The Role of Environmental Transmission in Recurrent Avian Influenza Epidemics

Avian influenza virus (AIV) persists in North American wild waterfowl, exhibiting major outbreaks every 2–4 years. Attempts to explain the patterns of periodicity and persistence using simple direct transmission models are unsuccessful. Motivated by empirical evidence, we examine the contribution of an overlooked AIV transmission mode: environmental transmission. It is known that infectious birds shed large concentrations of virions in the environment, where virions may persist for a long time. We thus propose that, in addition to direct fecal/oral transmission, birds may become infected by ingesting virions that have long persisted in the environment. We design a new host–pathogen model that combines within-season transmission dynamics, between-season migration and reproduction, and environmental variation. Analysis of the model yields three major results. First, environmental transmission provides a persistence mechanism within small communities where epidemics cannot be sustained by direct transmission only (i.e., communities smaller than the critical community size). Second, environmental transmission offers a parsimonious explanation of the 2–4 year periodicity of avian influenza epidemics. Third, very low levels of environmental transmission (i.e., few cases per year) are sufficient for avian influenza to persist in populations where it would otherwise vanish.


Parameters of the model
Here we discuss further details of the parameterization of the model. We provide a description of the parameters that we use in the main text and their corresponding values in Table 1 below. Parameters that deserve special consideration are the habitat carrying capacity N d , the direct transmissibility β, the exposure rate ρ, and the re-scaled environmental infectiousness αω † . These are as follows: • The value of N d is chosen such that the model in absence of infection yields ∼5,000 to 6,000 susceptibles, in agreement with observed duck population sizes in the wild [1].
• β is a study parameter. Defining R direct 0 = Sβ/(µ + γ), a range of β between 0 and 0.05 corresponds to a fairly large range of R direct 0 between 0 and 5.74. The value of 0.15 that we use for β in the simulations described in Figs. 2, 3 of the main text corresponds to an R direct 0 ∼ 1.6.
• The exposure rate ρ is given by the drinking rate of the duck (ranging from 2.5 × 10 3 to 2.5 × 10 4 liters/year [2]) divided by the volume of water that dilutes the virus. Since we are not modeling a particular pool of water, the choice of ρ remains somehow arbitrary. Therefore, in Sec. 4 of this supplement, we explore the robustness of our results with varying ρ.
• αω is a study parameter that we vary over six orders of magnitude, as little is known about this parameter.
In the following sections of this supplement, we present further numerical explorations that support our statements in the main text.   Table  1) versus the direct transmissibility β. The orbits are sampled yearly, at the end of the wintering season. Note that for a large range of β, the model shows numbers of infected ducks less than one (i.e., the area below the dotted red line) which elude a biological interpretation. All orbits sampled for Fig. 1 have at least one point below 22. If, we discard all orbits with points below (let us say) 10 as being plagued by the "atto-fox" phenomenon (as the fractional parts of these predictions may represent a significant percent from their corresponding integer parts), then we are left with a very narrow β-interval for which our continuous model has biological interpretation (i.e., the blue region). It is unlikely that this model with periodicity of 1-2 years would be validated by AIV epidemiological data from the wild habitat. In fact, prevalence data exhibits outbreaks with a 2-4 year periodicity and possibility of extinction between the outbreaks. Since we are particularly interested in the processes of extinction and persistence of AIV, we refined our study by constructing a stochastic model, where the host population variables are integer-valued.
3 Difference-of-Gaussians wavelet analysis B (which we chose to be equal to the viral persistence rate at the wintering site in the winter η w W ) and the viral persistence rate at the wintering grounds in the summer η w B . All the other parameters are listed in the Table 1. The simulation details are the same as for Fig. 3. The pattern in the low right panel is further discussed in the main text. Again, we notice that the displayed pattern is robust with changing parameters. B (which we chose to be equal to the viral persistence rate at the wintering site in the winter η w W ) and the viral persistence rate at the wintering grounds in the summer η w B . All the other parameters are listed in the Table 1. The simulation details are the same as for Fig. 3. The pattern in the low right panel is further discussed in the main text. Note that the displayed pattern remains robust with changing parameters.