Modelling the progression of pandemic influenza A (H1N1) in Vietnam and the opportunities for reassortment with other influenza viruses

Background A novel variant of influenza A (H1N1) is causing a pandemic and, although the illness is usually mild, there are concerns that its virulence could change through reassortment with other influenza viruses. This is of greater concern in parts of Southeast Asia, where the population density is high, influenza is less seasonal, human-animal contact is common and avian influenza is still endemic. Methods We developed an age- and spatially-structured mathematical model in order to estimate the potential impact of pandemic H1N1 in Vietnam and the opportunities for reassortment with animal influenza viruses. The model tracks human infection among domestic animal owners and non-owners and also estimates the numbers of animals may be exposed to infected humans. Results In the absence of effective interventions, the model predicts that the introduction of pandemic H1N1 will result in an epidemic that spreads to half of Vietnam's provinces within 57 days (interquartile range (IQR): 45-86.5) and peaks 81 days after introduction (IQR: 62.5-121 days). For the current published range of the 2009 H1N1 influenza's basic reproductive number (1.2-3.1), we estimate a median of 410,000 cases among swine owners (IQR: 220,000-670,000) with 460,000 exposed swine (IQR: 260,000-740,000), 350,000 cases among chicken owners (IQR: 170,000-630,000) with 3.7 million exposed chickens (IQR: 1.9 M-6.4 M), and 51,000 cases among duck owners (IQR: 24,000 - 96,000), with 1.2 million exposed ducks (IQR: 0.6 M-2.1 M). The median number of overall human infections in Vietnam for this range of the basic reproductive number is 6.4 million (IQR: 4.4 M-8.0 M). Conclusion It is likely that, in the absence of effective interventions, the introduction of a novel H1N1 into a densely populated country such as Vietnam will result in a widespread epidemic. A large epidemic in a country with intense human-animal interaction and continued co-circulation of other seasonal and avian viruses would provide substantial opportunities for H1N1 to acquire new genes.


Age dependent mixing
Age class specific contact frequency was derived from a survey of social contact patterns conducted in 2007 in 865 members of a community in one semi-rural district of north Vietnam. A contact was defined as:

Either
• a two-way conversation with three or more words in the physical presence of another person Or • physical skin-to-skin contact (for example a handshake, hug, kiss or contact sports).
Participants recorded every contact made during one day, the age of the contact, and the duration and location of each contact. If a person was contacted more than once in a day, the contact was recorded only once but the total time spent with that contact over the entire day was recorded. A contact intensity matrix by seven age classes was constructed by adjusting the daily contact frequency data by the size of each age class.
The matrix was corrected for reciprocity, i.e. where the contact frequency between age classes was not symmetrical the mean of the two values was used for both. The data were normalized so that the all age-class specific rates were relative to the maximum rate of 1. and it represents relative contact rates corrected for population size, meaning these are the relative contact rates one would observe if each of the age classes had the same number of individuals. The true contact patterns are different for each province since the age distribution in each province is somewhat different. This pattern is similar to one calculated from census data from Portland, Oregon (USA), in that the main areas of contact intensity are (1) within age-groups and (2) peaking in children and older adults [1].
Note that these contact patterns differ from the European data presented in Mossong et al, the main difference being the high intensity of contacts in the 50+ age groups in Vietnam [2]. The next-generation matrix computed for these data is different for each province in Vietnam (since age structure varies by province), and it is generally different than the Dutch next-generation matrix presented in Wallinga et al [3].
According to these next-generation matrices, infection patterns in Vietnam would primarily be driven by the 6-15 age group, while infection in the Netherlands would be evenly driven by 6-to 39-year-olds.

Internal migration by land
Age class specific frequency of travel outside of the Province of residence was A special direct connection was created between Hanoi and Hai Phong (the major port in north Vietnam) in the model, since a large amount of road traffic travels between these two Provinces along one major road but the two Provinces share no adjacent border, so connectivity was thought to be under-estimated in the model.
Province and age-class specific frequency of travel overland outside of the Province of residence was estimated by adjusting the frequency of travel survey data by the relative connectivity of each Province.
Internal migration by waterways and railways was ignored in the model since GSO data on the volume of traffic by type of transport indicated that waterway and railway travel together contributed less than 10% of all passenger volume in 2007.

