SEIRモデルを用いたクラスタ班の予測の確認

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1. SEIRϞσϧͰײછऀ਺ਪҠΛ༧ଌͯ͠Έͨ 2020-4-13 1 Introduction ɹ·ͣॳΊʹɺචऀ͸ײછ঱ͷઐ໳ՈͰ͸ͳ͘ɺཧ࿦Ո΍σʔλαΠΤϯςΟε τͷઐ໳Ոͱ΋ݴ͑ͣɺ͜ͷ෼ੳͷਖ਼֬ੑ͸อূͰ͖ͳ͍͜ͱΛྃ͝ঝ͍ͩ͘͞ɻ ͦͷ͏͑Ͱɺࠓճɺઌ೔ްੜলΫϥελ൝͔Βग़͞ΕͨײછऀਪҠͷࢼࢉ [1] Λ ࣗ෼ͳΓʹ௥ͬͯΈ·ͨ͠ɻ ݱঢ়ͷ೔ຊͷײછऀ਺Λ༧ଌ͢ΔϞσϧͱͯ͠͸ɺࢲ͕஌ΔݶΓɺްੜলΫ ϥελ൝ͷ੢Ӝ͞Μͷ΋ͷͱɺ౦େͷେڮ͞Μɺԣ඿ࢢཱେͷࠤ౻͞Μͷ΋ͷ͕ ͋Γ·͢ɻ͜ͷ 3 ͭͷϞσϧͷൺֱɾݕ౼ʹ͍ͭͯ͸຀໺͞ΜͷαΠτ [2] ͕ৄ ͍͠Ͱ͕͢ɺͦΕʹΑΕ͹ɺ੢Ӝ͞Μͷ΋ͷͱେڮ͞Μͷ΋ͷ͸ຊ࣭తʹ͸ಉ͡ Ͱɺࠤ౻͞Μͷ΋ͷ͸ɺະײછऀ͕ײછޙൃ঱͢Δ·Ͱͷ஗Εύϥϝʔλʔ τ ͷ ਺ࣈ͕େ͖͗͢ΔͷͰ͸ͳ͍͔ͱ͍͏ݟղͰ͢ʢࢲ΋ಉҙݟͰ͢ʣ ɻࠓճɺେڮ ͞Μͱಉ༷ ESIR ϞσϧΛ༻͍ͯɺ઀৮ස౓Λ 4/12 Ҏ߱ 8 ׂݮগͨ͠ΒͲ͏ͳ Δ͔ͱ͍͏γϛϡϨʔγϣϯΛߦ͍·ͨ͠ɻ 2 Model ɹ·ͣɺϞσϧʹ͸͔ͳΓେࡶ೺ͳԾఆΛ͍͔͓͍͓ͭͯ͘Γɺఆੑతͳڍಈ΍ Φʔμʔ͸ਖ਼͍͔͠΋͠Ε·ͤΜ͕ɺ༧ଌ਺஋ͷਖ਼֬ੑ͸͔ͳΓո͍͠Ͱ͢ɻϞ σϧ͸ SEIR ϞσϧΛϕʔεͱ͠ɺͨͩ͠਺ࣈͷऔΓѻ্͍एׯͷมߋΛՃ͑ɺS ʢະײછʣ ɺEʢײછࡁະൃ঱ऀʣ ɺIʢൃ঱ऀʣ ɺQʢִ཭ࡁʣͷ 4 ঢ়ଶ͔Β੒Δͱ ͠·ͨ͠ɻ·ͨɺ4 ͭͷঢ়ଶͷਓޱΛ଍ͨ͠΋ͷ͸૯ਓޱͰ͋Δ N ͱ౳͍͠΋ͷ ͱ͠·ͨ͠ʢਤ 1ʣ ɻ ঢ়ଶؒͷભҠʹ͍ͭͯ͸ҎԼͷΑ͏ʹߟ͑·͢ɻ·ͣ͸ɺඪ४తͳ SEIR Ϟσ ϧͱಉ༷ʹɺS ͔Β E ΁ͷભҠ͸ SɺI ͦΕͧΕͷ਺ʹൺྫ͢Δ΋ͷͱ͠ɺβ SI N ͱ ͠ɺ͞Βʹ E ͔Β I ΁ͷભҠ͸ E ͷ਺ʹൺྫ͢Δ΋ͷͱ͠ɺͦͷ܎਺͸ 1/(ൃ঱ ·ͰͷλΠϜεέʔϧ) ʹ 1/5 Ͱ͋Δͱ͠·ͨ͠ɻ͜͜ͰɺμΠϠϞϯυϓϦϯ ηε΍ΞϝϦΧ಺Պֶձͷݚڀʹ͓͍ͯɺײછ͔Βൃ঱·Ͱͷ೔਺͕ฏۉ 5 ೔Ͱ ͋Δ͜ͱΛ༻͍·ͨ͠12ɻ͞Βʹɺ[5] Ͱߟྀ͍ͯ͠Δִ཭ϓϩηεͱಉ༷ʹɺI 1μΠϠϞϯυϓϦϯηε߸ͷௐࠪͰ͸ײછ͔Βൃ঱·Ͱͷ೔਺͸ϫΠϒϧ෼෍ʹ͍ۙͱ͞Ε͍ͯ ΔͷͰ͕͢ɺେࡶ೺ʹࢦ਺෼෍ͱͳΔ΋ͷͱ͠·ͨ͠ɻѻ͍ͱͯ͠͸ɺ͜͜Ͱͷ΋ͷΑΓɺࠤ౻͞Μ [3] ͷ༻͍͍ͯΔΑ͏ͳ࣌ؒ஗ΕΛಋೖͨ͠ํ͕ΑΓਖ਼֬ͱࢥ͍·͕͢ɺ͜͜Ͱ͸ SEIR Ϟσϧͱಉ ༷ E ʹൺྫ͢ΔભҠ͕ੜ͡Δͱ͠·ͨ͠ɻ 2͜͜Ͱ͸ൃ঱લͷײછ͸ແࢹ͠·͕ͨ͠ɺίϩφ΢Οϧε͸ൃ঱લʢ1 3 ೔લʣʹ΋ײછྗΛ࣋ ͭͱ͍͏ใࠂ΋͋Γ [4]ɺ͜ͷ γ ͷ਺ࣈΛ 1/3 ͳ͍͠͸ 1/4 ʹ͠ͳ͚Ε͹͍͚ͳ͍͔΋͠Ε·ͤΜɻ 1
