Probabilistic indoor transmission modeling
for influenza (sub)type viruses
Szu-Chieh Chen
a,b, Chung-Min Liao
c,*
a
Department of Public Health, Chung Shan Medical University, Taichung, Taiwan 40242, ROC
bDepartment of Family and Community Medicine, Chung Shan Medical University Hospital, Taichung, Taiwan 40242, ROC c
Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan 10617, ROC Accepted 29 September 2009 KEYWORDS Influenza; Indoor transmission; Infection; Modeling; Vaccine
Summary Objectives: To use a probability based transmission modeling approach to examine the influenza risk of infection virus in indoor environments. This was based on 10 years of data gathered from influenza-like illness sentinel physician and laboratory surveillance, and experimental viral shedding data in Taiwan.
Methods: We integrated sentinel physician-reported cases and positive rates of influenza A (H1N1), A (H3N2), influenza B, and respiratory syncytial virus in Taiwan using the WellseRiley mathematical model. This model incorporates environmental factors such as room ventilation and breathing rates. We also linked vaccine match rate with related transmission estimations to predict the controllable potential using a control model characterized by basic reproduction number (R0) and proportion of asymptomatic infections (q).
Results: A quantitative framework was developed to better understand the infection risk and R0estimates of A (H1N1), A (H3N2), and B viruses. The viral concentration in human fluid was linked successfully with quantum generation rates to estimate virus-specific infection risks. Our results revealed that A (H3N2) virus had a higher transmissibility and uncontrollable potential than the A (H1N1) and B viruses.
Conclusions: Probabilistic transmission model can incorporate virus-specific data on experi-mental viral shedding, long-term sentinel physician and laboratory surveillance to predict virus-specific infection risks in Taiwan.
ª 2009 The British Infection Society. Published by Elsevier Ltd. All rights reserved.
Introduction
Influenza is one of the most important infectious diseases affecting humans. The continuous threat of pandemic
human influenza pandemics suggests an urgent need to con-duct long-term year-round viral surveillance of individual (sub)types in order to improve our understanding of the hu-man influenza.1 In the past, reliable estimates based on probabilistic transmission modeling for influenza virus
* Corresponding author. Tel.:þ886 2 23634512; fax: þ886 2 23626433. E-mail address:[email protected](C.-M. Liao).
0163-4453/$36ª 2009 The British Infection Society. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jinf.2009.09.015
(sub)types have been rare, especially those based on human influenza experimental data.2,3
Carrat et al.3 indicated that different influenza (sub)-type viruses exhibit only slight differences in viral shedding, although A (H3N2) infections gave consistently higher viral titers compared with A (H1N1) infections. The intrinsic dif-ferences between influenza (sub)types may determine their potential infection risk in healthy individuals. By drawing on evidence derived from experimental volunteers to investi-gate the natural history and dynamics of viral shedding of vi-rus (sub)types, the accuracy of the predicted outcomes of control measure programs can be ascertained.
Although influenza vaccination is still considered a com-monly used intervention for containing influenza trans-mission worldwide, human influenza A (H1N1), A (H3N2), and influenza B viruses continue to be predominant circu-lating strains globally.4e6 Hsieh et al.7 reported that the
match rate for influenza vaccines, based on World Health Organization (WHO) figures, were 82% for A (H1N1), 53% for A (H3N2), and 47% for influenza B virus for the period from 1997 to 2004 in Taiwan. This match rate was markedly lower than the 77% match seen in circulating strain vaccines worldwide during this time.
Currently, no simple control modeling has been used to take into account for the different influenza virus (sub)types in the R0eqcontrol model. The basic reproduction number R0 essentially determines the rate of spread of an epidemic and given an indication of the policy intensities required to control the epidemic.8,9qrepresents the proportion of asymptomatic infections which arises prior to the onset of symptoms for each influenza (sub)type viruses. Fraser et al.10have adopted
these two key variables of transmission to analyze the general properties of directly transmitted agents and determine the success rate of certain public health measures for containing early-stage outbreaks. By integrating the match rate and related transmission estimations, a more useful illustration of the controllable potential can be derived based on the concept of controllability.
