行政院國家科學委員會專題研究計畫 成果報告

行政院國家科學委員會專題研究計畫 成果報告

室內呼吸性傳染病之傳輸潛能機率風險分析(第 2 年) 研究成果報告(完整版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 97-2314-B-040-006-MY2

執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 中山醫學大學公共衛生學系（所）

計 畫 主 持 人 ： 陳詩潔

計畫參與人員： 碩士班研究生-兼任助理人員：游舒涵

報 告 附 件 ： 出席國際會議研究心得報告及發表論文

處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中 華 民 國 99 年 10 月 20 日

1

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計畫編號：NSC 97 － 2314 － B － 040 － 006 － MY2 執行期間： 97 年 08 月 01 日至 99 年 07 月 31 日

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執行單位： 中山醫學大學公共衛生學系

中 華 民 國 99 年 10 月 09 日

2

行政院國家科學委員會專題研究計畫期中報告 行政院國家科學委員會專題研究計畫期中報告 行政院國家科學委員會專題研究計畫期中報告 行政院國家科學委員會專題研究計畫期中報告 室內呼吸性傳染病之傳輸潛能機率風險分

室內呼吸性傳染病之傳輸潛能機率風險分 室內呼吸性傳染病之傳輸潛能機率風險分

室內呼吸性傳染病之傳輸潛能機率風險分析 析 析 析(2/2)

計畫編號計畫編號計畫編號計畫編號：：：：NSC 97－－－2314－－ －－B－－ －－040－－ －－006－－ －－MY2 －

執行期限執行期限

執行期限執行期限：：：：97/08/01~99/07/31 執行單位

執行單位 執行單位

執行單位：：：：中山醫學大學公共衛生學系中山醫學大學公共衛生學系中山醫學大學公共衛生學系中山醫學大學公共衛生學系 主主主主 持持持持 人人人人：：：：陳詩潔陳詩潔陳詩潔陳詩潔

摘要 摘要 摘要 摘要

本計畫提供一個預測性模式以模擬易 感族群之流感傳輸潛能，以整合流感病毒動 態模式、室內氣膠傳輸模式及飛沫粒徑分佈 特性。藉由流感病毒動態模式，人體宿主和 病原體之間交互作用可分為三個等級，包括 (A)上皮細胞等級、(B)人體免疫反應等級與 (C)病毒等級，可評估每日時變的病毒力價 產 生 率 與 飛 沫 粒 徑 大 小 ； 進 而 利 用 Wells-Riley 方程式推求重要的參數感染風

險(P)和基本再生數(R0)以量化流感傳輸潛

能。結果顯示，A 型(H1N1)流感反應第一個 的是干擾素(i)，開始於病毒產生兩個小時 內，並於感染後第三至第四天間達到高峰，

而毒殺性 T 細胞(Z)反應較慢，於感染後第 十天達到高峰。然而，在病毒層級中，病毒 力價在感染後第二天達最大值，第七天達到 5.96×10⁹的 virions 病毒粒子數。靈敏度分析 得知，受感染細胞產生病毒之速度(k) = 4000 d^-1 infected cell^-1 與未受感染上皮細胞感染 率(β) = 5×10^-10 d^-1 virion^-1使實驗性人體感染 數據和模式預測相關性達顯著(r = 0.99, p <

0.0001)，顯示 β 在病毒動態中扮演重要的角

色。結果指出流感病毒感染速率 P 和 R₀的

中位數和 95%信賴區間分別為 0.132 (0.09 – 0.19) 和 1.19 (0.76 – 1.86)。潛在的流感病毒

傳輸可藉由 R₀＞1 判斷。建議預測的模式可

作為進一步具體調查控制策略的工具，例如 個人防護面罩去改變顆粒大小和數量濃度 特徵以及減少呼出飛沫以降低室內環境中 的感染風險。

關鍵詞：流感，傳染風險，病毒動態，液滴，

模式，公共衛生。

Abstract

This proposal provides a predictive model that can integrate the modified influenza A (H1N1) virus dynamic model, indoor aerosol transmission model, and characterizing the droplet size distribution to estimate the transmission potential in a

