理論疫学 -methodological report-

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1. ཧ࿦Ӹֶ -methodological report- ༿৭1 2020 ೥ 10 ݄ 20 ೔ 1twitter: @mepbphhond
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3. ໨ ࣍ ୈ 1 ষ ͸͡Ίʹ 4 ୈ 2 ষ ং࿦ -Basic/Effective reproduction number- 5 ୈ 3 ষ Time interval ͷਪఆ 6 3.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1 Doubly interval-censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2.1 Estimation of the time interval distribution using doubly interval-censored likelihood . . . . . . . . . . . . . . . . . . . . . . 7 3.2.2 Estimation of the time interval distribution using Bayesian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.3 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ୈ 4 ষ ྲྀߦॳظஈ֊Ͱͷ cCFR ͱ R0 ͷਪఆ 9 4.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.1.1 CFR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.1.2 The factor of underestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.1.3 R0 of an infection from the growth rate of an outbreak (epidemic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2.1 Estimation of the delay distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2.2 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2.3 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ୈ 5 ষ Lockdown ͷఆྔతධՁ 13 5.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.1.1 Counterfactual model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.1.2 Hazard function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2.1 Reduced number of exported cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2.2 Reduced probability of major epidemic overseas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2.3 Time delay to major epidemic gained from the reduction in travel volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2.4 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ୈ 6 ষ Overdispersion ͷਪఆ 16 6.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.1.1 Overdispersion ͱ Rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2.1 Overdispersion ਪఆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2.2 Proportion responsible for 80% of secondary transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2.3 Simulation of the effect of the underreporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2.4 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 ୈ 7 ষ Intervention ͱײછऀ਺ͷධՁ 19 7.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.1.1 Richards model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.2.1 Backcalculation ͱ forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.2.2 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 ୈ 8 ষ Asymptomatic rate ਪఆ 21 8.1 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8.1.1 Statistical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8.1.2 Missing data and sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8.1.3 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2
4. ୈ 9 ষ Back-projection ʹΑΔ COVID-19 ͷײછऀ਺ٴͼ Rt ਪఆ 22 9.1 Back-projection ֓ཁ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 9.1.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 9.1.2 EM algorithm (EMS algorithm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 9.1.3 Back-projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 9.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 9.3 Figures for the estimated Rt and incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ୈ 10 ষ Competing risk model ʹΑΔ HIV 26 10.1 ४උ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 10.1.1 Competing risk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 10.1.2 Weibull distribution ͱ survival time analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 10.2 ࣮ࡍʹߦΘΕͨղੳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10.2.1 Derivation of likelihood using a mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10.2.2 Statistical model and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10.2.3 ·ͱΊ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ୈ 11 ষ ௥Ճ༧ఆ࿦จ 29 ୈ 12 ষ ਪఆجૅ 30 12.1 Heterogeneity of population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 12.2 Next generation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 12.3 ײછऀ਺ͷਪఆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 12.4 Back-calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 12.5 Zero-inflated poisson model (ZIP model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 12.6 Renewal equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 12.7 Particle filtering and particle Markov-chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ୈ 13 ষ ͓ΘΓʹ 34 3
5. ୈ1ষ ͸͡Ίʹ ຊߘ͸, ཧ࿦Ӹֶ theoretical epidemiology ͷ࿦จʹ͍ͭͯҹ৅తͰ͋ͬͨ΋ͷͷ method Λ·ͱΊͨ΋ͷͰ͋Δ. ͱ͸͍͑, ૬౰ؾ·͙Εʹ બΜͰ͍Δ. جຊతͳࣄ߲ͷҰ෦͸࠷ޙͷষͰ·ͱΊͯղઆͨ͠. 4
6. ୈ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-ϚοέϯυϦοΫ͕ఏҊͨ͠ҎԼͷΑ͏ͳৗඍ෼ํఔࣜܥΛجૅͱ͠, Ϟσϧ Λ༻͍ͯ෼ੳΛߦ͏ཧ࿦Ӹֶͷओཁͳख๏Ͱ͋Δ.                            dS dt = −βS I dI dt = βS I − 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 = βS I − γI = 0 (∗) ͜ͷͱ͖, ɹײછ঱ͷ఻೻͸ఆৗঢ়ଶͱͳΔ. (*) Λ੔ཧ͢Δͱ࣍ͷ௨Γ. βS I − 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 2Λجຊ࠶ੜ࢈਺ (Basic Reproduction Number) ͱ͍͍,t=0, ͭ·Γ໔ӸΛ࣋ͨͳ͍ूஂʹ͓͍ͯײછऀ਺͕ 0 ਓͷͱ͖ʹײછऀ ͕ 1 ਓੜͨ͡ͱ͖ͦͷਓʹΑͬͯೋ࣍ײછ͢Δฏۉͷਓ਺Λද͢. ैͬͯ, R0 < 1 ͷͱ͖ײછ͸֦େͤͣ R0 > 1 ͷͱ͖͸ײછ͕֦େ͢Δͱߟ ͑Δ͜ͱ͕Ͱ͖Δ. ·ͨ Re Λ࣮ޮ࠶ੜ࢈਺ (Effective Reproduction Number) ͱ͍͍, ೚ҙͷ̓ʹ͓͍ͯ, ͭ·Γ͋Δ࣌఺Ͱ̍ਓͷײછऀʹΑͬ ͯೋ࣍ײછ͢Δฏۉͷਓ਺Λද͢. ैͬͯ, Re < 1 ͷͱ͖ײછ͸֦େͤͣ Re > 1 ͷͱ͖͸ײછ͕֦େ͢Δͱߟ͑Δ͜ͱ͕Ͱ͖Δ. R0 ͸͋ΔϞ σϧʹ͓͍ͯॳظ৚݅ʹґଘ͢ΔͨΊਓޱີ౓΍஍ҬੑʹΑΓมಈ͢Δ. Re ͸ײडੑਓޱʹ͋Δਓʑͷߦಈม༰ (खચ͍͏͕͍΍ϚεΫΛ͢ Δ͜ͱ, ਓͱͷ઀৮ΛݮΒ͢ͳͲ) ΍໔ӸΛ࣋ͭ͜ͱʹΑͬͯ௿Լͤ͞Δ͜ͱ͕Ͱ͖Δ. ࣮ޮ࠶ੜ࢈਺ͷ஋ʹΑͬͯײછ঱Ӹֶʹ͓͚Δײછ ఻೻ͷఔ౓Λ೺Ѳ͢Δ. ͳ͓, D ͸ฏۉײછظؒΛද͢. Ҏ্Ͱ঺հͨ͠ϞσϧҎ֎ʹ΋ SEIR model ΍ SIS model, SIRS model ͳͲίϯύʔτϝϯτϞσϧ͸༷ʑଘࡏ͠, ߟ͍͑ͨײછ঱ͷੑ࣭ ʹ߹ΘͤͯϞσϧΛߏங͢Δ. ࠓճྫࣔͨ͠ SIR model ͸ Deterministic model Ͱ͋Δ͕, ΄͔ʹ΋ Stochastic model ΍ Statistical model, ·ͨ Network theory Λར༻ͨ͠ϞσϧͳͲ͍Ζ͍Ζͳ֯౓͔Βߟ͍͑ͯ͘. 1ճ෮ͱִ཭͞ΕΔׂ߹Λද͢. ೔ຊޠ͚ͩݟΔͱগࠞ͠ཚ͢Δ͔΋͠Ε·ͤΜͶ. R ͕ Recover Ͱ͸ͳ͘ Removed ͳ͜ͱͱ߹ΘͤΔͱཧղ͕ਐΉ͔΋͠Εͳ͍. 2R θϩͰ͸ͳ͘, R naught ͱಡΉ. 5
7. ୈ3ষ Time intervalͷਪఆ ໨త   ͜ͷষͰ͸ Linton NM, Kobayashi T, Yang Y, et al. 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. Published 2020 Feb 17. ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔ΛഎܠΛ֬ೝͨ͠ޙʹղઆͨ͠.   3.1 ४උ 3.1.1 Doubly interval-censoring ײછ೔͕ෆ໌ͳ৔߹, ௥੻ௐࠪͱ͍ͬͨϑΟʔϧυݚڀͰ͸જ෬ظؒΛਪఆ͢Δ͜ͱ͕Ͱ͖ͳ͍. ͜ͷ৔߹ʹ, interval-censoring Λߦ͏. ͯ͞, ͜͜Ͱ͸͋Δײછऀ͕ະײછऀʹ͏ͭ͢ͱ͖Λߟ͑Δ. ͜ͷͱ͖, ײછ೔ times of exposure ΋ൃ঱೔ illness onset ΋ෆ໌Ͱ͋Δ৔߹Λ ߟ͑ͯΈΑ͏.1 感染⽇ 発症⽇ 治癒 潜伏期間τ EL ER SL SR e s ͜͜Ͱ, EL ͸ະײછऀͷײછ೔ͱͯ͠ߟ͑ΒΕΔதͰ࠷΋ૣ͍೔, ER ͸࠷΋஗͍೔ͱ͢Δ. S L ͸ൃ঱೔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 ∫ S R,i S L,i g(e) f(s − e)dsde   ͜ΕΑΓ, ࠷໬ਪఆ maximum likelihood estimation(MLE) ʹΑͬͯײછ೔ͱൃ঱೔ͷ࠷໬ਪఆ஋ estimated maximum likelihood ͕ಘΒΕΔͷ Ͱજ෬ظ͕ؒΘ͔Δ. ͜ͷख๏Λ doubly interval-censoring ͱ͍͏. 3.2 ࣮ࡍʹߦΘΕͨղੳ ຊߘͷझࢫʹΑΓ, ݁࿦ʹ͍ͭͯΑΓ݁࿦ͷಋ͖ํ (Methods) Λத৺ʹղઆ͢Δ. ͜ͷ࿦จͰ͸, જ෬ظؒ΍ൃ঱͔ΒೖӃ·Ͱͷظؒ, ൃ঱ ͔Βࢮ๢·Ͱͷظؒ, ೖӃ͔Βࢮ๢·Ͱͷظؒͱ͍͏ time interval ͷਪఆʹ͍ͭͯ doubly interval-censored likelihood Λ࢖ͬͨख๏ͱ Bayesian framework Λ༻͍ͨख๏ͷ྆ํͰߦΘΕͨ. 1࣮ࡍ, ෆݦੑײછऀ΍௥੻ͷݶքͳͲʹΑΓײછ೔͔ൃ঱೔ͷͲͪΒ͔, ͋Δ͍͸ͲͪΒ΋͕ෆ໌Ͱ͋Δ͜ͱ΋Α͋͘Δ. 6