The monitoring of influenza activity has been under-taken by the Centers for Diseases Control, Taiwan (Taiwan CDC) since 1999.11,12Influenza-like illness (ILI) surveillance was carried out by sentinel primary care physicians and was based on integrated clinical and virological surveillance components.11e13 There are 13 contracted collaborating
laboratories distributed geographically in northern, cen-tral, southern, and eastern Taiwan. Based on the results of yearly ILI surveillance components, respiratory syncytial virus (RSV) appears to be the most frequent cause of respi-ratory tract infections in children.14,15
Recently, disease transmission via exhaled infectious droplets in the indoor environment has received substantial attentions.16,17Early research has held that the upper
respira-tory tract (nose, mouth and throat) is the primary location of droplet formation.18,19As such, the particle size distributions
of expired droplets play a key role in the evaluation of infec-tion risk. Duguid18indicated that the lognormal distribution could best describe the respiratory droplet with a geometric mean (GM) of 14 mm and a geometric standard deviation (GSD) of 2.6 for cough, and a GM of 8.1 mm with a GSD of 2.3 for sneeze. Papineni and Rosenthal20measured expired bio-aerosol droplets to be less than 2 mm in size with no droplets larger than 8 mm in the nose and mouth when breathing, coughing, and talking.
Our previous studies21,22have focused on the transmission and control measure modeling by integrating the WellseRiley mathematical model and a deterministic epidemiological susceptible-exposed-infected-recovery (SEIR) model. These models were used to estimate age group-specific infection risks in the indoor environments throughout summer and winter seasons. In this study, we integrated the sentinel physician-reported cases and positive rates of influenza A (H1N1), A (H3N2), B, and RSV in Taiwan with the WellseRiley mathematical model. This model allows for the incorpora-tion of environmental factors such as room ventilaincorpora-tion and breathing rates. The objective of this study was to employ the probabilistic transmission modeling approach to examine the virus-specific infection risk in the indoor environments. Our model incorporates data based on 10 years of ILI sentinel physician and laboratory surveillance, and experimental viral shedding data in Taiwan.
Materials and methods
ILI sentinel physician and laboratory surveillance Weekly-based ILI sentinel physician surveillance data in Taiwan were obtained from Taiwan CDC for the period from
Table 1 Epidemiological periods of influenza A (H1N1), A (H3N2), and influenza B virus. Infectious disease Incubation period (days) Latent period (days) Infectious period (days) Mean duration of illness (days) Mean duration of viral shedding (days)
Influenza 1e4a 1e3a 4e8a
1e3b 1.9c 4.1c
1e3d 1e3d 2e3d
A (H1N1) 4.4e5e 3.70 (1.73, 5.66)e
A (H3N2) 3.7e 4.50 (3.71, 5.28)e
B 4.1e 5.14 (4.48, 5.80)e
a Adopted from Anderson and May.8
b Adopted from Thomas and Weber.33
c Adopted from Mills et al.34
d Adopted from Anderson.35
e Adopted from Carrat et al.3expressing as mean with 95% CI.
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1999 to 2006. This data represent ILI cases reported in patients under outpatient and hospital care in all medical centers and teaching hospitals in Taiwan. The case definition of ILI included patients with fever (ear temper-ature over 38C), and respiratory symptoms and signs such
as myalgia, headache, and fatigue.11
Weekly data on influenza (sub)type isolates in various regions around Taiwan were obtained from Taiwan CDC for the period from 2002 to 2008. The positive rates of influenza A (H1N1), A (H3N2), and influenza B viruses were determined by the percentage of respiratory tract infections that were positive for influenza each week. Data relating to the positive rates of RSV were also obtained from Taiwan CDC for the period from 2004 to 2008.
The biological characteristics and pathogenesis of each influenza (sub)types is summarized inTable 1. Natural his-tory provides the baseline for estimating the proportion of asymptomatic infections (q) for each influenza (sub)-type. We assumed the values for incubation period, latent period, and mean duration of viral shedding to be a normal distribution. The estimation of q was thus calculated by di-viding the probability distribution of the asymptomatic in-fection period with the probability distribution of the mean duration of viral shedding. The Monte Carlo simula-tion was performed to quantify the uncertainty of q by us-ing Crystal Ball software (Version 2000.2, Decisioneerus-ing, Inc., Denver, CO, USA). Table Curve 2D (Version 5.01, AISN Software Inc., Mapleton, OR, USA) was used to per-form model fitting techniques.
Figure 1 (A) Weekly-based influenza-like illness (ILI) sentinels physician surveillance in Taiwan area for the year 1999e2006. (B) Weekly-based positive rate of influenza A (H1N1), A (H3N2), and influenza B viruses from 2002 to 2008. (C) The box and whisker plot of positive rate of influenza A (H3N2), A (H1N1), B, and respiratory syncytial virus (RSV). (D) Weekly-based positive rate of RSV from 2004 to 2008.
