Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

A Perspective on Multiple Waves of Influenza Pandemics

  • Anna Mummert,

    Affiliation Department of Mathematics, Marshall University, Huntington, West Virginia, United States of America

  • Howard Weiss,

    Affiliation School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, United States of America

  • Li-Ping Long,

    Affiliation Department of Basic Sciences, Mississippi State University, Mississippi State, Mississippi, United States of America

  • José M. Amigó,

    Affiliation Centro de Investigación Operativa, Universidad Miguel Hernández, Elche, Spain

  • Xiu-Feng Wan

    wan@cvm.msstate.edu

    Affiliation Department of Basic Sciences, Mississippi State University, Mississippi State, Mississippi, United States of America

A Perspective on Multiple Waves of Influenza Pandemics

  • Anna Mummert, 
  • Howard Weiss, 
  • Li-Ping Long, 
  • José M. Amigó, 
  • Xiu-Feng Wan
PLOS
x

Abstract

Background

A striking characteristic of the past four influenza pandemic outbreaks in the United States has been the multiple waves of infections. However, the mechanisms responsible for the multiple waves of influenza or other acute infectious diseases are uncertain. Understanding these mechanisms could provide knowledge for health authorities to develop and implement prevention and control strategies.

Materials and Methods

We exhibit five distinct mechanisms, each of which can generate two waves of infections for an acute infectious disease. The first two mechanisms capture changes in virus transmissibility and behavioral changes. The third mechanism involves population heterogeneity (e.g., demography, geography), where each wave spreads through one sub-population. The fourth mechanism is virus mutation which causes delayed susceptibility of individuals. The fifth mechanism is waning immunity. Each mechanism is incorporated into separate mathematical models, and outbreaks are then simulated. We use the models to examine the effects of the initial number of infected individuals (e.g., border control at the beginning of the outbreak) and the timing of and amount of available vaccinations.

Results

Four models, individually or in any combination, reproduce the two waves of the 2009 H1N1 pandemic in the United States, both qualitatively and quantitatively. One model reproduces the two waves only qualitatively. All models indicate that significantly reducing or delaying the initial numbers of infected individuals would have little impact on the attack rate. Instead, this reduction or delay results in a single wave as opposed to two waves. Furthermore, four of these models also indicate that a vaccination program started earlier than October 2009 (when the H1N1 vaccine was initially distributed) could have eliminated the second wave of infection, while more vaccine available starting in October would not have eliminated the second wave.

Introduction

An influenza pandemic occurs when a new strain of the influenza virus emerges, usually through antigenic shift, for which there is little or no immunity in the human population. The mutation is such that it is able to cause illness in a single individual and then spreads person-to-person worldwide. In the 20th century, the world experienced three influenza pandemics: the 1918 H1N1 "Spanish flu'', the 1957 H2N2 "Asian flu'', and the 1968 H3N2 "Hong Kong flu''. The first influenza pandemic of the 21st century occurred in 2009 and was caused by a swine-origin H1N1 influenza A virus [1].

During each of these four pandemics, the United States experienced multiple waves of infections, where the numbers of infections and deaths exhibited well-separated temporal peaks with a separation time-scale of months [1]. For example, the first wave of the 2009 pandemic in the United States began in March and peaked in late June and early July. There were markedly fewer cases throughout August, and the second larger wave peaked in late October, early November (Figure 1, Table S1 in File S1). While many countries such as the United States experienced at least two waves of infections during the 2009 pandemic, other countries such as China experienced only a single wave of infection.

thumbnail
Figure 1. The H1N1 positive tests reported to the CDC in the United States from April 24, 2009 (week 17) to November 27, 2009 (week 48).

https://doi.org/10.1371/journal.pone.0060343.g001

The underlying mechanisms leading to single or multiple waves of acute infectious diseases are not well understood. In this paper, we describe and explore several mechanisms that can produce multiple waves.

For several infectious diseases, including influenza, the timing of school vacations is thought to be a mechanism for multiple waves because children have reduced transmission due to far fewer contacts during vacations than when school is in session. Parts of either the summer or winter school breaks occurred during the gaps between the two waves of the past four influenza pandemics in the United States. Evidence supporting this potential mechanism for the second wave of the 2009 US H1N1 pandemic include the surge in hospital outpatient visits for influenza-like illness approximately two weeks after schools re-opened in the fall of 2009 [2]. However, school vacations clearly cannot be the sole cause of multiple waves [2], [3], [4], [5], [6]. For instance, during the 1968 pandemic in the United States, the second wave began in November, two months after school resumed. In addition, children have school vacations every year, while seasonal influenzas have only single waves of infections.

Another possible mechanism for multiple waves is due to rapid mutations of the H1N1 virus. Many RNA viruses, such as influenza, quickly acquire genetic variants through random mutations, some of which lead to non-synonymous mutations in protein sequences. The more genetic variants that there are, the higher the odds are that one of them carries a mutation upon which selection can act. Through random mutation and subsequent selection, RNA viruses evolve into a form better adapted for human-to-human transmission.

The protein sequence analyses for the 2009 H1N1 HA genes showed the average number of mutations increased slightly from April (2.75±0.71, based on the comparison with the A/California/07/2009 strain) to November of 2009 (4.26±1.53). The maximum number of mutations occurred in November 2009 (11 residues when compared with A/California/07/2009, versus two residues when compared with the major strain circulating in April 2009). Virus diversity, which we introduce in the next section, quantifies the extent of the mutations.

