Materials and methods

Natural history of influenza infection
Worldwide influenza pandemics have occurred at least three times in recorded history. The most serious pandemic is considered to have been the “Spanish flu” A (H1N1) in 1918–1919, which affected large parts of the world's population and was assessed to have killed at least 40 million people [14]. The much later “Asian flu” A (H2N2) in 1957–1958 and “Hong Kong flu” A (H3N2) in 1968–1969 also caused significant morbidity and mortality worldwide. To construct the model reported here, we used observational data from these previous pandemics for the latent and infectious periods, and illness attack and mortality rates as well as surveillance data for avian influenza A (H5N1) infection for the rate of crisis due to the unknown natural history of a future influenza infection.
Following the latent period, a person infected with influenza has infectivity during the infectious period; thereafter, he/she recovers with immunity or dies. The latent and infectious periods lasted 1–3 and 3–6 days with means of 1.9 and 4.1 days, respectively, in the “Asian flu” A (H2N2) [9], and the latent period ranged mainly from 1 to 4 days, with a maximum of 7 days for a few cases, in swine flu A (H1N1) in 2009 [15]. Based data on “Asian flu” A (H2N2), the latent and infectious periods were assumed to range over 1–3 and 3–6 days, respectively; the associated probabilities were tested at 30, 50, and 20%, and 30, 40, 20, and 10%, respectively.
Infected states were distinguished as symptomatic and asymptomatic infection. In this model, it was assumed that the rate of crisis was 67% on average, while the other cases were asymptomatic on the basis of the surveillance of avian influenza A (H5N1) infection [16], and that asymptomatic cases had half the infectivity of symptomatic cases [9]. A confirmative rate in all infections was defined by the ratio of patients who sought medical care and were diagnosed with a novel influenza. Because many cases of avian influenza A (H5N1) appeared to have severe symptoms with vomiting and diarrhea [17], the confirmative rate was assumed to be higher, at 60%.
The illness attack rate is different among different types of viruses [18], with that of the “Asian flu” A (H2N2) being much higher in children than in adults [19]; in contrast, the illness attack rate of the “Hong Kong flu” A (H3N2) was approximately the same among all age groups [20]. The illness attack rate of swine flu A (H1N1) virus has been reported to be twice as high in children as in adults [21]. A study of the sporadic transmissions of avian influenza A (H5N1) virus from birds to humans revealed that children are more susceptible to this virus than adults [2]. Because the age-specific illness attack rate of avian influenza A (H5N1) was similar to that of “Asian flu” A (H2N2) [22], this model adopted the estimated values in the “Asian flu” [10] as the age-specific illness attack rate. The mortality in “Spanish flu” A (H1N1) in 1918–1919 showed a W-shaped curve, i.e., it was high not among infants and elderly people but also among young adults (25–34 years), an age group which usually has a very low mortality with seasonal influenza [23, 24]. Although the profile for a future influenza pandemic is unclear, we assumed that a novel influenza infection would cause a high mortality similar to the situation of the “Spanish flu” A (H1N1) [24]. The age-specific illness attack and mortality rates in this model are presented in Table 1.
Table 1 Illness attack rate and mortality
Age (years) Illness attack rate (%)a Mortality (%)b
<1 32 2.25
1–4 0.70
5–14 46 0.16
15–24 0.60
25–34 29 1.00
35–44 0.58
45–54 0.32
55–64 0.42
65–74 0.67
75–84 1.20
>84 2.20
aRefer to [10]
bRefer to [24]
There are currently two kinds of antiviral drugs, oseltamivir and zanamivir, and both are regarded as being highly effective for both the treatment and prophylaxis of influenza infection [5, 25]. As treatment, a person must be treated with two tablets of oseltamivir a day for a course of 5 days for treatment; as prophylaxis, treatment consists of one tablet a day over a course of 7–10 days. Antiviral efficacy lasts only during the course, and no residual effect remains thereafter. The administration of antiviral drugs alleviates symptoms, reduces infectiveness, shortens the infectious period as a treatment effect, and also prevents infection as a prophylactical effect [26, 27]. Longini et al. [10] estimated the effects of administering the antiviral drugs that were also adopted in this model and found: (1) the illness attack rate in susceptible persons dosed with antiviral drugs for prophylaxis decreased to 0.30 on average compared with those without dosing (relative susceptibility = 0.30); (2) the probability of developing symptoms of influenza in infected persons dosed with antiviral drugs for prophylaxis decreased to 0.60 on average compared with those without dosing; (3) the infectiousness of infected persons dosed with antiviral drugs for treatment or prophylaxis decreased to 0.62 on average compared with those without dosing (relative infectiousness = 0.62); (4) the infectious period in infected persons dosed with antiviral drugs for treatment or prophylaxis was reduce to 1 day compared with those without dosing.