Internal migration by air
Internal migration by air travel was estimated using publicly available data on the frequency of flights and aircraft type between domestic airports in Vietnam. It was assumed that all flights were full since data on the number of passengers was not available. These data are publicly available for Vietnam Airlines  The overall hospitalization rate was set at 1% and the distribution of these hospitalizations across the age-classes was derived from data of the proportion of H1N1 cases hospitalized in Mexico and the U.S by age [9]. In the model the overall hospitalization rate was varied between 0.5% and 1.5% of all cases, since reported rates of 5-6% are likely to be biased by over-ascertainment of severe cases compared to mild cases. (Table S2).

Age dependent susceptibility to infection
Fraser et al found that the model that best fit the available data included both an agedependent contact intensity parameter and age dependent susceptibility to infection [4]. Studies of age specific serological reactivity against the novel H1N1 virus are consistent with an age-dependent susceptibility to infection or disease [10]. Since agedependent contact intensity is represented in our model by the contact matrix described on page 2 of this supplement, we sought to estimate the contribution of agedependent susceptibility. To do this we used data on the age distribution of cases in the US and data on age dependent contact frequency from a European study [2,9].
The relative frequency of contact by age class was factored out of the relative distribution of H1N1 cases by age in the US in order to derive an estimate of susceptibility by age class, independent of contact behaviour (Table S3).

Seasonality
We assumed no seasonal affect on transmissibility of H1N1 or on contact patterns

Contact between infectious humans and domestic pigs, chickens and ducks
To estimate the number of domestic pigs, chickens and ducks exposed to an H1N1 infected human we divided the total number of infected persons in each province by the average household size for each province to give a conservative estimate of the number of infected households. This is a conservative estimate since it assumes all human H1N1 cases are clustered by household. The estimated number of infected households was then multiplied by the proportion of households raising pigs, chickens and ducks and the average number of pigs, chickens and ducks present in households that raise these animals.

Interventions
We explored the potential impact of school closure by introducing a relative reduction in contact frequency among children in the age class 6 and 15 years. We explored the potential impact of broader social distancing measures by reducing contact frequency across all age classes.

Mathematical Model
An SEIR-model (Susceptible-Exposed-Infectious-Recovered) with a four-stage infectious period was used to model the core infection dynamics in each province.
The model equations are where Λ kl represents the force of infection on age class k in province l. Λ kl is defined by where A=7 is the total number of age classes and the τ-variables allow for stagespecific infectivities. The variable S kl represents the number of susceptible individuals in age class k currently in province l. E kl represents exposed individuals and I kl,s represents infected individuals in stage s (out of a total of four) of their infection.
The parameters τ k were all set to one since we could not find good information on shedding and infection duration at the time the model analysis was run. The parameters β k represent age-specific susceptibility and can be found in Table S3.
Parameters d l represent province-specific population density and were computed as outlined on page 1 of this supplement. N l is the population of province l. Parameters z ik are mixing rates between age class i and age class k (from contact rate matrix on page 2). The parameter ν is the recovery rate (1/ν is the duration of infection), and 1/e is the length of exposure before a host becomes infectious.
Migration and hospitalization were integrated stochastically into the above differential-equations model. Once a day, discrete individuals could move from one province to another according to the migration matrix outlined earlier; migration probabilities were balanced between provinces so the system behaved like a gravity is the number of infections generated at location l in age class k', by an infectious individual in location l of age class k. N kl is the number of individuals in age class k at location l, and P T is a proportionality constant that depends on the unknown probability of transmission given contact. A similar calculation can be done when the two locations are not equal, and a complete next-generation matrix can be filled in.
This assumes only one migration event during a course of infection, but this is a fair approximation since daily migration probabilities are small and infections are short. It is known that transmissibility varies from person to person and contact rates vary from community to community; thus, choosing a single R 0 value for an entire population can prove difficult. As a reference point, we use the R 0 value for a hypothetical Ho Chi Minh City with spatially uniform population density, spatially uniform interpersonal contact behavior, and no emigration. The R 0 values for the other provinces are smaller by some amount, depending on each province's population density and age structure.

Sensitivity analysis
The varied parameters with ranges were: 1.2 < R 0 < 3.1; 3.8 days < duration of infection < 5.5 days; 1.35% < daily probability of migration < 5.00%; 0.5 < traffic on small road relative to large road < 1.0; 0.5% < probability of hospitalization < 1.5%. 1000 parameter sets were drawn randomly from this range using Latin hypercube sampling and medians and quartile ranges are presented from these 1000 runs [11].
Daily probability of migration was derived from the Ha Nam survey -which gave a mean estimate of 1.35% of people moving province each day; this was used as the lower end of the modeled range.