Some of these mutations occurred in the receptor binding sites of HA genes [7], [8] and in other segments of the virus [9]. Some mutations first appeared after the first wave ended (e.g., HA-S220T NA-N248D in Japan) [10]. As the pandemic progressed, the number of mutations at the receptor binding site position 222 increased around the world [7], [11], [12], [13], [14]. The mutation D222G/N was observed in 90% of the blood samples of the A/H1N1/2009 viremia cases [8], [15], [16], [17], [18]. The D222G mutation has been shown to increase virulence and to increase the virus transmissibility in animal models, as well in human airway epithelial cell lines [19], [20], [21], [22]. Other mutations in the receptor binding sites of the HA gene [23], [24], and other mutations at other segments (e.g., PA, PB1-F2, PB2, NP, and NS1) have been found in clinical isolates and have been shown to increase the replication efficiency and pathogenesis in vitro in animal models. Increased virulence and pathogenesis of such mutations is a mechanism that may lead to a second wave of infection.

In this study, five mathematical models are formulated to explore complementary mechanisms which can produce two waves of acute infections. Four reproduce the two waves of the 2009 H1N1 influenza pandemic, quantitatively. The other reproduces the waves qualitatively. The first two mechanisms capture changing contact rates and changing (or evolving) virus transmissibility. In these models, the changes manifest as a time varying transmission rate. The transmission rate is the per capita rate of infection given contact, and depends in a highly complex way on both the contact rate between susceptible and infected individuals and the transmissibility of the infection. The first mechanism uses a periodic transmission rate to capture the seasonal contact rate. The second mechanism incorporates all sources of variability in contacts and transmissibility into the time-dependent transmission rate.

The third mechanism incorporates a heterogeneous population with individuals split into two weakly interacting sub-populations. The split could be based on demographics, geography, or variations in immunity, among others. The two populations only very weakly interact, that is, the transmission between the two groups is low. Two waves appear with each sub-population experiencing only one wave of infection.

The fourth mechanism is virus mutation which causes delayed susceptibility of some individuals. As the 2009 H1N1 pandemic progressed in the United States, the virus mutated rapidly and created new viable quasispecies. We hypothesize that some of these mutations increased the transmissibility of the influenza virus and that this increased viral transmissibility caused some individuals who were not able to be infected during the first wave to become infected during the second wave.

The fifth mechanism is waning immunity, where recovered individuals lose immunity to the influenza virus at which point they again become susceptible to infection. The second wave appears because some individuals become infected for a second time.

The five mechanisms are incorporated into appropriate extensions of the standard (Susceptible-Exposed-Infected-Removed) transmission model [47]. Models 1–4 reproduce the two waves during the 2009 pandemic in the United States, qualitatively and quantitatively. Model 5 reproduces the two waves qualitatively.

Countries such as China implemented border control strategies and experienced only a single wave of infections. We implement border control in each model to determine whether the US could also have experienced one wave with border control. In addition, the US vaccination production and distribution data is incorporated into the models to test the effect of increasing vaccine availability and altering the timing of the vaccination effort.

Results

Models 1 and 2: SEIR models with variable transmission rate

Models 1 and 2 explore the ability of a variable transmission rate to generate the multiple waves of infection in a homogenous population. To do this we use a time-dependent, or variable, transmission rate βt in a classical SEIR compartment transmission model (see Supplementary Information in File S1, equations (S1)–(S4)). We provide two strikingly different transmission rates, each of which can reproduce the two waves. Model 1 exploits seasonality. We demonstrate that the school year hypothesis of a higher transmission rate during the school year and a lower rate during the summer and school vacations can generate the two waves. Clearly, this type of seasonality alone provides no insight into why two waves occur only during pandemic years and not every year; there must be additional factors driving the pandemic waves. Therefore, we provide Model 2, also with a variable transmission rate that captures the intricate interplay between the contact rate and transmissibility of the infection.

Model 1: Periodic transmission rate

Many experts believe that contacts between school age children play a significant role in influenza transmission and in producing pandemic waves. The contact rate is thought to be significantly higher for school age children during school terms than during school vacations [2], [3], [4], [5], [6]. The presumption is that the first wave begins in the winter or early spring when school is in session and wanes when the children have significantly reduced contact during their summer breaks. The second wave begins soon after school resumes in late summer or early fall, when the contact rate increases.

To capture the seasonality of school contacts, the transmission rate βt is set to be the periodic function (with period one year)(Figure 2A), where there is higher transmission when students are in school, and lower transmission over the summer months.

thumbnail
Figure 2. Reproducing multiple waves using Model 1 with periodic time-dependent transmission rate, βt.

(A) The periodic transmission rate, which is low during the summer break and higher when school is in session. Summer is approximately June and July (weeks 23–31 and days 161–217). (B) The model generated disease prevalence, compared with the CDC data scaled for underreporting.

https://doi.org/10.1371/journal.pone.0060343.g002

The resulting model was simulated using parameters from Table S9 in File S1. Figure 2B shows the output of Model 1 superimposed on the derived CDC incidence data for the United States in 2009. The model clearly captures the two waves of infection.

Model 2: Derived time-dependent transmission rate

The transmission rate in this model captures all sources of variability of the contact rate between susceptible and infected individuals and the virus transmissibility. This variability could occur because of public health interventions, seasonality, or the evolution of the virus, among other factors.

The transmission rate βt is determined using a new algorithm [25], [26], [27] that ensures the output I(t) of the model perfectly fits a smooth interpolation of the reported cases by the Centers for Disease Prevention and Control (CDC)[28]. There is no error between the model output of I(t) and the data, and only the initial value β0 needs to be specified.

The resulting model was simulated using the parameters in Table S10 in File S1. Figure 3A shows the output of Model 2 superimposed on the derived CDC incidence data for the United States in 2009. Recall that the near perfect agreement was ensured by our choice of βt. Errors in the simulation are due to the accumulation of small numerical errors.

thumbnail
Figure 3. Reproducing multiple waves using Model 2 with extracted transmission rate, βt.