proposed susceptible population. By studying the influenza virus dynamic model, we explored the consequences of host-pathogen interactions involving in three levels: (A) epithelial cell level, (B) human immune response level, and (C) virus level. To evaluate the quantum generation rate varying with the day post infection and pathogen-carrying particle diameter, and to quantify exhaled bioaerosol infections for the critical key parameters of risk of infection (P) and basic reproduction number (R₀) based on Wells-Riley mathematical equation. Results show that the first responder to influenza A infection was interferon molecules (i) that the production began less than 2 h after viral introduction, and then reached a peak between days 3 – 4. The cytotoxic T-cell (Z) responder was much slower with a CTL activity peak at day 10. However, in virus level, a large number of virus concentrations appeared at day 2 and reached to a peak value of 5.96×10⁹ virons at day 7. Sensitivity analysis indicated that input values of Creation rate of viruses by an infected epithelial cell (k) = 4000 d^-1 infected cell^-1 and infection rate of an unprotected epithelial cell (β) = 5×10^-10 d^-1 virion^-1 resulted in a significant correlation between experimental human infection data and model predictions (r = 0.99, p < 0.0001), suggesting that β has an important role involved in the virus dynamics. The result indicated that the box and whisker plots of median with 95% CI of P and R0 were estimated to be 0.132 (0.09 – 0.19) and 1.19 (0.76 – 1.86) for influenza viruses. The potential transmission of infection for influenza viruses can be judged by R0 > 1. The proposed predictive model may serve as a tool for further investigation of specific control measure such as the personal protection masks to alter the particle size and number concentration characteristics and minimize the exhaled bioaerosol droplet to decrease the infection risk in indoor environment settings.

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Key words: Influenza; Transmission risk;

Virus dynamics; Droplet; Modeling; Public Health.

Introduction

A variety of mathematical and computational models have been proposed for elucidating the nonlinear transmission dynamics of epidemics and for enhancing our understanding of the within-host spread of diseases and the immune response (Nowak and May, 2000; Perelson, 2002; Van Kerkhove et al., 2010). Important results have been obtained from the mathematical modeling of virus dynamics for the HIV (Perelson et al., 1996; 1997; Nowak and Bangham, 1996), hepatitis B virus (HBV) (Marchuk et al., 1991;

Nowak et al., 1996), hepatitis C (Neumann et al., 1998) and influenza infections (Baccam et al., 2006; Hancioglu et al., 2007; Chang and Young, 2007).

When influenza virus infected a healthy person, there are many complex factors governing the process of influenza infection, the multiple mechanisms were interacted among immunology, cells dynamics, and virus generation rate. Computer simulation would be a useful tool to independently dissect the potential contribution and relative importance of each factor or to investigate unexpected scenarios that are difficult to replicate experimentally.

Viral kinetics can be used to express the competed results of human immunity ability with influenza virus generation. Baccam et al.

(2006) provided two models describing the kinetics of influenza A virus infection in human: a target cell-limited model and a target cell-limited model with delayed virus production. They used data from experimentally infected volunteers to estimate the reasonable parameters of biological characteristics appeared in the models. Their findings suggested that the model considering the delayed virus production was more realistic because of the infected cells begin producing influenza virus for nearly 5 hours.

Chang and Young (2007) developed a simple scaling law-based ordinary differential model for describing the time courses of the numbers of infectious viral particles, activated cytotoxic T-lymphocytes, interferon molecules,

infected cells, uninfected cells, and the subset of uninfected cells that are protected by interferon from viral infection. They found that the rise time, duration, and the severity of the influenza A infection could be expressed as a function of the initial viral load and the relevant parameters based on the developed scaling laws.

Experimental human influenza A virus infection can provide the local and systemic cytokine responses during the infection period (Hayden et al., 1998), even the safety and efficacy of the oral or intravenous neuraminidase inhibitor (Hayden et al., 1999;

Hayden et al., 1996; Fritz et al., 1999; Calfee et al., 1999). Different experimental trails for assessing the dose, form, and the challenge’s responses include daily viral titer, shedding virus, peak titer, days of shedding, and clinical symptom scores. Hence, by using the experimental human infection data to validate the viral dynamic models leads to increase the accuracy of the model prediction and a comparison of parameter sensitive analysis can also be achieved.

Transmission of exhaled infectious droplets in the indoor environment has been receiving substantial attentions. Duguid (1946) indicated that the lognormal distribution could best describe the respiratory droplet with a geometric mean (GM) 14 µm and a geometric standard deviation (GSD) 2.6 for cough and a GM 8.1 µm and a GSD 2.3 for sneeze.

Papineni and Rosenthal (1997) measured expired bioaerosol droplets (in nose and mouth breathing, coughing and talking) to be less than 2 µm in size, with no droplets larger than 8 µm. Hence, the particle size distribution may play a key role for evaluating the infection risk.

Hence, in this study, we sought to develop a mathematical model by combining a target cell-limited model with delayed virus production by Baccam et al. (2006) and a reduced population dynamic model of immune response by Chang and Young (2007).

