Mathematical Model

The model has two layers: a set of within-city models and a global model that links them through the air transportation network. We first describe the within-city model and then the global model. It is written in Python 2.6.2 ( http://www.python.org) using the SciPy and NumPy packages [12]. The code is available at http://www.csquid.org/software/globalmodel/.

Within each city, susceptibles are divided into subpopulations and risk groups. Subpopulation membership is used to determine transmission probabilities, and risk group membership is used to determine morbidity and mortality. In our simulations, we have two subpopulations, children (age years) and adults (age years), each with two risk groups (low and high risk). Since we focus on influenza transmission rather than morbidity and mortality, there is only a single risk group. The population of each city is divided into susceptible, infectious, and removed compartments, which have subcompartments to keep track of subpopulation, risk group, vaccination status, and symptom status (Text S1). Upon infection, each person is assigned to become asymptomatic or symptomatic. He or she is then assigned uniformly at random to one of the six viral load trajectories (Text S1 and Figure S1). On all trajectories, infection lasts six days but infectiousness varies with symptom status and viral load. In our simulations, infected persons become symptomatic with probability , and asymptomatic infecteds are half as infectious as symptomatic ones. Infected persons transmit infection according to a next-generation matrix scaled to achieve a within-city , the reproductive number that is during influenza season and may be lower at other times of the year according to local influenza seasonality (Text S1). In our simulations, the transmission probabilities are tuned to obtain a next-generation matrix proportional that from [13], where the child-to-child transmission is 1.8, adult-to-adult transmission is 0.2, and the child-to-adult and adult-to-child transmissions are 0.5 (see Eqn 9 in Text S1). This ensures that child-child influenza transmission is most intense, adult-adult transmission is least intense, and child-adult and adult-child transmission are intermediate. Using all of this information and the effects of vaccination (described below), a system of discrete time and state-space stochastic equations (Text S1) governs spread of influenza within cities in one-day time steps. Major parameters are summarized in Table S1.

In the model, vaccination can reduce susceptibility to infection (by per infectious contact), infectiousness following infection (by ), and the probability of becoming symptomatic after infection (). In our simulations, we use vaccine efficacy estimates for a well-matched seasonal influenza vaccine: and , in accordance with [14]. These efficacies are not reached immediately upon vaccination, and we define the vaccine efficacy ratio function to be the proportion of full vaccine efficacy attained days after vaccination. As in [15], this function is governed by three parameters: determines the shape of the increase in vaccine efficacy during the first 13 days, level is the maximum vaccine efficacy achieved after the first dose, and determines the shape of the increase in vaccine efficacy after the second dose (which is assumed to be given 21 days after the first dose). The underlying vaccine efficacy ratio function is:(1)

In the intervals and , the function is concave for negative , convex for positive , and linear for (Figure 1). , , and are obtained by multiplying by the , , and , respectively. In our simulations, we use , .

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Figure 1. Vaccine efficacy over time.

Vaccine efficacy rises over time, reaching maximum efficacy after 28 days. The model assumes that all individuals who receive their first dose vaccine will receive a second exactly 21 days later.

https://doi.org/10.1371/journal.pone.0019515.g001

The global model links together the 321 within-city models, allows travel between them, and controls seasonality. It records the total number of susceptibles, incident infections, infections, and recovereds in each time step, and (optionally) can store a matrix for each time step showing the number of travelers from each city to every other city. All cities are divided into two age groups: 0–14 years old and 15+ years old. The proportion of the population in each age group in each city is determined by the proportion of the population under 15 in the corresponding country [16]. All cities are assumed to have the same next-generation matrix at peak seasonal transmission. For each city , we have the average number of persons who travel to each other city in the model per day. We divide this by the population of to get a probability of traveling from to each other city in the model at each time step (Text S1). Symptomatic individuals are 75% less likely to travel ( see sensitivity analysis in Figure S4). For efficiency, only infected travelers are tracked in the model. Infected visitors to a city are put into the infected compartment corresponding to their vaccination time, symptom status, viral load trajectory, and day of illness. They progress through the infected compartments and travel to other cities just like the other infected persons in the destination city. Upon recovery, they return immediately to their home city. Because of the short infectious period of influenza, we assumed that infected travelers would not have the opportunity to return before recovering. A city's population may experience small and temporary fluctuations in size because of travel, but the number of travelers is much smaller than the population size.

