A 'small-world-like' model for comparing interventions aimed at preventing and controlling influenza pandemics
© Carrat et al; licensee BioMed Central Ltd. 2006
Received: 22 February 2006
Accepted: 23 October 2006
Published: 23 October 2006
With an influenza pandemic seemingly imminent, we constructed a model simulating the spread of influenza within the community, in order to test the impact of various interventions.
The model includes an individual level, in which the risk of influenza virus infection and the dynamics of viral shedding are simulated according to age, treatment, and vaccination status; and a community level, in which meetings between individuals are simulated on randomly generated graphs. We used data on real pandemics to calibrate some parameters of the model. The reference scenario assumes no vaccination, no use of antiviral drugs, and no preexisting herd immunity. We explored the impact of interventions such as vaccination, treatment/prophylaxis with neuraminidase inhibitors, quarantine, and closure of schools or workplaces.
In the reference scenario, 57% of realizations lead to an explosive outbreak, lasting a mean of 82 days (standard deviation (SD) 12 days) and affecting 46.8% of the population on average. Interventions aimed at reducing the number of meetings, combined with measures reducing individual transmissibility, would be partly effective: coverage of 70% of affected households, with treatment of the index patient, prophylaxis of household contacts, and confinement to home of all household members, would reduce the probability of an outbreak by 52%, and the remaining outbreaks would be limited to 17% of the population (range 0.8%–25%). Reactive vaccination of 70% of the susceptible population would significantly reduce the frequency, size, and mean duration of outbreaks, but the benefit would depend markedly on the interval between identification of the first case and the beginning of mass vaccination. The epidemic would affect 4% of the population if vaccination started immediately, 17% if there was a 14-day delay, and 36% if there was a 28-day delay. Closing schools when the number of infections in the community exceeded 50 would be very effective, limiting the size of outbreaks to 10% of the population (range 0.9%–22%).
This flexible tool can help to determine the interventions most likely to contain an influenza pandemic. These results support the stockpiling of antiviral drugs and accelerated vaccine development.
There are increasing concerns that an A/H5N1 influenza pandemic is imminent. Based on data from recent pandemics, 50 countries have developed pandemic preparedness plans and most industrialized countries are stockpiling antiviral drugs . An international workforce has been created to develop an H5N1 vaccine , and immunogenicity trials are promising [3, 4].
Public health decision-making will be based largely on experience with past pandemics, but models are needed to plan and evaluate interventions based on vaccination, antiviral prophylaxis/therapy, quarantine, and closure of public places. As the transmissibility and pathogenicity of emerging influenza viruses cannot be predicted, and neither can their pandemic potential, such models should be flexible enough to be adapted to a wide range of situations. They must deal with various types of populations and test different kinds of interventions, used together or in isolation.
Recent papers focus on the containment of an outbreak in a rural area of Southeast Asia, where a pandemic virus seems most likely to emerge [5, 6], or on strategies for mitigating the severity of a pandemic in the United States or Great Britain, where a virus is likely to spread secondarily [7, 8]. The authors used different methodologies, but the results of both studies showed that a nascent pandemic could be contained by using a combination of antiviral drugs and confinement measures. Another paper suggested that, in the United States, vaccination (particularly of children) could be very effective .
We have developed a model for simulating the spread of influenza virus infection in the community during a pandemic. The model includes not only individual parameters, which take into account the risk of infection and the dynamics of viral shedding according to age, treatment, and vaccination status, but also community parameters, in which meetings between individuals are simulated by the use of a complex random graph.
Individual-centered model of influenza infection, illness, and health-care use
A computer model was first developed to describe influenza infection and its consequences for a given individual. We used the classical four-stage model of infection, as follows: Susceptible (S – may be infected), Exposed (E – is infected but cannot transmit the disease), Infectious (I – is infected and can transmit the disease), and Recovered (R – can no longer transmit the disease and is immune to new infections).
The three basic parameters used to describe transitions between the different stages were the person-to-person transmission rate, which is assumed to vary with the age of susceptible and infectious individuals and with the time since infection; the length of the latent period (time between infection and onset of infectivity); and the length of the infectious period.
Parameters describing the transmissibility and pathogenicity of influenza virus.
Adapted from , and consistent (to a scale factor) with [10,15–21]
Children (0–18 years)
Adults (19–65 years)
Elderly (>65 years)
Proportion of asymptomatic individuals (children, adults, elderly people)
; also used in 
Relative infectivity of asymptomatic individuals
Assumption also used in 
As influenza virus infection is not always symptomatic, we postulated that 30% of infected individuals would not be sufficiently ill to be identifiable , and that these subjects would be half as infective as other subjects. For symptomatic individuals, we postulated that the duration and intensity of symptoms would be proportional to infectivity, based on the observation that the onset of symptoms after experimental infection coincides with a sharp increase in viral shedding [10, 15–21], i.e. the incubation period is equal to the latent period.