(A) The model generated disease prevalence generated for the entire range 0.54 ≤β0≤1.03, compared with the CDC data scaled for underreporting. (B) The extracted transmission rates recovered for three β0 values. Top green curve β0 = 0.73; middle blue curve β0 = 0.63; bottom red curve β0 = 0.54.

https://doi.org/10.1371/journal.pone.0060343.g003

Figure 3A was generated using β0 = 0.63, however any β0 in the range 0.54≤β0≤1.03 will produce the same model output. Any β0 outside of this range produces unrealistic outputs. Since βt depends on β0 in a complicated way, in Figure 3B we show representative graphs of βt across a range of values for β0.

In conclusion, Model 2 reproduces the data, and thus the two waves of infection. The mechanism of Model 2 is the construction of a time dependent transmission rate βt, capturing all of the fluxuations in transmission due to changing contact rates, virus transmissibility, environment, and so on. The resulting transmission function indicates that the second wave cannot be explained as simply a drop in contacts due to school closures. A drop in contact rate due to school closures would appear as a corresponding drop in βt. However, no such drop is seen in the transmission function during the summer months.

Model 3: Two weakly interacting sub-populations

For this model, we hypothesize that the population is split into two groups with minimal interaction. Each group experiences one wave of infection resulting in two waves for the entire population.

In Model 3, the total population is split into two sub-populations; population 1 is 22% of the total and the rest are in population 2. The SEIR model (Supplementary Information equations (S1)–(S4) in File S1) is extended to eight equations, one S, E, I, and R equation for each sub-population. The sub-populations weakly interact, meaning that the transmission rate between populations is small. This model has three constant transmission rates: β1 for the transmission among sub-population 1, β2 for the transmission among sub-population 2, and β3 for the transmission between the two sub-populations. This model assumes that only sub-population 1 is initially infected and sub-population 2 does not acquire the infection until some time during the summer at which point some infected individuals appear (perhaps arrive home from traveling) who are infected.

Model 3 was simulated using the parameters in Table S11 in File S1. Model 3 reproduced the two waves qualitatively (Figure 4B), with each sub-population experiencing only one wave (Figure 4A). Its attack rate is just below that of the 2009 pandemic H1N1 (20.08% compared with the CDC estimate of 25%). These simulations verify that two weakly interacting populations can experience two waves.

thumbnail
Figure 4. Reproducing multiple waves using Model 3 with two weakly interacting sub-populations.

(A) Infection wave of sub-population 1 (blue) and infection wave of sub-population 2 (black). (B) The model generated disease prevalence compared with the CDC data scaled for underreporting.

https://doi.org/10.1371/journal.pone.0060343.g004

Model 4: Virus mutation and delayed susceptibility

During the course of the 2009 influenza pandemic, the genetic diversity of the virus in both the United States and around the world increased significantly. For this model, we hypothesize that the increase in genetic diversity results in strains with higher transmissibility in the human population. This hypothesis supports the notion that some individuals who were not susceptible to the virus during the first wave become susceptible when new strains with greater transmissibility emerge as the pandemic progresses.

We incorporate this idea into the standard SEIR model (with constant transmission rate β) by including a reserved class N consisting of individuals who are initially non-susceptible (see equations (S5)–(S9) in the Supplementary Information, File S1). These individuals become susceptible as the viral genetic diversity increases at a rate that is proportional to the time-dependent genetic diversity d(t).

Genetic diversity of virus quasispecies is defined as the mean of pairwise genetic distances among the genomes of the viruses isolated in one month. In bioinformatics there are various measures of genetic diversity. Here, genetic diversity was measured using three distances: the p-distance, the patristic, and the CCV distance between nucleotide sequences. These distances were normalized to attain values between 0 and 1, and the genetic diversity function was generated assuming that the diversity was constant throughout the entire month. In the United States data both of these genetic diversities increased slowly during the first wave and dramatically during the second (Figure 5, Tables S2, S3, S4, S5, S6, S7, and S8 in File S1).

thumbnail
Figure 5. Dynamics of genetic diversity of the HA gene of 2009 H1N1 pandemic virus in the United States.

Genetic diversity is defined as the mean of pairwise genetic distances among the genomes of the viruses isolated in one month. The distances were normalized to attain values between 0 and 1. Two genetic distances were measured: (A) p-distance, (B) patristic distance, (C) CCV distance. The genetic diversity of NA gene and other internal genes are shown in Tables S1, S2, S3, S4, S5, S6, and S7 in File S1.

https://doi.org/10.1371/journal.pone.0060343.g005

Model 4 reproduced the two waves using all three distances (Figure 6), using the parameters in Table S12 in File S1. Here, we construct the genetic diversity function as a step function of order 0. However, Model 4 will also reproduce the second wave using linear, quadratic, etc., interpolations of the monthly distance data. Thus, the model is robust in terms of the distance measure used and the way that these distance measurements are combined to determine the diversity function d(t).

thumbnail
Figure 6. Reproducing multiple waves using Model 4 with virus mutation causing delayed susceptibility of some individuals.

The model generated disease prevalence using the (A) p-distance, with γ  = 0.11, (B) patristic distance, with γ = 0.09, (C) CCV distance, with γ = 0.08, compared with the CDC data scaled for underreporting.

https://doi.org/10.1371/journal.pone.0060343.g006

In conclusion, Model 4 qualitatively reproduces the US incidence data and quantitatively reproduces the attack rate for the 2009 pandemic H1N1. This model lends support to the supposition that virus mutations can drive the second wave through the addition of susceptible individuals who were not susceptible to previous virus quasispecies. The unknown factor of Model 4 is the initial fraction of reserve non-susceptible individuals.

Here the value N(0) = 0.35 was chosen so that the simulation recreated the data.

Model 5: Waning immunity

A fifth mechanism is known to produce two waves of infection – waning immunity [8],[36]. We incorporate waning immunity to the H1N1 virus with an SEIR model where individuals in the removed class lose immunity at a constant rate and rejoin the susceptible class (see Supplementary Information equations (S10)–(S13), in File S1).