Quantum generation rate for different influenza (sub)types
In this study, the ‘‘infectious dose’’ of virus was quantified by the concept of ‘‘quantum.’’ We adapted the concept based on Nicas et al.16 in order to estimate quantum for influenza virus (sub)types. This was achieved by quantifying the risk of secondary airborne infection based on the char-acteristics of emission of respiratory pathogens. We there-fore considered two parameters that might affect quantum estimation, particle size diameter and days post infection. We used a particle size diameter10 mm to estimate air-borne infection risk and defined quantum with the following equation,18,19
qðt; xÞZE Ct Nx nx; ð1Þ where q(t, x) is the quantum generation rate varying with the day post infection (t) and the particle size diameter x 10 mm (TCID50 h1), E is the expulsion event rate by sneeze (event hr1), Ctis the influenza virus (sub)type shed-ding in respiratory fluid (TCID50 ml1), Nxis the particle num-ber concentration in each particle size diameter x (ml1), and vxis the particle volume per expulsion event (ml).
The best-fitting model for viral shedding of influenza A (H1N1), A (H3N2), and B viruses were obtained from experimental data3as provided by 116, 41, and 8 partici-pants who shed influenza viruses, respectively. The sum of the total particle volumes at specific particle size diam-eter x can be expressed asNx vx. We adopted the
avail-able experimental data from Duguid18 to describe the relationship between the particle size diameter and droplet number concentration of sneeze. The relationship between the particle volume and the number of particles emitted per sneeze was adopted from Loudon and Roberts.19
WellseRiley mathematical equation
The WellseRiley mathematical equation was used to estimate the indoor airborne infection risk in an enclosed
space. Riley et al.23made two assumptions to quantify the indoor respiratory infections. The first assumption implies that an infectious droplet nucleus has an equal chance of
0 10 20 30 40
A
Particle number 4 105 3 105 2 105 1 105 0 Data Fitted Model 0 5 10 15 20 25 30 35 40B
C
Total article volume (ml)
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 Particle volume (ml) 0 10 20 30 40 1 10-7 8 10-8 6 10-8 Particle size diameter ( m)
Particle size diameter ( m)
Figure 2 (A) The original experimental data for sneeze from Duguid18shows the relationship between particle size diameter
and particle number concentration. (B) The size-dependent to-tal particle volume for sneeze which are estimated byFig. 2A
andFig. 2C, in that (C) was the best fitted model to the data
Duguid18and describing the relationship between the particle
size diameters corresponding to the particle initial volume per sneeze from diameter 0 to 40 mm.
Table 2 Optimal fitted equations of particle number, time-dependent virus concentration in respiratory fluid, size-dependent total particle volume per expulsion event of cough.
Type Fitted equationc r2
Particle number
Sneeze: NxZ2123 þ 367734expð0:5ðlnðx=7:11Þ=0:65Þ2Þ 0.99 (T1)a
Time-dependent virus concentration in respiratory fluid
A (H1N1) log (Ct) Z LN(0.05, 2.91, 2.13, 3.59, 1.82) 0.99 (T2)b
A (H3N2) log (Ct) Z LN(7.76, 10.98, 2.63, 14.98, 3.71) 0.98 (T3)b B log (Ct) Z LNs(703.71, 706.47, 3.04, 1998.50, 214.69) 0.84 (T4)b
Size-dependent particle volume
vxZ1:3852 þ 0:0341x3 0.97 (T5)a
Size-dependent total particle volume
Sneeze: NxvxZLNð232283:11; 31368754:02; 27:29; 53:33; 2:44852Þ 0.99 (T6)b
a Based on data from Duguid.18
b LN4ða; b; c; d; eÞZa þ bexpðln2lnð1 þ ðx cÞðe2 1Þ=ðdeÞÞ2=lnðeÞ2Þ.
c N
xis the particle numbers at specific particle diameter x (mm), Ctis the virus concentration in respiratory fluid (TCID50 ml1) at specific time t (d), vxis the particle volume at specific particle diameter x (mm), andNxvxis total particle volume at specific particle
diameter x (mm).