For case and contact tracing, and for access to interventions (treatment, prophylaxis, etc.), patients must be seen by a physician. We postulated that most symptomatic subjects would seek medical advice (90%), and that 40% of those who consulted would do so within the first day after onset, 30% the second day, and 30% after the second day. These rates were chosen to be higher than those observed during a seasonal influenza epidemic , as public awareness would be higher in a pandemic situation and as antiviral treatment would be available only from a physician. Finally, we postulated that 80% of individuals who consulted a physician would remain confined to their home for one week.
We postulated that 5% to 13% of symptomatic subjects (depending on age) would be hospitalized for serious complications and that 20% to 30% of those hospitalized would die. The case-fatality rates thus ranged from 1% to 4%, in keeping with data collected during previous pandemics [22, 23] The average hospital stay was set at 12 days, based on French national statistics on hospitalization for pneumonia and influenza . We postulated that transmission could not occur between patients or from patients to hospital staff, owing to strict application of preventive measures.
The community model was based on a complex random graph realistically describing meetings between individuals. We first generated a set of individuals based on a particular demographic profile (gender, age groups, and household sizes) adapted from French national census data , in which each individual is assigned to a household and a place of occupation (for example, a school for a child, or a workplace for a working adult). Households and places of occupation were assigned to districts, and children were preferentially assigned to schools located in the district where they lived; 20% of working adults were assigned to workplaces located in other districts. In the reference simulation, 23% of individuals were children, 67% were adults (80% in employment), and 10% were elderly.
Two types of bidirectional graphs were generated. First, a fully connected graph was generated for each household, as we assumed that every household member would make daily meetings with all other household members (if any).
For schools, workplaces and other locations (nursing homes, hospital, etc), meetings between individuals were modeled with the Barabasi-Albert (BA) random graph . The BA graph was developed in the late 1990s to describe systems in which the probability that a node will have a given number of connections with other nodes does not depend on the size of the system. This type of graph can correctly describe systems such as links on the worldwide web and citations in scientific journals [27, 28]. It can also provide a realistic representation of social contacts: the first application of this method was to describe the network of movie actors ).
BA graphs are built up from a small initial numbers of nodes (three, for example), in two steps: a growth step, in which a new node with m connections is added; and a preferential attachment step, in which the nodes to which the new node connects are chosen. The probability Π that the new node will be connected to node i depends on the connectivity k i of that node, such that . The probability density P(k) that a node in the network is connected to k other nodes is independent of the size of the system and has a power law distribution, that is P(k) ~ Ak-γ, where [l.c. gamma] is 3 and coefficient A is proportional to the square average connectivity of the network (A ~ m2). The average connectivity of a BA graph is 2m.
Parameters describing the community model simulating the spread of influenza.
1 to 6
5 classes; 20 children and 2 adults per class
Each class is modeled using a BA graph (m = 2); supplementary random links between individuals belonging to different classes.
Children living in the district
13 to 15 classes; 30 students and 3 teachers per class
Children and students are linked to teachers.
One college for 5 districts
6 to 3000 according to Zipf distribution 
BA graph (m = 6)
80% from the district; 20% from outside the district
45 elderly people, 50 employees per nursing home
BA graph (m = 6)
BA graph (m = 1)
Simulation process and empirical calibration
Each simulation started with the generation of a network of 10,000 individuals and one infected individual. In order to deal with heterogeneities of susceptibility or connectivity between individuals, we proceeded as follows: we first randomly chose one infected individual and then simulated the first generation of secondary infections. Then each individual infected during the first generation was used as the initial infective in a new simulation where the network and the population were reset to their initial values. The selection of an individual from the first generation ensures proper sampling of the initial infected individual in a heterogenous contact network .
A discrete time step (half a day) was chosen. At each time point, meetings between infectious and susceptible individuals were derived from the graph, and transmission of influenza virus during each meeting was simulated by comparing a uniform random number with the calculated probability of transmission. The per-meeting probability of transmission was calculated as the product of infectivity (depending on time since infection) and the relative susceptibility of the contact, and was adjusted for other parameters (vaccination, treatment, etc.). The simulations stopped after the maximal length of the infectious period following the last transmission event.
A critical parameter in the epidemiology of infectious diseases is the basic reproductive number (R0). R0 is defined as the average number of secondary infections produced by a single infected person in a fully susceptible population. In our model, analytical calculation of R0 is not feasible . For this reason, we proceeded by simulation, randomly choosing one infective subject as described above, and then counting the number of secondary infections.
The observed rates of seroconversion and illness due to the pandemic strains that circulated during the 20th century were used to calibrate the model, and particularly to scale infectivity. During the 1957 pandemic, serological infection rates as high as 75% were observed among children and 25% among adults . In 1918, during the first pandemic wave, the attack rate of clinical influenza was maximum in children (40%) and then fell gradually with age, reaching 9% in people aged 75 years or more. An average attack rate of 34% was reported during the 1957 pandemic, with an age distribution similar to that observed during the first pandemic wave of 1918 . The age distribution of attack rates during the 1968 pandemic was noticeably different, with values decreasing less markedly with age, ranging between 41% and 43% in children, but remaining above 30% in all other age groups . Most of these rates were obtained from studies of families with children (which tend to overestimate the true attack rates in the general population), but served as benchmarks for empirical calibration of our model. The shape and length of the pandemic curve were also consistent with those reported in cities during the 1918 pandemic .