This model was simulated using the parameters in Table S13 in File S1. It qualitatively reproduces the two waves, but always (necessarily) with a smaller amplitude second wave (Figure 7).

thumbnail
Figure 7. Reproducing multiple waves using Model 5 with waning immunity.

The model generated disease prevalence, compared with the CDC data scaled for underreporting.

https://doi.org/10.1371/journal.pone.0060343.g007

The effect of initial infections on the second wave

The models were used to investigate the consequences of a smaller number of initially infected individuals in the United States and to identify factors that may affect the on-set of waves.

We assumed that an early intervention, e.g., border control, would have reduced the initial number of imported cases from Mexico. We included this in our models by reducing the initial fraction of infected individuals, I(0). For Model 3, we reduce the number of individuals of both sub-population 1 and 2. The simulation outputs (Figure 8) illustrate the result of two different intervention levels. The first plot for each model shows the resulting number of infected individuals, assuming that interventions reduced the initial number of infected individuals by 25% of the reported value adjusted for underreporting, while the second plot corresponds to a simulation output that reduced the number of individuals initially infected by 90%.

thumbnail
Figure 8. Effects of reduced and delayed initial infected individuals on influenza prevalence using (A) Model 1 with periodic transmission rate (B) Model 2 with extracted transmission rate, βt (C) Model 3 with two weakly interacting sub-populations (D) Model 4 with virus mutation causing delayed susceptibility of some individuals (E) Model 5 with waning immunity.

The model generated disease prevalence with the control strength indicated, compared with the CDC data scaled for underreporting, where 25% strength of control means that the initial fraction of infected individuals is 75% of the adjusted reported fraction.

https://doi.org/10.1371/journal.pone.0060343.g008

A value, called the attack rate (Itotal), was computed for all models. The attack rate is the total fraction of individuals who become infected during the disease outbreak. With no intervention, the attack rate is Itotal = 0.2457, 0.2258, 0.2008, and 0.2307 for Models 1, 2, 3, and 4, respectively. The attack rate barely changes with the addition of interventions, even strong interventions. For example, with a 90% effective intervention, the attack rate is Itotal = 0.2379, 0.2324, 0.1978, and 0.2132 for Models 1, 2, 3, and 4, respectively. In addition, in the simulations of Models 1, 2, and 4, interventions of over 80% result in a loss of the second wave. For Model 5, individuals may lose immunity to the disease and become infected a second time. Thus the notion of attack rate is not as well defined. We compute instead the total number of individuals who become infected once throughout the outbreak as 0.7395, and 0.7361 with 90% effective intervention.

The effect of vaccinations on the second wave

The models were used to investigate the consequences of a vaccination program started earlier or with an increased availability of vaccine in the United States on the outbreak progression. We assumed that vaccination would have reduced the number of susceptible (and reserve non-susceptible) individuals by the number of available vaccines, with a vaccine efficacy of 90% [29]. We have accounted for individuals receiving the vaccine according to the dates and amounts of distributed vaccine, along with a two week delay [30]. During the delay the immune system is generating antibodies and is still vulnerable to infection. We investigated both the scenarios of earlier available vaccine (0–5 months earlier) and the scenarios of more available vaccine (1–5 times as much available). Figure 9 illustrates the result of different vaccination efforts for the models.

thumbnail
Figure 9. Effects of timing and amount of available vaccine on influenza prevalence using (A) Model 1 with periodic transmission rate (B) Model 2 with extracted transmission rate, βt (C) Model 3 with two weakly interacting sub-populations (D) Model 4 with virus mutation causing delayed susceptibility of some individuals (E) Model 5 with waning immunity.

The model generated disease prevalence with the vaccination strategy indicated, compared with the CDC data scaled for underreporting. Here original timing begins vaccination on October 14, 2009, and one month earlier begins vaccination thirty days prior; original amount available is the daily amount of shipped vaccine with 90% efficacy, and twice as much available is twice the shipped amount with 90% efficacy.

https://doi.org/10.1371/journal.pone.0060343.g009

Earlier vaccinations can both eliminate the second wave of infections and significantly reduce the attack rate. Simulations indicate that vaccinations must be distributed approximately 2 months prior to the actual distribution dates to eliminate the second wave in Models 1, 2, 4, and 5. A larger availability of vaccine can only reduce the attack rate when combined with earlier vaccination. For example, with the original amount of vaccine distributed in October 2009, the attack rate is Itotal = 0.2142, 0.2072, 0.1931, and 0.2143 for Models 1, 2, 3, and 4, respectively, while with five times the original amount distributed at the same time, the attack rate is Itotal = 0.1788, 0.1845, 0.1689, and 0.1931 for Models 1, 2, 3, and 4, respectively. For Model 5, the total number who are infected once with the original vaccine distribution is 0.6951, while it is 0.6670 with five times the original amount distributed with the original timing.

Table 1 provides a summary of the five models, including the number of peaks of infection resulting from border control and vaccination, the attack rate with border control, and a reference for the model, if available.

thumbnail
Table 1. Summary of border control and vaccination on multiple waves of 2009 H1N1 influenza pandemics.

https://doi.org/10.1371/journal.pone.0060343.t001

Discussion

We study five mechanisms that can produce two waves of infection, and may explain the two waves experienced in the United States during the 2009 pandemic H1N1. These mechanisms are implemented and explored in mathematical models. Four models perform well, recreating the two waves (Figures 2, 3, 5 and 7) qualitatively and quantitatively, while one recreates the two waves only qualitatively. The two waves can be reproduced using the models individually or in any combination.

Two novel mechanisms (Models 2 and 4) are proposed. Models 1 and 2 incorporate a variable transmission rate, which was assumed constant in Models 3, 4 and 5. Model 4 explicitly incorporates the genetic diversity of the influenza quasispecies. We deem these models (2 and 4) complementary since changes in the transmission rate and transmission patterns among sub-populations could have been due, in part, to the emergence of more transmissible H1N1 mutants, which increases the genetic diversity of the quasispecies.