Seasonality of influenza in the model

Transmissibility for each city rises and falls in an annual cycle in the model (Figure 2). In the model, a country is always either in-season and transmission is high (i.e., ) or out-of-season (i.e., ). For cities north of the Tropic of Cancer, influenza transmission was assumed to be high () from September 15 to June 1 each year and low otherwise (), except for cities that are known to deviate from this pattern. Likewise, influenza transmission was assumed to be high from April 15 to October 15 for cities south of the Tropic of Capricorn (Figure S2). For temperate regions, we assumed that transmission was outside of influenza season. For regions known to have year-round transmission, we assumed that transmission was outside of influenza season. In other parts of the tropics, we assumed that transmission was outside of influenza season.

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Figure 2. Seasonality in the model affects epidemic dynamics.

The maps in the top row depict the relative transmissibility of influenza in the model on (A) July 10 and (B) November 10. Each city in the model is plotted with a dot size proportional to the city's population and colored red when influenza is highly transmissible, blue when influenza is least transmissible, and orange for intermediate levels of transmissibility. The influenza season in the temperate northern and temperate southern hemispheres occurs during their respective winters, hence the large proportion of red dots in the south on July 10 and red dots in the north on November 10. The Tropics of Cancer and Capricorn are plotted as dashed horizontal lines. Seasonality in the tropics does not follow this pattern, and may have multiple peaks, often corresponding to the rainy season. The maps in the bottom row show the prevalence of influenza in each city in a simulation in which the pandemic began in Mexico City on April 1. The size of each red dot is proportional to the prevalence of influenza in each city. (C) In July, prevalence is high in several cities in South America. (D) In November, prevalence is high across the temperate northern temperate regions. Large epidemics can only occur when the seasonal transmissibility in a city permits.

https://doi.org/10.1371/journal.pone.0019515.g002

Seasonality of influenza transmission is likely to be caused by a variety of factors, from annual weather cycles to social factors [17]. We collected data on influenza season in various parts of the world from the literature (Table S2). For large regions, influenza epidemics peak about two months after they start [18], so if the timing of the peak of influenza is known for a country, then we assumed that the influenza season started about two months earlier. If epidemic curves were shown and there was an obvious peak of influenza activity, we defined the influenza season to cover the peak as well as the periods elevated activity before and after the peak. For regions in the tropics for which there was no influenza activity data available, we assumed that influenza season coincided with the rainy season. A few countries, such as China, India, and Brazil, are known to have different influenza seasons in different regions, and we tried to infer the season for each city when possible. The seasonality used for each city in the model is summarized in Figure S3.

Mapping global influenza transmission

To better understand the global transmission of influenza, we identified clusters of cities within which transmission occurs rapidly. To do this, we started with the epidemic percolation network (EPN) of the global model (Text S2). The EPN is a directed random graph that represents the final outcomes of a stochastic epidemic model [19], [20]. Informally, the EPN is a graph where we draw directed edges from each person to all persons would infect if the population were entirely susceptible. If an epidemic begins with the infection of person , all persons who can be reached from by following a series of edges—the out-component of node in the EPN—will be infected. Thus, the EPN gives us a final outcome of the epidemic model for any given set of initial infections.

The EPN for the entire global model would include hundreds of millions of nodes and edges. To map the global spread of infection, we collapsed the full EPN into a city-to-city EPN with a directed edge from each city to all cities that can be reached directly from (Text S2). The weight of each edge is proportional to the expected number of directed edges in the EPN pointing from persons in city to persons in city , assuming that all cities are transmitting at their peak seasonal . This resulted in a network with 321 nodes and 53,534 weighted links. We simplified this network with an information-theoretic clustering algorithm [21] based on finding a two-level code that minimizes the expected code length required to describe the path of a random walker on the city-to-city EPN (Text S2). We used the Map Generator software package at www.mapequation.org [22] to perform the clustering algorithm and generate the map.