More importantly, the modified influenza virus dynamic model was used to explore the sensitive analysis and to perform the model validation against the experimental human infection data within an individual. We believed that this present framework could be

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incorporated explicitly into control measure modeling schemes.

Materials and Methods

Mathematical model of modified influenza virus dynamics

By studying a model of influenza virus dynamics that builds on past well-developed models of Baccam et al. (2006) and Chang and Yang (2007), we explored the consequences of host-pathogen interactions involving in three levels: (i) epithelial cell level, (ii) human immune response level, and (iii) virus level. The essential features of the present model are depicted in Fig. 1. The roles of immune response properties of epithelial cells act as the interferon (IFN) and cytotoxic T-lymphocytes (CTLs) immunity.

Briefly, in epithelial cell level (Fig. 1A), uninfected cells (X) can be infected by free virus particles (V) and can transmit to infected cells (Y). During the virus transform and replication by cell DNA, Y will take almost 6 to 8 hr for transmission to the state of producing virus infected cells (J). In human immune response level (Fig. 1B), cytotoxic T-cell (Z) can be induced and activated by infected cells (Y) and produced virus infected cells (J). Interferon molecules (i) also play the role in immune response mechanism and can protect cells (XR) from increasing infected cells. The present model used the free virions in epithelial cells (V) to represent the virus level (Fig. 1C).

Fig. 1. Schematic representation showing the interaction pathways of influenza virus infects human lung epithelial cell among (A) epithelial cell level, (B) human immune response level, and (C) virus level. The definition of symbols and detailed descriptions are explained in texts and Table 1.

The system of ordinary differential equations corresponding to the model in Fig. 1 and based on previous work (Baccam et al., 2006; Chang and Yang, 2007) is as follows,

(

^{X X}

)

^V

dt dX dX

− R

−

−

=λ β , (1)

(

X X

)

V aY qY pYZ

dt dY

R − − −

−

=β , (2)

pJZ aJ dt qY

dJ = − − , (3)

uV dt kJ

dV = − , (4)

(

^{Y J}

)

^{Z bZ}

dt c

dZ = + − , (5)

(

R

)

R R

R i X X X

dt

dX =γ − −α , (6)

(

^{i i}

)

ⁱ

V dt Γ di

MI

I − −α

= ^∗ , (7)

where λ is the equilibrium creation rate of epithelial cells (d^-1); β is the infection rate of an unprotected epithelial cell per virion (d^-1 virion^-1); a is the reciprocal of infected epithelial cell lifetime (d^-1); q is the transition rate from Y to J (d^-1); p is the infected epithelial cell CTL-induced destruction rate (d^-1 CTL^-1); k is the creation rate of viruses by an infected epithelial cell (d^-1 infected cell^-1);

u is the reciprocal of influenza A virus lifetime

(d^-1); c is the CTL-induced creation rate of CTL per infected epithelial cell (d^-1 infected cell^-1); b is the reciprocal of CTL lifetime (d^-1);

γ is the rate constant for induction of resistive

state by IFN (inflamed) (d^-1 IFN^-1); α_R is the rate of virus resistant epithelial cell decay (d^-1);

Γ

_I is the induction rate for IFN production (d^-1 virion^-1); i^＊is the effective creation rate number for IFN; and α_MI is the rate of loss of IFN-producing macrophages (d^-1). Model simulations were performed by using ode45 solver in MATLAB (The MathWorks Inc., Natick, Massachusetts, USA).

dX

IFN-protecte Cells (XR)

B

C

β(X-XR)V

γγγγi(X-XR)

pY Z

pJ Z λ

λλ λ

qY

kJkJ

cYZ cJZ

α α α αRXR

Uninfected cells (X)

Infected cells

Producing virus infected

cells (J)

bZ

Γ Γ Γ ΓIV(i^*-i) α

α α

αMIi Interferon

molecules (i)

uVuV

Free virus particles (V)

A

Cytotoxic T-cells (Z)

dY dJ

5

Parameter estimation and sensitivity analysis

The definition, symbols, input values, and expected physiological ranges of parameters in the present modified influenza virus dynamic model were summarized in Table 1. The input parameters of biological characteristics were derived from experimentally infected volunteers (Murphy et al., 1980; Bocharov and Romanyukha, 1994, Beauchemin et al, 2005) in that the physiological ranges for individual differences can also be estimated.