8. 3.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 3 ষ Time interval ͷਪఆ 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 ∫ S R,i S L,i g(e) f(s − e)dsde (3.1) ͜͜Ͱ, f(t) Λͦͷ··࠾༻͢Δͱબ୒όΠΞε1͕͔͔ΔͨΊ, ӈଆ੾அ right truncation Λߟྀͯ͠ҎԼͷࣜΛ༻͍ͨ. f′(s − e, e) = f(s − e) ∫ T−e 0 re−ru 1 − e−ru F(T − e − u)du (3.2) ͜͜Ͱ, r ͸ࢦ਺ؔ਺త૿Ճ཰Ͱ F(·) ͸ f(·) ͷྦྷੵ෼෍ؔ਺. T ͸࠷৽ͷ؍ଌ࣌ࠁ2Λࣔ͢. ͯ͞, ैͬͯ໬౓͸ҎԼͷΑ͏ʹͳΔ. ·ͨ͜ͷͱ͖ΠϯςάϥϧҎԼΛ ιi ͱͯ࣍͠ͷΑ͏ʹද͢. L′(Θ|D) = ∏ i ∫ ER,i EL,i ∫ S R,i S L,i g(e) f′(s − e, e)dsde ιi = ∫ ER,i EL,i ∫ S R,i S L,i g(e) f′(s − e, e)dsde (3.3) S L,i > ER,i ͷͱ͖, s′ = s − e ͱஔ׵ͯ࣍͠ͷΑ͏ʹมܗͰ͖Δ. ιi = ∫ ER,i EL,i de g(e) ∫ S R,i −e S L,i −e f′(s′, e)ds′ = ∫ ER,i EL,i g(e) { F′(S R,i − e, e) − F′(S L,i − e, e) } de (3.4) ER,i > S L,i > EL,i Ͱ͋Δͱ͖, ҎԼͷΑ͏ʹͳΔ. ιi = ∫ S l,i EL,i g(e){F′(SR,i − e, e) − F′(S L,i − e, e)}de + ∫ ER,i S L,i g(e)F′(S R,i − e, e)de (3.5) ࠷ޙʹ, EL,i > S L,i Ͱ͋Δͱ͖, ҎԼͷΑ͏ʹͳΔ. ιi = ∫ ER,i EL,i g(e)F′(SR,i − e, e)de (3.6) ͜ͷΑ͏ʹ৔߹෼͚Λͯ͠ time interval ΛͦΕͧΕٻΊͨ.3 f(s − e) ͸ΨϯϚ෼෍, ର਺ਖ਼ن෼෍, ϫΠϒϧ෼෍Λߟ͑ͨ. 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)4ʹΑͬͯϞϯςΧϧϩੵ෼͔Βඇ੾அର਺໬౓ʹର͢Δࣄޙ༧ଌ෼෍ (time interval distribution) Λਪఆ͢Δ. ͳ͓, ର਺ਖ਼ن෼෍ͱ ϫΠϒϧ෼෍͸ͲͪΒ΋ඪ४ਖ਼ن෼෍ʹै͏. ΨϯϚ෼෍͸ shape parameter ͸ฏۉ 3, ඪ४ภࠩ 5 ͷਖ਼ن෼෍ʹै͍, inverse scale parameter ͸, location parameter Λ 0, scale parameter Λ 5.0 ʹͱΔίʔγʔ෼෍ͱ͠֊૚ϕΠζΛߟ͑ͨ.5ͳ͓, ࣄલ෼෍ͷબ୒͸ Stan developer community ʹΑΔਪ঑6ʹैͬͨ. ·ͨ, times of exposure ͱ illness onset ͷࣄલ෼෍͸ҎԼͷΑ͏ʹఆࣜԽͨ͠. ei = EL,i + (ER,i − EL,i )˜ ei si = ˆ S L,i + (S R,i − ˆ S L,i )˜ si (3.7) ͜ͷͱ͖, ei > S L,i ͳΒ͹ ˆ S L,i = ei , ei = S L,i ͳΒ͹ ˆ S L,i = S L,i Ͱ͋Δ. ei < S L,i ͷͱ͖͸ ˜ ei ͱ˜ si ͸࣍ͷ෼෍ʹै͏. ˜ ei ∼ nomal(mean = 0.5, S D = 0.5), ˜ si ∼ nomal(mean = 0.5, S D = 0.5) 1ੜଘ࣌ؒղੳͷ࿩. ଧͪ੾Γ censoring (੾அ truncation) ͕ߟྀ͞ΕΔ΂͖ิਖ਼Ͱ͢. ࠓճͷ right truncation ͸જ෬ظ͕ؒ௕͍৔߹ʹΤϯυϙΠϯτͷઃఆʹΑΓ؍ଌ͞Εͣ, population of interest ͱ sampled population ͕ҟͳͬͯ͠·͍ selection bias ͕ൃੜ͢Δঢ়ଶΛ͍͏. Inverse probability weighting methods for Cox regression with rightʖtruncated data (https://doi.org/10.1111/biom.13162). 2͜ͷͱ͖͸ 2020 ೥ 1 ݄ 31 ೔. 3࣮ࡍͷσʔλΛ༻͍ͨܭࢉ͸ R Λ࢖༻ͨ͠. 4No U-Turn Sampler(NUTS) Λ࣮૷ͨ͠ Stan ʹΑΓཚ਺ੜ੒͸ߦΘΕͨ. 5Ұൠ࿦ͱͯ͠֊૚ϕΠζ͸࠷໬๏͕೉͘͠ AIC ͰධՁͣ͠Β͍ͷͰ, MCMC Λ༻͍ͯ WAIC Λߟ͑Δࣄ͕ଟ͍. 6Stan developer team. Prior choice recommendations. https://github.com/stan- dev/stan/wiki/Prior-Choice-Recommendations. Assessed 6 February 2020. 7
9. 3.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 3 ষ Time interval ͷਪఆ ͜ͷͱ͖ Stan ʹΑΔඇ੾அର਺໬౓Λ࣍ͷΑ͏ʹఆٛͨ͠. logL(Θ) = ∑ i distribution lpd f(si − ei |Θ) (3.8) ͜͜Ͱ, lpd f ͸֬཰ີ౓ؔ਺ͷର਺Ͱ͋Γ, lognomal lpd f ͔ gamma lpd f ͔ weibull lpd f ͷ͍ͣΕ͔Λද͢. ·ͨ, જ෬ظؒʹؔͯ͠͸ଧ ͪ੾Γ໬౓Λߟ͑ͨ. MCMC ͸, the simulation phase ͱͯ͠ 5000 ճΛ burn in ͯ͠ the tuning phase Ͱ 10000 ճΛ࠾༻͠, ͦΕΛ̐ܥྻ෼ߦͬͨ. Ҏ্͔Βࣄޙ༧ଌ෼෍Λਪఆ֤͠ϞσϧΛ WAIC ͰධՁ͠࠷దͳϞσϧΛબ୒ͨ͠. ͳ͓, WAIC ͸ҎԼͷ௨Γ. WAIC = −2(lppd − pWAIC ) (3.9) ͜͜Ͱ, lppd = ∑ i log Pr(yi ), pWAIC = ∑ i V(yi ) (3.10) Ͱ͋Δ. ͳ͓, Pr(·) ͓Αͼ V(·) ͸ର਺໬౓ͷฏۉͱ෼ࢄΛͦΕͧΕද͢. 3.2.3 ·ͱΊ Ҏ্ͷ̎௨Γͷਪఆ๏͔Β time interval ͸ਪఆ͞Εͨ. ࣮ࡍʹ࿦จ͕ൃද͞Εͨͱ͖ʹಘΒΕͨৄ͍݁͠Ռ౳ͷߟ࡯ʹ͍ͭͯ͸, ࿦จͷ Results ΍ Discussion ΛಡΜͰ΋Β͍͍ͨ. ҎԼʹ࿦จͷ Abstract ͷΈࡌ͓ͤͯ͘. 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 govern 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 approximated 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 period, 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.   8
10. ୈ4ষ ྲྀߦॳظஈ֊ͰͷcCFRͱR0 ͷਪఆ ໨త   ͜ͷষͰ͸, Jung SM, Akhmetzhanov AR, Hayashi K, et al. 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. Published 2020 Feb 14. ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆͨ͠.   4.1 ४උ 4.1.1 ͱ 4.1.2 Ͱ͸, ͜ͷষͰղઆ͢Δ࿦จͷ४උͱͯ࣍͠ͷઌߦ࿦จͷ಺༰ʹ͍ͭͯҰ෦ղઆ͢Δ.   Nishiura H, Klinkenberg D, Roberts M, Heesterbeek JA. Early epidemiological assessment of the virulence of emerging infectious diseases: a case study of an influenza pandemic. PLoS One. 2009;4(8):e6852. Published 2009 Aug 31.   4.1.1 CFR ެऺӴੜ্ֶͷॏཁͳࢦඪͷ 1 ͭͱͯ͠, ࣬පͷॏ঱౓Λද͢ CFR ͕͋Δ. CFR ͱ͸͋Δ࣬පʹΑΔࢮ๢਺Λ͋Δ࣬පͷጶױ਺Ͱׂͬͨ஋ Ͱக໋཰ case-fatality risk ͷུͰ͋Δ. े෼ʹ௕͍؍࡯ظؒΛऔͬͨ৔߹͸ࢮ๢཰ death rate1Λጶױ཰ morbility2Ͱׂͬͨ஋ͱͳΔ. ཧ૝͸෼฼ ͕ײછऀͷ૯਺Ͱ͋Δ΂͖͕ͩ͢΂ͯͷײછऀΛ਺্͑͛Δ͜ͱ͸ෆՄೳͳͷͰ਍அ͞Εͨ঱ྫͷΈΛࢦ͢͜ͱ͕ଟ͍. Outbreak ͷॳظஈ֊3 Ͱ͸৘ใ͕֬ఆ਍அʹݶఆ͞ΕΔ͜ͱ͕ଟ͍. ͕ͨͬͯ͠, ࿦จ಺Ͱ͸֬ఆ਍அͷΈΛର৅ͱ͠ confirmed CFR(cCFR) ͱݺͿ. cCFR ͸฼ूஂ ͷײછ͕े෼ʹ֬ೝ͞Ε͍ͯͳ͍ͨΊ, CFR ΛաେධՁ͢Δ܏޲ʹ͋Δ. ͔͠͠ COVID-19 ͷΑ͏ͳෆ࣮֬ੑͷߴ͍ঢ়گԼͰ͸ symptomatic CFR (sCFR) ͷ্ݶΛࣔ͢ࢦඪͱͯ͠ॏཁࢹ͞ΕΔ. ޿͘༻͍ΒΕ͍ͯΔ crude estimate of the cCFR ͸ t ࣌఺ʹ͓͚Δྦྷੵࢮ๢਺ͱͷൺͰٻΊ ΒΕΔ͕, ຆͲ͕աখධՁ͞Εͯภͬͨ cCFR Λ΋ͨΒ͢܏޲ʹ͋Δ.4 Ҏޙ, bias ʹݴٴ͢Δͱ͖ʹ͸ biased cCFR ͱ unbiased cCFR Λ༻͍Δ. CFR ͸̏ͭͷΞϓϩʔν͕ଘࡏ͢Δ. (1) ͋Δ࣌఺Ͱͷྦྷੵ঱ྫ਺ʹର͢Δྦྷੵࢮ๢਺ͷׂ߹ (2) clinical outcomeʢࢮ๢͔ճ෮) ͕Θ͔͍ͬͯΔױऀͷྦྷੵ਺ʹର͢Δྦྷੵࢮ๢਺ͷׂ߹ (3) onset ͔Β death ·Ͱͷ time delay Λௐ੔ͨ͠ྦྷੵ঱ྫ਺ʹର͢Δྦྷੵࢮ๢਺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, ͋Δಛఆͷ outbreak Ͱͷ࣮ݱ஋ͱΈͳ͞ΕΔ ͷ̏ͭͰ͋Δ. t ࣌఺Ͱͷྦྷੵࢮ๢਺ͱྦྷੵ֬ఆ঱ྫ਺ΛͦΕͧΕ Dt ͱ Ct ͱ͢Δ. ͜ͷͱ ͖, bt ͸࣍ͷ௨Γ. bt = Dt Ct (4.1) ͜Ε͕ unbiased cCFR ʹൺ΂ͯաখධՁͰ͋Δ͜ͱΛࣔ͢. t ೔໨ͷ৽ن֬ఆ঱ྫ਺Λ ct , ࢮ๢͢Δͱ͖ͷ onset ͔Β death ·Ͱͷ࣌ؒʹ͍ͭ ͯͷ৚݅෇͖֬཰ີ౓ؔ਺ Λ fs ͱ͢Δͱ, Ct ͱ Dt ͸ҎԼͷΑ͏ʹදͤΔ. Ct = t ∑ i=0 ci Dt = pt t ∑ i=0 ∞ ∑ j=0 ci−j fj (4.2) ͜͜Ͱ, pt ͸ײછͯ͠ࢮ๢͢Δׂ߹Λද͢ͷͰ͜ͷਪఆ஋͸ π Λද͢. (4.1), (4.2) Λ΋ͱʹ bt ͸࣍ͷΑ͏ʹදͤΔ. 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஋. 3͜ͷষͰղઆ͢Δ࿦จ͸ύϯσϛοΫલͰ͋Γ outbreak ॳظஈ֊Ͱ͋ͬͨ. 4ੜଘ࣌ؒղੳ্ͷ໰୊Ͱ͋Δ. ৄ͘͠͸࣍ͷ࿦จΛࢀর͞Ε͍ͨ. Methods for estimating the case fatality ratio for a novel, emerging infectious disease. (DOI: 10.1093/aje/kwi230) 9
11. 4.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 4 ষ ྲྀߦॳظஈ֊Ͱͷ cCFR ͱ R0 ͷਪఆ bt = pt t ∑ i=0 ∞ ∑ j=0 ci−j fj t ∑ i=0 ci (4.3) ͔͜͜ΒΘ͔ΔΑ͏ʹ, time delay Λද̎͢ॏ࿨Λ༻͍ͨ෼ࢠͷ஋͕෼฼ΑΓখ͍ͨ͞Ί, 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৘ใ͔Βਪఆ͞Ε Δͱ͍͑Δ. ut = t ∑ i=0 ∞ ∑ j=0 ci− j fj t ∑ i=0 ci (4.5) ͱ͓͘ͱ, ut ͸ the factor of underestimation ͱͳΔ. ͕ͨͬͯ͠, pt = bt ut Ͱ͋Δ. ͜͜·Ͱ͸, ཭ࢄ࣌ؒʹ͍ͭͯٞ࿦͖͕ͯͨ͠࿈ଓ࣌ؒʹ͍ͭͯߟ͑ͯΈΑ͏. ྲྀߦͷॳظஈ֊Ͱ growth rate ͕ r ͷࢦ਺ؔ਺తͳ૿ՃΛೝ ΊΒΕΔͱ͖৽ن֬ఆ঱ྫ਺ͷظ଴஋͸ҎԼͷ௨Γ. E(ct ) = c0ert (4.6) ͳ͓, r ͸಺ࡏతͳೋ࣍ײછ͚ͩͰͳ͘ impoted ͳײછ΋ؚ·ΕΔ. (4.6) Λ༻͍ͯ, the factor of underestimation ͸࣍ͷΑ͏ʹͳΔ. u = ∫ t 0 erτ ∫ ∞ 0 e−rs f(s)dsdτ ∫ t 0 erτdτ = ∫ ∞ 0 e−rs f(s)ds (4.7) 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਺ؔ਺తͳ૿ՃʯΛԾఆͯٞ͠࿦ΛਐΊ͕ͨ͜Εͷ৴ጪੑΛ͔֬ΊͯΈΑ͏. ̎ষ Ͱઆ໌ͨ͠ྲྀೖྲྀग़ͷͳ͍࠷΋جຊతͳ SIR model ʹ͍ͭͯߟ͑Δ. ྲྀߦॳظʹ͓͍ͯ͸ S (t) ≈ N0 ͱΈͳ͢͜ͱ͕Ͱ͖ΔͷͰҎԼͷ͕ࣜߟ ͑ΒΕΔ. dI dt ≈ βN0I − γI ≈ (βN0 − γ)I (4.8) ͕ͨͬͯ͠, (βN0 − γ) ͸ఆ਺ͳͷͰྲྀߦॳظʹ͓͍ͯ͸ࢦ਺ؔ਺૿Ճ͢Δͱͯ͠Α͍. ͯ͜͞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′ ͱ͢Δ2ͱ serial interval Ts = D + D′ ʹࢦ਺ؔ਺૿ՃΛ͢ Δͱߟ͑ R0 ͸࣍ͷΑ͏ʹදͤΔ. R0 = 1 + ΛTs (4.10) 4.2 ࣮ࡍʹߦΘΕͨղੳ cCFR ͷਪఆʹ͍ͭͯ͸, ࣌ؒ஗ΕΛߟྀͨ͠ real-time Ͱͷਪఆ͕ߦΘΕͨ. 1right censoring ͷ࿩. ঱ྫ֬ೝ͞Ε t ࣌఺Ҏ߱ʹࢮ๢͢ΔՄೳੑ͕͋Δ. 2SEIR model ʹ͓͚Δ I ͷظؒΛ D, E ͷظؒΛ D′. 10
