Université Pierre et Marie Curie-Paris6, INSERM, UMR-S 707, Paris F-75012, France

Assistance Publique Hôpitaux de Paris, Hôpital Saint-Antoine, Paris F-75012, France

Centre de Géostatistique de l'Ecole des Mines de Paris, Fontainebleau F-77300, France

Abstract

Background

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.

Methods

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.

Results

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%).

Conclusion

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.

Background

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

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

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.

Methods

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.

In order to obtain a biologically realistic description of the person-to-person transmission process, we assumed that infectivity varies with time since infection and is proportional to the degree of viral shedding by infected individuals (Table

Parameters describing the transmissibility and pathogenicity of influenza virus.

Parameter

Baseline values

Sources

Adapted from [13], and consistent (to a scale factor) with [10,15–21]

Latent period

0.5 days

Peak

2.5–3 days

Duration

<10 days

[5,13]

Children (0–18 years)

1.15

Adults (19–65 years)

1

Elderly (>65 years)

1

30%

[48]; also used in [6]

50%

Assumption also used in [6]

Infectivity profiles of individuals according to time since influenza infection

**Infectivity profiles of individuals according to time since influenza infection**. A latent period of 0.5 days was postulated. Black dots represent infectivity in children and grey dots, infectivity in adults or elderly subjects.

As influenza virus infection is not always symptomatic, we postulated that 30% of infected individuals would not be sufficiently ill to be identifiable

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

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

Community model

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

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

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 _{i }of that node, such that ^{-γ}, where [l.c. gamma] is 3 and coefficient ^{2}). The average connectivity of a BA graph is 2

Various BA graphs were generated for the various locations simulated here (Table

Parameters describing the community model simulating the spread of influenza.

Place

Size

Graph

Assignment

Meetings

Households

1 to 6

Fully connected

D

Schools

Elementary

5 classes; 20 children and 2 adults per class

Each class is modeled using a BA graph (

Children living in the district

WD

Secondary

13 to 15 classes; 30 students and 3 teachers per class

Children and students are linked to teachers.

One college for 5 districts

WD

Workplaces

6 to 3000 according to Zipf distribution [49]

BA graph (

80% from the district; 20% from outside the district

WD

Nursing homes

45 elderly people, 50 employees per nursing home

BA graph (

WD

District

All individuals

BA graph (

WE

BA Barabasi-Albert; D every day; WD every working day; WE every weekend

Figure

Connectivity distribution of the simulated population

Connectivity distribution of the simulated population.

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 (_{0}). _{0 }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 _{0 }is not feasible

Figure _{0 }was 2.07 and the disease generation time, which represents the mean interval between infection of a given person and infection of all the people that this individual infects, was 2.44 (SD 1.48) days.

Distribution of the basic reproductive number _{0}

**Distribution of the basic reproductive number R _{0}**. There were 8000 simulations. Superscripts indicate the numbers of simulations generating a number of secondary infections greater than 10.

We then explored the sensitivity of the basic reproductive number to the number of meetings and to the per-meeting probability of transmission. Parameters describing the meetings (mean weighted connectivity between 1 and 7) and per-meeting transmissibility (0.1 to 3 times the reference value) were varied on a 10 [multiplication sign] 10 grid with 40 simulations for each combination of parameters. Normal regression analysis with a log link was performed with the mean number of secondary cases as the response variable and weighted connectivity and per-meeting transmissibility as predictors. As expected, the mean weighted connectivity and the per-meeting transmissibility correlated independently with the basic reproductive number (Figure

Sensitivity analysis of the basic reproductive number _{0}

**Sensitivity analysis of the basic reproductive number R _{0}**. The figure shows the isopleth of

The observed rates of seroconversion and illness due to the pandemic strains that circulated during the 20^{th }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

Results

Reference scenario

Two hundred realizations were simulated for each scenario. Three patterns were observed. No secondary infections were generated in 20% of simulations (see above). In 23% of simulations, a limited number of infections occurred and the epidemics always affected fewer than five subjects per 1000 (Figure

Simulation of the reference scenario for a flu pandemic (no intervention)

**Simulation of the reference scenario for a flu pandemic (no intervention)**. The top figure describes the distribution of the numbers of secondary cases following introduction of a single infected individual into the population (200 simulations), and the bottom figure describes the infection curves of simulated outbreaks. The bold line is the average of the simulated outbreaks.

Reference scenario for a flu pandemic after one initial case (no intervention). Estimates are cumulative numbers per 100 inhabitants, unless otherwise specified.

Outcomes

Outbreak

All simulations

Mean

Minimum-Maximum

Mean

Infections

Children (0–18 years)^{a}

76.5

71.9–79.7

43.6

Adults (19–65 years) ^{a}

39.9

34.8–44.0

22.8

Elderly (>65 years) ^{a}

25.3

20.8–30.1

14.4

Physician visits

31.2

28.0–33.7

17.8

Hospital admissions

1.74

1.30–2.30

0.99

Deaths

0.36

0.17–0.55

0.21

Lost workdays^{b}

137

118–150

78

^{a}Per 100 individuals of a given age

^{b}Per 100 working adults

Intervention scenarios

We first simulated the effectiveness of neuraminidase inhibitors in individuals who sought medical advice and were treated for five days. We assumed that treatment reduced infectivity and clinical severity (including the risk of complications and death) by 28%

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.

