹதԝ஋Λ༻͍͚ͨͩͰݪཧతʹ͸ฏۉ஋Ͱ΋࠷ස஋Ͱ΋Α͍. 2Quilty, B.J.; Clifford, S.; CMMID nCoV working group; Flasche, S.; Eggo, R.M. Effectiveness of airport screening at detecting travellers infected with novel coronavirus (2019-nCoV). Eurosurveillance 2020, 25, 2000080. 34 ষͰߟ͑ͨΑ͏ʹ͜ͷͱ͖ R0 = 1 + rD. 21
  23. doubling time td ͸, td = ln 2 r =

    4.95 ೔ͱͯ͠ܭࢉ͞ΕΔ. ͕ͨͬͯ࣍͠ͷ 2 ͕ࣜ੒Γཱͭ. H0 (σ0 ) = 1 − e−Λ(σ0) = 1 2 ↔ Λ(σ0 ) = C(erσ0 − 1) = ln 2 H1 (σ1 ) = 1 − e−(1−ϵr)Λ(σ1) = 1 2 ↔ Λ(σ1 ) = C(erσ1 − 1) = ln 2 1 − ϵr (5.14) Ҏ্ΑΓ, travel volume ͷݮগʹΑΓੜͨ࣌ؒ͡஗Ε time delay σ1 − σ0 ͸ ࣍ͷ௨Γ. σ1 − σ0 = ln ( C(1 − ϵr ) + ln 2 C(1 − ϵr ) + ln 2(1 − ϵr ) ) td ln 2 = ln    C 1 − πm 1 − πm + ln 2 C 1 − πm 1 − πm + ln 2 1 − πm 1 − πm    td ln 2 (5.15) ͳ͓ܭࢉ͸ JMP Version 14.0 Λ༻͍ͯߦΘΕ৴པ۠ؒ͸ϓϩϑΝΠϧ໬౓ ͔Βࢉग़͞Εͨ. 5.2.4 ·ͱΊ 3 ͭͷ؍఺͔Β Lockdown ʹ൐͏ travel volume ͷݮগ͕༩͑ΔӨڹΛఆྔ తʹධՁͨ͠. ࣮ࡍʹ࿦จ͕ग़͞Εͨͱ͖ʹಘΒΕͨৄ͍݁͠Ռ౳ͷߟ࡯ʹ ͍ͭͯ͸, ࿦จͷ Results ΍ Discussion ΛಡΜͰ΋Β͍͍ͨ. ࣍ͷϖʔδʹ ࿦จͷ Abstract ͷΈࡌ͓ͤͯ͘. 22
  24. Abstract   The impact of the drastic reduction in

    travel volume within mainland China in January and February 2020 was quantified with respect to re- ports of novel coronavirus (COVID-19) infections outside China. Data on confirmed cases diagnosed outside China were analyzed using sta- tistical models to estimate the impact of travel reduction on three epi- demiological outcome measures: (i) the number of exported cases, (ii) the probability of a major epidemic, and (iii) the time delay to a major epidemic. From 28 January to 7 February 2020, we estimated that 226 exported cases (95% confidence interval: 86,449) were prevented, cor- responding to a 70.4% reduction in incidence compared to the counter- factual scenario. The reduced probability of a major epidemic ranged from 7% to 20% in Japan, which resulted in a median time delay to a major epidemic of two days. Depending on the scenario, the estimated delay may be less than one day. As the delay is small, the decision to control travel volume through restrictions on freedom of movement should be balanced between the resulting estimated epidemiological impact and predicted economic fallout.   23
  25. ୈ6ষ Heterogenerous populationͰͷਪఆ 6.1 Heterogeneity of population 2 ষͰड़΂ͨΑ͏ʹײછ঱ͷྲྀߦΛਤΔࢦඪʹ࠶ੜ࢈਺ reproduction