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4 S.-C. Chen, C.-M. Liao
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 WellseRiley mathematical equation to estimate the transmission potential of influenza (sub)type viruses in a hospital setting,24
PZD SZ1 exp Iqmaxpt Q 1V Qt 1 exp Qt V ; ð2Þ where P is the probability of infection for susceptible population varied with influenza (sub)type, S is the number of susceptible individuals, D is the number of positive cases
among S individuals susceptible to the infection, I is the number of sources of infection, qmaxis the maximum value of the modeling results of q(t,x)(TCID50 h1), p is the pulmonary ventilation rate of susceptible individuals (m3d1), t is the exposure time (d), Q is the fresh air supply rate that removes the infectious aerosol in volume per unit of time (m3h1), and V is the volume of the ventilated space (m3). To model the respiratory infection risk, we incorporated I Z 1 and S Z n 1 into Eq. (2) to estimate the R0for quantifying the average number of successfully secondary infection cases generated by a typical primary infected case within an entirely susceptible population as,
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 A (H1N1)
A
r2=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 6Day post infection t (d) Particle size diameter x (μm) q(t,x)
B
h 0 5 DI C T 1-l m 0 5 DI C T( di ul f yr ot ar i ps er ni n oi t ar t ne c n oc s ur i V ) el ac s g ol ( ) 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 A (H3N2) r2=0.98C
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 7 8 9 10 h 0 5 DI C T1-D
q(t,x) 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 15Date post infection (d)
B
E
r2=0.84 0 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 0 0.5 1 1.5 2 2.5 3 3.5F
q(t,x) h 0 5 DI C T1-Figure 3 (A, C, and E) represented the viral dynamics of influenza A (H1N1), A (H3N2), and influenza B viruses, respectively, and (B, D, and F) illustrated the quantum generation rate q(t,x)for sneeze in that t expressed the day post infection (day) and x expressed the particle size diameter (mm).
R0Zðn 1Þ 1 exp qmaxpt Q 1V Qt 1 exp Qt V ; ð3Þ
where n represents the total number in the ventilation airspaces. The virus-specific R0 values can then be esti-mated by using Eq.(3).
Control measure effect of influenza vaccination We adopted the R0eq control curve to formulate the control measure effect. The concepts of our control model were based on those elaborated by Fraser et al.10and used
the two key parameters of R0and q to predict the level of control policy required to achieve outbreak containment.
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0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50A (H1N1) A (H3N2) B
0.0 0.4 0.8 1.2 1.6 2.0(
n
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A (H1N1) A (H3N2) B
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Air change per hour (h-1)
n oi tc ef ni f o ks i R (P ) 0.000 0 1 2 3 4 5 6 0.200 0.400 0.600 0.800 A(H3N2) A(H1N1) B
D
E
0.0 0.2 0.4 0.6 0.8 1.0 Median 25%-75% 2.5%-97.5%A (H1N1) A (H3N2) B
A
θ
Mean duration of viral shedding (days)
B
0.00 0.01 0.01 0.02 0.02 0.03 0.96 2.00 3.04 4.08 5.12 6.16 yti li b a b or P A(H1N1) Mean =3.70 day 0.00 0.01 0.01 0.02 0.02 0.03 4.21 4.57 4.94 5.30 5.66 6.02 B Mean =5.14 day 0.00 0.01 0.01 0.02 0.02 0.03 3.42 3.85 4.29 4.72 5.15 5.58 Mean =4.50 day A(H3N2)Figure 4 The box and whisker plots of (A) proportion of asymptomatic infectious (q) was presented in that (B) show the proba-bility of mean duration of viral shedding for influenza A (H1N1), A (H3N2), and influenza B viruses, respectively. The risk of infection (P), and basic reproduction number (R0) for influenza A (H1N1), A (H3N2), and influenza B viruses were also illustrated in that (D) estimated the P values varied with air change rate from 0 to 6 per hour.
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6 S.-C. Chen, C.-M. Liao
By following the parameter estimates for R0and q, the R0eq critical control line can be constructed from the control measure efficacy and R0estimate based on the WellseRiley equation.24 Here the R0eq control curve can be written as,10
R0Z h
ð1 3Þ þ 3qi1; ð4Þ
where 3 is the efficacy of influenza vaccination.
Factors affecting the efficacy of influenza vaccination include vaccine coverage rate, vaccine implement methods, and predictive epidemic strains. However, we focused only on the vaccine match rates of WHO recommended influenza vaccines for strains currently circulate in Taiwan.