Study area and population structure of the IBM
Sapporo city, the capital of Hokkaido in Japan, was chosen as the targeted area because Sapporo city has a number of characteristics that make it suitable for simulating the spread of infection, including relatively small daily influxes and outflows of people compared with other major cities in Japan [28]. The National Census of Japan [28, 29] was used as the source of data on the age structure, resident population for each ward, household composition, places of schooling and work, and mode of transportation. The School Basic Survey [30] was the source of data on the number of schools and the enrollment rate at college, while the Employment Status Survey [31] provided the employment rate.
Sapporo city, which had a population of 1,880,863 in 2005 [28], is one of the ordinance-designated metropolis cities in Japan and is divided into ten wards as administrative units [32]. There is a substantial difference in the density of population among wards. There are also many commuters to school and work between wards during the daytime, which leads to the mass movement of people on a daily basis [28]. To simulate the situations of an influenza epidemic on a realistic basis, we adopted an IBM that can reflect the heterogeneous population structure for the targeted area, in which each individual was assigned his/her personal information, such as age, household, residence district, social activity group, and casual contact group. The age structure in Sapporo city [28] is shown in Fig. 1. The distribution of household size was determined on the basis of the National Census of Japan data [28], where each household was assumed to consist of at least one adult aged ≥19 years.
Fig. 1 Age distribution of Sapporo city [28]
We specified five categories of social activity groups that an individual may belong to according to age: playgroup (3–6 years), elementary school (7–12 years), high school (13–18 years), college group (19–22 years), and work group (19–64 years). The age-specific rates of the enrollment of college students and employment were determined based on data in the School Basic Survey [30] and the Employment Status Survey [31]. The size of social activity groups was assumed to be smaller than their actual size so that individuals in a group could come into close contact with each other and transmit the influenza virus among them.
Two kind of untraceable casual contact groups were considered: a low-risk group who came into contact with other people at markets, in the neighborhoods, etc. and a high-risk group who used public transportation facilities, such as crowded trains and buses. The ratio of high-risk casual contact groups was set at 38% on average in all casual contact groups based on a census of the mode of transportation [29]. All casual contact groups were composed afresh twice a day with a size of 20 individuals on average.

Contact rate in the IBM
We assumed that individuals would mix with each other randomly (in the household, during social activities, and in casual contact groups) and that individuals had a possibility of becoming infected when an infected person belonged to any of their groups. The average contact rate in a household was set to be higher than that for other groups, followed by that in social activity groups. The average contact rate in a playgroup was set to be higher than that in other social activity groups because children spent more time in close proximity to each other at a day nursery. The usual average contact rate in the low-risk casual contact group was set to be quite low because such contacts occur accidentally during outings [12]. On the other hand, the average contact rate in high-risk casual contact groups, such as while commuting on crowded trains or buses, was assumed to be 30-fold higher than that in the low-risk casual contact group. This was assessed by the average commuting time to school and work (an hour per day) [33]. The average contact rates for every group are tabulated in Table 2. Because it is difficult for symptomatic patients to follow their usual behavioral pattern [34], it was assumed that a symptomatic patient had a different behavioral pattern from the day after onset; a symptomatic patient retreats to the home, thereby exposing only his/her family [35].
Table 2 Daily average contact rates in the household, during social activities, and in casual contact groups
Group Infected Susceptible Contact rate (on average)
Household Child Child 0.6a
Child Adult 0.3a
Adult Child 0.3a
Adult Adult 0.4a
Social activity group
 Playgroup Child (3–6 years) Child (3–6 years) 0.25a
 Elementary school Child (7–12 years) Child (7–12 years) 0.0435a
 High school Child (13–18 years) Child (13–18 years) 0.0375a
 College Adult (19–22 years) Adult (19–22 years) 0.0315a
 Work Adult (19–64 years) Adult (19–64 years) 0.0575a
Casual contact group
 Low Anyone Child (0–4 years) 0.0000181a
Anyone Child (5–18 years) 0.0000544a
Anyone Adult (19–64 years) 0.000145a
Anyone Adult (65–years) 0.0002175a
 High Anyone Child (0–4 years) 0.000543
Anyone Child (5–18 years) 0.001632
Anyone Adult (19–64 years) 0.00435
Anyone Adult (65–years) 0.006525
aRefer to [12]

Control strategies
This article explored four kinds of interventions with the aim of analyzing their effectiveness in terms of suppressing an influenza epidemic: administration of antiviral drugs for prophylaxis through two methods of medication, TAP and STAP; school closure; restraint; combinations of these interventions were also analyzed. It was assumed that all interventions would be started 14 days after the introduction of the initial patient.