12. 4.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 4 ষ ྲྀߦॳظஈ֊Ͱͷ cCFR ͱ R0 ͷਪఆ 4.2.1 Estimation of the delay distribution t ࣌఺Ͱͷጶױ਺1Λ i(t) = i0ert 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୒ͨ͠.2 f(t; θ) Λ parameter θd = {ad , bd } ͷର਺ਖ਼ ن෼෍ͱ͢Δͱ, the factor of underestimation ͸ҎԼͷ௨Γ. u(r, θd ) = ∫ ∞ 0 e−rs f(s; θd )ds (4.12) ࣌ؒ஗ΕΛௐ੔͞Εͨྦྷੵጶױ਺͸ u(r, θd )I(t) ʹै͏. t ೔·Ͱʹใࠂ͞Εͨྦྷੵࢮ๢਺ 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 scale3ͱ͢ Δ. ͜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਺໬౓͸ͦΕͧΕҎԼͷ௨Γ. logLE ({r, i0 }|{E(t), te ≤ t ≤ T}) = T ∑ t=te log ( binom(E(t)|size = ˜ u(r, ˜ θeI(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, ˜ θeI(t), prob = CFR(t)) ) (4.18) ͨͩ͠, 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 ೔·ͰมԽͤ͞Δײ౓෼ੳ sensitivity analysis Λߦͬͨ. ࣍ʹ 2 ͭ໨ͷγφϦΦʹ͍ͭͯ. ॳΊͯ expoted case ͕֬ೝ͞Εͨ 2020 ೥ 1 ݄ 13 ೔Λى఺ͱͯ͠ܭࢉΛͨ͠. ྆γφϦΦͱ΋, cut-off times, the detection window of virus T, ෢׽ࠃࡍۭߓͷ catchment population n ͷେ ͖͞Λม͑ͯ 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.4 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 ͨ͠.5 ·ͨ, sequential fitting Λߦͬͨ. ͜Ε͸·ͣ໬౓ Le ͚ͩΛߟ͑, ࣍ʹ ͦͷͱ͖ಘΒΕͨਪఆύϥϝʔλͷฏۉΛ༻͍ͯ໬౓ LΣ Λ fitting ͤͨ͞. ࠷ޙʹ, ϓϩϑΝΠϧ໬౓Λ༻͍ͯಋग़͞Εͨ৴པ۠ؒ6ͱ߹Θͤ ͯ, ࠷໬ਪఆΑΓ pointwise estimates Λܭࢉ͢Δ͜ͱͰ, fitting ͨ͠ਪఆ஋Λݕূͨ͠. ͦͷ݁Ռ, ݁Ռ͕Ұக͍ͯ͠Δ͜ͱ͕Θ͔ͬͨ. 14.1.2 ʹ͓͚Δ E(c(ct)) ͱಉ͡. 2ର਺ਖ਼ن෼෍, ΨϯϚ෼෍, ϫΠϒϧ෼෍Λ̎छྨͷํ๏Ͱݕ౼͠·ͨ͠ΑͶ. 3scale Λ θ ͱ͢Δͱ inverse scale λ = 4 θ Λ༻͍ͯΨϯϚ෼෍͸ f(x) = λk Γ(k) xk−1e−λ. 4Python PyMC3 package ͷ No-U-Turn Λ࢖༻. 5ΨϯϚ෼෍Λར༻ͯ̎ͭ͠ͷϞʔϝϯτΛҰகͤͯ͞࿈ଓԽͨ͠. 6Delta ๏ͱϓϩϑΝΠϧ໬౓Λ༻͍Δํ๏͕໬౓͔Β৴པ۠ؒΛಘΔํ๏. ࠓճ͸ৄࡉ͸શͯল͍ͨ. ͳ͓, ৴པ۠ؒͷਪఆ͸ bootstrap ๏ͳͲ΋༻͍ΒΕΔ. 11
13. 4.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 4 ষ ྲྀߦॳظஈ֊Ͱͷ cCFR ͱ R0 ͷਪఆ 4.2.3 ·ͱΊ ࠷໬๏ʹΑͬͯ, ྲྀߦॳظஈ֊Ͱ cCFR ͳͲͷύϥϝλΛਪఆͦ͠ͷ݁ՌΛ༻͍ͯ R0 Λਪఆͨ͠. ࣮ࡍʹ࿦จ͕ग़͞Εͨͱ͖ʹಘΒΕͨৄ ͍݁͠Ռ౳ͷߟ࡯ʹ͍ͭͯ͸, ࿦จͷ Results ΍ Discussion ΛಡΜͰ΋Β͍͍ͨ. ࣍ͷϖʔδʹ࿦จͷ Abstract ͷΈࡌ͓ͤͯ͘. Abstract   The exported cases of 2019 novel coronavirus (COVID-19) infection that were confirmed outside of China provide an opportunity to estimate the cumulative incidence and confirmed case fatality risk (cCFR) in mainland China. Knowledge of the cCFR is critical to characterize 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.   12
14. ୈ5ষ LockdownͷఆྔతධՁ ໨త   ͜ͷষͰ͸, Anzai A, Kobayashi T, Linton NM, et al. Assessing the Impact of Reduced Travel on Exportation Dynamics of Novel Coronavirus Infection (COVID-19). J Clin Med. 2020;9(2):601. Published 2020 Feb 24. ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆͨ͠.   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 Inference ʹ͓͚Δ൓ࣄ࣮Ϟσϧ counterfactual model ͱ͸ͳʹ͔ʹ͍ͭͯ؆୯ʹ৮ΕΔ. ߏ଄తҼՌϞσϧ͸࣮ࡍʹ࣮ߦ͞Ε͍ͯͳ͍ࣄ৅ͷޮՌΛਪఆͰ͖Δ͕, ͜ͷͱ͖࣮ࡍʹ࣮ߦͯ͠ͳ͍ࣄ৅͸൓ࣄ࣮ͱݺͿ. ൓ࣄ࣮Ϟσϧ ʹΑͬͯҼՌޮՌ causal effect ͕Θ͔Δ͕, ൓ࣄ࣮Λ̍ඪຊ͔ΒಘΔ͜ͱ͸Ͱ͖ͣඪຊ਺Λଟͯ͘͠ूஂϨϕϧͰͷ൓ࣄ࣮Λݕ౼͢Δ. ͜ͷ ͱ͖ಘΒΕΔͷ͸ causal effect Ͱ΋ಛʹฏۉҼՌޮՌ average causal effect Ͱ͋Δ. ࣮ݧݚڀͷ৔߹͸ causal effect ΛಘΔʹ͸ަབྷόΠΞε cofounding bias Λճආ͢ΔͨΊʹཧ૝తͳ randomized experiments Λߦ͏͜ͱ͕ େ੾͕ͩ, ؍࡯ݚڀͷ৔߹͸ͦΕͰ͸આಘྗʹ͚ܽΔ. ؍࡯ݚڀʹ͓͍ͯ average causal effect ΛಘΔʹ͸ (1) Exchangeability, (2) Consistency (3)Positivity ͷ̏ཁૉΛຬͨ͢͜ͱ͕ඞཁͰ͋Δ.1 झࢫ͔Β֎ΕΔͷͰඞཁ࠷௿ݶ͜͜·ͰͰࢭΊ͓ͯ͘. 5.1.2 Hazard function ੜଘ࣌ؒղੳʹ͓͚Δϋβʔυؔ਺ hazard function ʹ͍ͭͯղઆ͢Δ. Hazard function λ(t) ͸ t ࣌఺ʹ͸ੜଘ͍͕ͯͨ͠ δ ͚ͩܦͬͨͱ͖ ʹ͸ࢮ๢͢Δةݥ౓Λද͢. ·ͨ, ͜ͷྦྷੵ෼෍ؔ਺ F(t) ͸ੜଘؔ਺ survival functionS (t), ͭ·Γ࣌ࠁ t Ͱੜଘ͍ͯ͠Δ֬཰Λ༻͍ͯҎԼͷ Α͏ʹදͤΔ. 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༻͍ͯ࣍ͷΑ͏ʹදͤΔ. 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 ೔ʹ͓͍ͯ֬ೝ͞Εͨ. ྲྀߦ։࢝೔Λ 2019 ೥ 12 ݄ 1 ೔ (0 ೔໨) ͱ͢Δͱ, ෢׽ࢢ͸ 53 ೔໨͔Β lockdown ʹೖͬͨ. COVID-19 ͷજ෬ظؒ͸໿ 5 ೔2ͳͷͰ travel volume ͷݮগͷӨڹ͸ 58 ೔໨͔ΒղऍՄೳͱͳΔ. λΠͰ֬ೝ ͞Εֳͨ֎ॳͷ঱ྫ͸ 43 ೔໨ʹ౰ͨΔ. 1ଞʹ΋ measurement error/misclassification, model specification, selection bias ͷΑ͏ͳԾఆ͕ஔ͔ΕΔ. 2̏ষͷ࿦จͷ݁ՌͰ͋Γ, ຊߘʹ΋ͦΕ͕ࡌͬͨ Abstract ͸̏ষͰҾ༻ͨ͠. 13
15. 5.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 5 ষ Lockdown ͷఆྔతධՁ Lockdown ʹΑΔ reduced number of expoted cases ͷਪఆͷͨΊʹ counterfactual model Λߟ͑Δ. Lockdown ͕ͳ͍ͱ͖͸ t ೔໨ͷࠃ֎Ͱͷ ൃ঱ൃੜ཰ c(t) Λஔ͘ͱ, ϙΞιϯճؼʹΑͬͯҎԼͷ model ͕ਪఆ͞ΕΔ. ͜Ε͸ 57 ೔໨·Ͱͷσʔλ͔Β fitting ͢Δ. E (c(t)) = c0ert (5.5) ͜ͷͱ͖, c0 ͸ॳظ஋Ͱ r ͸தࠃࠃ֎Ͱͷࢦ਺ؔ਺త૿Ճ཰. h(t) Λ t ೔໨Ͱͷ؍ଌ஋ͱ͢ΔͱҎԼͷࣜʹΑΓ reduced number of exported cases ͕ಘΒΕΔ. V = 67 ∑ 58 (h(t) − E(c(t))) (5.6) ͜ͷͱ͖, ύϥϝʔλͷਪఆ஋ͱͦͷڞ෼ࢄߦྻ͔ΒϞσϧΛཱͯͨ. ڞ෼ࢄߦྻΛ༻͍ͯ 95%CI ͕ಘΒΕΔ. 5.2.2 Reduced probability of major epidemic overseas ͋ΔҰ࣍ײછ঱ྫ͔Βੜͨ͡ೋ࣍ײછ঱ྫ਺ͷ෼෍͕ R0 ͱෛͷೋ߲෼෍ʹै͍ the dispersion parameter Λ k ͱԾఆ͢Δ. 1 ͜ͷͱ͖, the probability of extinction (ײછ঱͕͋Δ 1 ਓ͔Β 2 ࣍ײછ͕ى͖ͣࣗવফ໓ (ઈ໓) ͢Δ֬཰) Λ π ͱ͢Δͱෛͷೋ߲෼෍ʹ͓͚Δظ଴஋ µ ʹͭ ͍ͯײછ͕ੜͨ͡ͱ͖ͷ 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 ͕ى͖Δ֬཰ p ͸ҎԼͷ௨Γ. p = 1 − πn (5.8) ͜͜Ͱ counterfactual ͳγφϦΦͱݱ࣮ͷγφϦΦʹ͍ͭͯݕ౼͢Δ. ۩ମతʹ͸, ҎԼͷ m ͱm Ͱ, travel volume ͷݮগ͕ى͖ͳ͔ͬͨ৔߹ ͕ counterfactual 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 ͷݮগʹΑͬͯಘͨ࣌ؒ஗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. ͦͷͨΊʹ travel volume ʹΑΔྲྀߦൃੜϦεΫͷ૬ରతݮগ ϵr Λ࣍Ͱ ༩͑Δ. 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·Ͱ ͷ࣌ؒͷதԝ஋ΛͦΕͧΕ σ0 ͱσ1 ͱ͢Δͱ, Hi (σi ) = 0.5 ͕੒Γཱͪ2, Αͬͯ Λ(σ0) = ln 2 ͱ Λ(σ1) = ln 2 1 − ϵr ͕Θ͔Δ. ͳ͓, r ͸ COVID-19 ͷ৔߹͸ 0.14 ͱਪఆ͞ΕΔ [5]. ͜ͷΑ͏ʹੈ୅͕࣌ؒࢦ਺ؔ਺ʹै͏ͱ͖3ഒՃ࣌ؒ doubling time td ͸, td = ln 2 r = 4.95 ೔ͱͯ͠ܭࢉ͞Ε Δ. ͕ͨͬͯ࣍͠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) 1ͳͥϙΞιϯ෼෍Ͱ͸ͳ͘ෛͷೋ߲෼෍Λબ୒ͨ͠ͷͩΖ͏͔. ͜Ε͸ҎԼͷΑ͏ͳཧ༝ʹΑΔ. ೔ຊʹ͓͍ͯ COVID-19 ͸ 8 ׂͷਓ͕ 2 ࣍ײછΛىͣ͜͞࢒Γ 2 ׂ͕ײ છ֦େʹد༩͍ͯ͠Δ͜ͱ͕Θ͔͍ͬͯΔ. ͜ͷΑ͏ͳͱ͖͸ฏۉ λ ͱͯ͠ϙΞιϯ෼෍ΛԾఆ͢Δʹ͸෼ࢄ͕େ͖͘ͳͬͯ͠·͏. ͜ΕΛ overdispersion ͱ͍͏. dispersion parameter ͕̍ͷͱ͖ෛͷೋ߲෼෍͸ϙΞιϯ෼෍ʹҰக͢Δ. ෛͷೋ߲෼෍ΛϙΞιϯ෼෍ͷظ଴஋ λ ͷࣄલ෼෍͕ΨϯϚ෼෍Ͱ͋Δͱ͢Δߟ͑ํͰѻͬͨ. 2͜ͷੑ࣭Λར༻͢ΔͨΊʹதԝ஋Λ༻͍͚ͨͩͰݪཧతʹ͸ฏۉ஋Ͱ΋࠷ස஋Ͱ΋Α͍. 34 ষͰߟ͑ͨΑ͏ʹ͜ͷͱ͖ R0 = 1 + rD. 14
16. 5.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 5 ষ Lockdown ͷఆྔతධՁ 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 ͷݮগ͕༩͑ΔӨڹΛఆྔతʹධՁͨ͠. ࣮ࡍʹ࿦จ͕ग़͞Εͨͱ͖ʹಘΒΕͨৄ͍݁͠Ռ ౳ͷߟ࡯ʹ͍ͭͯ͸, ࿦จͷ Results ΍ Discussion ΛಡΜͰ΋Β͍͍ͨ. ࣍ͷϖʔδʹ࿦จͷ Abstract ͷΈࡌ͓ͤͯ͘. Abstract   The impact of the drastic reduction in travel volume within mainland China in January and February 2020 was quantified with respect to reports of novel coronavirus (COVID-19) infections outside China. Data on confirmed cases diagnosed outside China were analyzed using statistical models to estimate the impact of travel reduction on three epidemiological 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, corresponding to a 70.4% reduction in incidence compared to the counterfactual 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.   15