To identify the most significant sensitive parameters in this modified influenza virus dynamic model, we performed a sensitivity analysis for four parameters involving creation rates λ and k with infection rate β and transition rate q, respectively. This qualitative analysis can show robustness of the effects of free virons dynamics to variations in key parameters and model assumptions. The

sensitivity analysis was performed by varying the key creation and destruction rates of λ, q, k, and β ranged from 0 – 6.57×10⁷ d^-1, 2 – 6 d^-1, 67 – 6700 d^-1 infected cell^-1, and 3×10^-14 – 6×

10^-10 d^-1 virion^-1, respectively.

Experimental influenza infection survey for model validation

To perform the model validation, 10 published experimental influenza infection studies were used as the study data. Briefly, the participants were inoculated intranasally (0.25 ml per nostril) with inoculation dose ranged from 10^4.5 to 10⁷ tissue culture infected dose (TCID₅₀) of H1N1 virus on day 0. Nasal washings were collected before viral inoculation for detection of virus infection and collected (days 0, 1, 2, 3, 4, 5, 6, 7 and 8) for virus isolation.

Table 1. Definition, symbols, input values, and expected physiological ranges of parameters in the present modify influenza virus dynamic model

Symbols Definition (unit) Input

values^b Range^a X₀ Equilibrium number of normal epithelial cells in upper six branches

(＃) 10⁹ -

λ Equilibrium creation rate of epithelial cells (d^-1) 6.25×10⁷ -

d Reciprocal of epithelial cell lifetime (d^-1) 0.0625 -

β Infection rate of an unprotected epithelial cell per virion (d^-1 virion^-1) 10^-10 3×10^-14 – 6×10^-10 a Reciprocal of infected epithelial cell lifetime (d^-1) 1 0.5-2

q Transition rate from Y to J (d^-1) 4^c 2 – 6^c

p Infected epithelial cell CTL-induced destruction rate (d^-1 CTL^-1) 10^-10 4×10^-12 – 5×10^-10

V₀ Initial virus particles (virions) 10⁷ -

k Creation rate of viruses by an infected epithelial cell (d^-1 infected

cell^-1) 340 67 – 6700

u Reciprocal of influenza A virus lifetime (d^-1) 2 2 – 4

Z₀ Initial number of influenza A specific CTLs in upper six branches

(＃) 7×10⁶ 0.72×10⁶ – 7.2×10⁶

c CTL-induced creation rate of CTL per infected epithelial cell (d^-1

infected cell^-1) 3.6×10^{-8 a} -

b Reciprocal of CTL lifetime (d^-1) 0.5 -

γ Rate constant for induction of resistive state by IFN (inflamed) (d^-1

IFN^-1) 10^-9 10^-8 – 10^-10

αR Rate of virus resistant epithelial cell decay (d^-1) 1 - ΓI Induction rate for IFN production (d^-1 virion^-1) 8×10^-10 -

i^＊ Effective creation rate number for IFN 10¹⁰ 7.7×10⁷ –7.7×10¹⁰

αMI Rate of loss of IFN-producing macrophages (d^-1) 0.5 0.3 – 0.5

aAll of the estimated physiological ranges shown are based on the parameter estimates in Bocharov and Romanyukha (1994), except for the value of c which was directly obtained from Beauchemin et al. (2005).

bAdopted from Murphy et al. (1980)(cited in Chang and Young (2007)).

cAdopted from Baccam et al. (2006)

The age group, size of study subgroups, the number of infected population, even the number of shedding virus and shedding

durations were all recorded. We estimated the daily-based average viral titers (logTCID50

ml^-1) based on the published study data at

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specific day 0 (time for inoculating) to day 8.

Statistical analysis was performed by using free virions of the present modified virus dynamics and daily-based average viral titers based on the experimental influenza infections.

The data management and statistical analyses were performed by using SAS Version 9.1.3 for Windows (SAS Institute Inc., Cary, North Carolina, USA). Diger 4 software (Golden Software Inc., Golden, Colorado, USA) was used for data collection from the published studies.

Quantum generation rate for different influenza subtype

In this study, the “infectious dose” for influenza (sub)type viruses is quantified by the concept of “quantum” that was emitted by an infected person. For estimating quantum for influenza (sub)type viruses, we adapted the concept based on Nicas et al. (2005) by quantifying the risk of secondary airborne infection by the characteristics of emission of respiratory pathogens. Hence, we considered two parameters of particle size diameter and the day post infection that might impact the quantum estimations. We focused on the particle size diameter ≤ 10 µm to estimate the airborne infection (Duguid, 1946, Loudon and Robert, 1967) and defined the quantum as,

(

^,

)

t x x

q t x =E C× ×N ×v , (8) where q(t, x) is the quantum generation rate varying with the day post infection (t) and particle size diameter x (x≤10µm) (TCID50 h^-1), E is the expulsion event rate by sneeze (event hr^-1),

C

_t is the influenza (sub)type viruses shedding in respiratory fluid (TCID50 ml^-1),

N

_xis the particle number concentration in each particle size diameter x (ml^-1), and

v

xis the particle volume per expulsion event (ml).