Model 1 is an SEIR component model of transmission with a time-dependent transmission rate reflecting the contact rate of school age children. It produces two waves with realistic parameter values, but provides no insight into why two waves of infection appear only with pandemics and not every year.

Model 2 is an SEIR model of transmission with a time-dependent transmission rate. Such a time-dependent transmission rate can account for temporal changes in virulence, seasonality, and contact rates, among others. The transmission rate is obtained from an algorithm that guarantees that the model output perfectly agrees with the infection data, even if the assumptions of the model do not apply. Thus one must be careful applying this algorithm and not over fit the data. We do not use our models to predict numbers of infections; we use them to exhibit possible mechanisms for two waves. In addition, we believe that the assumptions of a time-dependent transmission rate along with the structure of an SEIR model are reasonable for influenza, thus the potential weakness of over fitting is avoided.

The extracted transmission rate for Model 2 seems to be at odds with the common belief that summer affords fewer interactions amongst children [2], [3], [4], [5], [6] and therefore causes a drop in the transmission rate. For all graphs in Figure 3B, there was no significant drop in βt after the school year ended; the transmission rate fluctuated around a nearly constant level in the first wave from mid-May through the end of August. It increased when the fall school term and the second wave began. Although at first surprising, there is no contradiction with a decrease in the number of infected individuals and a (basically) constant transmission rate, as is seen over the summer months. A constant transmission rate implies that the ratio of new infections to the number of susceptible individuals remains constant, which happens over the summer when both the numbers of new infections and of susceptible individuals are decreasing.

From days 150 to 250, the transmission rate βt oscillated with mean period 19.9. These oscillations appear more rapid than the contact rate and environmental factors can account for. We do not see how this oscillation can be a manifestation of reporting conventions. However, these oscillations seem essential to reproduce the reported data. Figure S1A in File S1 shows a 21-day Gaussian filtered (smoothed) βt function, and the corresponding model simulation (Figure S1B in File S1), which is a substantially lower quality fit to the data. All attempts at smoothing the oscillations using different types of filters resulted in a worse fit to the data. Presumably, smoothing the data would eliminate these oscillations, but this goes in a very different direction.

Model 3 consists of two weakly coupled SEIR models of transmission, with a constant transmission rate. We have found that this model is the least robust, since it is the most sensitive to changes in the model parameter values. By construction, the model will always exhibit two waves regardless of the initial number of infected individuals (e.g., with a strong border control).

Model 4 involves virus mutation causing delayed susceptibility of some individuals. This new mechanism posits that as the genetic structure of the influenza virus changed during the pandemic, more of the general population became susceptible to infection. Such a dynamic of transmissibility of influenza A virus has been hypothesized to generate the second wave [31]. Model 4 assumes that the initially non-susceptible individuals become susceptible at a rate proportional to the genetic diversity of the flu quasispecies. The effects of vaccination and border control are considered for Model 4 (Figure 8D, Figure 9D) using the CCV distance, and similar results hold for both the p-distance and the patristic distance.

Model 5 incorporates waning immunity to the influenza virus causing some individuals to become infected again. This model qualitatively reproduces the two waves of the 2009 H1N1 pandemic in the US; however, the first wave will always be larger than the second. In addition, waning immunity requires an unrealistically short period of immunity (six months) to accurately reproduce the timing of the two waves of the 2009 H1N1 pandemic. For these reasons, we believe that this mechanism played at most a minor role in the second wave.

While all combinations of all five models will generate the two waves, it is impossible to determine the actual mechanism(s) from any model. This is a fundamental limitation of mathematical modeling. Models can suggest plausible mechanisms, but they cannot prove them. There is so little data from other countries and other pandemics that any type of statistical investigation trying to narrow down the true explanation is impossible.

Although age structure probably plays a role in flu transmission, our models do not explicitly include such structure and they do not need to. We are able to illustrate that all five mechanisms can reproduce two waves without explicit age structure in the model. We point out that the two populations in Model 3 could be two age groups, though other groups are also possible.

The transmission rate during the summer school vacation

For all graphs in Figure 3B, there was no significant drop in βt after the school year ended; the transmission rate fluxuated around a nearly constant level in the first wave from mid-May through the end of August. It increased when the fall school term and the second wave began. Although at first surprising, there is no contradiction with a decrease in the number of infected individuals and a (basically) constant transmission rate, as is seen over the summer months. A constant transmission rate implies that the ratio of new infections to the number of susceptible individuals remains constant, which happens over the summer when both the numbers of new infections and of susceptible individuals are decreasing.

The effects of border control

During the 2009 H1N1 pandemic, countries including China, Japan, Hong Kong, Singapore, and Australia implemented various border control strategies to prevent arriving airline passengers from infecting the countries' citizens [32], [33], [34], [35]. Health authorities screened arriving passengers for flu-like symptoms using thermal scanners. In China, passengers suspected of being infected, along with passengers seated nearby, were placed into quarantine for seven days. China's border control began when the first cases of H1N1 were confirmed in California; China continued screening passengers for more than two months.

There has been debate as to whether the United States should have implemented some form of border control with Mexico after the first novel influenza cases appeared. The results from simulations indicate that strong intervention at the beginning of the outbreak would have had negligible impact on the attack rate in the United States, but it could have resulted in a single wave of infections. Our simulations imply that by significantly decreasing the initial number of infected individuals, the two waves would collapse into a single wave of infections, and the peak number of infections would occur slightly earlier with one wave. This suggests that China's strong border control could be a potential mechanism explaining their one wave of infection.