17. ୈ6ষ Overdispersionͷਪఆ ໨త   ͜ͷষͰ͸, Endo A; Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Abbott S, Kucharski AJ, Funk S. Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. Wellcome Open Res. 2020;5:67. Published 2020 Jul 10. ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆͨ͠.   6.1 ४උ 6.1.1 Overdispersion ͱ Rt ײછμΠφϛΫεΛޠΔ্Ͱ Rt ΍ R0 ͕Α͘஫໨͞ΕΔ͕ɺͦΕ͚ͩͰ͸ෆे෼ͳ৔߹͕͋Δ. Rt ͸ t ࣌఺ʹ͓͚ΔҰ࣍ײછऀ͕ײછੑظؒ ʹੜΈग़͢ೋ࣍ײછऀ਺ͷฏۉΛද͢. ͔͠͠, COVID-19 ͷΑ͏ʹຆͲͷҰ࣍ײછऀ͸গͳ͍ਓ਺ʹ͔͠ײછͤͣ͞, ୅ΘΓʹ super-spreader ͕ଘࡏ͍ͯ͠Δ৔߹͸Ͳ͏Ͱ͋Ζ͏͔. ͜ΕʹΑͬͯ, super-spreader Λ཈͑Δ͜ͱʹूத͢Δ͔Ͳ͏͔͕ํ਑ͱܾͯ͠ఆ͞ΕΔ͕ Rt Λ͍͘Β ோΊͨͱ͜ΖͰฏۉΛදͦ͢Ε͔Β͸Ұ෦ͷ super-spreader ͷଘࡏ͸ݟ͑ͯ͜ͳ͍. ͜Ε͸, ϙΞιϯ෼෍ͷظ଴஋ λ ͕ΨϯϚ෼෍ʹै͏ͱ͍ ͏ղऍͱҰக͠, Αͬͯෛͷೋ߲෼෍ negative binomial distribution ͱͯ͠ߟ͑ΒΕΔ. ͜ͷͱ͖ͷ dispersion parameter k Λ overdispersion ͱ͍ ͏. COVID-19 Ϋϥελʔରࡦ൝͕શ਺೺Ѳ͔Β super spreading events ͷ controll ʹํ਑స׵ͨ͠ͷ͸͜ͷ overdispersion Λࠜڌͱ͍ͯ͠Δ. 6.2 ࣮ࡍʹߦΘΕͨղੳ 6.2.1 Overdispersion ਪఆ Offspring distribution ͸ NB ʹै͍, final cluster size ͷ PMF ͸ҎԼͷΑ͏ʹදͤΔ [7]. c(x; s) = P(X = x; s) = ks kx + x − s ( kx + x − s x − s ) ( R0 k )x−s ( 1 + R0 k )kx+x−s (6.1) ͕ͨͬͯ͠, ongoing epidemic Λද͢ PMF ͸࣍ͷ௨Γ. co (x; s) = P(X ≥ x; s) = 1 − x−1 ∑ m=0 c(m; s) (6.2) ྲྀߦ͕ਐߦ͍ͯ͠Δࠃͷू߹ A ͱ, final size ʹୡͯ͠ऩଋ͍ͯ͠Δࠃͷू߹ B1Λ͋Θͤͯ, k ʹ͍ͭͯͷ໬౓ΛཱͯΔ. L(R0 , k) = ∏ i∈A P(X = xi ; si ) ∏ i∈B P(X ≥ xi ; si ) (6.3) 17 days without any new confirmation of cases. 16
18. 6.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 6 ষ Overdispersion ͷਪఆ R0 ΛมԽͤ͞ͳ͕Β k ΛಘΔ. ͜ͷͱ͖, ෆ࣮֬ੑͱͯ͠ 95%CrI Λߟ͑Δ. ͨͩ͠, prior distribution ͱͯ͠ k ͷٯ਺1ʹ͸ඪ४ภࠩ 10 ͷ weakly-informed half-normal distribution, R0 ʹ͸ฏۉ 3, ඪ४ภࠩ 5 ͷ weakly-informed nomal distribution Λ༩͑ͨ. ͜ͷͱ͖ͷ CrI ಋग़ʹ͸ MCMC ͱͯ͠, R ͷ”LaplacesDemon”package ʹΑΔ hit-and-run Metropolis algorithm ͕༻͍ΒΕͨ.2 k → ∞ ͷͱ͖, NB ͸ Poisson distribution ʹҰக͢Δ. Αͬͯ, offspring distribution ʹ NB ͱ Poisson branching process ͷ྆ऀΛߟ͑3WBIC Λ༻͍ͯ൚ԽੑೳΛධՁͨ͠. ͜ͷ݁Ռ, NB ͷ΄͏͕ WBIC ͕খ͘͞ͳΓ࠾୒͞Εͨ. . ͜ͷϞσϧͰ͸, k → 0 ͱ R0 → 0 ʹΑͬͯઆ໌͞Εͯ͠·͏ͨΊ R0 ͱ k ͷ joint estimation ͸࣮ݱͰ͖ͳ͍. 6.2.2 Proportion responsible for 80% of secondary transmission 2 ࣍ײછͷ 80%ʹؔ༩͢Δײછऀͷׂ߹ p80% Λਪఆ͢Δ. ͜ͷͱ͖, (6.2) ͱಉ༷ʹͯ͠, 1 − p80% = ∫ X 0 NB ( ⌊x⌋; k, k R0 + k ) dx (6.4) ͱදͤΔ. ͨͩ͠, NB ( x; k, k R0 + k ) ͸ฏۉ R0 ͷ negative binomial PMF Ͱ͋Γ, X ͸ҎԼͷؔ܎ࣜΛຬͨ͢. 1 − 0.8 = 1 R0 ∫ X 0 ⌊x⌋NB ( ⌊x⌋; k, k R0 + k ) dx (6.5) ͜ͷӈล͸؆୯ͷͨΊҎԼͷΑ͏ʹมܗ͢Δ. 1 R0 ∫ X 0 ⌊x⌋NB ( ⌊x⌋; k, k R0 + k ) dx = ∫ X−1 0 NB ( ⌊x⌋; k + 1, k R0 + k ) dx (6.6) ͜ΕʹΑΓ, p80% ͱ 95% CrI ͕ಘΒΕΔ. 6.2.3 Simulation of the effect of the underreporting ͜ͷਪఆͷ limitation ͱͯ͠ dataset ͷ underreporting rate ʹӨڹΛड͚͏Δ͜ͱ͕͋ΔͨΊ, ͦͷӨڹʹ͍ͭͯݕূͨ͠. ·ͣ, reporting probability ͕͢΂ͯͷ cases Ͱ౳͍͜͠ͱΛԾఆ͠ binomial sampling ʹΑͬͯ observed data ͕ಘΒΕΔ͜ͱΛߟ͑ͨ. ͜ͷͱ͖, si ͱ x0 i Λ country i ʹ͓͚Δ observed cases ͱ true cases ͷਓ਺ͱ͠ qi Λ reporting probability ͱͯ͠ҎԼͷΑ͏ʹॻ͚Δ. si ∼ Binom(x0 i , qi ) (6.7) 1ٯ਺ʹϧʔτΛ͚ͭΔ΂͖ͳͲٞ࿦ͷ༨஍͕͋Δ. 2Thinning ͸ 500 ຖͰ 10,000 ճߦΘΕͨ. 3k = 1010 ͱͯ͠ղੳ͸ߦΘΕͨ. 17
19. 6.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 6 ষ Overdispersion ͷਪఆ 6.2.4 ·ͱΊ Abstract   Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmis- sion chains of human-to-human transmission suggest high basic reproduction number R0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribu- tion. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals ( 10%). A joint estimation yielded likely ranges for R0 and k (95% CrIs: R0 1.4-12; k 0.04-0.2); however, the upper bound of R0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on super- spreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.   18
20. ୈ7ষ Interventionͱײછऀ਺ͷධՁ ໨త   ͜ͷষͰ͸, Nishiura H. Backcalculating the Incidence of Infection with COVID-19 on the Diamond Princess. J Clin Med. 2020;9(3):657. Published 2020 Feb 29. ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆͨ͠.   7.1 ४උ 7.1.1 Richards model ϩδεςΟ οΫۂઢ, ϩδεςΟ οΫํఔࣜΛΑΓҰൠԽͨ͠ generalized logistic function Λผ໊ Richards model ͱ͍͏. ϩδεςΟ οΫۂ ઢ͕มۂ఺Λڥʹରশͳͷʹରͯ͠ Richards model ͸ରশੑ͕ͳͯ͘΋දͤΔΑ͏ʹ͞Ε͍ͯΔ. J(t) = K {1 + g exp(−r(t − ti ))}1 g (7.1) ͨͩ͠, ϩδεςΟ οΫํఔࣜ͸࣍ͷ௨Γ. J(t) = K 1 + (K/J(0) − 1) exp(−rt) (7.2) g ͸ y ্࣠, ·ͨ͸ t = ti ʹ͓͚Δײછऀ਺ΛఆΊΔ parameter Ͱ͋Δ. ͜ͷ Richards model ͸ 4 ͭͷ parameter ͕͋Δ1. 7.2 ࣮ࡍʹߦΘΕͨղੳ 7.2.1 Backcalculation ͱ forecasting ײછऀ਺ it ͱજ෬ظؒͷ PDF ft Ͱ৞ΈࠐΈ convolution Λͯ͠ backcalculation ͢Δͱ, ൃ঱೔ͷ incidence ͷظ଴஋Λߟ͑Δ͜ͱͰ, ײછऀ ਺ it Λ step function ͱ MLE ͔ΒಘΔ͜ͱ͕Ͱ͖Δ. ͜͜Ͱ, COVID-19 ʹؔͯ͠͸ ft ͸ฏۉ 5, ඪ४ภࠩ 3 ͷର਺ਖ਼ن෼෍ʹ fit ͢Δ͜ͱ͕Θ ͔͍ͬͯΔ. E(ct ) = t−1 ∑ s=1 it−s fs (7.3) Richards model Λ dataset ʹ fitting ͤ͞Δ͜ͱͰ parameter ͕ಘΒΕΔ. E(Ct ) = K {1 + s exp(−a(t − ti ))}1 s (7.4) ͜͜Ͱ, հೖ intervention ͕ೖΔલ·Ͱͷ dataset ͱೖͬͨޙ΋ؚΊͨ dataset ͷ 2 ͭΛ༻͍ͯͦΕͧΕ fitting ͤͯ͞ forecasting ͢Δͱ, intervention ʹΑͬͯ๷͙͜ͱͷͰ͖ͨײછऀ਺͕ಘΒΕ, intervention ͷධՁʹͭͳ͕Δ. 1x ࣠ʹԊͬͨฏߦҠಈΛ͢Δ߲Λߟྀ͢Δͱ 5 ͭͷ parameters. 19
21. 7.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 7 ষ Intervention ͱײછऀ਺ͷධՁ 7.2.2 ·ͱΊ Abstract   To understand the time-dependent risk of infection on a cruise ship, the Diamond Princess, I estimated the incidence of infection with novel coronavirus (COVID-19). The epidemic curve of a total of 199 confirmed cases was drawn, classifying individuals into passengers with and without close contact and crew members. A backcalculation method was employed to estimate the incidence of infection. The peak time of infection was seen for the time period from 2 to 4 February 2020, and the incidence has abruptly declined afterwards. The estimated number of new infections among passengers without close contact was very small from 5 February on which a movement restriction policy was imposed. Without the intervention from 5 February, it was predicted that the cumulative incidence with and without close contact would have been as large as 1373 (95% CI: 570, 2176) and 766 (95% CI: 587, 946) cases, respectively, while these were kept to be 102 and 47 cases, respectively. Based on an analysis of illness onset data on board, the risk of infection among passengers without close contact was considered to be very limited. Movement restriction greatly reduced the number of infections from 5 February onwards.   20
22. ୈ8ষ Asymptomatic rateਪఆ ໨త   ͜ͷষͰ͸, Mizumoto K, Kagaya K, Zarebski A, Chowell G. Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan, 2020. Euro Surveill. 2020;25(10):2000180. ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆͨ͠.   8.1 ࣮ࡍʹߦΘΕͨղੳ 8.1.1 Statistical modeling ༻͍ͨ dataset ͸ right-censored ͞Ε͍ͯͯ, ͭ·Γ঱ঢ়͕ग़͍ͯͳ͍ਓ͸ asymptomatic ·ͨ͸·ͩൃ঱͍ͯ͠ͳ͍ਓͷ 2 छྨΛؚΉ. ײછ ͨ͠Մೳੑͷ͋Δ time interval Λ [ai , bi ] ͱ͠, c ͸ൃ঱͕؍࡯͞Εͯଧͪ੾Γʹͳͬͨ೔Λද͢. ·ͨ, i ͕ײછͨ͠೔Λ Xi ͱ͠, infection to onset ͱ͍͏ time delay Λજ෬ظؒ incubation period Di , FD ͸ͦͷ CDF ͱ͢Δ. ͜͜Ͱ, c ࣌఺ʹ঱ঢ়͕ग़͍ͯͳ͍֬཰͸ asymptomatic rate p Λ༻͍ͯҎԼͷΑ͏ʹදͤΔ. g(x, p) =        p + (1 − p)(1 − FD (c − x)) (if they do not have symptoms) (1 − p)FD (c − x) (if they do have symptoms) (8.1) (8.2) ͜ͷͱ͖, likelihood ͸ҎԼͷ௨Γ. L(X, p) = ∏ i g(Xi , p) (8.3) a b Infection onset D c X FD ͸જ෬ظؒͷ CDF Ͱ͋Γ, PDF ͸ Weibull distribution (mean = 6.4days, S D = 2.3days) Ͱ͋Δ [11]. p ͱ Xi ͷ joint estimation Λߦͬͨ. Xi ͷࣄલ෼෍Λ [ai , bi ] ͷҰ༷෼෍ͱͯ͠༩͑Δ. rstan Λ༻͍ͯ, HMC (Hamiltonian Monte carlo) ͷ NUTS ͔Β parameters ͱͦͷ 95% CrI Λਪ ఆͨ͠. ͜ͷͱ͖, 20,000 ճͷ iteration Λ burn in ͨ͠ޙ, 100,000 ճΛ 3 ܥྻಘͨ. 8.1.2 Missing data and sensitivity analysis Reported date ͕ෆ໌ͳ cases ͕ 76 ྫத 35 ྫແ঱ީͱͯ͋ͬͨ͠. ܽଛॲཧͱͯ͠, ແ࡞ҝׂ෇ΛͦΕͧΕߦ͍, ͦΕΛ 200 ճ܁Γฦͨ͠. ͜ΕʹΑΓ, uncertainty bounds ΛಘΔͨΊͷ percentile points Λಘͨ. ײ౓෼ੳͰ͸, જ෬ظؒͷฏۉΛ 4.4 days − 8.4 days ͱͯ͠มԽͤ͞Δ ͜ͱͰ, p ΁ͷӨڹΛௐ΂ͨ. 8.1.3 ·ͱΊ Abstract   On 5 February 2020, in Yokohama, Japan, a cruise ship hosting 3,711 people underwent a 2-week quarantine after a former passenger was found with COVID-19 post-disembarking. As at 20 February, 634 persons on board tested positive for the causative virus. We conducted statistical modelling to derive the delayadjusted asymptomatic proportion of infections, along with the infections ʟtimeline. The estimated asymptomatic proportion was 17.9% (95% credible interval (CrI): 15.5–20.2%). Most infections occurred before the quarantine start.   21
