ISSN: 1080-6059
Volume 13, Number 4–April 2007
Research
Effectiveness of Interventions to Reduce Contact Rates during a Simulated Influenza Pandemic
Michael J. Haber,* 
David K. Shay,† Xiaohong M. Davis,† Rajan Patel,‡ Xiaoping Jin,† Eric Weintraub,† Evan Orenstein,§ and William W. Thompson†
*Emory University Rollins School of Public Health, Atlanta, Georgia, USA;
†Centers for Disease Control and Prevention, Atlanta, Georgia, USA; ‡Amgen, Inc., Thousand
Oaks, California, USA; and §Yale University, New Haven, Connecticut, USA
Suggested citation for this article
Supplemental Materials Appendix:
Details of the Simulation Model
The Model
The following parameters describe how persons made contact with others. For
individual 
from stratum 
and
mixing group of type 
and stratum 
, we denoted
by 
the group of all persons with whom made contact on a day of type 
,
where 
for weekdays and 
for
weekend days. These groups were referred to as "contact groups." The size, 
,
of and the average total duration in
minutes, 
, of all the contacts made by with
each member of on 1 day were specified as input
parameters. At the beginning of each simulation, the initial contact groups 
were
determined for by selecting at random 
persons
of stratum from each mixing group other than
the household to which belonged. For households (
)
the contact groups consisted of all household members (other than ) in the corresponding stratum. If a
mixing group had fewer than members of
stratum, then the contact group consisted of
the entire mixing group.
Influenza transmission was determined by contact parameters and transmission
rates. The rate of viral transmission per minute of contact from an infected
person in stratum to a susceptible person in stratum 
(where 
) was
denoted by 
. The probability that transmission
occurred during a contact of 
minutes was
.
On each day of the simulated outbreak, the model calculated for each
susceptible person the probability of becoming infected that day, based on the
contacts made with all persons in each contact group. Consider a susceptible
person from stratum and
a person 
in one of 's
contact groups, 
. Define 
if was
not infectious and 
if was
infectious. The probability that escaped
infection from that day was
.
To remain uninfected, must have escaped infection from
all the members of all her/his contacts groups. Hence the probability that became
infected on this day was: 
.
This probability was compared with a random number, 
,
drawn from the interval [0,1]. The person A became infected if 
.
Each newly infected person entered a latent period, at the conclusion of which the person became infectious to others, based on values estimated by Elveback et al. (1). We assumed that the probability of symptoms developing, given influenza infection, was 0.67 and that an infected person who did not become ill was 50% less infectious than one who did. An ill person with severe symptoms withdrew to home, made contacts only with household members, and the duration of these contacts was decreased by 50%. We assumed that in 50% of adults and 75% of children severe symptoms developed and the person withdrew. An ill person could require hospitalization or die from influenza complications. The probabilities of hospitalization and death were determined on the basis of the distribution of age-specific hospitalizations and deaths in an average seasonal (nonpandemic) influenza season (2–4) and on the total hospitalization and death rates expected in a pandemic that is similar to the Asian influenza pandemic of 1957–58, for which the overall illness attack rate was estimated at 0.33, with an influenza death rate of 0.58/1,000 persons (5). A list of the initial settings of all the parameters used in these models is provided below.
The simulated epidemic started with a small number of infective persons. The transmission process continued until no further infected persons remained in the community. At the end of each simulated epidemic, the program determined the proportion of persons who became ill as well as the proportions of hospitalizations and deaths in the community. We ran 200 simulations and calculated the means of the above 3 proportions over these simulated epidemics.
Baseline Values of Parameters
A. Influenza-related Parameters (based on Longini et al. [5])
The following parameters describe how persons made contact with others. For
individual from stratum and
mixing group of type and stratum , we denoted
by the group of all persons with whom made contact on a day of type ,
where for weekdays and for
weekend days. These groups were referred to as "contact groups." The size, ,
of and the average total duration in
minutes, , of all the contacts made by with
each member of on 1 day were specified as input
parameters. At the beginning of each simulation, the initial contact groups were
determined for by selecting at random persons
of stratum from each mixing group other than
the household to which belonged. For households ()
the contact groups consisted of all household members (other than ) in the corresponding stratum. If a
mixing group had fewer than members of
stratum, then the contact group consisted of
the entire mixing group.
Influenza transmission was determined by contact parameters and transmission
rates. The rate of viral transmission per minute of contact from an infected
person in stratum to a susceptible person in stratum (where ) was
denoted by . The probability that transmission
occurred during a contact of minutes was.