2. Figure 1: ঢ়ଶؒͷભҠ ͔Β Q ΁ͷભҠ͸ I ʹൺྫ͢Δ΋ͷͱ͠ɺδI ͱ͠·ͨ͠ɻ͜͜Ͱɺൃ঱͔ͯ͠Β ִ཭͞ΕΔ·ͰͷλΠϜεέʔϧΛɺ༗঱ऀִ͕཭͞ΕΔ 6 ೔ఔ౓ [10] ͱແ঱ঢ় ऀ͕࣏༊͢Δ·Ͱͷ 14 ೔ఔ౓ͷؒ͘Β͍ͱΈͳ͠ɺ10 ೔ͱͯ͠ɺγ = 1/10 ͱ͠ ·ͨ͠3ɻ͋ΘͤͯɺԼهͷΑ͏ͳඪ४తͳ SEIR Ϟσϧͱಉ͡ํఔࣜΛಘ·͢ɻ ˙ S = −β(t) S(t)I(t) N ˙ E = β(t) S(t)I(t) N − γE(t) ˙ I = γE(t) − δI(t) ˙ Q = δI(t) (1) 3 Simulation ɹ·ͣ͸ɺݱ࣮ͷσʔλͱͷϑΟ οςΟϯάͰ͕͢ɺްੜলͷײછऀ౷ܭ [6] ʹؚ ·ΕΔΑ͏ʹͳΔͷ͸ Qʢִ཭ࡁʣͱͳͬͨ࣌఺Ͱͷ͜ͱͱͳΓ·͢ͷͰɺQ Λ ྦྷੵײછऀͷάϥϑͱϑΟ οςΟϯά͠·͢ɻ্هͷύϥϝʔλʔΛॴ༩ͱͯ͠ɺ kaz-ogiwara ͞Μ͕ఏڙ͍ͯ͠Δ೔ຊͷײછऀσʔλ [7] Λ༻͍ͯɺβ ͱ I0 (0 ೔࣌ ఺Ͱͷ I) Λਪܭͨ݁͠ՌɺͦΕͧΕ 0.29,23 ͱͳΓ·ͨ͠4ɻ͜͜Ͱ͸ෆਖ਼֬Ͱ ͸͋Γ·͕͢ɺ؍ଌͨ͢͠΂ͯͷ Q ʹ͍ͭͯ୯७ʹ࠷খೋ৐๏Λ༻͍·ͨ͠ɻ 5ɹ ͞Βʹɺ઀৮ස౓ͷ௿ݮͰ͕͢ɺ౦ژ౎ͷࣗॗͷ։࢝͸ 4/11 ͔ΒͰ͕͢ɺपΓ ͷ౎ಓ෎ݝͱͷ݉Ͷ߹͍΋ߟ͑ͯ 4/12 ͔Β 8 ׂݮͱͳΔΑ͏ʹ͠·ͨ͠ɻ 3͜Ε΋ຊདྷ͸ Q Λ༗঱ঢ়ऀͱແ঱ঢ়ऀʹ෼͚ͯɺͦΕͧΕัଊ཰͕ҧ͏΋ͷͱͯ͠ѻ͏ํ͕๬ ·͍͠Ͱ͕͢ɺେࡶ೺ʹͻͱ͘͘Γʹ͠·ͨ͠ɻͦͷͨΊɺ࣮ࡍͷྦྷੵײછऀ਺Λ Q ʹϑΟ οτͤ͞ Δ࣌ɺແ঱ঢ়ऀΛΧ΢ϯτͯ͠͠·͍ͬͯΔ෼ͣΕ͕ੜ͡·͢ɻ 4ਤ 2 Ͱ͸ྦྷੵײછऀ਺ͷϑΟ οτ͕ͦΕͳΓʹྑ͍Α͏ʹ΋ݟ͑·͕͢ɺ୯ʹࢦ਺ؔ਺Ͱ૿Ճ͠ ͍ͯΔͱ͍͏͚ͩͰ͋ΓɺϞσϧͷਖ਼͠͞Λࣔ͢΋ͷͰ͸͋Γ·ͤΜɻ 5ຊདྷ͸ɺࠤ౻͞Μͷ΍ΒΕ͍ͯΔΑ͏ͳܗͰ [3] ϊΠζΛೖΕͯɺ࣮ݱ֬཰͕Ұ൪ߴ͍ͱ͖ͷύ ϥϝʔλʔΛ༻͍Δ΂͖ͩͱࢥ͍·͢ɻ 2
3. ݁Ռͱͯ͠ɺྦྷੵײછऀ਺͸ਤ 2 ͷΑ͏ʹɺ৽نײછऀ਺ʢ೔࣍ʣ͸ਤ 3 ͷ Α͏ʹͳΓ·ͨ͠ɻ੢Ӝ͞Μͷ݁ՌΑΓ΍΍൵؍తͰɺ৽نײછऀ਺ʢ೔࣍ʣ͕ 100 ਓఔ౓ʹ໭Δ·Ͱʹ 2 ͔݄ఔ౓Λཁ͢Δͱ͍͏݁ՌʹͳΓ·ͨ͠ɻ·ͨɺࢀ ߟ·Ͱʹ઀৮ස౓ 70% Χοτͷ৔߹͸ 100 ೔ܦͬͯ΋৽نױऀ਺ 100 ਓͷϨϕ ϧʹ໭Γ·ͤΜͰͨ͠ɻཧ༝ͷҰͭͱͯ͠͸ɺ੢Ӝ͞ΜͷࢼࢉΑΓ΋ݱࡏͷঢ়گ ͸΍΍ѱԽ͓ͯ͠Γɺࣗॗཁ੥ͷग़ͨ࣌఺Ͱͷ৽نײછऀ਺͕ 500 ਓ͔Β 700 ਓ ఔ౓ʹ૿͍͑ͯΔ͜ͱ͕͋Γ·͕͢ɺ্هͷಋग़ʹ͓͚Δେࡶ೺ͳԾఆ͕ݪҼͰ ͋Δ͔΋͠Ε·ͤΜɻͱ͸͍͑ɺϞσϧͷఆੑతͳ;Δ·͍͸ະัଊऀΛߟྀʹ ೖΕͯ΋มΘΒͣɺ઀৮ස౓Λ 