The best fitted model for viral shedding of influenza H1N1 viruses were obtained from experimental data (Carrat et al., 2008) in that influenza data were provided by 116 participants who shed influenza viruses. The sum of the total particle volume at specific particle size diameter x can be expressed

as

N

_x×

v

_x . We adopted the valuable experimental data from Duguid (1946) to describe the relationship between particle size diameter and droplet number concentration of sneeze. The relationship between particle initial volume and number of particles emitted per sneeze was adopted from Loudon and Roberts (1967).

Wells-Riley mathematical equation

The Wells-Riley mathematical equation was used to estimate indoor airborne infection risk in an enclosed space. Riley and Nardell (1989) made two assumptions to quantify the indoor respiratory infections. The first assumption implies that an infectious droplet nucleus has an equal chance of being anywhere within a building’s airspace. The second assumption implies that the quantum concentration and the outdoor air supply rate remain constant with time.

We modified the Wells-Riley mathematical equation to estimate the transmission potential of influenza viruses in a hospital setting,



























 



 



−

−

−

−

−

=

=

V

Qt Qt

V Q

pt Iq S

P D

1 exp ^max 1 1 exp

(9)

where P is the probability of infection for susceptible population varied with influenza viruses, D is the number of cases among S persons susceptible to the infection, S is the number of susceptible, I is the number of sources of infection, qmax expressed the maximum value of our modeling results of

(

^,

)

q t x (TCID50 h^-1), p is the pulmonary ventilation rate of susceptible individuals (m³ d^-1), t is the exposure time (d), Q is the fresh air supply rate that removes the infectious aerosol in volume per unit of time (m³ h^-1), and V is the volume of the ventilated space (m³). For modeling the respiratory infection, we incorporate initial I =1 and S = n – 1 into Eq. (9) to estimate basic reproductive number (R0) for quantifying the average number of successfully secondary infection cases generated by a typical primary infected case in an entirely susceptible population,

7

( )



































 



 



 



−

−

−

−

−

−

=

V

Qt Qt

V Q

pt n q

R

₀ 1 1 exp ^max 1 1 exp

(10)

where the symbol of n represents the total number in our modeling ventilation airspaces.

The R0 of different influenza viruses thus can be estimated by using Eq. (10).

Results and discussion Influenza virus dynamics

We found that uninfected cells (X) decreased by free virus infection from the initial values of 10⁹ cells; then IFN-protected cells (X_R) increased rapidly at days 2 – 3 and reached a peak value of 8.2×10⁸ cells at day 3.4, showing the 89% protection by interferon at that moment (Fig. 2A). On the other hand, the peak values of infected cells (Y) and producing virus infected cells (J) occurred at days 2.5 and 7 with 9.55×10⁶ cells and 3.54×10⁷ cells, respectively (Fig. 2A).

In human immune response level, the first responder to influenza A infection was interferon molecules (i) that the production began less than 2 h after viral introduction, and then reached a peak between days 3 – 4 (Fig. 2B). The cytotoxic T-cell (Z) responder was much slower with a CTL activity peak at day 10 (Fig. 2B). In virus level, a large number of virus concentration appeared at day 2 and reached to a peak value of 5.96×10⁹ virons at day 7 (Fig. 2C). Our simulations showed that the virus replication in human epithelial lung cells was much more rapid than that of initial viral load at V₀ = 10⁷ virons.

Human experimental viral titer concentration

Table 2 summaries the experimental human influenza A (H1N1) infection data. The test age groups were young health man and women in that the number of infected population ranged from 8 – 59 persons with shedding durations ranged from 2 – 5.1 days.

In order to estimate the daily-based average viral titers, we used approximate values for Barroso et al. (2005). The overall patterns started after day 1 and approached to peak viral titers during day 2 with 10^3.51 TCID50

ml^-1 then slowly decreased to less than 10¹ TCID50 ml^-1 at day 6.

Fig. 2. Model influenza A variables in the three levels among (A) X (uninfected cells), XR (IFN-protected cells), Y (infected cells), and J (producing virus infected cells) in epithelial cell level, (B) Z (cytotoxic t-cells) and I (interferon molecules) in human immune response level, and (C) V (viral titers) with an initial viral load of 10⁷ virons. Input values of parameters are presents in Table 1.