Network structure of global influenza transmission

The clustering algorithm identified 13 clusters connected by 146 directed edges. Table 1 summarizes characteristics of the clusters, and Table S3 lists the cities in each cluster. “Flow” is the steady-state proportion of time that a random walker on the city-to-city EPN spends within the cluster, “outflow” is the steady-state probability that a random walker within the cluster jumps to a city in a different cluster. To measure the relative importance of the cities within each cluster to the global transmission of influenza, we divided the cluster's flow by the number of cities it contains, normalizing so the average value over all clusters equals one. This is called the “per-city flow” in the table. Figure 3 shows the 13 nodes in their approximate geographical locations and the 36 edges across which the most inter-cluster influenza transmission occurs, which account for 90% of all transmission between clusters.

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Figure 3. Map showing influenza transmission clusters.

Clusters represent groups of cities within which transmission of influenza is rapid; transmission between clusters is slower. The map shows all 13 clusters and the 36 directed edges across which the most inter-cluster transmission occurs, which account for 90% of all inter-cluster influenza transmission. The area of each cluster is proportional to the steady-state proportion of time a random walker on the city-to-city EPN spends in the cluster. The proportion of each cluster contained in its border ring equals the probability that a random walker within the cluster jumps to a city in a different cluster, so the proportion contained in the interior is equal to the probability that a random walker within the cluster jumps to another city in the cluster. The width of each edge is proportional to the steady-state proportion of jumps between clusters that cross it. For emphasis, the color of cluster interiors and border rings gets darker with increasing area and the color of the edges gets darker with increasing width.

https://doi.org/10.1371/journal.pone.0019515.g003

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Table 1. Global transmission cluster characteristics.

https://doi.org/10.1371/journal.pone.0019515.t001

The map captures several important features of global spread of influenza. The major population centers in the temperate northern hemisphere are highly connected, resulting in a narrow epidemic curve with a single peak. Connections between clusters in the tropics and the temperate southern hemisphere are not as dense, resulting in wider epidemic curves with multiple peaks. The Hong Kong and Southeast Asia cluster plays a larger role in the spread of influenza than would be expected based on the number of cities it contains. The North/Central America+Caribbean and Europe+North/West Africa clusters have high flow per city but relatively low outflow probabilities. The Hong Kong and Southeast Asia cluster has both high flow per city and a high outflow probability. Its tropical location allows it to serve as a bridge between the Northern and Southern Hemispheres. The map indicates that pandemics starting within season in the temperate northern hemisphere, North and West Africa, or the Caribbean would quickly spread throughout those regions but diffuse much more slowly to other parts of the globe. Epidemics originating in China are linked to North America and Europe primarily through Japan and Southeast Asia.

Global patterns of pandemic influenza spread

To investigate the most plausible global patterns for pandemic influenza spread, we model the initial outbreak to occur at different key geographic locations, times of the year, and values of . In Figure 4, we show plots of the global spread for pandemics starting in Hong Kong, Ho Chi Minh City, Cairo and Mexico City. Hong Kong was the first city to experience a large epidemic in the Hong Kong influenza pandemic in 1968. Both Egypt and Vietnam have experienced considerable avian influenza A(H5,N1) human cases where reassortment events could lead to a pandemic strain.

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Figure 4. Prevalence of influenza over time in simulated pandemics with various starting locations, dates, and transmissibilities.

In each plot, the infection prevalence (%) for the cities in the three regions of the globe are plotted: North (cities north of the Tropic of Cancer, in red), South (cities south of the Tropic of Capricorn, in blue), and tropics (cities between the two tropics, in green). Each plot shows the results from a single simulation. (A) Simulated pandemics with were started in Hong Kong (first column), Ho Chi Minh City (second column), Mexico City (third column), and Cairo (fourth column). The epidemics were started on March 1, April 1, May 1, and June 1 and plotted in the first, second, third, and fourth rows, respectively. (B) Plots of simulated pandemics with , organized as in panel A. Note that the y-axis has a different scale than panel A.

https://doi.org/10.1371/journal.pone.0019515.g004

The scenario most similar to the Hong Kong pandemic of 1968–1969 is shown in the plot for Hong Kong starting on June 1 in Figure 4A. In this case, we see two peaks in the temperate southern hemisphere countries because the first wave out of Hong Kong does not go to completion before the end of the influenza season in many of these countries (Movie S2). If initial spread comes out of Hong Kong earlier in the year (i.e., March or April), then most of the temperate southern hemisphere peaks during the winter of the first year. In all these cases, a single peak occurs in December in the temperate northern hemisphere countries. The epidemics in the tropics tend to peak in multiple clusters between August and January. A similar pattern occurs if the pandemic starts in Ho Chi Minh City rather than Hong Kong.