On each day of the simulated outbreak, the model calculated for each
susceptible person the probability of becoming infected that day, based on the
contacts made with all persons in each contact group. Consider a susceptible
person from stratum and
a person in one of 's
contact groups, . Define if was
not infectious and if was
infectious. The probability that escaped
infection from that day was.
To remain uninfected, must have escaped infection from
all the members of all her/his contacts groups. Hence the probability that became
infected on this day was: .
This probability was compared with a random number, ,
drawn from the interval [0,1]. The person A became infected if .
Each newly infected person entered a latent period, at the conclusion of which the person became infectious to others, based on values estimated by Elveback et al. (1). We assumed that the probability of symptoms developing, given influenza infection, was 0.67 and that an infected person who did not become ill was 50% less infectious than one who did. An ill person with severe symptoms withdrew to home, made contacts only with household members, and the duration of these contacts was decreased by 50%. We assumed that in 50% of adults and 75% of children severe symptoms developed and the person withdrew. An ill person could require hospitalization or die from influenza complications. The probabilities of hospitalization and death were determined on the basis of the distribution of age-specific hospitalizations and deaths in an average seasonal (nonpandemic) influenza season (2–4) and on the total hospitalization and death rates expected in a pandemic that is similar to the Asian influenza pandemic of 1957–58, for which the overall illness attack rate was estimated at 0.33, with an influenza death rate of 0.58/1,000 persons (5). A list of the initial settings of all the parameters used in these models is provided below.
The simulated epidemic started with a small number of infective persons. The transmission process continued until no further infected persons remained in the community. At the end of each simulated epidemic, the program determined the proportion of persons who became ill as well as the proportions of hospitalizations and deaths in the community. We ran 200 simulations and calculated the means of the above 3 proportions over these simulated epidemics.
Baseline Values of Parameters
A. Influenza-related Parameters (based on Longini et al. [5])
Pandemic illness rates by age group (used for calibration of transmission rates and contact parameters): 0–4 years, 36%; 5–18 years, 62%; 19–64 years, 25%; >65 years, 21%; overall rate, 33%.
Probability of illness given infection = 0.67.
Relative infectiousness of infected persons who do not become ill = 0.50.
Rate of withdrawal due to "severe" symptoms: in children, 0.75; in adults, 0.50.
Relative contact rate when withdrawn due to "severe" symptoms = 0.50.
B. Transmission Rates
We assumed that the transmission rate (transmission probability per 1 minute of contact) might vary by the ages of the infected and susceptible persons but not by the mixing group or by weekday versus weekend day. The values of the transmission rates, which are presented in Appendix Table 1, were determined in a calibration process so that the above illness attack rates were obtained.
C. Probabilities of Hospitalization and Death, given Illness, by Age Group
For the purpose of estimating the hospitalization and death probabilities, we used 9 age groups. We started with data on influenza-related hospitalization and death rates for an average seasonal influenza epidemic (2–4). We then adjusted these rates so that we obtained the predicted overall rates for a pandemic (247 and 70 per 100,000, respectively, based on Meltzer et al. [6]). To determine the conditional probabilities for ill persons we divided these rates by the expected pandemic illness rates listed in section A. The conditional probabilities are presented in Appendix Table 2.
D. Contact Frequencies and Durations
D.1. Persons Who Reside at Home
Four age strata are included in the simulation models: 0–4; 5–18; 19–64; and >65 years. There are 5 types of mixing
groups: households, daycare centers, schools, workplaces, and the community
(for contacts of long-term care facility [LTCF] residents, see section D.2).
For a given mixing group and type of day, and for each combination of 2 strata 
we
needed to determine: (i) the number of persons from
stratum contacted in 1 day by a person
from stratum , 
, and (ii) the average
total duration per day (in minutes) of all the contacts with 1 person, 
.
These numbers are symmetric: 
and 
.
D.1.a. Weekdays
Contacts in the household: We assumed that each member of the household
contacted every other member, so we did not specify 's. Appendix Table 3 presents values for the 's.
Contacts in daycare centers: 
, 
.
All other contact parameters are zero.
Contacts in schools: 
, 
.
All other contact parameters are zero.
Contacts in workplaces: 
, 
.
All other contact parameters are zero.
Contacts in the community: Appendix Table 4 presents the values of (
). For simplicity, we assume that no
contacts occur between children and adults in the community.
D.1.b. Weekend days
On a weekend day, contacts are made only in households and in the community.
The weekend values of the 's in
households and in the community are twice the corresponding weekday values. The
community weekend values of the 's
are twice the corresponding weekday values.