8 ׂݮগͤ͞Δ͜ͱʹΑͬͯऩଋ͸Մೳͩͱ͍͏ ݁ՌͱͳΓ·ͨ͠ɻ Figure 2: ྦྷੵײછऀ਺ Figure 3: Ұ೔౰ͨΓͷ৽نݕࠪཅੑ਺ References [1] ೔ຊܦࡁ৽ฉ 4/11 ʮ઀৮ 7 ׂݮʯͰ͸ऩଋ·Ͱ௕ظԽɹ๺େڭत͕ܯ৊ https://www.nikkei.com/article/DGXMZO57961860R10C20A4CZ8000/ [2] http://jun-makino.sakuraweb.com/articles/corona/face.html [3] COVID-19 ৘ใڞ༗ ʕ COVID19-Information sharing, ౎ಓ෎ݝ͝ͱͷγ ϛϡϨʔγϣϯʹΑΔݕ౼, [4] Wei, Wycliffe E. ”Presymptomatic Transmission of SARS-CoV-2ʕ Singapore, January 23–March 16, 2020.” MMWR. Morbidity and Mortality Weekly Report 69 (2020). [5] Sato, Aki-Hiro, et al. ”An epidemic simulation with a delayed stochastic SIR model based on international socioeconomic-technological databases.” 2015 IEEE International Conference on Big Data (Big Data). IEEE, 2015. [6] ް ੜ ࿑ ಇ ল, ৽ ܕ ί ϩ φ ΢ Π ϧ ε ײ છ ঱ ʹ ͭ ͍ ͯ, https://www.mhlw.go.jp/stf/seisakunitsuite/bunya/0000164708 00001.html https://www.fttsus.jp/covinfo/pref-simulation/ 3
4. [7] https://github.com/kaz-ogiwara/covid19 [8] Mizumoto, Kenji, et al. ”Estimating the asymptomatic proportion of coro- navirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan, 2020.” Eurosurveillance 25.10 (2020). [9] Lauer, Stephen A., et al. ”The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application.” Annals of internal medicine (2020). [10] Wei Su, https://covid-2019.live/ 4