For pandemics starting in Mexico City or Cairo, with , two peaks occur in the temperate southern hemisphere if the pandemic starts on April 1, but only one peak occurs if the pandemic starts on March 1. The epidemics in the temperate northern hemisphere and the tropics are roughly the same as when the pandemic started in Hong Kong or Ho Chi Minh City. If the pandemic strain starts spreading on May 1 or June 1, there is no pandemic at all. In this case, we are past the temperate northern hemisphere influenza season and infected travelers arrive too late in the temperate southern hemisphere influenza season to sustain transmission there.

Figure 4B shows simulations of pandemics having initial spread in the same four cities for a more transmissible virus with . In these cases, the pandemic is much larger and faster than when and the patterns of spread are similar regardless of when and where the pandemic spread starts. The case where pandemic spread starts in Mexico City on April 1 is closest to the pandemic H1N1 2009 situation. In this case, there is only a single first peak in the temperate southern hemisphere in July and a large peak in the temperate northern hemisphere in late October. As in the case when , there is no subsequent pandemic when spread starts in Mexico City on June 1.

Effect of a global vaccination strategy

For our modeling, we assume that pandemic vaccine is available 180 days after the appearance of the pandemic strain. For pandemic H1N1 2009, substantial quantities of vaccine became available in October, 2009, roughly five to six months after the recognition of the pandemic strain in late April, 2009. In the US, the epidemic peaked in October, just as the vaccine was arriving. We assumed that all vaccine was delivered and administered at once, and more realistic rollouts would result in a slower and possibly less efficacious global mass vaccination campaign.

We used the per capita GDP from 2007 [23] to determine how much vaccine each country would be able to obtain. Wealthy countries (per capita GDP $25,000 in year 2000 dollars) cover 50% of their populations. Other developed countries (per capita GDP $10,000) cover 25% of their populations. The remaining countries cover 10%, many relying on the World Health Organization's vaccine distribution plan. See Table S6 for a summary of vaccine coverage in the model for individual countries.

To reduce mortality and morbidity, vaccine should first be distributed to children and individuals at high risk of complications from influenza infection [24], [25]. We assume that 10% of children are and 17% of adults are at high risk [26]. In the simulations, we prioritize high-risk children, followed by high-risk adults, healthy children, then healthy adults. We assume that a maximum of 50% of any group will get vaccinated. Therefore, if a country can cover 50% of its population, then 50% of each of these risk groups is covered. If a country can cover 25% of its population, then 50% of the high-risk children and adults are covered, 50% of the healthy children, and about 10–13% of healthy adults. If a country can cover 10% of its population, then 50% of the high-risk children and adults are covered, about 5% of healthy children, and no healthy adults. This strategy results in lower overall attack rates as compared to a strategy in which everyone has the same priority (Figure S8).

Table 2 shows the results for such a vaccination campaign for several of the pandemic scenarios shown in Figure 4. The row for the pandemic starting in Mexico City on April 1, with gives the scenario closest to pandemic H1N1 2009. In this case, the model predicts about 1.2 billion eventual influenza illnesses with no vaccination and about 930 million had vaccination been carried out as described above. The biggest impact of vaccination would have been in the temperate northern hemisphere, reducing the illness attack rate from 16% to 9%. Such a vaccination campaign would have little effect on the epidemic in the temperate southern hemisphere, and a small effect in the tropics. Movie S3 gives a dynamic view of the effect of such vaccination. This vaccination plan has the biggest beneficial effect in the temperate northern hemisphere if the pandemic strain begins spread from Mexico City or Cairo in March or April. For the temperate southern hemisphere, this vaccination strategy would have the biggest beneficial effect if the is lower, at , and if the beginning of pandemic spread is in Hong Kong in May or June, or Cairo in April. In general, the tropics benefit most only when the other regions gain a benefit as well.

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Table 2. Potential global impact of mass influenza vaccination (averages from 10 simulations for each scenario).

https://doi.org/10.1371/journal.pone.0019515.t002