In practice, the efficacy of such border control measures to reduce transmission is uncertain. These screenings will miss asymptomatic individuals [35], and in particular infected individuals with no fever. However border control has been shown to delay the local community transmission [32], [33], [34], which may allow sufficient time for health interventions, such as vaccinations. On the other hand, border control may only delay transmission and could still result in a second wave, as seen in the simulations.

The effects of vaccination

The first doses of 2009 H1N1 vaccine were available in the United States in early October 2009, which was just before the peak of the 2009 H1N1 pandemic. Conventional wisdom is that this vaccination had little effect on the course of the disease spread. Though we were unable to find any empirical evidence in the literature, our simulations support this idea. Simulations of all models indicate that the actual vaccine schedule did not significantly reduce the total number of infections and even a significantly larger amount of vaccine available in October would have little effect on the attack rate. However, an earlier vaccination program would significantly decrease the attack rate and can eliminate the second wave of infections. Simulations show that the attack rate would be significantly reduced had the vaccine been distributed two months earlier than it was (with the same availability).

Recall that by design, Model 3 will produce two waves. Extremely strict border control (99.9% control) can prevent the second wave; however starting the vaccination program five months earlier with the original amount of vaccine cannot prevent the second wave.

Materials and Methods

Data sets

The epidemiologic dataset of 2009 weekly H1N1 positive tests reported to the CDC was downloaded from the CDC online weekly reports (www.cdc.gov/flu/). Clearly, these weekly numbers are highly affected by the testing efforts. An estimated 25% of the U.S. population was infected with the pandemic H1N1 throughout the two waves [36]. Using this along with the CDC compiled data, the influenza confirmed case data was adjusted to reflect underreporting by a scale factor of 700∶1. The authors are aware that a team of investigators estimated an underreporting scale factor of 79∶1 at the beginning (April through July) of the pandemic [32]. This is not a contradiction, since, very quickly, many physicians and hospitals stopped testing and the underreporting factor skyrocketed. Since we could find so little information about the underreporting rate during the entire pandemic in the U.S., we chose a uniform scaling factor.

The genomic sequence dataset was downloaded from the Influenza Virus Resource Database [37], which was updated on November 16, 2011. This dataset includes 1,502 HA genes of 2009 pH1N1 pandemic influenza viruses (229 were isolated in April of 2009; 273 in May; 247 in June; 74 in July; 46 in August; 119 in September; 195 in October; 214 in November; and 105 in December). All HA genes were fully or nearly fully sequenced, and only a single HA sequence for the same strain was selected.

Measurement of genetic distance and genetic diversity

The genetic diversity of the influenza virus quasispecies is defined as the mean of the pairwise genetic distances among the genes of the virus. We employ three notions of genetic distance: the p-distance, the patristic distance, and the CCV distance [38], [39]. P-distance measures the proportion of nucleotide sites between two sequences. This is the simplest measure and does not take into consideration multiple substitutions at the same site, substitution rate biases, or differences in the evolutionary rates among sites [40]. The p-distances were measured using the software package MEGAN [41]. The patristic distance encodes the total amount of genetic change that exists between genetic sequences by summing the lengths of branches in the phylogenetic tree between the two viruses. The patristic distances were calculated using PATRISTIC [42] using the maximum likelihood phylogenetic trees constructed by GARLI [43]. CCV is a method for measuring genetic distance based on the probability of strings [Wan et al. 2007], and this method has been shown to be effective in measuring the genetic distance between influenza genes [Wan et al. 2007b; Wan et al. 2008a; Wan et al. 2008b; Long et al. 2012].

Model parameter selection

Every model parameter value is consistent with the range commonly found in the literature. Within the common ranges, parameters for Models 1–4 were found that quantitatively and qualitatively reproduce the two waves of the 2009 H1N1 pandemic in the US. For Model 5, no parameters were found from the common ranges that reproduce the two waves quantitatively.

The incubation period for influenza can range between one and four days. We follow modeling convention and use a one day incubation period in all models. Adults shed influenza virus for up to 11 days; however, the amount shed is significantly lower after the first 3 to 5 days. Several previous modeling studies of pandemic H1N1 have assumed the infectivity period (1/v) is the lower bound of 3 days, as is done in all models here.

The transmission rate β (or the initial transmission rate for Models 1 and 2) is chosen to ensure that the basic reproduction number R0 is in the allowable range for influenza. The basic reproduction number is the average number of secondary cases caused by one infected individual into an entirely susceptible population. However, it is well known that approximately 20% of the US population had prior immunity to the pandemic influenza. Therefore, it is unreasonable to compute R0 assuming that the entire population was susceptible. We use instead the standard definition where it is not assumed that all individuals are initially susceptible. For the 2009 H1N1 influenza, R0 is estimated to be in the range 1.2–2.25 [44]. For Model 1, the transmission rate at the start of the outbreak (t = 140) is 0.5267, resulting in an R0 of 1.58. For Model 2, simulations restrict β0 to the range 0.54 to 1.03. Figure 3 shows β0 in the range 0.54 to 0.74, which gives a basic reproduction number in the range 1.39 to 1.88. For Model 3, at the beginning of the infection β1 = 2.1, which gives a basic reproduction number of R0 = 1.386, and for the second wave, β2 = 0.565, which gives a basic reproduction number of R0 = 1.322. For Model 4, β = 0.9, and so the basic reproduction number is R0 = 1.20. For Model 5, β = 0.63, and so the basic reproduction number is R0 = 1.89.

For Models 1, 3, 4, and 5, the initial fraction of infected individuals, I(0) (I1(0) for Model 3) is set by the US incidence data and the initial fraction of exposed individuals E(0) is set to be 0. For Models 1, 4 and 5, the initial fraction of removed individuals R(0) is set to 0.2, corresponding to elderly individuals with acquired immunity [45]; in Model 3, this value is set to 0. For Model 3, the infection in sub-population 2 begins at day t = 210 and I2(210) is the US incidence data scaled by 0.06.