Sensitivity analysis

To evaluate the variability of parameters that significantly contribute to the free virons, sensitivity analyses were performed. The time course of free virons was used to show the model sensitivity to variations in parameters k, and β, respectively, (Fig. 3). We found that the infection rates of an uninfected cell to infected cells increased with increasing the input parameters of infection rate β from 6×10^-10 to 3×10^-14 d^-1 virons^-1. Similarly, free virons in epithelial cell increased with increasing the value of creation rate of virus by an infected epithelial cell k (Fig. 3).

Here Pearson correlation analysis was used to determine the optimal parameter

0.0E+00 2.0E+08 4.0E+08 6.0E+08 8.0E+08 1.0E+09 1.2E+09

0 1 2 3 4 5 6 7 8 9 10

0.0E+00 2.0E+07 4.0E+07 6.0E+07 8.0E+07 X (Uninfected cells)

XR (IFN-protected cells) Y (Infected cells) J (Producing virus infected cells)

1.2×10⁹ 1.0×10⁹ 8.0×10⁸ 6.0×10⁸ 4.0×10⁸ 2.0×10⁸ 0.0 Uninfected cells (X), IFN-protected cells (XR)

8.0×10⁷

6.0×10⁷

4.0×10⁷

2.0×10⁷

0.0 Infected cells (Y) Producing virus infected cells (J) A. Epithelial cell level X (Uninfected cells)

XR (IFN-protected cells) Y (Infected cells) J (Producing virus infected cells)

0.0E+00 2.0E+09 4.0E+09 6.0E+09 8.0E+09 1.0E+10 1.2E+10

0 1 2 3 4 5 6 7 8 9 10

Z (Cytotoxic T-cells) I (Interferon molecules)

Days post infection (day) 0.0E+00

1.0E+09 2.0E+09 3.0E+09 4.0E+09 5.0E+09 6.0E+09 7.0E+09

0 1 2 3 4 5 6 7 8 9 10

7.0×10⁹ 6.0×10⁹ 5.0×10⁹ 4.0×10⁹ 3.0×10⁹ 2.0×10⁹ 1.0×10⁹

Free virons (V) C. Virus level 1.2×10¹⁰

1.0×10¹⁰ 8.0×10⁹ 6.0×10⁹ 4.0×10⁹ 2.0×10⁹ 0.0

B. Human immune response level

Z (Cytotoxic T-Cells) i (Interferon molecules)

Cytotoxic T-cells (Z), Interferon molecules (i)

0.0

8

inputs with best statistical significances between the daily-based average viral titers and the prediction of free virons that varied with physiological ranges of parameter. Our results indicated that input values of k = 4000 and β = 5×10^-10 resulted in a significant correlation between experimental human infection data and model predictions (r = 0.99,

p < 0.0001) (Table 3).

This sensitivity analysis thus suggested that infection rate of an unprotected epithelial cell (β) has an important role involved in the virus dynamics, whereas creation rate of virus by an infected epithelial cell (k) is the second sensitive parameter to the model.

Table 2. Summary of experimental human influenza A (H1N1) infection data Age

group

Size of study subgro ups (ni)

Virus Inocul. Dose (TCID50)

Infected N (% of subgroup

size)

Shedding virus N (% of infected)

Mean virus shedding

duration (day)

Reference

18-25 9 A/Californi a/10/78/H1 N1

10^4.5 8 (89) 8 (100) NA Treanor et al.

(1987)

18-45 16 A/Kawasak

i/86/H1N1

10⁷ 16 (100) 16 (100) 2.8 Hayden et al.

(1994) 18-33 59 A/Texas/91/

H1N1

10⁵ 49 (83) 49 (100) 2 Hayden et al.

(1996) 19-40 19 A/Texas/36/

91/H1N1

10⁵ 19 (100) 19 (100) NA Hayden et al.

(1998) 18-40 14 A/Texas/36/

91/H1N1

10⁶ 14 (100) 14 (100) NA Murphy et al.

(1998) 19-33 8 A/Texas/36/

91/H1N1

10⁵ 8 (100) 8 (100) 4.6 Fritz et al. (1999)

19-35 8 A/Texas/36/

91/H1N1

10⁵ 8 (100) 8 (100) NA Calfee et al. (1999)

18-40 13 A/Texas/36/

91/H1N1

10⁶ 13 (100) NA 4.5 Hayden et al.

