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Unlike local transmission of pandemic influenza A (H1N1-2009), which was frequently driven by school children, most cases identified in long-distance intranational and international travelers have been adults. The present study examines the relationship between the probability of temporary extinction and the age-dependent next-generation matrix, focusing on the impact of assortativity. Preferred mixing captures as a good approximation the assortativity of a heterogeneously mixing population. We show that the contribution of a nonmaintenance host (i.e., a host type which cannot sustain transmission on its own) to the risk of a major epidemic is greatly diminished as mixing patterns become more assortative, and in such a scenario, a higher proportion of non-maintenance hosts among index cases elevates the probability of extinction. Despite the presence of various other epidemiological factors that undoubtedly influenced the delay between first importations and the subsequent epidemic, these results suggest that the dominance of adults among imported cases represents one of the possible factors explaining the delays in geographic spread observed during the recent pandemic.

Since it was first identified in early 2009, a novel strain of influenza A (H1N1-2009) has caused a global pandemic. Although the rapid international spread created various epidemiological challenges, such as quantifying the strain's transmission potential and virulence during the very early stages of the pandemic [

A parsimonious simplification of the complexity of an age-structured contact network can be obtained by approximating the network by an appropriately quantified age-dependent next-generation matrix. This is accomplished by using the next-generation matrix, the square matrix with generic entry

The present study investigates the relationship between the age-dependent next-generation matrix,

In addition to these issues, the present study investigates the role of the age of cases importing infection to a local area by long-distance travel either intranationally or internationally on the resulting growth of a local epidemic. As a practical example, the age-dependent transmission of the H1N1-2009 pandemic is considered, and we first present our study motivations in the next section.

Figure

Epidemiology of the pandemic influenza A (H1N1-2009) in Hong Kong and the Netherlands. (a) Introduction of imported confirmed cases in Hong Kong followed by an increase in local (indigenous) confirmed cases from May–June, 2009 [

Figure

Both Figures

We employ a multitype branching process to approximate the probability of extinction of the clade of infection emanating from a single index case [

Letting the random vector

Let

In the two-host population, that is, a population consisting of children and adults, the probabilities of extinction given a single child or adult infected individual,

The probability of extinction is investigated for the following three different scenarios. First, to gain an overview of the extinction probabilities

Second,

Preferential mixing given a single child index case. The hypothetical population consists of 10 children (left) and 10 adults (right) with equal susceptibility and infectiousness. We consider an introduction of a single child index case who has a potential to cause 8 secondary transmissions. Panels (a)–(c) illustrate contacts generated by the child index case with different

Third, to clarify the practical implications of the predomination of adults among travelers, we examine the sensitivity of the probability of extinction to the proportion of adult travelers over various

Figure

Probability of extinction given an introduction of single child or adult index case. The probabilities of extinction are calculated, assuming that a single child or adult index case is introduced into a fully susceptible large population. The probability approximately accounts for heterogeneous transmission among and between child and adult populations. The estimates of the next-generation matrices are extracted from A. Fraser et al. [

Figure

Assortativity and the probability of extinction. The probability of extinction in a fully susceptible population given a single adult or child index case is measured as a function of

Figure

The impact of the age specificity of index cases on the probability of extinction. All panels, (a)–(c), examine the sensitivity of the probability of extinction in a fully susceptible large population given 10 index cases with

The present study investigated the relationship between the next-generation matrix and the probability of extinction, employing a simple model that may be viewed as an approximation to a full network model. The modelled heterogeneous mixing accounted for assortativity via an assumption of preferred mixing, and the probability of extinction was derived from a multidimentional branching process model. As a practical example, the age dependency in the transmission of pandemic influenza A (H1N1-2009) was considered, dividing the population into children and adults. Through quantitative illustrations, it has been shown that the probability of extinction given an adult index case increases with

Three practical implications are drawn from our exercise. First, the importance of assortativity in appropriately capturing the probability of extinction highlights a critical need to account for this aspect when quantifying the next-generation matrix in an approximately modelled heterogeneous population. Whenever the statistical inference of the next-generation matrix is made for directly transmitted diseases, the estimation framework should ideally account for assortative mixing. Whereas the social contact survey revealed that the age-dependent contact pattern is highly assortative [

Second, as was highlighted with an application, accounting for the age specificity in the surveillance of international and interregional mobility patterns and its use for statistical inference of epidemic dynamics are of utmost importance. For example, global airline transportation is one of the most well-studied networks, and this has been analyzed for H1N1-2009 [

Third, as a disease control implication, although adults dominate imported cases, it should be remembered that the more important target host is still children. If stringent border control measures, for example, travel reduction and movement restrictions among all incoming passengers [

Although the role of heterogeneously mixing population in the spread of infectious diseases has been examined using stochastic modelling approaches, past studies tended to focus on final epidemic size and its relevance to disease control policy [

Unlike local transmission of the H1N1-2009 which was frequently driven by school children, imported cases were predominantly adults. This study examined the relationship between the age-dependent next-generation matrix and the probability of extinction, focusing on the role of nonmaintenance hosts and the impact of assortativity on the epidemic extinction. The preferred mixing assumption captures assortativity in a much simpler way than full contact network models, allowing analysis in place of Monte Carlo calculations. The contribution of nonmaintenance hosts to the risk of a major epidemic is diminished as the mixing pattern becomes more assortative, so that an increase in the proportion of nonmaintenance hosts among index cases increases the probability of extinction, if temporary in the face of repeat importations. These results helped us to formulate a hypothesis that the dominance of adults in imported cases was one of the possible causes of observing substantial delay in interregional and international spread of the 2009 influenza pandemic. The importance of capturing the assortativity in estimating the next-generation matrix was highlighted.

The work of H. Nishiura is supported by the Japan Science and Technology Agency (JST) PRESTO program. A. R. Cook is grateful to the National University of Singapore for supporting his research. B. J. Cowling is supported by the Harvard Center for Communicable Disease Dynamics from the US National Institutes of Health Models of Infectious Disease Agent Study program (grant no. 1 U54 GM088558).