For Model 2, the initial values S(0), E(0), I(0), and R(0) are determined by the algorithm which computes the time-dependent transmission rate βt. Therefore the choice of β0 determines all four of the initial values. For β0 = 0.63 the initial conditions agree reasonably well with the initial conditions of the other models.

For Model 4, N(0) was chosen to ensure that the basic reproduction rate R0 was within the known bounds. With β0 = 0.9, the value of N(0) can range from 0 to 0.35. In addition, model simulations show that to generate two waves, the value of N(0) must be larger than 0.25. With the chosen value of N(0) = 0.35 the value of S(0) is forced to be 0.44.

Supporting Information

File S1.

Supplementary Equations, Tables, and Figures.

https://doi.org/10.1371/journal.pone.0060343.s001

(DOC)

Acknowledgments

We are grateful to Marc Lipsitch and Rustom Antia for critical comments on a preliminary draft of this manuscript, and to Jialiang Yang for his assistance in the genetic distance measurements.

Author Contributions

Conceived and designed the experiments: XFW HW. Performed the experiments: AM HW XFW LPL. Analyzed the data: AM HW XFW JA. Contributed reagents/materials/analysis tools: XFW. Wrote the paper: AM HW XFW.

References

  1. 1. Miller MA, Viboud C, Balinska M, Simonsen L (2009) The signature features of influenza pandemics-implications for policy. N Engl J Med 360: 2595–2598.
  2. 2. Chao DL, Halloran ME, Longini IM Jr (2010) School opening dates predict pandemic influenza A(H1N1) outbreaks in the United States. J Infect Dis 202: 877–880.
  3. 3. Chowell G, Echevarria-Zuno S, Viboud C, Simonsen L, Tamerius J, et al. (2011) Characterizing the epidemiology of the 2009 influenza A/H1N1 pandemic in Mexico. PLoS Med 8: e1000436.
  4. 4. Monto AS, Koopman JS, Longini IM Jr (1985) Tecumseh study of illness. XIII. Influenza infection and disease, 1976–1981. Am J Epidemiol 121: 811–822.
  5. 5. Cauchemez S, Ferguson NM, Wachtel C, Tegnell A, Saour G, et al. (2009) Closure of schools during an influenza pandemic. Lancet Infect Dis 9: 473–481.
  6. 6. Cauchemez S, Valleron AJ, Boelle PY, Flahault A, Ferguson NM (2008) Estimating the impact of school closure on influenza transmission from Sentinel data. Nature 452: 750–754.
  7. 7. Melidou A, Gioula G, Exindari M, Chatzidimitriou D, Diza E, et al. (2010) Molecular and phylogenetic analysis of the haemagglutinin gene of pandemic influenza H1N1 2009 viruses associated with severe and fatal infections. Virus Res 151: 192–199.
  8. 8. Chan PK, Lee N, Joynt GM, Choi KW, Cheung JL, et al. (2011) Clinical and virological course of infection with haemagglutinin D222G mutant strain of 2009 pandemic influenza A (H1N1) virus. J Clin Virol 50: 320–324.
  9. 9. Pan C, Cheung B, Tan S, Li C, Li L, et al. (2010) Genomic signature and mutation trend analysis of pandemic (H1N1) 2009 influenza A virus. PLoS One 5: e9549.
  10. 10. Morlighem JE, Aoki S, Kishima M, Hanami M, Ogawa C, et al. (2011) Mutation analysis of 2009 pandemic influenza A(H1N1) viruses collected in Japan during the peak phase of the pandemic. PLoS One 6: e18956.
  11. 11. Wang B, Dwyer DE, Soedjono M, Shi H, Matlho K, et al. (2011) Evidence of the circulation of pandemic influenza (H1N1) 2009 with D222D/G/N/S hemagglutinin polymorphisms during the first wave of the 2009 influenza pandemic. J Clin Virol 52: 304–306.
  12. 12. Drews SJ, Pabbaraju K, Wong S, Tokaryk KL, May-Hadford J, et al. (2011) Surveillance of autopsy cases for D222G substitutions in haemagglutinin of the pandemic (H1N1) 2009 virus in Alberta, Canada. Clin Microbiol Infect 17: 582–584.
  13. 13. Miller RR, MacLean AR, Gunson RN, Carman WF (2010) Occurrence of haemagglutinin mutation D222G in pandemic influenza A(H1N1) infected patients in the West of Scotland, United Kingdom, 2009–10. Euro Surveill 15.
  14. 14. Valli MB, Selleri M, Meschi S, Zaccaro P, Vincenti D, et al. (2011) Hemagglutinin 222 variants in pandemic (H1N1) 2009 virus. Emerg Infect Dis 17: 749–751.
  15. 15. Tse H, To KK, Wen X, Chen H, Chan KH, et al. (2011) Clinical and virological factors associated with viremia in pandemic influenza A/H1N1/2009 virus infection. PLoS One 6: e22534.
  16. 16. Puzelli S, Facchini M, Spagnolo D, De Marco MA, Calzoletti L, et al. (2010) Transmission of hemagglutinin D222G mutant strain of pandemic (H1N1) 2009 virus. Emerg Infect Dis 16: 863–865.
  17. 17. Mak GC, Au KW, Tai LS, Chuang KC, Cheng KC, et al. (2010) Association of D222G substitution in haemagglutinin of 2009 pandemic influenza A (H1N1) with severe disease. Euro Surveill 15.
  18. 18. Kilander A, Rykkvin R, Dudman SG, Hungnes O (2010) Observed association between the HA1 mutation D222G in the 2009 pandemic influenza A(H1N1) virus and severe clinical outcome, Norway 2009–2010. Euro Surveill 15.
  19. 19. Chutinimitkul S, Herfst S, Steel J, Lowen AC, Ye J, et al.. (2010) Virulence-associated substitution D222G in hemagglutinin of 2009 pandemic influenza A(H1N1) virus affects receptor binding. J Virol.