Based on the R0eqcontrol curve, when a given infectious agent is situated below the R0eqcurve (as denoted by A1), the outbreak is always eventually controlled. Conversely, when an infectious agent lies above the curve (as denoted by A2), additional control measures are required to control the spread. Therefore, The uncontrollable ratio can be estimated from the ratio of A1/(A1þ A2) and can be used to assess the effectiveness of adopted control measures.
Results
Data reanalysis of sentinel physician and laboratory surveillance
Fig. 1Ashows the epidemics of influenza virus in winter sea-sons as annual trends.Fig. 1Aalso indicates that influenza A (H3N2) had observable activity during all periods, whereas influenza A (H1N1) was a dominant strain in the winters of 2005e2006 and 2007e2008. The gray bands represent the winter seasons (weeks 49e52 and weeks 1e9), which featured an average weekly reported ILI cases of 1126. Influenza B virus was a dominant strain in the winters of 2004e2005 and 2006e2007 (Fig. 1B). The uncertainties of positive rates for A (H1N1), A (H3N2), and type B are given inFig. 1C.Fig. 1Calso indicates that median positive rates (95% confidence interval (CI)) of 1.81 (0.04e11.84%), 0.32 (0.01e9.86%), 1.43 (0.12e16.30%), and 1.43 (0.01e1.56%)
for influenza A (H3N2), A (H1N1), influenza B virus and RSV, respectively. Fig. 1D shows that the weekly-based averaged positive rate of RSV were estimated to be 0.42% during 2004e2008.
Quantum generation rate for different influenza (sub)types
Fig. 2A shows the relationship between the particle size diameter and particle number of a sneeze event, as adop-ted from Duguid.18The best-fitting equation was presented inTable 2(Eq. (T1)) with r2
Z 0.99. Moreover,Fig. 2Bshows the correlation between particle size diameter and size-dependent total particle volume as adopted from Loudon and Roberts19 (Eq. (T6)), in that the size-dependent parti-cle volume (Fig. 2C) was fitted by Eq. (T5) (Table 2).
In regard of the time-dependent viral concentration present in respiratory fluid (Ct), it was revealed that the in-fluenza A (H1N1) and A (H3N2) curves sharply increased at day 1, reached the maximum values at day 2, and returned to the baseline values at days 7e8 (Fig. 3A, C). Hence, we integrated the frequency of a sneeze event per hour (E ) with E Z 5 h1, time-dependent virus concen-tration in respiratory fluid for different influenza (sub)types (Eqs. (T2)e(T4)), and size-dependent total particle vol-umes (Eqs. (T5) and (T6)) in order to simulate dynamics of the quantum generation rate (q(t, x)).
Fig. 3B, D, and F reveals the interesting response surfaces of the influenza (sub)type-specific quantum gener-ation rates by Eq.(1). Results indicated that the maximum quantum generation rate (qmax) was estimated to be 5.25 TCID50 h1at x Z 10 mm and day 2 post infection with influ-enza A (H1N1). Influinflu-enza A (H3N2) and type B were esti-mated to be nearly 9.22 TCID50 h1 and 3.33 TCID50 h1 at x Z 10 mm and day 3 post infection, respectively. These results implied that type A influenza was the most virulent. The size-dependent particle number concentration of sneeze activity may explain why the qmaxall appeared at the particle size diameter x Z 10 mm. Fig. 2Aalso shows that the number of exhaled particles reaches almost to 3 105e4 105for one sneeze event.
Table 3 Input parameters used in WellseRiley mathematical equation to estimate the basic reproduction number (R0).
