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 ײછ͕ෆ໌ͳ߹, ௐࠪͱ͍ͬͨϑΟʔϧυݚڀͰજ෬ظؒΛਪఆ͢Δ͜ͱ͕Ͱ͖ͳ͍. ͜ͷ߹ʹ, interval-censoring Λߦ͏. ͯ͞, ͜͜Ͱ͋Δײછऀ͕ະײછऀʹ͏ͭ͢ͱ͖Λߟ͑Δ. ͜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 ະײછऀͷײછͱͯ͠ߟ͑ΒΕΔதͰ࠷ૣ͍, ER ࠷͍ͱ͢Δ. S L ൃͱͯ͠ߟ͑ΒΕΔதͰ࠷ૣ͍, SR ࠷͍ͱ͢Δ. ·ͨજ෬ظؒʹؔͯ͠ τ = s − e ͱදͤΔ. ͜͜Ͱ, ҎԼͷؔΛఆٛ͢Δ. f(τ) : જ෬ظ͕ؒτͱͳΔ֬ີؔ g(t) : t ࣌ʹ͓͍ͯײછ͢Δ֬ (ײછऀ͕ະײછऀʹײછͤ͞Δ֬) h(t) : ̓࣌ʹ͓͍ͯൃ͢Δ֬ ͕ͨͬͯ͠, 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) ʹΑͬͯײછͱൃͷ࠷ਪఆ estimated maximum likelihood ͕ಘΒΕΔͷ Ͱજ෬ظ͕ؒΘ͔Δ. ͜ͷख๏Λ doubly interval-censoring ͱ͍͏. 3.2 ࣮ࡍʹߦΘΕͨղੳ ຊߘͷझࢫʹΑΓ, ݁ʹ͍ͭͯΑΓ݁ͷಋ͖ํ (Methods) Λத৺ʹղઆ͢Δ. ͜ͷจͰ, જ෬ظؒൃ͔ΒೖӃ·Ͱͷظؒ, ൃ ͔Βࢮ·Ͱͷظؒ, ೖӃ͔Βࢮ·Ͱͷظؒͱ͍͏ time interval ͷਪఆʹ͍ͭͯ doubly interval-censored likelihood Λͬͨख๏ͱ Bayesian framework Λ༻͍ͨख๏ͷ྆ํͰߦΘΕͨ. 1࣮ࡍ, ෆݦੑײછऀͷݶքͳͲʹΑΓײછ͔ൃͷͲͪΒ͔, ͋Δ͍ͲͪΒ͕ෆ໌Ͱ͋Δ͜ͱΑ͋͘Δ. 6
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··࠾༻͢ΔͱબόΠΞε1͕͔͔ΔͨΊ, ӈଆஅ 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 ࢦؔత૿ՃͰ F(·) f(·) ͷྦྷੵؔ. 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 ͱஔͯ࣍͠ͷΑ͏ʹมܗͰ͖Δ. 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, ରਖ਼ن, ϫΠϒϧΛߟ͑ͨ. 3.2.2 Estimation of the time interval distribution using Bayesian framework Bayes ਪఆʹΑΓ time interval distribution ΛٻΊΔ. ࣄલରਖ਼ن, ΨϯϚ, ϫΠϒϧΛߟ͑Ϛϧίϑ࿈ϞϯςΧϧϩ๏ (MCMC)4ʹΑͬͯϞϯςΧϧϩੵ͔Βඇஅରʹର͢Δࣄޙ༧ଌ (time interval distribution) Λਪఆ͢Δ. ͳ͓, ରਖ਼نͱ ϫΠϒϧͲͪΒඪ४ਖ਼نʹै͏. ΨϯϚ shape parameter ฏۉ 3, ඪ४ภࠩ 5 ͷਖ਼نʹै͍, inverse scale parameter , location parameter Λ 0, scale parameter Λ 5.0 ʹͱΔίʔγʔͱ͠֊ϕΠζΛߟ͑ͨ.5ͳ͓, ࣄલͷબ Stan developer community ʹΑΔਪ6ʹैͬͨ. ·ͨ, times of exposure ͱ illness onset ͷࣄલҎԼͷΑ͏ʹఆࣜԽͨ͠. 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 ࣍ͷʹै͏. ˜ ei ∼ nomal(mean = 0.5, S D = 0.5), ˜ si ∼ nomal(mean = 0.5, S D = 0.5) 1ੜଘ࣌ؒղੳͷ. ଧͪΓ censoring (அ truncation) ͕ߟྀ͞ΕΔ͖ิਖ਼Ͱ͢. ࠓճͷ right truncation જ෬ظ͕͍ؒ߹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) Λ࣮ͨ͠ Stan ʹΑΓཚੜߦΘΕͨ. 5Ұൠͱͯ͠֊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
ʹΑΔඇஅରΛ࣍ͷΑ͏ʹఆٛͨ͠. logL(Θ) = ∑ i distribution lpd f(si − ei |Θ) (3.8) ͜͜Ͱ, lpd f ֬ີؔͷରͰ͋Γ, lognomal lpd f ͔ gamma lpd f ͔ weibull lpd f ͷ͍ͣΕ͔Λද͢. ·ͨ, જ෬ظؒʹؔͯ͠ଧ ͪΓΛߟ͑ͨ. MCMC , the simulation phase ͱͯ͠ 5000 ճΛ burn in ͯ͠ the tuning phase Ͱ 10000 ճΛ࠾༻͠, ͦΕΛ̐ܥྻߦͬͨ. Ҏ্͔Βࣄޙ༧ଌΛਪఆ֤͠ϞσϧΛ WAIC ͰධՁ͠࠷దͳϞσϧΛબͨ͠. 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(·) ରͷฏۉͱࢄΛͦΕͧΕද͢. 3.2.3 ·ͱΊ Ҏ্ͷ̎௨Γͷਪఆ๏͔Β time interval ਪఆ͞Εͨ. ࣮ࡍʹจ͕ൃද͞Εͨͱ͖ʹಘΒΕͨৄ͍݁͠Ռͷߟʹ͍ͭͯ, จͷ Results Discussion ΛಡΜͰΒ͍͍ͨ. ҎԼʹจͷ 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