(1999) 19-40 14 A/Texas/36/

91/H1N1

10⁵ 14 (100) 14 (100) 5.1 Kaiser et al. (1999) 18-45 18 A/Texas/36/

91/H1N1

10⁶ 17 (94) 17 (100) 3.2 Barroso et al.

(2005)

Fig. 3. Sensitivity analysis of λ (equilibrium creation rate of epithelial cells (d^-1)), q (transition rate from Y to

J (d^-1)), k (creation rate of viruses by an infected epithelial cell (d^-1 infected cell^-1)) and β (Reciprocal of epithelial cell lifetime (d^-1)) are presented. The expected physiological ranges of those four parameters are present in Table 1.

Table 3. Optimal Pearson correlation analysis between experimental human infection data and modeling results of modified virus dynamic model

Parameter

Input value (Original

value)

Optimal r p-value

λ 6.25×10⁷

(6.25×10⁷) - 0.43 0.2459

q 2 (4) - 0.46 0.2077

k 4000 (340) 0.996 < 0.0001

β 5×10^-10

(10^-10) 0.999 < 0.0001

Day post infection (day)

0.0E+00 5.0E+09 1.0E+10 1.5E+10 2.0E+10 2.5E+10 3.0E+10

0 1 2 3 4 5 6 7 8

β=3×10^(-14) β=4.24×10^(-12) β=-10^(-11) β=10^-10 β=1.5×10^(-10) β=2×10^(-10) β=3×10^(-10) β=4×10^(-10) β=5×10^(-10) β=6×10^(-10) β=3×10^-14 β=4.24×10^-12 β=1×10^-11 β=1×10^-10 β=1.5×10^-10 β=2×10^-10 β=3×10^-10 β=4×10^-10 β=5×10^-10 β=6×10^-10 B

0.00E+00 1.00E+11 2.00E+11 3.00E+11 4.00E+11 5.00E+11 6.00E+11

0 1 2 3 4 5 6 7 8

k=67 k=200 k=340 k=600 k=1000 k=2000 k=3000 k=4000 k=5000 k=6000 k=6700

A

k=67 k=200 k=340 k=600 k=1000 k=2000 k=3000 k=4000 k=5000 k=6000 k=6700 6×10¹¹

5×10¹¹ 4×10¹¹ 3×10¹¹ 2×10¹¹ 1×10¹¹ 0.0

Free virons (V)

3.0×10¹⁰ 2.5×10¹⁰ 2.0×10¹⁰ 1.5×10¹⁰ 1.0×10¹⁰ 5.0×10⁹ 0.0

Free virons (V)

9

Model validation

To test this prediction, we performed the model validation with derived optimal estimates of creation rate k and infection rate β values. Fig. 4A shows the time course of predicted free virons in epithelial cell with the optimal parameters of k = 4000 and β = 5×10^-10 against the experimental data of daily-based average viral titers. Generally, the results were in agreement with the experimental data trend, except on days 6 – 8 where the data experienced a decreasing fashion. Furthermore, this study also compared the predictive capacities among the free virons among target-cell limited model with delay virus production (Baccam et al., 2006), the immune response model (Chang and Young, 2007), and our modified virus dynamic model (Fig. 4B). Our results indicated that the best-fitted time course of simulated free virons by the present modified virus dynamic model well captured the observed dynamics (r² = 0.99, p < 0.0001) than those of the target-cell limited model with delay virus production (r² = 0.85, p = 0.0004) and the immune response model (r²

<

0.1, p = 0.955) (Fig. 4B). Therefore, the performance of the published models was relative lower than that of the present proposed model as revealed by model comparisons against the experimental data.

Fig. 4. (A) Mapping of the daily-based average viral titers and simulated free virons for model validation. (B)

Furthermore, this study compares the prediction accuracy of free virons among target-cell limited model with delay virus production (Baccam et al., 2006), immune response model (Chang and Young, 2007), and our modified model.

Quantum generation rate varied with influenza virus

Fig. 5A indicates the relationship between particle size diameter and particle number for sneeze that adopted from Duguid et al. (1946). Moreover, Fig. 5B shows the correlation between the particle size diameter and size-dependent total particle volume that adopted from Loudon and Roberts (1967). The analysis of the time-dependent virus concentration in respiratory fluid (

C

_t ) revealed that the influenza A (H1N1) curve sharply increased at day 1, reached the maximum values during the day 2, and return to the basic values at days 7 to 8 (Figs. 5A, 3C). Hence, we integrated the frequency of sneeze per hour (E) with E = 5 h^-1, time-dependent virus concentration in respiratory fluid for influenza viruses, and size-dependent total particle volume to simulate the dynamics of quantum generation rate (^{q t x}

(

^,

)

^).