Outcomes

Outbreak;

All simulations;

Mean

Minimum-Maximum

Mean

Infections

Children^{a}

72.5

67.9–76.4

38.4

Adults^{a}

36.5

30.6–41.8

19.3

Elderly^{a}

22.3

18.6–26.0

11.8

Physician visits

28.0

24.3–31.2

14.9

Hospital admissions

0.98

0.66–1.18

0.52

Deaths

0.21

0.12 – 0.32

0.11

Lost workdays^{b}

125

107–141

66

Treatment units (doses)^{}

- c

243

215–269

129

^{a}Per 100 individuals of a given age

^{b}Per 100 working adults

^{}

- c

Several randomized controlled trials have demonstrated the preventive effectiveness of neuraminidase inhibitors (see

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.

Outcomes

Prophylaxis

(Outbreak,

Prophylaxis + treatment

(Outbreak,

Mean

Minimum-Maximum

Mean

Minimum-Maximum

Infections

Children^{a}

61.8

57.0–66.2

58.5

53.1–64.5

Adults^{a}

30.2

25.6–35.0

28.8

24.2–33.4

Elderly^{a}

16.8

11.2–21.2

14.9

11.3–18.9

Physician visits

23.8

21.8–27.0

22.2

19.6–24.8

Hospital admissions

1.18

0.70–1.57

0.77

0.59–0.98

Deaths

0.25

0.13–0.43

0.16

0.06–0.28

Lost workdays^{b}

106

94–122

100

85–116

Treatment units (doses)^{}

- c

196

179–210

393

356–435

^{a}Per 100 individuals of a given age

^{b}Per 100 working adults

^{}

- c

We then examined the impact of 10-day confinement to home of all members of households in which a case was identified by a physician, combined with prophylaxis of household contacts and treatment of the index case. This strategy would increase effectiveness by comparison with similar scenarios not involving confinement: coverage of 70% of affected households would be sufficient to reduce the risk of an outbreak by 52%, restricting it to 17% of the population (range 0.8%–25%) (Figure

Impact of interventions on infection curves of simulated outbreaks

Impact of interventions on infection curves of simulated outbreaks.

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

Reactive mass vaccination would significantly reduce the frequency, size, and mean duration of outbreaks (Figure

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.

Outcomes

Interval (days)

0

(Outbreak,

28

(Outbreak,

Mean

Minimum-Maximum

Mean

Minimum-Maximum

Infections

Children^{a}

8.6

0.7–31.5

62.6

4.6–77.7

Adults^{a}

3.0

0.4–11.9

29.7

1.4–40.2

Elderly^{a}

1.6

0.0–5.8

18.1

0.5–27.8

Physician visits^{b}

1.6

0.0–10.3

23.9

1.3–31.2

Hospital admissions

0.15

0.01–0.5

1.34

0.06–1.92

Deaths

0.033

0–0.14

0.29

0.02–0.51

Lost workdays^{}

- c

11

1–44

103

5–137

Vaccination

69.8

69–71

61.6

44–71

^{a}Per 100 individuals of a given age

^{b}Physician visits for 'flu-like' illness (excludes visits for influenza vaccination)

^{}

- c

Finally, we simulated an intervention in which schools and workplaces are closed when a threshold number of infections (5/1000 subjects in our example) has been reached in the population and are reopened 10 days after the last observed case of infection. This strategy could be used if vaccines and/or antiviral drugs were in short supply or ineffective. Table

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.

Outcomes

Closing schools

(Outbreak,

Closing schools and workplaces

(Outbreak,

Mean

Minimum-Maximum

Mean

Minimum-Maximum

Infections

Children^{a}

10.3

1.6–23.1

2.0

0.6–3.8

Adults^{a}

9.9

0.6–23.7

0.8

0.4–1.5

Elderly^{a}

6.1

0.3–14.5

0.8

0.1–3.2

Physician visits

6.4

0.6–14.1

0.7

0.4–1.4

Hospital admissions

0.36

0.03–0.9

0.04

0.0–0.11

Deaths

0.081

0–0.3

0.009

0.0 – 0.04

Lost workdays^{b}

324

80–464

1885

977–5484

Duration of closure (days)

101

13–107

27

14–78

^{a}Per 100 individuals of a given age

^{b}Per 100 working adults

Discussion

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 _{0 }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 _{0 }ranged from 1.4 to 2.4 _{0 }(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

**PDF file One-way sensitivity analysis figures**. A PDF file showing one-way sensitivity analyses of the reference scenario (Figure 7a) and a "combined interventions" scenario including the coverage of 70% of affected households, with treatment of the index patient, prophylaxis of household contacts, and confinement to home of all household members (Figure 7b), to parameters governing the natural history of influenza infection or healthcare use. The red curves describe simulated outbreaks with the parameter values used in the manuscript.

Click here for file

**Word file One-way sensitivity analysis table**. A Word file showing one-way sensitivity analyses of the reference and the "combined interventions" scenarios to parameters governing the natural history of influenza infection or healthcare use.

Click here for file

Conclusion

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.

Abbreviations

BA Barabasi-Albert

SD standard deviation

Competing interests

F Carrat has received fees for consultancies from Roche, GlaxoSmithKline, Chiron, and Aventis in the past five years. The other authors declare that they have no competing interests.

Authors' contributions

FC conceived and supervised the study and drafted the manuscript. JL, HL, and VS developed the simulation model and carried out the simulations. CL and HW conceived the study and helped draft the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was partly supported by INSERM and by the European Union Framework 6^{th }programme – scientific support to policy, INFTRANS project.

Pre-publication history

The pre-publication history for this paper can be accessed here: