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Conceived and designed the experiments: EM JDM JMM CM. Performed the experiments: EM JDM JMM CM. Analyzed the data: EM JDM JM CM JMM. Contributed reagents/materials/analysis tools: EM JDM JMM CM. Wrote the paper: JDM JM JMM.

John Mathews has advised the Australian Government on influenza as an employee (1999–2004) and as a member of the National Pandemic Influenza Advisory Committee (NIPAC). He has provided advice on influenza as a part-time consultant to Government (2004–2006) and industry (2007). He has had access to information restricted by confidentiality considerations. All other authors have no conflict of interest to declare.

The clinical attack rate of influenza is influenced by prior immunity and mixing patterns in the host population, and also by the proportion of infections that are asymptomatic. This complexity makes it difficult to directly estimate _{0}_{0}

On the island of Tristan da Cunha (TdC), 96% of residents reported illness during an H3N2 outbreak in 1971, compared with only 25% of RAF personnel in military camps during the 1918 H1N1 pandemic. Monte Carlo Markov Chain (MCMC) methods were used to estimate model parameter distributions.

We estimated that most islanders on TdC were non-immune (susceptible) before the first wave, and that almost all exposures of susceptible persons caused symptoms. The median _{0}_{0}

Our model implies that the RAF population was partially protected before the summer pandemic wave of 1918, arguably because of prior exposure to interpandemic influenza. Without such protection, each symptomatic case of influenza would transmit to between 2 and 10 new cases, with incidence initially doubling every 1–2 days. Containment of a novel virus could be more difficult than hitherto supposed.

Reports of past influenza pandemics show marked variation in clinical attack rates between populations. In the 1918–19 H1N1 pandemic, rates of clinical illness were less than 20% in some urbanised communities, but more than 60% in isolated communities such as Western Samoa

Our flexible model, which we here apply to outbreaks of H3N2 from 1968–71

Epidemic curves from single wave outbreaks with low rates of symptomatic influenza provide little or no information to separate the effects of viral exposure (and hence magnitude of _{0}_{0}_{0}

The population of Tristan da Cunha, a remote island in the South Atlantic, had been free of influenza for 8–9 years when H3N2 was introduced by ship from South Africa in 1971

_{0}_{0}

1–

2/_{e}

2/_{w}

1/_{i}

Each compartment corresponds to a class of individuals in the population, and the arrows indicate the flows of individuals from class to class over time. As a result of exposure to the force of infection (

The mean generation time or serial interval for our SEIR model is _{e}_{i}_{e}_{0}_{0}_{0}/T_{i}

We used MATLAB v7.3 to fit deterministic epidemic curves and for Monte Carlo Markov Chain (MCMC) simulations to estimate parameter distributions; we used the negative binomial distribution to calculate each likelihood ^{2} days) and serial interval (log normal distribution with a mode of 2.6 days and variance of (0.33)^{2} days), based on estimates from the literature _{i}_{e}

_{0}

Observed and fitted (median parameters) incidences for the H3N2 outbreak on the island of Tristan da Cunha in 1971 (cases per day, starting from 15^{th} August). Error bars (+/− one SD) are calculated using the negative binomial variance (See

Estimated quantity | RAF camps (1918) | Tristan da Cunha (1971) |

_{0} | 2.88 (2.26, 4.28) | 6.44 (3.73, 10.69) |

0.51 (0.34, 0.65) | 0.84 (0.62, 0.99) | |

0.38 (0.28, 0.60) | 0.91 (0.72, 1.00) | |

0.55 (0.41, 0.70) | 0.49 (0.39, 0.57) | |

2/_{w} | 68 (56, 95) | 12 (9, 17) |

2/_{e} | 1.30 (fixed) | 1.36 (0.82, 1.87) |

1/_{i} | 1.00 (fixed) | 0.98 (0.30, 1.83) |

2/ | 2.30 (fixed) | 2.34 (1.56, 3.26) |

Initial doubling time in fully susceptible population (days) | 1.25 (0.85, 1.69) | 0.62 (0.52, 0.73) |

Initial doubling time in actual population (days) | 3.93 (3.69, 4.19) | 0.72 (0.63, 0.81) |

Initial transmissions per day per transmitter in fully susceptible population | 2.88 (2.26, 4.28) | 6.76 (3.84, 16.35) |

Initial transmissions per day per transmitter in actual population | 1.46 (1.43, 1.50) | 5.59 (3.20, 13.33) |

The 1918 pandemic is known to have been caused by H1N1; the 1971 outbreak on Tristan da Cunha was caused by H3N2.

Parameter values (median, 95% credibility intervals) were estimated by MCMC simulation (See

Due to the weekly reporting of influenza in RAF camps, our MCMC algorithm was unable to distinguish between a range of possible solutions, leading to wide credibility intervals on the estimates for most parameters (See _{e}_{i}_{e}_{i}_{0}

Observed and fitted (median parameters) incidences of influenza reported from RAF camps in UK during the 1918 pandemic of H1N1 (cases per week, starting from week of June 8). Error bars (+/− one SD) are calculated using the negative binomial variance.

Initial doubling times for influenza were estimated as 0.72 days on Tristan da Cunha and 3.93 days in RAF camps. In fully susceptible populations, the estimated doubling times would be 0.62 days and 1.25 days respectively.

For Tristan da Cunha, there were initially 5.59 (3.20, 13.33) transmissions per day for each transmitter, corresponding to 6.76 (3.84, 16.35) if the population were fully susceptible. For RAF personnel, the estimates were 1.46 (1.43, 1.50) and 2.88 (2.26, 4.28) per day respectively.

Our flexible model explains multiple waves of influenza by incorporating biological effects that have been overlooked in some earlier pandemic models. The model allows for the possibility that asymptomatic infection

Seasonality _{0}

Asymptomatic influenza infections are known to be immunising _{0}

Our results suggest that prior immunity was important in protecting against clinical attack in the 1918 H1N1 pandemic, but do not explain the origins of that immunity. However, heterosubtypic immunity likely provides at least some protection against influenza A of novel subtype

One result from our Tristan da Cunha model could seem counter-intuitive: exposure in the first wave did not always protect against re-infection in the second wave several weeks later, and protection apparently waned much more quickly than in the RAF population (

Our analyses have provided an economical explanation for the time course of the observed data in two contrasting outbreaks. Rather than providing inconsistent evidence, we suggest that the two outbreaks provide complementary evidence about how “immunity” to influenza can evolve over different time scales from different starting points. The dynamics of multiple-wave outbreaks on these different time-scales are at least partly due to the past exposure history of the population. We did not expect, and did not observe, comparable estimates for the waning time of immunity in the RAF and TdC populations.

Our inferred values of 2–10 for _{0}_{0}_{0}_{0}_{0}

For the RAF outbreaks, with data reported only at weekly time steps, there is little information to allow MCMC estimation to separate the effects of changing serial interval from the effects of changing _{0}_{e}_{i}_{e}_{i}_{e}_{i}_{e}_{i}

The estimates for the latent period from TdC simulations (See

Our flexible model, with host immunity and asymptomatic immunising infections as the constraints on observable disease spread, adequately reproduces the observed epidemiology of influenza in disparate populations, and leads to additional insights into virus behaviour. Less flexible models can explain single wave outbreaks with a range of _{0}_{0}_{e}_{0}

What might our findings mean for pandemic planning? The bad news is that the pandemic doubling time in a fully susceptible population could be as short as 1 or 2 days, and that _{0}_{0}_{e}

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We thank the University of Melbourne and the Murdoch Childrens Research Institute for supporting this work. We also thank Terry Nolan, Ray Watson, Niels Becker, Ian Gust, Alan Hampson, Lorena Brown, Ian Barr, Moira McKinnon, Katrina Scurrah, Paul Pallaghy, Sunetra Gupta, Neil Ferguson and the anonymous reviewers for advice.

The corresponding author had full access to all the data in the study and had final responsibility for the decision to submit for publication.