  20. 20. Belser JA, Jayaraman A, Raman R, Pappas C, Zeng H, et al. (2011) Effect of D222G mutation in the hemagglutinin protein on receptor binding, pathogenesis and transmissibility of the 2009 pandemic H1N1 influenza virus. PLoS One 6: e25091.
  21. 21. Abed Y, Pizzorno A, Hamelin ME, Leung A, Joubert P, et al. (2011) The 2009 pandemic H1N1 D222G hemagglutinin mutation alters receptor specificity and increases virulence in mice but not in ferrets. J Infect Dis 204: 1008–1016.
  22. 22. Liu Y, Childs RA, Matrosovich T, Wharton S, Palma AS, et al. (2010) Altered receptor specificity and cell tropism of D222G hemagglutinin mutants isolated from fatal cases of pandemic A(H1N1) 2009 influenza virus. J Virol 84: 12069–12074.
  23. 23. Wang W, Castelan-Vega JA, Jimenez-Alberto A, Vassell R, Ye Z, et al. (2010) A mutation in the receptor binding site enhances infectivity of 2009 H1N1 influenza hemagglutinin pseudotypes without changing antigenicity. Virology 407: 374–380.
  24. 24. Zheng B, Chan KH, Zhang AJ, Zhou J, Chan CC, et al. (2010) D225G mutation in hemagglutinin of pandemic influenza H1N1 (2009) virus enhances virulence in mice. Exp Biol Med (Maywood) 235: 981–988.
  25. 25. Pollicott M, Wang H, Weiss H (2009) Recovering the Time-dependent Transmission Rate from Infection Data. Available from: arXiv:09073529.
  26. 26. Hadeler KP (2011) Parameter identification in epidemic models. Math Biosci 229: 185–189.
  27. 27. Mummert A (2012) Studying the Recovery Procedure for the Time-dependent Transmission Rate(s) in Epidemic Models. Journal of Mathmatical Biology Available from DOI 10.1007/s00285-012-0558-1
  28. 28. CDC (2010) 2009 H1N1 Flu. CDC.
  29. 29. Greenberg ME, Lai MH, Hartel GF, Wichems CH, Gittleson C, et al. (2009) Response to a monovalent 2009 influenza A (H1N1) vaccine. N Engl J Med 361: 2405–2413.
  30. 30. CDC (2010) 2009 H1N1 Vaccine Doses Allocated, Ordered, and Shipped by Project Area CDC.
  31. 31. Jackson C, Vynnycky E, Mangtani P (2010) Estimates of the transmissibility of the 1968 (Hong Kong) influenza pandemic: evidence of increased transmissibility between successive waves. Am J Epidemiol 171: 465–478.
  32. 32. Wood JG, Zamani N, MacIntyre CR, Beckert NG (2007) Effects of internal border control on spread of pandemic influenza. Emerg Infect Dis 13: 1038–1045.
  33. 33. Bishop JF, Murnane MP, Owen R (2009) Australia's winter with the 2009 pandemic influenza A (H1N1) virus. N Engl J Med 361: 2591–2594.
  34. 34. Cowling BJ, Lau LL, Wu P, Wong HW, Fang VJ, et al. (2010) Entry screening to delay local transmission of 2009 pandemic influenza A (H1N1). BMC Infect Dis 10: 82.
  35. 35. Miller E, Hoschler K, Hardelid P, Stanford E, Andrews N, et al. (2010) Incidence of 2009 pandemic influenza A H1N1 infection in England: a cross-sectional serological study. Lancet 375: 1100–1108.
  36. 36. CDC (2010) Updated CDC Estimates of 2009 H1N1 Influenza Cases, Hospitalizations and Deaths in the United States, April 2009 – April 10, 2010. CDC.
  37. 37. Bao Y, Bolotov P, Dernovoy D, Kiryutin B, Zaslavsky L, et al. (2008) The influenza virus resource at the National Center for Biotechnology Information. J Virol 82: 596–601.
  38. 38. Long LP, Yuan C, Cai Z, Xu H, Wan X-F (2012) Mixture model analysis reflecting dynamics of the population diversity of 2009 pandemic H1N1 influenza virus. In Silico Biology 11: 225–236.
  39. 39. Wan XF, Wu X, Lin G, Holton SB, Desmone RA, et al. (2007) Computational identification of reassortments in avian influenza viruses. Avian Dis 51: 434–439.
  40. 40. Nei M, Kumar S (2000) Molecular Evolution and Phylogenetics. New York: Oxford University Press.
  41. 41. Huson DH, Auch AF, Qi J, Schuster SC (2007) MEGAN analysis of metagenomic data. Genome Res 17: 377–386.
  42. 42. Fourment M, Gibbs MJ (2006) PATRISTIC: a program for calculating patristic distances and graphically comparing the components of genetic change. BMC Evol Biol 6: 1.
  43. 43. Zwickl DJ (2006) Genetic algorithm approaches for the phylogenetic analysis of large biological sequence datasets under the maximum likelihood criterion: The University of Texas at Austin.
  44. 44. Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, et al. (2009) Pandemic potential of a strain of influenza A (H1N1): early findings. Science 324: 1557–1561.
  45. 45. Hancock K, Veguilla V, Lu X, Zhong W, Butler EN, et al. (2009) Cross-reactive antibody responses to the 2009 pandemic H1N1 influenza virus. N Engl J Med 361: 1945–1952.
  46. 46. Dietz K (1976) The incidence of infectious diseases under the influence of seasonal fluctuation. Lect Notes Biomath 11: 1.
  47. 47. Towers S, Feng Z (2009) Pandemic H1N1 influenza: predicting the course of a pandemic and assessing the efficacy of the planned vaccination programme in the United States. Euro Surveill 14: 19358.