Fig. 5. (A) The original experimental data fro sneeze from Duguid (1946) shows the relationship between particle size diameter and particle number concentration. (B) The size-dependent total particle

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12

0 1 2 3 4 5 6 7 8

Viral titer (Log TCID50 ml-1) A

-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11 1.0E+12

0 1 2 3 4 5 6 7 8

Day post infection

Viral titer (Log TCID50 ml-1 ) Baccam et al. (2006)

Chung and Young (2007) This study (β = 5×10^-10) B

1.0×10¹¹ 1.0×10¹⁰ 1.0×10⁹ 1.0×10⁸ 1.0×10⁷ 1.0×10⁶ 1.0×10¹² 1.0×10¹¹ 1.0×10¹⁰ 1.0×10⁹ 1.0×10⁸ 1.0×10⁷ 1.0×10⁶ 1.0×10¹²

This study (k = 4000) This study (β = 5×10^-10) Mean estimation of experimental data

Simulated free virons (V) Simulated free virons (V)

0 100000 200000 300000 400000

0 10 20 30 40

A

Particle number

4×10⁵

3×10⁵

2×10⁵

1×10⁵

0

Data Fitted Model

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

0 5 10 15 20 25 30 35 40

B

C

Particle size diameter (µm) 4.0×10^-3

3.5×10^-3 3.0×10^-3 2.5×10^-3 2.0×10^-3 1.5×10^-3 1.0×10^-3 0.5×10^-3

0

0.E+00 2.E-08 4.E-08 6.E-08 8.E-08 1.E-07

0 10 20 30 40

1×10^-7 8×10^-8 6×10^-8 4×10^-8 2×10^-8

Total particle volume (ml) Particle volume

10 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Date post infection (day)

A

r

²

=0.99

Fitted Model Data

Standard Error

0 1 2 3 4 5 6 7 8 9

2 3 4 5 6 7 8 9

0 1 2 3 4 5 6

Day post infection t (d) Particle size

diameter x (µµµµm)

q(t,x)

B

T C ID 50 h

-1

V ir u s con ce n tr at ion i n r es p ir at or y fl u id (T C ID 50 m l

-1

) (l og sc al e)

volume for sneeze which are estimated by Fig. 2A and Fig. 2C, in that (C) was the best fitted model to the data

Duguid (1946) and describing the relationship between the particle size diameters corresponding to the particle initial volume per sneeze from diameter 0 to 40 µm.

Fig. 6. Panels A, C, and E represented the viral dynamics of influenza A subtype H1N1, subtype H3N2, and influenza B viruses, respectively. Panels B, D, and F illustrate the quantum generation rate^{q t x}

(

^,

)

for sneeze events in which t represents the number of days after infection (day) and x expresses the particle size diameter (µm).

Fig. 6B revealed the interesting response surface of the influenza type-specific quantum generation rate by Eq. (8). This results indicate that the maximum quantum generation rate (q_max) is estimated to be 5.25 TCID50 h^-1 (at x = 10 µm, day 2 post infection) for influenza A (H1N1). The size-dependent particle number concentration of sneeze activity may explain why the qmax all appeared at the particle size diameter x = 10.

Observably, the exhaled particle number almost attended to 3×10⁵ – 4×10⁵ for one sneeze (Fig. 6A).

Risk of infection and basic reproduction number

Based on Eqs. (9) and (10), R0 in Wells-Riley mathematical equation can be estimated. The result indicates that the box and whisker plots of median with 95% CI of risks of infection (P) and basic reproduction numbers (R₀) are estimated to be 0.132 (0.09 – 0.19), 1.19 (0.76 – 1.86) for A (H1N1) (Fig. 7).

Potential transmission infection of three influenza viruses can be judged by R₀ > 1.

Fig. 7. The box and whisker plots of (A) the risk of infection (P) and (B) basic reproduction number (R0) for influenza A subtype H1N1, respectively.

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王芊樺、陳詩潔陳詩潔陳詩潔、游舒涵、龔冠瑜、謝孟陳詩潔 桓、王凱淞。咳嗽與說話飛沫粒徑特性研 究。2010 環境與室內空氣品質研討會 (中華 民國環境工程學會)，屏東。中華民國九十 九年十一月十二日、十三日。

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陳詩潔、劉靜宜、李思萱、周筱函、蔡一鳴、

龔冠瑜。2009。流感病毒動態於人體肺部之 研析。2009 台灣公共衛生學會。台北。中 華民國九十八年十二月五、六日。