    num- ber ͕͋Δ. ͍·, ૝ఆ͢Δूஂ͕ۉҰ homogeneous Ͱ͋Δͱ͖, basic re- production number Λ R0 ͱ͢Δͱ׆ಈࣗॗʹΑΔ઀৮཰ͷ௿Լ p Λ༻͍ͯ effective reproduction number Re ͸ҎԼͷΑ͏ʹදͤΔ. Re = (1 − p)R0 (6.1) ͜͜ͰײછΛऩଋͤ͞ΔͨΊʹ͸ Re < 1 ͱ͢Ε͹ྑ͍ͷͰ, p > 1 − 1 R0 (6.2) ߦಈࣗॗʹΑͬͯਓͱͷ઀৮Λ 8 ׂݮΒ͢ͱ͍͏ͱ͖, ʮ8 ׂʯ͸͜ͷܭࢉʹ ΑͬͯಘΒΕΔ.1 ͔͠͠ R0 Λܭࢉ͢Δʹ͋ͨͬͯ homogenerous population ΛԾఆͨ͠. ͜ ͷ৔߹ R0 ≈ 2.5 ͱͳΓ઀৮཰͸ 6 ׂݮͰྑ͍ܭࢉʹͳͬͯ͠·͏. ͕࣮ͩࡍ ͸, sexual worker ͳͲ͍ΘΏΔ໷ͷ֗͸೉͘͠ੑత઀৮ sexual mixing ʹհ ೖͰ͖ͳ͍ࣄ͕ଟ͘, ͦͷଐੑʹ͋Δ͔ͳ͍͔ʹΑͬͯ R0 ͸ҟͳΔͷ͕ݱ࣮ Ͱ͋Δ. ͜ͷΑ͏ͳෆۉҰूஂ heterogenerous population ͷ৔߹͸࣍ʹઆ໌ ͢Δ࣍ੈ୅ߦྻ next generation matrix ʹΑͬͯଐੑผʹ R0 ΛٻΊ, ͦͷ୅ ද஋ͱͯ͠ಘͨ R0 Λ૝ఆ͢Δ.2 6.2 Next generation matrix ( IX,t+1 IY,t+1 ) = K ( IX,t IY,t ) (6.3) 1HIT = 1 − 1 R0 ͱ͠ herd immunity Λද͢. ͜Ε͸ϫΫνϯΧόʔ཰ͷࢉग़ͳͲʹ΋ར ༻͞ΕΔ. 2ޙ΄Ͳઆ໌͢Δ͕, ͓ͦΒ࣍͘ੈ୅ߦྻʹΑͬͯ net reproduction number Λ࣮ࡍ͸ܭࢉ ͍ͯ͠ΔͱࢥΘΕΔ. 24
  26. ͱ͠, K ͸ҎԼͷΑ͏ʹ͢Δ. K = ( RXX RXY RY X

    RY Y ) (6.4) ͜͜Ͱ, ߦΛ recipients, ྻΛ transmitters Ͱ͋Γ, ࠓճ͸ X ͸ sexual workers, Y ͸ͦ͏Ͱͳ͍৬छͷਓʑͱ͢Δ.1 (6.3) ͸ t ੈ୅ͷײછऀ਺͔Β t+1 ੈ୅ͷ ײછऀ਺ͷਪҠΛද͓ͯ͠Γ, K ͷ͜ͱΛ࣍ੈ୅ߦྻ next generation matrix ͱ͍͏. t → ∞ Ͱ͸ੈ୅ؒͷײછऀ਺ͷൺ͸ҰఆͰ͔ͭ, ಉੈ୅಺ͷଐੑผ ͷײછऀ਺ͷൺ΋ҰఆͰ͋ΔͷͰ R0 ͸ K ͷݻ༗஋ͱͯ͠ܭࢉ͞ΕΔ. ࠷΋ جຊతͳ৔߹ʹ͓͍ͯ͸ K ͸࣍ͷΑ͏ʹදͤΔ. K = ( RXX RXY RY X RY Y ) = ( βXX NX D βXY NX D βY X NY D βY Y NY D ) (6.5) ͳ͓, β ͸઀৮཰, N ͸ population size ͱ͢Δ. Next generation matrix ͔Β net reproduction numberRN = R0 S (ͨͩ͠ S ͸ײडੑਓޱ) ΛٻΊΔʹ͸ ࣍ͷ௨Γ. ( RXX RXY RY X RY Y ) = ( βXX SX D βXY SX D βY X SY D βY Y SY D ) (6.6) ͍ΘΏΔʮ8 ׂݮʯʹ͍ͭͯ͸͜ΕΛ༻͍ͯ 8 ষͰղઆ͢ΔͷͰ͜͜ͰࢭΊ ͓ͯ͘. 1ͭ·Γ RXY Ͱ͋Ε͹ͦΕ͸ Y ͔Β X ʹײછͤ͞Δ basic reproduction number Λࢦ͢. 25
  27. ୈ7ষ ײછऀ਺ͷਪఆ ෆݦੑײછͳͲͰൃ঱ਓ਺͸೺ѲͰ͖ͯ΋ײછਓ਺͸ϑΟʔϧυӸֶௐࠪ Ͱ͸ಘʹ͍͘͜ͱ͕ଟ͍. ͜ͷͱ͖ҎԼͰઆ໌͢Δٯܭࢉ back-calculation ʹ Αͬͯײછऀ਺Λਪఆ͢Δ͜ͱ͕͋Δ. Ϋϥελʔରࡦ൝΋͜ΕΑΓ R