23. ୈ9ষ Back-projectionʹΑΔCOVID-19ͷײછऀ਺ٴͼRt ਪఆ ໨త   ͜ͷষͰ͸, github ʹΫϥελʔରࡦ൝੢ӜνʔϜʹΑͬͯެ։͞Εͨ https://github.com/contactmodel/COVID19-Japan-Reff ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔ղઆͨ͠.   9.1 Back-projection ֓ཁ 9.1.1 Framework Back-projection ͱ͸, reported cases ΍ lab confirmed cases ͔Β onset cases ΍ infection cases Λਪఆ͢Δ͜ͱΛࢦ͢1. ͜ͷͱ͖, ৞ΈࠐΈ convolution ʹΑͬͯද͞ΕΔ໬౓͔Β MLE ͷࢹ఺ͱ Bayesian framework ͷࢹ఺Ͱ Rt ͸ਪఆ͞ΕΔ͕, likelihood ͷෳࡶ͞ނʹײછऀ਺͸ Expectation-Maximization algorithm (EM algorithm) Λ༻͍, ͞Βʹฏ׈Խ smoothing Λߦ͏ (EMS algorithm). ͜ͷࡍ, Bayesian framework Ͱ͸ smoothing ͸ϋΠύʔύϥϝʔλͱ͞ΕΔ. ͳ͓, back-projection Λ༻͍ͨ಺༰͸ [15][16][17] ͕ৄ͍͠. 9.1.2 EM algorithm (EMS algorithm) ӅΕม਺͕͋Δ໬౓Λ࠷େԽ͢Δͱ͖ʹ EM algorithm2͕ߟ͑ΒΕΔ. ໨ඪ͸΋ͪΖΜ, argmax θ p(x | θ) (9.1) ໬౓ؔ਺Λ෼ղ͢Δͱ ln p(x | θ) = L(q, θ) + KL(q∥p) (9.2) ͱදͤΔ (ূ໌͸ޙ൒ʹهड़). ͨͩ͠ L ͸ӅΕม਺ͷ PDF q(z) Λ༻͍ͯ L(q, θ) = ∫ q(z) ln ( p(x, z | θ) q(z) ) dz = ln p(x | θ) − KL(q∥p) (9.3) ͱ͍͑, KL(q∥p) ͸ q(z) ͱӅΕม਺ʹؔ͢Δࣄޙ෼෍ p(z | x, θ) ͷ KL divergence Ͱ͋Δ. KL divergece ͕ 0 Ҏ্ͳ͜ͱ͔Β L(q, θ) ͸ෆ׬શ σʔλͷର਺໬౓ͷԼݶͱ͞ΕΔ (Gibbs’ inequality). ͜ͷͱ͖ʹର਺໬౓Λ࠷େԽ͢Δ͜ͱΛߟ͑Δͱ, KL divergence Λ 0 ͱ͢Δͱ͖ͷظ ଴஋ΛٻΊ , ͦͷظ଴஋Λ࠷େԽ, ͢ͳΘͪෆ׬શσʔλͷର਺໬౓ͷԼݶΛ࠷େԽͤ͜͞ͷҰ࿈ͷૢ࡞Λஞ࣍తʹߦͬͯऩଋ͢Δͱ͖ͷ parameter ΛಘΔ. E step ʹ͍ͭͯ͸ KL divergence ͕ 0 ͱͳΔ৚݅Λߟ͑Ε͹Α͍ͷͰ q(z) = p(z | x, θ) Ͱ͋Δ͔Β ln p(x | θ) = L(q, θ) = ∫ q(z) ln ( p(x, z | θ) q(z) ) dz = ∫ p(z | x, θ) ln ( p(x, z | θ) p(z | x, θ) ) dz = Q(θ, θt ) + constant (9.4) Ͱ͋Γ, Q(θ, θt ) ͸׬શσʔλͷର਺໬౓ͷظ଴஋Ͱ͋Δ. • Expectation step ӅΕม਺ Z ͷ͋Δ໬౓͸௚઀࠷େԽͰ͖ͳ͍ͨΊ, ׬શσʔλ3ͷର਺໬౓Λߟ͑Δ. ӅΕม਺ʹؔ͢Δࣄޙ෼෍͸ಘΔ͜ͱ͕Ͱ͖Δͷ ͰͦΕΛ༻͍ͯલड़ͷର਺໬౓ͷظ଴஋ΛٻΊΔ. Q(θ, θt ) = 1 T T ∑ k=1 (∫ p(z | x; θt ) ln p(x, z; θ)dz ) (9.5) ͨͩ͠, p(x; θ) = ∫ p(x, z; θ)dz (9.6) Ͱ͋Δ. 1༧ଌͱҰݴʹݴͬͯ΋, projection ͱ forecasting ͷ 2 छྨ͋Γ۠ผ͞Ε͍ͯΔ. 2ࠞ߹Ψ΢ε෼෍ͳͲ͸ཧղΛ͢͢ΊΔ্ͰΘ͔Γ΍͍ͩ͢Ζ͏͕, ͦΕҎ֎ͷ෼෍ʹ͍ͭͯ΋ EM algorithm Λߟ͑Δ͜ͱ͕Ͱ͖Δ. 3؍ଌσʔλ X ͱજࡏม਺ Z ͷͲͪΒ΋͕͋Δσʔλ. ෆ׬શσʔλͱ͸؍ଌσʔλͷΈͷ͜ͱΛࢦ͢. 22
24. 9.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 9 ষ Back-projection ʹΑΔ COVID-19 ͷײછऀ਺ٴͼ Rt ਪఆ • Maximization step E step Ͱಘͨظ଴஋ͷ࠷େͱͳΔ parameter ΛಘΕ͹ྑ͍ͷͰ, θt+1 = argmax θ Q(θ, θt ) (9.7) ͜͜ʹฏ׈ԽΛߟྀ͢Δ͜ͱͰ EMS algorithm ͱͳΔ. 9.1.3 Back-projection ͜͜Ͱ͸ྫͱͯ͠ onset cases µt ͔Β infection cases θs Λ back-projection ͢Δ͜ͱΛߟ͑Δ. The time of infection Λ incubation period Ͱ৞Έ ࠐΈ convolution Λ͢Δ͜ͱͰ onset date ͕ಘΒΕΔ. Back-projection Ͱ͸جຊతʹ 12 ষͷਪఆجૅʹ͓͚Δ backcalculation ͷߟ͑Λ༻͍Δ. infection cases ͕ඇఆৗ Poisson աఔʹै͏໬౓ͱͳΔ poisson variables Ͱ͋Δͱ͖, onset cases ΋ poisson distribution ʹै͏. µt = t ∑ s=1 θs ft−s (9.8) µ ∼ τ ∏ t=1        t ∑ s=1 θs ft−s        at exp       − t ∑ s=1 θs ft−s        (9.9) θ ∼ Pois(µt ) (9.10) Αͬͯର਺໬౓ؔ਺͸ҎԼͷ௨Γ. ℓ(θ; a) = τ ∑ t=1 (−µt + at ln µt ) + constant (θ1 , θ2 , ..., θτ ≥ 0) (9.11) ͜Ε͸, ͢΂ͯͷ parameters ͕ඇෛͰ͋Γ, ͜ͷΑ͏ʹજࡏม਺͕ଘࡏ͢Δ͜ͱ͔Β͜ͷ໬౓Λ༻͍ͨ MLE ͸ద͓ͯ͠Βͣ EM algorithm Λ ༻͍Δ. ͜͜Ͱ, ٞ࿦ΛਐΊ΍͘͢͢ΔͨΊʹ at ΛҎԼͷΑ͏ʹஔ͘. at = t ∑ s=1 nst (9.12) ͜͜Ͱ͍͏ nst ͸ s ೔ʹײછͨ͠ਓͰ t ೔ʹ diagnose ͞Εͨ cases Λද͢. ɹ nst ͸؍ଌ஋Ͱ͋Δ͕, nst ͷ෼෍ʹै͏ random variable Nst Λஔ ͘ͱ, ͜ͷ conditional distibution ͸ mean ͕ at θs ft−s µt ͷ binomial distribution Ͱ͋Δ. ͜ͷͱ͖׬શσʔλ complete data ͷର਺໬౓ؔ਺͸࣍ͷ ௨Γ. ℓc (θ; n) = τ ∑ t=1 t ∑ s=1 (−θs ft−s + nst ln θs ) + constant (θ1 , θ2 , ..., θτ ≥ 0) (9.13) ͜ͷͱ͖, E step ͸ҎԼͷ௨Γ. Q(θ, θ( j)) = τ ∑ t=1 t ∑ s=1       −θs ft−s + at θ( j) s ft−s µ( j) t ln θs        (θ1 , θ2 , ..., θτ ≥ 0) (9.14) ͕ͨͬͯ͜͠ΕΛ࠷େԽ͢Δͱ͖ͷ parameter ͷਪఆ஋Λ M step ͱͯ͠༩͑Ε͹Α͍ͷͰ࣍ͷ௨Γ. θ(j+1) s = θ(j) s Fτ−s τ ∑ t=s at ft−s µ( j) t (s = 1, 2, ..., τ) (9.15) Fτ−s = τ−s ∑ d=0 fd = τ ∑ t=1 t ∑ s=1 ft−s (9.16) ͜ͷͱ͖, ਪఆ஋͸ at ʹ sensitive Ͱ͋Δ͕࣮ࡍͷײછऀ਺͸࣌ؒܦաͱͱ΋ʹ׈Β͔Ͱ͋Δ͸ͣͳͷͰฏ׈Խ smoothing Λߦ͏. ͜͜Ͱ, λ( j+1) s (s = 1, 2, ..., τ) Λ (9.15) Ͱಘͨਪఆ஋ͱ͢Δͱ S step ͸ҎԼͷ௨Γ. θ(j+1) s = k ∑ i=0 wi λ(j+1) (s+i−k)/2 (9.17) ͨͩ͠, s < 0 Ͱλ(j+1) s = 0, s ≥ τͰλ( j+1) s = λ( j+1) τ ͱ͠, ∑ k i=0 wi = 1 Ͱ i = 0, 1, ..., k Ͱ symmetrical ͱ͢ΔՃॏฏۉΛߟ͑ͨ. Smoothing Λߦ͏͜ ͱ͸ਪఆ஋ͷෆ҆ఆੑΛमਖ਼͢Δ͚ͩͰͳ͘ EM algorithm ͷऩଋ·Ͱʹඞཁͳ൓෮ճ਺ iteration Λେ͖͘ݮগͤ͞Δ͜ͱ͕Ͱ͖Δ. ͜ͷΑ ͏ʹ back-projection ʹ͓͚Δ EM algorithm ͸͕ࣜ୯७Ͱ͋Δ͜ͱ͔Β computation ͕༰қͰ͋Γ, EMS algorithm Ͱ͋ͬͯ΋ಉ༷1Ͱ͋Δ. 9.2 ࣮ࡍʹߦΘΕͨղੳ Back-projection ʹΑͬͯ, • lab confirmation date ͔Β illness onset date Λਪఆͨ࣍͠ʹ, • illness onset date ͔Β time of infection 1͕ͨͬͯ͠, R ͷ”surveillance”package Ͱ back-projection Λߦ͏͜ͱ͕Ͱ͖Δ. 23
25. 9.3Figures for the estimated Rt and incidence ୈ 9 ষ Back-projection ʹΑΔ COVID-19 ͷײછऀ਺ٴͼ Rt ਪఆ Λਪఆͨ͠. ͜ͷͱ͖, domestic cases ͱւ֎͔Βྲྀೖ͖ͯͨ͠ imported cases Λ໌֬ʹ෼͚ͯ onset date Λਪఆ͢Δ. ͜ΕΛ༻͍ͯ࠶ੜํఔࣜ renewal equation ͔Β Rt Λಘͨ. Back-projection ʹ͓͍ͯ༻͍ͨ෼෍ʹ͍ͭͯड़΂Δ. Incubation period ͸ previous study ͷ݁Ռ [1] ΑΓ mean ͕ 5.6 days, SD ͕ 3.9 days ͷ lognormal distribution ͱͨ͠. Illness onset ͔Β reporting ·Ͱͷ time delay ͸ right-truncation Λߟྀͯ࣍͠ͷΑ͏ʹ໬౓ؔ਺Λߟ͑ͨ. L(θ | Sk , Ok , T) = K ∏ k=1 h(S k − Ok | θ) H(T − OK | θ) (9.18) • Sk : ใࠂ೔ reporting date • Ok : ൃ঱೔ date of ilness onset • T: ؍ଌ೔ the latest calender day of observation • h: PDF of Weibull distribution • H: CDF of Weibull distribution • θ: scale parameter ͱ shape parameter લड़ͨ͠Α͏ʹ onset date Λ back-projection ͰಘΔͨΊʹ্ड़ͷ reporting delay Ͱ৞ΈࠐΉ. ͦͷޙ, ͜͜Ͱಘͨ onset date distribution Λ incubation period Ͱ৞ΈࠐΜͰ back-projection Λ΋͏Ұ౓ߦ͏. ͯ͞, ͜ΕΑΓ࣮ߦ࠶ੜ࢈਺ effective reproduction number Rt Λߟ͑Δ. Renewal equation ΑΓ, i(t) = ∫ ∞ 0 A(t, τ)i(t − τ)dτ (9.19) (A(t, τ) = R(t)g(τ)) (9.20) ͨͩ͠, τ ͸ײછྸ infection age1Ͱ, g(t) ͸ੈ୅࣌ؒ generation time ͷ PDF Ͱ͋Δ. ͳ͓, COVID-19 ͸ generation time (infection to infection) ͕ۙࣅతʹ serial interval (onset to onset) ͱͰ͖ΔͨΊ previous study ͷ݁Ռ [19] ΑΓ mean ͕ 4.8 days, SD ͕ 2.3 days ͷ Weibull distribution Λ༻͍ͨ2. ·ͱΊͯ࣍ͷΑ͏ʹදͤΔ. i(t) = R(t) ∫ ∞ 0 g(τ)i(t − τ)dτ (9.21) ͜ΕΛ༻͍ͯ, Rt Λߟ͑Δ. ͳ͓, discrete time ͱ͢Δ. E(idomestic (t)) = Rt t−1 ∑ τ=1 itotal (t − τ)g(τ) F(T − t) F(T − t + τ) (9.22) ͜ͷͱ͖, F ͸ײછ͔Βใࠂ·Ͱͷ time delay ͷ CDF Ͱ͋Γ, right-truncation Λߟྀͯ͠௥Ճ͞Ε߲ͨͰ͋Δ. Right-censored ͳ৔߹ͱશͯ؍ ࡯͞Εͨ৔߹ʹର͢Δׂ߹Λࣔ͢. Ҏ্ΑΓ, domestic ͷײછऀ਺Λ Poisson(E(idomestic (t))) ͱߟ͑ͯ࣍ͷΑ͏ʹ໬౓͕දͤΔ. L(Rt ;Cdomestic (t)) = T ∏ t=1 exp(−E(idomestic (t)))(E(idomestic (t)))idomestic(t) idomestic (t)! (9.23) Rt ͸ MLE ͰಘΔ͜ͱ΋ Bayesian method Λ༻͍Δ͜ͱ΋Ͱ͖Δ͕, MLE ͷ৔߹͸ 95%CI Λ profile likelihood ͔Βಘͯ, Bayesian framework ͷ৔߹͸ 95%CrI Λ MCMC Λ༻͍ͯಘΔ. ղੳ͸ RStan Λ༻͍ͯߦΘΕͨ. ͳ͓, ຊདྷ͸ back-projection ʹ͓͍ͯ reporting delay Λߟྀ͢΂ ͖Ͱ͋Γ, back-projection ͷෆ࣮֬ੑ΋Ճ͑ͨ Rt ΛಘΔ΂͖Ͱ͋Δ͕, ݁Ռʹେ͖ͳࠩ͸ͳ͍͜ͱ͕ଟ͍. 9.3 Figures for the estimated Rt and incidence ੨৭͸࣮ߦ࠶ੜ࢈਺ Rt ͷਪఆ݁Ռʹجͮ͘ਪҠΛࣔ͠, ബ͍੨৭͸ 95%CrI Λද͢. ࠇ৭ͷ๮άϥϑ͸ imported cases ͷ the time of infection ͷਪఆ݁Ռʹجͮ͘ਪҠͰ͋Γ, ԫ৭͸ domestic cases Λࣔ͢. ͳ͓, ਤ 9.1 ͸શͯͷ case Λ plot ͕ͨ͠, ਤ 9.2 ͸ unknown illness onset date Ͱ ͋Δ cases Λআ֎ͯ͠ plot ͨ͠΋ͷͰ͋Δ. 1ײછੑͷ͋Δظؒͷ͜ͱ. 2͜ͷਪఆ͸ 3 ষͷ಺༰ [1] ͱಉ͡Ͱ͋Δ. 24