Reference scenario for a flu pandemic after one initial case (no intervention). Estimates are cumulative numbers per 100 inhabitants, unless otherwise specified.
n = 114
n = 200
Children (0–18 years) a
Adults (19–65 years) a
Elderly (>65 years) a
Lost workdays b
Treatment with neuraminidase inhibitors of 90% of individuals consulting a physician for 'flu-like' symptoms. Estimates are cumulative numbers per 100 inhabitants, unless otherwise specified.
Outbreak; n = 106
All simulations; n = 200
0.12 – 0.32
Lost workdays b
Treatment units (doses) c
Household contact prophylaxis with antiviral drugs, with or without treatment of the index cases. The interventions are applied in 70% of households in which one member consults a physician. Estimates are cumulative numbers per 100 inhabitants, unless otherwise specified.
(Outbreak, n = 90)
Prophylaxis + treatment
(Outbreak, n = 98)
Lost workdays b
Treatment units (doses) c
We also modeled a scenario in which mass vaccination would begin a certain time after identification of the first case (0, 14, 28 days) and in which the target level of vaccine coverage would be achieved within 14 days. We postulated that individual protective immunity would be achieved two weeks after vaccination and that vaccination would reduce susceptibility by 80% during each meeting (leaky vaccine, meaning that vaccinated individuals would respond by acquiring partial immunity, rather than acquiring either complete immunity or no immunity at all ). Mass vaccination could take place in schools, workplaces, nursing homes, hospitals, and physicians' offices. We assumed that vaccination would lead to the loss of 0.04 workdays per working adult .
Reactive vaccination of 70% of the susceptible population according to the interval between implementation and identification of the first case in the community. Estimates are cumulative numbers per 100 inhabitants, unless otherwise specified.
(Outbreak, n = 121)
(Outbreak, n = 122)
Physician visits b
Lost workdays c
Impact of closing institutions when >5 infections per 1000 subjects are observed in the community. Estimates are cumulative numbers per 100 inhabitants, unless otherwise specified.
(Outbreak, n = 111)
Closing schools and workplaces
(Outbreak, n = 101)
0.0 – 0.04
Lost workdays b
Duration of closure (days)
Using a realistic description of influenza infection in the individual subject, we show that an influenza pandemic with a burden comparable to that of 20th-century pandemics might be mitigated by combining measures aimed at reducing meeting frequency and virus transmissibility. This conclusion is based on several assumptions [5, 6] and would be influenced by average infectivity, variability of infectivity , and the frequency and patterns of meetings between individuals , as these two dimensions govern the basic reproductive number. We found that an average R0 of 2.07 can provide attack rates and pandemic curves consistent with those reported in previous pandemics, including the devastating 1917/1918 pandemic. This value was consistent with that reported in previous studies, where R0 ranged from 1.4 to 2.4 [7, 8]. It should be noted that a random choice of the initial infected individual would lead to a strong underestimation of R0 (1.4 in our model). Findings would also be most sensitive to parameters governing the natural history of influenza illness or health-care use. One-way sensitivity analysis showed that the lengths of the latent or incubation periods or the proportion of physician visits occurring during the first day of illness might strongly modify the dynamics of the epidemic or the effectiveness of interventions (see Additional files 1 and 2). Changes that occur during epidemics, such as increased virus fitness for human-to-human transmission  and spontaneous changes in meeting rates in response to the perceived risk, must also be considered. The severity of an epidemic would also be highly sensitive to the efficacy of preventive or therapeutic treatments or vaccination, efficiency of case identification, timely implementation of control measures, population coverage, and public compliance . The number of unknown parameters is too large for meaningful sensitivity analysis. In addition, the characteristics of the next pandemic influenza virus strain cannot be reliably predicted, and neither can the effectiveness of control measures. For example, a critical factor not included in this work is the possible emergence of resistance to antiviral drugs [43, 44]. However, it could be useful to collect pandemic-independent information on patterns of social meetings and the precise mechanism by which influenza usually spreads during winter epidemics in temperate countries. The choice of the BA scale-free network for describing person-to-person meetings within places of occupation may be questionable , but BA networks generate broad heterogeneity in meeting patterns, which may contribute to generating 'superspreading' events . The 20/80 rule, which suggests that 20% of individuals are responsible for 80% of transmission events, can be tested on epidemiological datasets .
This flexible tool can help to determine the interventions most likely to contain an influenza pandemic. At present, our results support the stockpiling of antiviral drugs and accelerated development of vaccines.
This work was partly supported by INSERM and by the European Union Framework 6th programme – scientific support to policy, INFTRANS project.
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