4.2.3 ·ͱΊ ࠷๏ʹΑͬͯ, ྲྀߦॳظஈ֊Ͱ cCFR ͳͲͷύϥϝλΛਪఆͦ͠ͷ݁ՌΛ༻͍ͯ R0 Λਪఆͨ͠. ࣮ࡍʹจ͕ग़͞Εͨͱ͖ʹಘΒΕͨৄ ͍݁͠Ռͷߟʹ͍ͭͯ, จͷ Results Discussion ΛಡΜͰΒ͍͍ͨ. ࣍ͷϖʔδʹจͷ 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
ͷݮগʹΑΓੜͨ࣌ؒ͡Ε 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༻͍ͯߦΘΕ৴པ۠ؒϓϩϑΝΠϧ͔Βࢉग़͞Εͨ. 5.2.4 ·ͱΊ 3 ͭͷ؍͔Β Lockdown ʹ͏ travel volume ͷݮগ͕༩͑ΔӨڹΛఆྔతʹධՁͨ͠. ࣮ࡍʹจ͕ग़͞Εͨͱ͖ʹಘΒΕͨৄ͍݁͠Ռ ͷߟʹ͍ͭͯ, จͷ Results Discussion ΛಡΜͰΒ͍͍ͨ. ࣍ͷϖʔδʹจͷ 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
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
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
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, ͭ·Γঢ়͕ग़͍ͯͳ͍ਓ asymptomatic ·ͨ·ͩൃ͍ͯ͠ͳ͍ਓͷ 2 छྨΛؚΉ. ײછ ͨ͠Մೳੑͷ͋Δ time interval Λ [ai , bi ] ͱ͠, c ൃ͕؍͞ΕͯଧͪΓʹͳͬͨΛද͢. ·ͨ, i ͕ײછͨ͠Λ Xi ͱ͠, infection to onset ͱ͍͏ time delay Λજ෬ظؒ incubation period Di , FD ͦͷ CDF ͱ͢Δ. ͜͜Ͱ, c ࣌ʹঢ়͕ग़͍ͯͳ͍֬ 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 ͷࣄલΛ [ai , bi ] ͷҰ༷ͱͯ͠༩͑Δ. 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 ྫແީͱͯ͋ͬͨ͠. ܽଛॲཧͱͯ͠, ແ࡞ҝׂΛͦΕͧΕߦ͍, ͦΕΛ 200 ճ܁Γฦͨ͠. ͜ΕʹΑΓ, uncertainty bounds ΛಘΔͨΊͷ percentile points Λಘͨ. ײੳͰ, જ෬ظؒͷฏۉΛ 4.4 days − 8.4 days ͱͯ͠มԽͤ͞Δ ͜ͱͰ, p ͷӨڹΛௐͨ. 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
͍ͯௐͨ͠. 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 ͷใࠂͰ͋Δ. ಉ༷ʹͯ͠৽ن AIDS diagnoses ͷؔҎԼͷ௨Γ. L2 = constant × 2017 ∏ t=1985 E(at )wt exp(−E(at )) (10.18) ͨͩ͠, wt t ʹ͓͚Δ AIDS disgnoses ͷใࠂͰ͋Δ. ͕ͨͬͯ͠શମͷ L , L = L1L2 (10.19) Ͱ༩͑ΒΕΔ. ্هͷʹؔͯ͠, Negative binomial distribution ʹ HIV diagnoses ͱ AIDS cases ͕ै͏߹ߟྀ͞Ε AIC ͰධՁͨ͠. 95%CI ʹؔͯ͠, parameter ͷ CI profile likelihood ͔Βಋग़͞Ε, ࠷ਪఆͷ CI parametric bootstrap ๏Λ༻͍ͯಘͨ. Bootstrap ๏ʹؔ ͯ͠, parameter multinomial distribution ͔Β resampling ͞Ε, ͜ͷ multinomial distribtion ͷฏۉ θ ͱ SD σ ͷϕΫτϧٯ Hessian ߦྻ σ2 = diag(H−1(θ)) ͷର֯ΛऔΔڞࢄߦྻ͔Βಘͨ. 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). ͜ͷ࣏ྍʹΑΓେ෯ʹજ෬ظ͕ؒԆ͞ΕΔ. 28
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
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
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
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