26. 9.3Figures for the estimated Rt and incidence ୈ 9 ষ Back-projection ʹΑΔ COVID-19 ͷײછऀ਺ٴͼ Rt ਪఆ ਤ 9.1: ਤ 9.2: 25
27. ୈ10ষ Competing risk modelʹΑΔHIV ໨త   ͜ͷষͰ͸, Nishiura H. Estimating the incidence and diagnosed proportion of HIV infections in Japan: a statistical modeling study. PeerJ. 2019;7:e6275. Published 2019 Jan 15. ʹ͓͍ͯͲͷΑ͏ͳ͜ͱ͕ߟ͑ΒΕ͍ͯΔ͔എܠΛ֬ೝͨ͠ޙʹղઆͨ͠.   10.1 ४උ 10.1.1 Competing risk model Competing risk model ͷ McKendrick equation ͸ҎԼͷ௨Γ. ( ∂ ∂t + ∂ ∂τ ) h(t, τ) = −(α(t) + ρ(τ))h(t, τ) (10.1) ͨͩ͠, t ͸ײછͨ͠ͱ͖ͷ calender time Ͱ͋Γ, τ ͸ײછྸ infection age 1Λද͢. ͜ͷͱ͖, λ(t) h(t, 0) Ͱ͋Δ. ͜ͷ model ʹ͓͍ͯ, જ෬ ظؒͷ PDF f(τ) ͸ survival function ͱ hazard function ͷੵͱͯ࣍͠ͷΑ͏ʹදͤΔ. f(τ) = ρ(τ) exp ( − ∫ τ 0 ρ(y)dy ) (10.2) (10.1) Λղ͘ͱ࣍ͷ௨Γ. h(t, τ) = λ(t − τ) exp ( − ∫ t t−τ α(x)dx − ∫ τ 0 ρ(y)dy ) (10.3) 10.1.2 Weibull distribution ͱ survival time analysis Weibull distribution ͷ PDF ͸࣍ͷ௨Γ. f(t) = k η ( t η ) k−1 exp       − ( t η ) k        (10.4) ͜͜Ͱ, k ͸ shape parameter (Weibull coefficient) Ͱ η ͸ scale parameter Ͱ͋Δ. ฏۉ͸ ηΓ ( 1 + 1 m ) Ͱ͋Γ, k = 1 Ͱࢦ਺෼෍ͱҰக͢Δ. Weibull distribution Λ༻͍ͨੜଘ࣌ؒղੳ, ͢ͳΘͪ Weibull model Λߟ͑Δͱ, hazard function Λ(t) ͱ survival function S (t) ͸ͦΕͧΕ࣍ͷΑ͏ʹද ͤΔ. Λ(t) = k η ( t η ) k−1 (10.5) S (t) = exp       − ( t η ) k        (10.6) ͳ͓, Weibull distribution ͸෺ମͷڧ౓΍ମੵΛࣔ֬͢཰෼෍ͱͯ͠ఏҊ͞Εͨ΋ͷͳͷͰ, λ(t) Λ࣌ؒґଘͷނো཰, S (t) Λނো͠ͳ͍֬཰ ͱͯ͠৴པ౓ͱݺͿ͜ͱ΋͋Δ. 1ײછ͔Βͷܦա࣌ؒͷ͜ͱ. පظྸ disease age ͸ൃ঱͔Βͷܦա࣌ؒ. 26
28. 10.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 10 ষ Competing risk model ʹΑΔ HIV 10.2 ࣮ࡍʹߦΘΕͨղੳ 10.2.1 Derivation of likelihood using a mathematical model HIV ͷ diagnosis ͕ى͖Δൃੜ཰ incidence Λ u(t) ͱ͢ΔͱҎԼͷΑ͏ʹදͤΔ. u(t) = ∫ t 0 α(t)h(t, s)ds = ∫ t 0 α(t)λ(t − s) exp ( − ∫ t t−s α(x)dx − ∫ s 0 ρ(y)dy ) ds (10.7) ಉ༷ʹͯ͠, AIDS ͷ incidence Λ a(t) ͱ͢ΔͱҎԼͷΑ͏ʹද͞ΕΔ. a(t) = ∫ t 0 ρ(t)h(t, s)ds = ∫ t 0 ρ(t)λ(t − s) exp ( − ∫ t t−s α(x)dx − ∫ s 0 ρ(y)dy ) ds (10.8) ͜ͷ (10.7) ͱ (10.8) ͸ competing risk model Λද͠, ͜ͷ̎ࣜΛ༻͍ͯ HIV ͷ incidence ΛܭࢉͰ͖, extended backcalculation ͱݺ͹ΕΔ. 10.2.2 Statistical model and estimation (10.7), (10.8) Λ discrete model ʹॻ͖௚͢ͱҎԼͷ௨Γ. ͨͩ͠, Tayler ల։ͷ 1 ࣍ۙࣅΛߟ͑ͨ. ut = t ∑ s=1 λt−s αt t−1 ∏ x=t−s+1 (1 − αx ) s−1 ∏ y=1 (1 − ρy ) (10.9) at = t ∑ s=1 λt−s ρt t−1 ∏ x=t−s+1 (1 − αx ) s−1 ∏ y=1 (1 − ρy ) (10.10) Epi curve ͕ࣄલʹΘ͔͍ͬͯͳ͍ͷͰ λt ͱ αt ͸ step function ͱͯͦ͠ΕͧΕҎԼͷΑ͏ʹද͢. λt =                          λ1 for t < 1989 λ2 for 1989 ≤ t < 1993 . . . λ9 for 2013 ≤ t (10.11) αt =                          α1 for t < 1989 α2 for 1989 ≤ t < 1993 . . . α9 for 2013 ≤ t (10.12) ͜͜Ͱ, જ෬ظؒʹ͍ͭͯ discrete model Λߟ͑Δͱ, (10.2) ΑΓҎԼͷ PMF ΛಘΔ. ρs s−1 ∏ y=1 (1 − ρy ) (10.13) ·ͨ, HIV ͷ incubation period ͸ Weibull distribution Ͱද͞ΕΔͷͰ, discrete Weibull model ͱ continuous version Λߟ͑ͯ1 ρs = S (t) − S (t + 1) S (t) = 1 − exp ( − ( t+1 η ) k ) exp ( − ( t η ) k ) (10.14) t−1 ∏ y=1 (1 − ρy ) = exp       − ( t η ) k        (10.15) t ೥ʹ͓͚Δ·ͩ਍அ͞Ε͍ͯͳ͍ HIV infection ͷׂ߹ h(t, s) ͸্هͷ discrete model ͱಉ༷ʹͯ͠ҎԼͷΑ͏ʹهड़Ͱ͖Δ. xt = t ∑ s=1 λt−s t−1 ∏ x=t−s+1 (1 − αx ) s−1 ∏ y=1 (1 − ρy ) (10.16) ਍அͷ͍ͭͨ HIV infection ͷׂ߹͸ ∑ (a+u)/ ∑ (x+a+u) ·ͨ͸ ∑ u/ ∑ (x+u) ͱ͍͑Δ. લऀ͸ HIV positive (HIV diagnoses + AIDS diagnoses) ͷׂ߹Λࣔ͠, ޙऀ͸ະ਍அؚΊͨྦྷੵ HIV positive ͷ͏ͪ HIV diagnosis ͷׂ߹Λࣔ͢. લऀ͸ྦྷੵ AIDS ײછऀ਺͔Β AIDS ࢮ๢ऀ਺ΛҾ 1t ͷ୯Ґ͕ year ͳͷͰ discrete model ͕੒ཱ͢Δ. 27
29. 10.2 ࣮ࡍʹߦΘΕͨղੳ ୈ 10 ষ Competing risk model ʹΑΔ HIV ͍ͯௐ੔ͨ͠. ޙऀͷܽ఺͸ ART ʹ͋Δ AIDS Λআ֎ͯ͠Δ͜ͱͩ1. ͜ΕΒׂ߹ΛಘΔͨΊʹ, λt ͱαt Λ࠷໬ਪఆʹΑΓٻΊΔ. ͜ͷͱ͖, HIV infection ͸ඇఆৗ Poisson աఔʹΑͬͯੜΈग़͞Ε HIV diagnoses ͱ AIDS cases ͸ Poisson distribution ʹै͏. Αͬͯ, HIV diagnoses ͷ ໬౓ؔ਺͸ҎԼͷ௨Γ. L1 = constant × 2017 ∏ t=1985 E(ut )rt exp(−E(ut )) (10.17) ͨͩ͠, rt ͸ t ೥ʹ͓͚Δ HIV disgnoses ͷใࠂ਺Ͱ͋Δ. ಉ༷ʹͯ͠৽ن AIDS diagnoses ͷ໬౓ؔ਺͸ҎԼͷ௨Γ. L2 = constant × 2017 ∏ t=1985 E(at )wt exp(−E(at )) (10.18) ͨͩ͠, wt ͸ t ೥ʹ͓͚Δ AIDS disgnoses ͷใࠂ਺Ͱ͋Δ. ͕ͨͬͯ͠શମͷ໬౓ L ͸, L = L1L2 (10.19) Ͱ༩͑ΒΕΔ. ্هͷ໬౓ʹؔͯ͠͸, Negative binomial distribution ʹ HIV diagnoses ͱ AIDS cases ͕ै͏৔߹΋ߟྀ͞Ε AIC ͰධՁͨ͠. 95%CI ʹؔͯ͠, parameter ͷ CI ͸ profile likelihood ͔Βಋग़͞Ε, ࠷໬ਪఆ஋ͷ CI ͸ parametric bootstrap ๏Λ༻͍ͯಘͨ. Bootstrap ๏ʹؔ ͯ͠͸, parameter ͸ multinomial distribution ͔Β resampling ͞Ε, ͜ͷ multinomial distribtion ͷฏۉ θ ͱ SD σ ͷϕΫτϧ͸ٯ Hessian ߦྻ σ2 = diag(H−1(θ)) ͷର֯੒෼ΛऔΔڞ෼ࢄߦྻ͔Βಘͨ. ղੳ͸ R ͱ JMP Λ༻͍ͯߦΘΕͨ. 10.2.3 ·ͱΊ Abstract   Background: Epidemiological surveillance of HIV infection in Japan involves two technical problems for directly applying a classical back- calculation method, i.e., (i) all AIDS cases are not counted over time and (ii) people diagnosed with HIV have received antiretroviral therapy, extending the incubation period. The present study aimed to address these issues and estimate the HIV incidence and the proportion of diagnosed HIV infections, using a simple statistical model. Methods: From among Japanese nationals, yearly incidence data of HIV diagnoses and patients with AIDS who had not previously been diag- nosed as HIV positive, from 1985 to 2017, were analyzed. Using the McKendrick partial differential equation, general convolution-like equations were derived, allowing estimation of the HIV incidence and the time-dependent rate of diagnosis. A likelihood-based approach was used to obtain parameter estimates. Results: Assuming that the median incubation period was 10.0 years, the cumulative number of HIV infections was estimated to be 29,613 (95% confidence interval (CI): 29,059, 30,167) by the end of 2017, and the proportion of diagnosed HIV infections was estimated at 80.3% (95% CI [78.7%–82.0%]). Allowing the median incubation period to range from 7.5 to 12.3 years, the estimate of the proportion diagnosed can vary from 77% to 84%. Discussion: The proportion of diagnosed HIV infections appears to have not yet reached 90% among Japanese nationals. Compared with the peak incidence from 2005–2008, new HIV infections have clearly been in a declining trend; however, there are still more than 1,000 new HIV infections per year in Japan. To increase the diagnosed proportion of HIV infections, it is critical to identify people who have difficulty accessing consultation, testing, and care, and to explore heterogeneous patterns of infection.   1ART(antiretroviral therapy). ͜ͷ࣏ྍʹΑΓେ෯ʹજ෬ظ͕ؒԆ௕͞ΕΔ. 28