    Λ༻ ͍ͯ༧ଌײછऀ਺ͱͦͷ༧ଌײછ೔Λಘ͍ͯΔ. 7.1 Back-calculation f(t) : જ෬ظ͕ؒ t Ͱ͋Δ֬཰ີ౓ؔ਺ ρ(t) : t ࣌఺ʹൃੜͨ͠ײછऀ਺ b(t) : t ࣌఺ʹൃੜͨ͠ൃ঱ऀ਺ ͱఆٛ͢Δͱ࣍ͷؔ܎͕ࣜ੒Γཱͭ. b(t) = ∫ t o ρ(u)f(t − u)du (7.1) 潜伏期間 t-1 潜伏期間 t-u 発症 T=t 感染 T=0 潜伏期間 t 感染 T=1 発症 T=t 感染 T=u 発症 T=t b1(t)=ρ(0)f(t): b2(t)=ρ(1)f(t-1): bu(t)=ρ(u)f(t-u): ্ͷਤ͸આ໌ͷͨΊ (7.1) ͷࣜΛ཭ࢄܕͱΈͯ࡞੒ͨ͠΋ͷͰ͋Δ. ͜Ε͸ൃ ঱ऀ਺Λܭࢉ͢ΔࣜͰ͋Δ͕, ࣮ࡍʹ͸ϑΟʔϧυݚڀ͔Β f(t) ͱ b(t) ͕ಘ ΒΕ͍ͯΔ. Αͬͯײછ࣌ࠁΛ f(t) ͔ΒϊϯύϥϝτϦοΫਪఆ͠ (7.1) Λ ٯʹղ͘͜ͱͰײછऀ਺ΛٻΊΔ. ͜ΕΛٯܭࢉ back-calculation1ͱ͍͏. 1௕ظతͳ༧ଌʹ͸޲͔ͣۙະདྷ༧ଌʹ༻͍ΒΕΔ. 26
  28. ୈ8ষ ʮ8ׂݮʯͷҙຯ 6 ষͰѻͬͨ net reproduction number Λѻ͏ next generation