Targeted antiviral prophylaxis
In the TAP intervention strategy, patients with symptoms as well as persons in close contact with them are treated with antiviral drugs in accordance with the “Pandemic Influenza Preparedness Action Plan” [5]. TAP is regarded as an important intervention in the early phase of an influenza epidemic in the “Guidelines for the Prevention and Control of Pandemic Influenza” [25]. In this model, it was assumed that a patient with symptoms would be treated with an antiviral drug just being diagnosed with a novel influenza infection, that all members of his/her household would be dosed with an antiviral drug for prophylaxis on that same day of diagnosis or thereafter, and that a person belonging to the same social activity group as the diagnosed patient would be traced, if possible, and treated with an antiviral drug within a few days of contact [12]. In this strategy, no members of the casual contact group could be traced. One of the advantages of TAP intervention is the small amount of antiviral drugs that are distributed for prophylaxis.
The model limits the possible number of symptomatic cases for whom members of the social activity group can be traced for close contact to ten persons for each ward per day because of the limited capacity of the healthcare center (only one center for each ward in Sapporo city) [32]. During the swine flu A (H1N1) outbreak in Japan, not all of the persons in close contact with an infected person could be traced and, moreover, some traced persons refused to take antiviral drugs [36]. Various situations with respect to the proportion of identifiable close contact persons (30, 50, and 70%) and necessary tracing periods (2, 4, and 6 days) were examined.

School-age TAP
In the STAP intervention strategy, antiviral drugs are distributed to the children of a school once a child is diagnosed with a novel influenza at a medical institution to prevent infection among school-aged children, who have a high attack rate. Naturally, all household members of the infected person would be dosed with an antiviral drug for prophylaxis from the day of diagnosis. Proportions for the coverage of distribution to children in the affected schools of 30, 50, and 70% were investigated, as was the necessary tracing period of 2, 4, and 6 days.

School closure
School closure is regarded as an important intervention measure because an influenza epidemic can be spread widely among children at school [5]. In this model, when a person is diagnosed with a novel influenza at a medical institution in a ward, school closure will be implemented from the day following the diagnosis for all schools in this ward. The “Guidelines for the Prevention and Control of Pandemic Influenza” [25] states that a judgment based on the epidemiological investigation will end the period of school closure; however, in our model, school closure was assumed to be continued until no patients were detected in the ward.

Restraint
Requests to residents to refrain from going out unnecessarily have been noted as a containment policy [5]. Restraint was generally recognized as a preventive measure by residents in Hong Kong when they experienced an outbreak of severe acute respiratory syndrome (SARS) [34]. However, many people in Japan are unwilling to practice a voluntary policy of restraint, and in one study, only 46.2% of people expressed willingness to practice restraint in a “Survey of pandemic behavior: to stay at home or not.” [37]. When a person is diagnosed with a novel influenza at a medical institution in a ward, restraint will be performed in this ward from the day following diagnosis. Various rates of restraint (10, 30, and 50%) were investigated. The period of restraint was assumed to continue for 2 weeks, and a person who agreed to restraint once was not required to be restrained again for at least 1 month.

Combined intervention
We investigated five scenarios with different combinations of interventions for their effects on suppression; these included the administration of antiviral drugs for treatment (TAP/STAP) with/without school closure and with/without restraint (Table 3). In TAP and STAP, the necessary tracing period and the coverage were set to 4 days and 50%, respectively; for restraint, the rate was set to 30%. TAP should be terminated when the number of patients reaches ten persons for each ward per day because the “Pandemic Influenza Preparedness Action Plan” [5] showed that the administration of medicine to persons in close contact with a patient should be stopped after an influenza infection has spread widely.
Table 3 Scenarios with different combinations of intervention measures
Scenario TAP STAP School closure Restraint
1 ○ ○
2 ○ ○
3 ○ ○
4 ○ ○ ○
5 ○ ○ ○
TAP targeted antiviral prophylaxis, STAP school-age targeted antiviral prophylaxis

Method of stochastic simulations
The stochastic model was programmed using Intel Visual Fortran to work on any computer using the Microsoft Windows platform, where the random number generators in the IMSL libraries were used. The time step of the stochastic process was adopted to be 1 day, and 100-trial simulations were carried out over 360 days, or until the epidemic was eradicated.