30. ୈ11ষ ௥Ճ༧ఆ࿦จ ߋ৽ස౓͕ۃΊͯ஗͍ͨΊ͜͜ʹಡΈऴ͑ͯؾʹೖͬͨ࿦จΛࢥ͍ग़ͨ͢ͼྻڍ͍ͯ͘͠. • Roberts MG. Epidemic models with uncertainty in the reproduction number. J Math Biol. 2013;66(7):1463-1474. • Endo A, Leeuwen EV, Baguelin M, 2019. Introduction to particle markov-chainmonte carlo for disease dynamics modellers. Epidemics 29, 100363. • Dibble CJ, O’Dea EB, Park AW, Drake JM. Waiting time to infectious disease emergence. J R Soc Interface. 2016;13(123):20160540. • Pei S, Kandula S, Yang W, Shaman J. Forecasting the spatial transmission of influenza in the United States. Proc Natl Acad Sci U S A. 2018;115(11):2752-2757. doi:10.1073/pnas.1708856115 • Li R, Pei S, Chen B, et al. Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Science. 2020;368(6490):489-493. • Zagheni E, Billari FC, Manfredi P, Melegaro A, Mossong J, Edmunds WJ. Using time-use data to parameterize models for the spread of close-contact infectious diseases. Am J Epidemiol. 2008;168(9):1082-1090. • Munasinghe L, Asai Y, Nishiura H. Quantifying heterogeneous contact patterns in Japan: a social contact survey. Theor Biol Med Model. 2019;16(1):6. Published 2019 Mar 20. • Ferguson NM, Donnelly CA, Anderson RM. Transmission dynamics and epidemiology of dengue: insights from age-stratified sero-prevalence surveys. Philos Trans R Soc Lond B Biol Sci. 1999;354(1384):757-768. • Quilty BJ, Clifford S, Flasche S, Eggo RM; CMMID nCoV working group. Effectiveness of airport screening at detecting travellers infected with novel coronavirus (2019-nCoV) [published correction appears in Euro Surveill. 2020 Feb;25(6):]. Euro Surveill. 2020;25(5):2000080. • Blumberg S, Funk S, Pulliam JR. Detecting differential transmissibilities that affect the size of self-limited outbreaks. PLoS Pathog. 2014;10(10):e1004452. Published 2014 Oct 30. • Ionides EL, Nguyen D, Atchad´ e Y, Stoev S, King AA. Inference for dynamic and latent variable models via iterated, perturbed Bayes maps. Proc Natl Acad Sci U S A. 2015;112(3):719-724. doi:10.1073/pnas.1410597112 • Endo A, Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Leclerc QJ et al. Implication of backward contact tracing in the presence of overdispersed transmission in COVID-19 outbreaks [version 1; peer review: awaiting peer review]. Wellcome Open Res 2020, 5:239 (https://doi.org/10.12688/wellcomeopenres.16344.1) • aaa 29
31. ୈ12ষ ਪఆجૅ 12.1 Heterogeneity of population 2 ষͰड़΂ͨΑ͏ʹײછ঱ͷྲྀߦΛਤΔࢦඪʹ࠶ੜ࢈਺ reproduction number ͕͋Δ. ͍·, ૝ఆ͢Δूஂ͕ۉҰ homogeneous Ͱ͋Δͱ͖, basic reproduction number Λ R0 ͱ͢Δͱ׆ಈࣗॗʹΑΔ઀৮཰ͷ௿Լ p Λ༻͍ͯ effective reproduction number Re ͸ҎԼͷΑ͏ʹදͤΔ. Re = (1 − p)R0 (12.1) ͜͜ͰײછΛऩଋͤ͞ΔͨΊʹ͸ Re < 1 ͱ͢Ε͹ྑ͍ͷͰ, p > 1 − 1 R0 (12.2) ߦಈࣗॗʹΑͬͯਓͱͷ઀৮Λ 8 ׂݮΒ͢ͱ͍͏ͱ͖, ʮ8 ׂʯ͸͜ͷܭࢉʹΑͬͯಘΒΕΔ.1 ͔͠͠ R0 Λܭࢉ͢Δʹ͋ͨͬͯ homogenerous population ΛԾఆͨ͠. ͜ͷ৔߹ R0 ≈ 2.5 ͱͳΓ઀৮཰͸ 6 ׂݮͰྑ͍ܭࢉʹͳͬͯ͠·͏. ͕࣮ͩࡍ͸, ৬ۀ΍೥ྸ܈ͳͲͦͷଐੑʹ͋Δ͔ͳ͍͔ʹΑͬͯ R0 ͸ҟͳΔͷ͕ݱ࣮Ͱ͋Δ. ͜ͷΑ͏ͳෆۉҰूஂ heterogenerous population ͷ৔߹͸࣍ʹઆ໌͢Δ࣍ੈ୅ߦྻ next generation matrix ʹΑͬͯଐੑผʹ R0 ΛٻΊ, ͦͷ୅ද஋ͱͯ͠ಘͨ R0 Λ૝ఆ͢Δ.2 12.2 Next generation matrix        IX,t+1 IY,t+1        = K        IX,t IY,t        (12.3) ͱ͠, K ͸ҎԼͷΑ͏ʹ͢Δ. K =        RXX RXY RYX RYY        (12.4) ͜͜Ͱ, ߦΛ recipients, ྻΛ transmitters Ͱ͋Γ, ࠓճ͸ X ͸͋Δ৬छͷਓʑͰ, Y ͸ͦ͏Ͱͳ͍৬छͷਓʑͱ͢Δ.3 (6.3) ͸ t ੈ୅ͷײછऀ਺ ͔Β t + 1 ੈ୅ͷײછऀ਺ͷਪҠΛද͓ͯ͠Γ, K ͷ͜ͱΛ࣍ੈ୅ߦྻ next generation matrix ͱ͍͏. t → ∞ Ͱ͸ੈ୅ؒͷײછऀ਺ͷൺ͸Ұఆ Ͱ͔ͭ, ಉੈ୅಺ͷଐੑผͷײછऀ਺ͷൺ΋ҰఆͰ͋ΔͷͰ R0 ͸ K ͷݻ༗஋ͱͯ͠ܭࢉ͞ΕΔ. ࠷΋جຊతͳ৔߹ʹ͓͍ͯ͸ K ͸࣍ͷΑ͏ ʹදͤΔ. K =        RXX RXY RYX RYY        =        βXX NX D βXY NX D βYX NY D βYY NY D        (12.5) ͳ͓, β ͸઀৮཰, N ͸ population size ͱ͢Δ. Next generation matrix ͔Β net reproduction numberRN = R0S (ͨͩ͠ S ͸ײडੑਓޱ) ΛٻΊΔ ʹ͸࣍ͷ௨Γ.        RXX RXY RYX RYY        =        βXXS X D βXYS X D βYXS Y D βYYS Y D        (12.6) 12.3 ײછऀ਺ͷਪఆ ෆݦੑײછͳͲͰൃ঱ਓ਺͸೺ѲͰ͖ͯ΋ײછਓ਺͸ϑΟʔϧυӸֶௐࠪͰ͸ಘʹ͍͘͜ͱ͕ଟ͍. ͜ͷͱ͖ҎԼͰઆ໌͢Δٯܭࢉ back- calculation ʹΑͬͯײછऀ਺Λਪఆ͢Δ͜ͱ͕͋Δ. Ϋϥελʔରࡦ൝΋͜ΕΑΓ R Λ༻͍ͯ༧ଌײછऀ਺ͱͦͷ༧ଌײછ೔Λಘ͍ͯΔ. 12.4 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 (12.7) 1HIT = 1 − 1 R0 ͱ͠ herd immunity Λද͢. ͜Ε͸ϫΫνϯΧόʔ཰ͷࢉग़ͳͲʹ΋ར༻͞ΕΔ. 2ޙ΄Ͳઆ໌͢Δ͕, ͓ͦΒ࣍͘ੈ୅ߦྻʹΑͬͯ net reproduction number Λ࣮ࡍ͸ܭࢉ͍ͯ͠ΔͱࢥΘΕΔ. 3ͭ·Γ RXY Ͱ͋Ε͹ͦΕ͸ Y ͔Β X ʹײછͤ͞Δ basic reproduction number Λࢦ͢. 30
32. 12.5Zero-inflated poisson model (ZIP model) ୈ 12 ষ ਪఆجૅ 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) ͕ಘΒΕ͍ͯΔ. Αͬͯײછ࣌ࠁΛ f(t) ͔ΒϊϯύϥϝτϦοΫਪఆ͠ (7.1) Λٯʹղ͘͜ͱͰײછऀ਺ΛٻΊΔ. ͜ΕΛٯܭࢉ back-calculation1ͱ͍͏. ৄ͘͠͸ 9 ষΛࢀর͞Ε͍ͨ. 12.5 Zero-inflated poisson model (ZIP model) ྫ͑͹, ݸਓ͕Ұ೥ؒͰژ౎ʹ๚ΕΔճ਺ͷ෼෍Λߟ͑Δͱ, Ұൠతʹ͸ Poisson distribution ʹै͏. ͔͠͠, ݱ࣮ʹ͸ 0 ճͷਓ͕େ੎͍Δ ͳͲ୯७ʹ Poisson distribution ͷΈΛߟ͑Δͷ͕೉͍͠৔߹͕͋Δ. ͜ͷΑ͏ͳ৔߹ͷ modeling ͱ͍͔ͯͭ͘͠ߟ͑ΒΕ͍ͯΔ͕2, ͦͷத ͷҰ͕ͭमਖ਼θϩա৒ϙΞιϯϞσϧ zero-inflated poisson model Ͱ͋Δ. ͜Ε͸, Poisson distribution ʹՃ͑ͯ, with-zero (WZ) model Λߟ͑ Δ. ͭ·Γ, 0 ͱ͍͏ observed case ͸, ߦಈʹΑͬͯແ࡞ҝʹ 0 ͱܾ·ͬͨ৔߹ͱ, ߦಈ͢Δલʹ 0 Λબ୒͍ͯͨ͠৔߹ͷ 2 ͭͷ֬཰ͷ࿨ͱ͠ ͯߟ͑Δͱ͍͏͜ͱͩ. ZIP model Ͱ͸, ͜ͷ, બ୒ͤͣʹ (ߦಈͤͣʹ)0 ճͱܾ·͍ͬͯΔׂ߹Λະ஌਺ͱͯ͠ඇબ୒཰ π ͱݺͿ. pt =            π + (1 − π) exp(−λt ) (k = 0) (1 − π) λk t exp(−λt ) k! (k ≥ 1) (12.8) (12.9) 12.6 Renewal equation 6 ষͰѻͬͨ net reproduction number Λѻ͏ next generation matrix Λߟ͑Δ.        RXX RXY RYX RYY        =        βXXS X D βXYS X D βYXS Y D βYYS Y D        (12.10) ׆ಈࣗॗʹΑͬͯ RXY , RYX , RYY ͸ӨڹΛड͚Δͱߟ͑ΒΕΔ. ͜͜Ͱ, ӨڹΛड͚ͨ৔߹ βXY (βYX ), βYY , S Y ͕খ͘͞ͳΔͱߟ͑ΒΕΔ. খ͞ ͘ͳͬͨͦΕΒͷม਺Λ༻͍ͯݻ༗஋Λܭࢉͯ͠ Rn ͕ 1 ΑΓখ͍͞৔߹Λߟ͑Δͱ, ׆ಈࣗॗΛड͚ೖΕΔ܏޲ʹ͋Δਓʑͷ β ͱ S ͷੵ͕ 8 ׂݮΔඞཁ͕͋Δͱ͍͏͜ͱͩ. ผͷઆ໌ͷ࢓ํͰಉ͜͡ͱΛߟ͑Α͏. i(t) = ∫ ∞ 0 A(t, s)i(t − s)ds (12.11) ͜Ε͸࠶ੜ࢈ํఔࣜͱ͍͍, i(t) ͸࣌ࠁ t Ͱͷ৽نײછऀ਺, A(s, t) ͸ ࣌ࠁ̓ʹҰਓͷײછऀ͕ൃੜͦ͠ͷਓ͕࣌ࠁ s ʹೋ࣍ײછΛҾ͖ى͜͢ ֬཰ ͱͨ͠. ͜Ε͸࣍ͷΑ͏ʹ΋දͤΔ. i(t) = βS (t) ∫ ∞ 0 f(s)i(t − s)ds (12.12) f(t) ͸ೋ࣍ײછ͕ى͜Δ·Ͱͷִ࣌ؒؒ t ͷ֬཰ີ౓ؔ਺. ͔͜͜ΒΘ͔ΔΑ͏ʹ৽نײછऀ਺͸઀৮཰ β ͱײडੑਓޱ S ͷੵʹൺྫ͢Δ. Ҏ্ͷ 2 ௨Γͷઆ໌͔ΒΘ͔ΔΑ͏ʹ, ͍ΘΏΔʮ8 ׂݮʯͱ͸, ʮଞਓͱͷ઀৮཰ͱײडੑਓޱͷੵΛ 8 ׂݮΒ͢ʯ͜ͱͰ͋Δ. Ϋϥε λʔରࡦ൝͸઀৮཰ΛҎԼͷࣜ3ͰಘΒΕΔ઀৮ස౓ߦྻ time of exposure matrix ͷ࠷େݻ༗஋͔Β઀৮཰Λࢉग़͠, ײडੑਓޱʹؔͯ͠͸ meta-population Λ༻͍ͯ NTT ͷۭؒσʔλΛղੳͯ͠ಘ͍ͯΔ. tij = H ∑ h=1 Z ∑ z=1 (ki hz kj hz ∑ n l=1 kl hz ) (12.13) 1௕ظతͳ༧ଌʹ͸޲͔ͣۙະདྷ༧ଌʹ༻͍ΒΕΔ. 2੾அϞσϧ truncation model (zero-altered poisson model, ZAP model ͱ΋͍͏) ͱ, ZIP model ͕جຊͰ͋Δ. Poisson distribution ʹݶΒͣʹ, ZIP model Ҏ֎ʹࠜຊ͸ͦΕͱಉ ͡Ͱ͋Δ zero-inflated negative binomial (ZINB) model ΍, zero-inflated binomial (ZIB) model ͳͲଞʹ΋༷ʑͳݚڀ͕ਐΜͰ͍Δ [12]. 3tij ͸೥ྸ܈ 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.agheni E, Billari FC, Manfredi P, Melegaro A, Mossong J, Edmunds WJ. Using time-use data to parameterize models for the spread of close-contact infectious diseases. Am J Epidemiol. 2008;168(9):1082-1090. doi:10.1093/aje/kwn220 31