    matrix Λ ߟ͑Δ. ( RXX RXY RY X RY Y ) = ( βXX SX D βXY SX D βY X SY D βY Y SY D ) (8.1) ׆ಈࣗॗʹΑͬͯ RXY , RY X , RY Y ͸ӨڹΛड͚Δͱߟ͑ΒΕΔ. ͜͜Ͱ, Ө ڹΛड͚ͨ৔߹ βXY (βY X ), βY Y , SY ͕খ͘͞ͳΔͱߟ͑ΒΕΔ. খ͘͞ͳͬ ͨͦΕΒͷม਺Λ༻͍ͯݻ༗஋Λܭࢉͯ͠ Rn ͕ 1 ΑΓখ͍͞৔߹Λߟ͑Δ ͱ, ׆ಈࣗॗΛड͚ೖΕΔ܏޲ʹ͋Δਓʑͷ β ͱ S ͷੵ͕ 8 ׂݮΔඞཁ͕͋ Δͱ͍͏͜ͱͩ. 1 ผͷઆ໌ͷ࢓ํͰಉ͜͡ͱΛߟ͑Α͏. i(t) = ∫ ∞ 0 A(t, s)i(t − s)ds (8.2) ͜Ε͸࠶ੜ࢈ํఔࣜͱ͍͍, i(t) ͸࣌ࠁ t Ͱͷ৽نײછऀ਺, A(s, t) ͸ ࣌ࠁ̓ ʹҰਓͷײછऀ͕ൃੜͦ͠ͷਓ͕࣌ࠁ s ʹೋ࣍ײછΛҾ͖ى֬͜͢཰ ͱͨ͠. ͜Ε͸࣍ͷΑ͏ʹ΋දͤΔ. i(t) = βS(t) ∫ ∞ 0 f(s)i(t − s)ds (8.3) f(t) ͸ೋ࣍ײછ͕ى͜Δ·Ͱͷִ࣌ؒؒ t ͷ֬཰ີ౓ؔ਺. ͔͜͜ΒΘ͔Δ Α͏ʹ৽نײછऀ਺͸઀৮཰ β ͱײडੑਓޱ S ͷੵʹൺྫ͢Δ. Ҏ্ͷ 2 ௨Γͷઆ໌͔ΒΘ͔ΔΑ͏ʹ, ͍ΘΏΔʮ8 ׂݮʯͱ͸, ʮଞਓͱ ͷ઀৮཰ͱײडੑਓޱͷੵΛ 8 ׂݮΒ͢ʯ͜ͱͰ͋Δ. Ϋϥελʔରࡦ൝͸ ઀৮཰ΛҎԼͷࣜ2ͰಘΒΕΔ઀৮ස౓ߦྻ time of exposure matrix ͷ࠷େ ݻ༗஋͔Β઀৮཰Λࢉग़͠, ײडੑਓޱʹؔͯ͠͸ۭؒ౷ܭֶΛ༻͍ͯ NTT ͷۭؒσʔλΛղੳͯ͠ಘ͍ͯΔ. ͜Ε͸ 2 ࣍ݩσʔλͳͷͰλϫʔϚϯγϣ ϯͷΑ͏ͳ৔߹ີू͍ͯ͠Δͱ൑அ͞ΕΔ͜ͱ͕͋Γ, ͜Ε΋ۭؒ౷ܭֶͰ ͳΜͱ͔ௐ੔͍ͯ͠ΔΑ͏ͩ. tij = H ∑ h=1 Z ∑ z=1 (ki hz kj hz ∑ n l=1 kl hz ) (8.4) 1ͳ͓, ࣍ੈ୅ߦྻͷଐੑͷ۠෼͸հೖͰ͖Δ৬छ͔Ͳ͏͔Ͱ͋Γ, ͜Ε͸ݸਓతͳ༧૝Ͱ͢ ͚Ͳ͓ͦΒ͘ଞʹ΋ҩྍैࣄऀ΍Πϯϑϥؔ࿈ͷ৬छͷਓ΋ͦͷଐੑͱͯ͠෼͚ͯ͋ΔͩΖ͏. ͦͷ৔߹ 3 × 3 Ҏ্ͷߦྻͱͯ͠ߟ͑Δ. 2tij ͸೥ྸ܈ i ͱ j ͷ઀৮ස౓, h ͸ location, z ͸࣌ؒ, ki hz ͸ z ͱ h ʹ͓͚Δ i ͷਓޱ. i ܈ ͱ j ܈ͷ઀৮཰͸ࣗ෼ͷ઀৮਺ͱ૬खͷ઀৮ൺ཰ͷੵ.Using Time-Use Data to Parameterize Models for the Spread of Close-Contact Infectious Diseases(2008 Nov 1;168(9):1082-90. doi: 10.1093/aje/kwn220. Epub 2008 Sep 18.) 27
  29. ୈ9ষ ͓ΘΓʹ ຊߘ͸੢Ӝݚ͓Αͼްੜ࿑ಇলΫϥελʔରࡦ൝ͷ෼ੳ͕ͲͷΑ͏ʹߦΘ Ε͍ͯΔ͔Λ͋Δఔ౓ՄࢹԽ͠, গ͠Ͱ΋਎ۙʹײͯ͡΋Β͏ͱ͍͏ݐલͷ ΋ͱॻ͔Εͨ. ϚχΞοΫͳ࿩΋͕͋ͬͨ͋͑ͯলུͤͣղઆͨ͠. ෼ذա ఔΛར༻ͨ͠ model

    ͳͲཧղෆ଍ͳ఺΋ු͖ூΓʹͳΔͷΛ࣮ײ͕ͨ͠, ͦ Ε΋ؚΊͯ࡞੒ͯͯ͠ΊͪΌΊͪΌָ͔ͬͨ͠. ͳ͓, ຊߘ͸ LaTeX Ͱॻ͔ Ε, ਤ͸͢΂ͯ draw.io Λ༻͍ͯ࡞੒ͨ͠. Stay home! 3 ີճආ! खચ͍͏͕͍! 28