D.2. LTCF residents
Each LTCF resident made contacts with 4 other residents for an average of 120 minutes (on weekdays and on weekend days) and with 2 members of the LTCF staff for 120 minutes (weekdays and weekend days). In addition, this person has contact with 1 family member for 60 minutes on weekdays and with 2 family members for 120 minutes each on weekend days.
To illustrate the computation of the daily infection probabilities, we assume that person A is a susceptible school-aged child (stratum 2) who lives in a household with 2 parents (ages 19–64, stratum 3) and a younger preschool child (stratum 1). We now calculate the probability that A will become infected on a given weekday. For this illustration we make the simplifying assumption that every person with whom A makes contact is infectious on this day. Person A makes contact in the household, in his or her school and in the community.
In the household, A makes contacts lasting a total of 
60
minutes (Appendix Table 3) with the preschool
child. The per-minute transmission rate from infectious younger child to person
A is 
0.00062 (Appendix Table
1). Therefore the probability that A escapes infection from that child is 
.
Person A also makes contact with his or her parents. The total duration of the
contacts with each parent is 
120 minutes (Appendix Table 3), and the transmission rate from the infected
parent is 
0.00053 (Appendix Table
1). Therefore, the escape probability from each parent is exp(-0.00053
120)
= 0.9384. The probability that A escapes infection
from all the household members is 0.96350.9384
= 0.8485.
In the school, person A makes contact with 10 other schoolchildren 
, where the total duration of the
contacts that each child makes is 120 minutes 
. The
per-minute transmission rate is 
0.00061. Therefore the
escape probability from all school contacts is [exp(-0.00061120)]
= 0.4809.
In the community, person A makes contact with 1 preschool child (
, Appendix Table 4) lasting a total of 30 minutes (
, Appendix Table 4), and with 2 school-aged children (
),
for a total of 60 minutes each (
. The per-minute
transmission rates from the preschool child and from each school-aged child are 0.00062 and 0.00061,
respectively. Hence the escape probability from all the community contacts is
[exp(-0.0006230)][exp(-0.00061
)] = 0.9123.
Thus, the overall probability that person A becomes infected on this day is
.
(This very high daily probability of infection is the result of the assumption
that all the persons with whom A makes contact on this day are infectious.) To
determine if A actually becomes infected, a random number between 0 and 1 is
generated, and if this number does not exceed 0.6277, then the simulation
program determines that A becomes infected on this day.
Appendix References
- Elveback LR, Fox JP, Ackerman E, Langworthy A, Boyd M, Gatewood L, et al. An influenza simulation model for immunization studies. Am J Epidemiol. 1976;103:152–65.
- Thompson WW, Shay DK, Weintraub E, Brammer L, Cox N, Anderson LJ, et al. Mortality associated with influenza and respiratory syncytial virus in the United States. JAMA. 2003;289:179–86.
- Thompson WW, Shay DK, Weintraub E, Brammer L, Bridges CB, Cox NJ, et al. Influenza-associated hospitalizations in the United States. JAMA. 2004;292:1333–40.
- Thompson WW, Weintraub E, Shay D, Brammr, Cox N, Fukuda K. Estimation of influenza-associated deaths and hospitalizations in the United States. In: Osterhous ADME, Cox N, Hampson AW, editors. Options for the control of influenza IV. Philadelphia: Elsevier Science, International Congress Series, 2004;1263:316–20.
- Longini IM, Halloran ME, Nizam A, Yang Y. Containing pandemic influenza with antiviral agents. Am J Epidemiol. 2004;159:623–33.
- Meltzer MI, Cox NJ, Fukuda K. The economic impact of pandemic influenza in the United States. Emerg Infect Dis. 1999;5:659–71.
Appendix Tables
Appendix Table 1. Transmission rates (γij) from an infectious person in age group j to a susceptible person in age group i.
Appendix Table 2. Age-specific conditional probabilities of hospitalization and death, given influenza infection
Appendix Table 3. Duration of contacts with household members
Appendix Table 4. Number of contacted persons and total duration of all contacts with 1 person in the community
Suggested Citation for this Article
Haber MJ, Shay DK, Davis XM, Patel R, Jin X, Weintraub E, et al. Effectiveness of interventions to reduce contact rates during a simulated influenza pandemic. Emerg Infect Dis [serial on the Internet]. 2007 Apr [date cited]. Available from http://www.cdc.gov/EID/content/13/4/581.htm
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Michael J. Haber, Department of Biostatistics, Rollins School of Public Health, Emory University, Atlanta, GA 30322, USA; email: mhaber@sph.emory.edu
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