33. 12.7Particle filtering and particle Markov-chain Monte Carlo ୈ 12 ষ ਪఆجૅ 12.7 Particle filtering and particle Markov-chain Monte Carlo I introduce particle filtering, which is also known as sequential Monte Carlo (SMC) to obtain the estimated unobserved cases and corresponding parameters. Particle Markov-chain Monte Carlo (PMCMC) [7] is useful to shadow the hidden state with time evolution process and jointly estimate the corresponding parameters. Because PMCMC includes SMC as sub-algorithm, I will explain what I adopt for SMC and that is Sampling Importance Resampling (SIR) algorithm for SMC and the detail is below: ਤ 12.1: Schematic representation of a hidden Markov process as a state space model. 0.1) Approximate the initial marginal likelihood for t = 1 sampling {x(i) 1 } ∼ p(x1) ˆ p(y1) = P ∑ i=1 p(y1 |x(i) 1 ) (12.14) 0.2) Obtain the posterior samples (particles){X(i) 1 } by resampling {x(i) 1 } with the following weights and filtering distribution (δ(·) denotes Dirac’s delta function): w(i) 1 = p(y1 |x(i) 1 ) ˆ p(y1) (12.15) p(X1 |y1) ≃ P ∑ i=1 w(i) 1 δ(X1 − x(i) 1 ) (12.16) 1.1) Generate samples at time t(≥ 1) for i = 1, ..., P by applying the proposal distribution { (x(i) t , w(i) t ) : i ∈ {1, ...P} } x(i) t ∼ π(xt |x(i) t−1 , y1:t) = p(x(i) t |x(i) t−1 ) (12.17) 1.2) Approximate the incremental marginal likelihood as ˆ p(yt |y1:t−1) = 1 P P ∑ i=1 p(yt |x(i) t ) (12.18) 1.3) Obtain the posterior samples {X(i) t } by resampling {x(i) t } with the following weights w(i) t : ˆ w(i) t = w(i) t−1 p(yt |x(i) t )p(x(i) t |x(i) t−1 ) π(xt |x(i) 1:t−1 , y1:t) = w(i) t−1 p(yt |x(i) t ) P ∑ i=1 w(i) t = 1 (12.19) w(i) t = ˆ w(i) t ∑ P j=1 ˆ w(j) t (12.20) Xt ∼ p(Xt |y1:t) ≃ P ∑ i=1 w(i) t δ(Xt − x(i) t ) (12.21) 2) Compute an estimate of the effective number of particles (effective sample size) and let ES S be the criterion for judging whether I peform resampling or not ES S = 1 ∑ P i=1 ( w(i) t ) 2 (12.22) This is the outline for SMC with the bootstrap filter (BF). We can obtain the estimated unobserved state through the SMC and also can obtain the parameters using the marginal likelihood through PMCMC (here ϕ denotes {θ, λ, ϵ}): L(ϕ; y1:t) = p(y1:t; ϕ) = p(yt , y1:t−1; ϕ) = t ∏ k=2 p(yk |yk−1; ϕ)p(y1; ϕ) = ∫ p(y1 |x1; ϕ)p(x1; ϕ)dx1 t ∏ k=2 ∫ p(yk |xk; ϕ)p(xk |y1:k−1; ϕ)dxk (12.23) 32
34. 12.7Particle filtering and particle Markov-chain Monte Carlo ୈ 12 ষ ਪఆجૅ I will scratch the codes referencing to Endo et al. 2019 [5] or use R package ”pomp” to implement PMCMC. The summary of PMCMC is given below: 1) For step n = 0, choose an initial parameter value ϕ(0) and run SMC to generate samples {X1:t } then the approximated marginal likelihood denotes ˆ p(y1:t; ϕ(0)). Randomly choose one trajectory x(0) 1:t from {X1:t }. 2) For step n ≥ 1, propose a new parameter value ˆ ϕ(n) by sampling from the proposal distribution q( ˆ ϕ(n); ϕ(n−1)). 3) Run SMC to generate particles { ˆ X1:t } and the approximated marginal likelihood ˆ p(y1:t; ˆ ϕ(n)) and randomly choose one trajectory ˆ x(n) 1:t from { ˆ X1:t } as a candidate for a MCMC sample for this step. 4) Compare the marginal likelihood with that in the previous step ˆ p(y1:t; ϕ(n−1)). With probability min      1, ˆ p(y1:t; ˆ ϕn) ˆ p(y1:t; ϕn−1) q(ϕn−1; ˆ ϕn) q( ˆ ϕn; ϕn−1)       (12.24) update ϕ(n) = ˆ ϕ(n) and x(n) 1:t = ˆ x(n) 1:t , otherwise keep the value from the previous step. 5) Repeat 2) - 4) until the Markov-chain converges. 33
35. ୈ13ষ ͓ΘΓʹ ຊߘ͸ LaTeX Ͱॻ͔Ε, ਤ͸͢΂ͯ draw.io Λ༻͍ͯ࡞੒ͨ͠. 34
36. ࢀߟจݙ [1] Linton NM, Kobayashi T, Yang Y, et al. Incubation Period and Other Epidemiological Characteristics of 2019 Novel Coronavirus Infec- tions with Right Truncation: A Statistical Analysis of Publicly Available Case Data. J Clin Med. 2020;9(2):538. Published 2020 Feb 17. doi:10.3390/jcm9020538 [2] Jung SM, Akhmetzhanov AR, Hayashi K, et al. 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. Published 2020 Feb 14. doi:10.3390/jcm9020523 [3] Nishiura H, Klinkenberg D, Roberts M, Heesterbeek JA. Early epidemiological assessment of the virulence of emerging infectious diseases: a case study of an influenza pandemic. PLoS One. 2009;4(8):e6852. Published 2009 Aug 31. doi:10.1371/journal.pone.0006852 [4] Anzai A, Kobayashi T, Linton NM, et al. Assessing the Impact of Reduced Travel on Exportation Dynamics of Novel Coronavirus Infection (COVID-19). J Clin Med. 2020;9(2):601. Published 2020 Feb 24. doi:10.3390/jcm9020601 [5] Endo A, van Leeuwen E, Baguelin M. Introduction to particle Markov-chain Monte Carlo for disease dynamics modellers. Epidemics. 2019;29:100363. doi:10.1016/j.epidem.2019.100363 [6] Endo A; Centre for the Mathematical Modelling of Infectious Diseases COVID-19 Working Group, Abbott S, Kucharski AJ, Funk S. Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China. Wellcome Open Res. 2020;5:67. Published 2020 Jul 10. doi:10.12688/wellcomeopenres.15842.3 [7] Blumberg S, Funk S, Pulliam JR. Detecting differential transmissibilities that affect the size of self-limited outbreaks. PLoS Pathog. 2014;10(10):e1004452. Published 2014 Oct 30. doi:10.1371/journal.ppat.1004452 [8] Nishiura H. Backcalculating the Incidence of Infection with COVID-19 on the Diamond Princess. J Clin Med. 2020;9(3):657. Published 2020 Feb 29. doi:10.3390/jcm9030657. [9] Mizumoto K, Kagaya K, Zarebski A, Chowell G. Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan, 2020. Euro Surveill. 2020;25(10):2000180. doi:10.2807/1560- 7917.ES.2020.25.10.2000180 [10] Backer JA, Klinkenberg D, Wallinga J. Incubation period of 2019 novel coronavirus (2019-nCoV) infections among travellers from Wuhan, China, 20-28 January 2020. Euro Surveill. 2020;25(5):2000062. doi:10.2807/1560-7917.ES.2020.25.5.2000062 [11] Wallinga J, Lipsitch M. How generation intervals shape the relationship between growth rates and reproductive numbers. Proc Biol Sci. 2007;274(1609):599-604. doi:10.1098/rspb.2006.3754 [12] Kucharski AJ, Russell TW, Diamond C, et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling study [published correction appears in Lancet Infect Dis. 2020 Mar 25;:]. Lancet Infect Dis. 2020;20(5):553-558. doi:10.1016/S1473-3099(20)30144- 4 [13] https://github.com/contactmodel/COVID19-Japan-Reff [14] Nishiura H. Estimating the incidence and diagnosed proportion of HIV infections in Japan: a statistical modeling study. PeerJ. 2019;7:e6275. Published 2019 Jan 15. doi:10.7717/peerj.6275 [15] Becker NG, Watson LF, Carlin JB. A method of non-parametric back-projection and its application to AIDS data. Stat Med. 1991;10(10):1527- 1542. doi:10.1002/sim.4780101005 [16] Becker NG. Uses of the EM algorithm in the analysis of data on HIV/AIDS and other infectious diseases. Stat Methods Med Res. 1997;6(1):24- 37. doi:10.1177/096228029700600103 [17] Dietz K, Seydel J, Schwartl¨ ander B. Back-projection of German AIDS data using information on dates of tests. Stat Med. 1994;13(19-20):1991- 2008. doi:10.1002/sim.4780131910 [18] Nishiura H, Linton NM, Akhmetzhanov AR. Serial interval of novel coronavirus (COVID-19) infections. Int J Infect Dis. 2020;93:284-286. doi:10.1016/j.ijid.2020.02.060 35