Symbol Meaning Value Remark
n People in the ventilated airspace LN (10, 1) Assumed
I Number of infectors 1 Assumed
V Volume of the shared airspace Uniform (250e350) m3 Assumed
t Exposure time Uniform (0.25e0.33) d Estimated
p Pulmonary ventilation rate of susceptible individuals N (11.16, 0.20) m3d1 Estimateda f Fraction of indoor air as exhaled
breath (f Z np/Q)
0.003875 Estimated
Q Fresh air supply rate (based on Q Z 1 ACH) N (1.5, 0.3) h1 Assumed qA (H1N1) Quantum generation rate of influenza A (H1N1) virus 5.251 TCID50 h1 Estimatedb qA (H3N2) Quantum generation rate of influenza A (H3N2) virus 9.218 TCID50 h1 Estimatedb qB Quantum generation rate of influenza B virus 3.325 TCID50 h1 Estimatedb
E Number of sneeze per hr 5 h1 Assumed
a Adopted from ICRP.36
b Estimated by qðt; xÞZE C
t Nx vx, in that this study presents the maximum estimations for different influenza (sub)type virus,
Risk of infection and basic reproduction number The proportion of asymptomatic infections (q) ranged from 0.01 to 0.82 (95% CI), 0.01 to 0.47, and 0.01 to 0.40 for in-fluenza A (H1N1), A (H3N2), and B viruses, respectively (Fig. 4A). The probability distributions of the mean duration of viral shedding for the three strains are shown inFig. 4B with means of 3.7, 4.5, and 5.14 day, respectively. Result indicated that influenza A (H1N1) virus had higher q than the other two strains. It also showed a negative relationship with the longer mean duration of viral shedding. By using Eqs. (2) and (3), the R0 of the WellseRiley mathematical equation can be estimated (Table 3).
This study used a set of assumed values for the number of individuals in the ventilated airspace (n), the volume of the shared airspace (V), exposure time (t), fresh air supply rate (Q), and number of sneeze events per hour (E ) to simulate a hospital setting. Other values such as the qA (H1N1), qA (H3N2),and qBwere estimated by Eq.(1). The result indicated that the box and whisker plots of median with 95% CI of infec-tion risks (P) and R0were estimated to be 0.132 (0.09e0.19), 1.19 (0.76e1.86); 0.157 (0.108e0.229), 1.41 (0.92e2.19); and 0.120 (0.08e0.178), 1.07 (0.69e1.69) for A (H1N1), A (H3N2) and type B viruses, respectively (Fig. 4Cand E). The potential transmission of infection for the three influenza viruses can be judged by R0> 1. Using the WellseRiley math-ematical equation, we also analyzed variation in the air change rate. The results demonstrated that environmental parameters contribute insignificant to the effects of virus-specific infection risk (Fig. 4D).
Virus-specific R0eqrelationships
Fig. 5 shows the virus-specific R0eq critical control lines obtained by combining the two key parameters estimates of R0and q (Eq.(4)). The only one control measure we considered was the year-round mass vaccination for hospital worker ulations, the elderly (aged 65 and over), and other at risk pop-ulations. Vaccine efficacy was adopted from the vaccine match rates described by WHO match rates for the relevant strains circulating in Taiwan. The match rates were estimated to be 82%, 53%, and 47% for A (H1N1), influenza A (H3N2), and type B viruses, respectively (Fig. 5).
The estimates for the uncontrollable ratios were 14.9% for A (H1N1), 31.5% for A (H3N2), and 10% for type B. This indicated that the influenza A (H3N2) virus features a more significant uncontrollable potential. To further ascertain the effects of different match rates on controllable potentials, we modeled the R0eq critical control lines by assuming match rates of 60e90%, 40e70%, and 30e60% for influenza A (H1N1), influenza A (H3N2), and influenza B viruses, respectively. The results showed uncontrollable ratios ranging from 12 to 24% for A (H1N1) (Fig. 5A), 12 to 57% for A (H3N2) (Fig. 5B), and 1.3 to 37% for influenza B (Fig. 5C).
Discussion
The nature of its interaction with the human immune system not only determines the patterns and processes of
evolutionary change in influenza viruses, but also reflects how these viruses interact with each other.25Moreover, in light of analyses of their genomic and epidemiological dynamics, the long-term monitoring of positive rate of influenza (sub)type viruses may play a specific role in reducing the year-round predominance of these (sub)types. Our results showed medians of positive rate to be 1.81% (95% CI 0.04e11.84), 0.32% (0.01e9.86), and 1.43% (0.12e 16.30) for A (H3N2), A (H1N1), and type B viruses, respec-tively, for the period from 2002 to 2008. The A (H3N2) showed higher potential positive rate than the other influ-enza strains. Intensive studies have been carried out to characterize the laboratory-based surveillance and molecu-lar epidemiology of influenza (sub)type viruses in Tai-wan.6,13,26Lin et al.13 reported that 1.5% of isolates were
A (H1N1), 21.5% were A (H3N2), and 77.0% were type B viruses during the 2006e2007 period in Taiwan. The
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 90% 80% 70% 60%
A
A (H1N1) 82%Basic reproduction number (
R0 ) A (H3N2)
B
53% 70% 60% 50% 40% 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 60% 50% 40% 30%C
BProportion of asymptomatic infectious ( ) 47%
θ
Figure. 5 (A)e(C) represent the associated with the esti-mates of 95% CI of basic reproduction number (R0) and the 95% CI of the proportion of asymptomatic infection (q) for influ-enza A (H1N1), A (H3N2), and influinflu-enza B viruses, respectively. We assumed that the match rates of WHO-commended vaccine compositions with circulating strains in Taiwan are (A) 60e90%, (B) 40e70%, and (C) 30e60%, respectively. An area above the curve (denoted as A2) means additional control measures would be required to control the spread.
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appearance of type B virus in Taiwan has also been dis-cussed recently.6,26
The phenomena associated with these three (sub)type viruses are also similar to that of other countries. In Thailand, the positive rates of influenza A and B viruses in 2004 were 85.52% and 14.47%, respectively (461 influenza A positive, 78 influenza B positive, out of a total of 539 infected specimens), and 65.77% and 34.22%, respectively in the 2005 (492 influenza A positive, 256 type B positive, out of a total of 748 infected specimens).27 Most of the influenza-positive isolates in Taiwan were influenza A, which is consistent with the WHO reports on worldwide in-fluenza activity.28
Carrat et al.3reviewed a number of volunteer challenge
studies, including experimental influenza infections, and discussed the dynamics of viral shedding, symptoms, and the relationship between viral shedding and illness in in-fected volunteers. Based on the findings of Carrat et al.,3 the maximum quantum generation rate (qmax) for A (H3N2) virus was estimated to be 9.22 TCID50 h1at droplet diameter of 10 mm and day 3 post infection. The estimates of q were proportional to the virus shedding patterns in respiratory fluids. This implied that influenza A virus was much more virulent than influenza B. Webster et al.29 also indicated that influenza B virus normally exists at a lower prevalence and causes a milder disease severity than that of influenza A viruses, especially when compared to the A (H3N2) virus.
Dilution ventilation with fresh air in the WellseRiley mathematical equation plays a crucial role in influenza infection. Natural or mechanical ventilations can both contribute to air change per hour (ACH). In the hospital setting, many spaces generally utilize mechanical ventila-tion. Our results demonstrated a negative correlation with virus-specific infection risk (P) and ACH rates.
There are several limitations to be noted in this study. Firstly, the WellseRiley airborne infection model assumes conditions to be in steady state and infection constitutes a one-hit process. Secondly, the study also did not take into account the fact that the proximity of susceptible individuals to an infectious source is likely to influence their infection risk. Lastly, the proposed WellseRiley
mathematical equation also did not factor in the deposition or settling of droplet particles from the air.30
In this study, the virus-specific R0values were estimated to be 1.19 (95% CI 0.76e1.86), 1.41 (0.92e2.19), and 1.07 (0.69e1.69) for A (H1N1), A (H3N2) and type B viruses, respectively. Recently reported virus-specific estimates of R0 and q are listed in Table 4. Massad et al.31 analyzed the 1918 A (H1N1) pandemic outbreak in the city of Sa˜o Paulo, indicating that the estimated R0value of 2.68 is com-parable to estimates carried out for other influenza strains, such as estimates of 1.5 to 2.5 for the A (H3N2). Rvachev and Longini32estimated an R0of 1.89 for the first wave of pandemic A (H3N2) which commenced in July 1968 in Hong Kong. Our findings in the present study regarding vi-rus-specific R0 estimates seem to be consistent with past research. However, at the time of writing, no relevant ex-isted which related to influenza B virus. There is also insuf-ficient data in the literature concerning the natural history of various strains of influenza to allow for accurate q esti-mates. Constructing the R0eq relationship allows rapid mathematical prediction and comparisons between differ-ent vaccine efficacies to be drawn.
In conclusion, by integrating the experimental viral shedding characteristics, quantum generation rate, and WellseRiley mathematical model, we were able to develop a quantitative framework describing infection risk and basic reproductive numbers of A (H1N1), A (H3N2), and B viruses. In the present investigation, the application of a R0eq critical control line was also able to simulate the controllable level, with vaccine efficacy values derived from match rate figures. Virus-specific infection risks were estimated by linking viral concentrations in human fluids with quantum generation rates. From this we were able to conclude that A (H3N2) virus exhibits a higher trans-missibility and uncontrollable potential than A (H1N1) and B viruses.
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10 S.-C. Chen, C.-M. Liao
