Contact Heterogeneity, Rather Than Transmission Efficiency, Limits the Emergence and Spread of Canine Influenza Virus

Host-range shifts in influenza virus are a major risk factor for pandemics. A key question in the study of emerging zoonoses is how the evolution of transmission efficiency interacts with heterogeneity in contact patterns in the new host species, as this interplay influences disease dynamics and prospects for control. Here we use a synergistic mixture of models and data to tease apart the evolutionary and demographic processes controlling a host-range shift in equine H3N8-derived canine influenza virus (CIV). CIV has experienced 15 years of continuous transfer among dogs in the United States, but maintains a patchy distribution, characterized by sporadic short-lived outbreaks coupled with endemic hotspots in large animal shelters. We show that CIV has a high reproductive potential in these facilities (mean R0 = 3.9) and that these hotspots act as refugia from the sparsely connected majority of the dog population. Intriguingly, CIV has evolved a transmission efficiency that closely matches the minimum required to persist in these refugia, leaving it poised on the extinction/invasion threshold of the host contact network. Corresponding phylogenetic analyses show strong geographic clustering in three US regions, and that the effective reproductive number of the virus (Re) in the general dog population is close to 1.0. Our results highlight the critical role of host contact structure in CIV dynamics, and show how host contact networks could shape the evolution of pathogen transmission efficiency. Importantly, efficient control measures could eradicate the virus, in turn minimizing the risk of future sustained transmission among companion dogs that could represent a potential new axis to the human-animal interface for influenza.

1 Details of the epidemic model for CIV in animal shelters 1.1 Mean field, single shelteṙ where λ = β(I + ωW ) N (6) and the "dot notation" on the left hand side is short for the time derivative of a state variable, e.g. X ≡ dX/dt.
The base model without vaccination is a special case where α = 0, and V = W = 0

R 0 from mean field absent vaccination
We use the spectral radius method [1]. Let x represent the state vector for the system, so that the ith element of x corresponds to the value of the ith state variable. Let x 0 represent the disease free state. Let F i (x) be the rate of appearance of new infected individuals into class i and V i (x) represent the rate at which infected individuals leave that class. Define the matrices and where i and j cover only the classes of infected individuals. Then where ρ is the spectral radius of the resulting matrix.
Absent vaccination, we have and This leads to 1.4 R 0 from mean field, full model Vaccine coverage, C, at the disease free equilibrium is Then and Letting R + 0 represent the R 0 under vaccination, we have This can be made more familiar by substituting the vaccine-free R 0 in place of β/(γ + δ) and defining K as the effective coverage, adjusted for vaccine performance, to yield

Prevalence
Prevalence at endemic equilibrium is given by

Seroprevalence
Time to seroconversion is close to duration of viral shedding at around 7 days, so we assume recovered individuals have seroconverted and infected individuals have not [2]. Equilibrium (longterm average) seroprevalence in a shelter where CIV is endemic is thus given by

Metapopulation model
We have individual level data for 124,519 dogs from 13 animal shelters, spanning 2008 to 2013. Each row in the resulting data frame consists of the arrival and departure dates of one dog from a specified shelter. The data also contained a column giving information on outcome type, allowing us to exclude dogs that were admitted as the result of euthanasia requests, as their residence times were systematically lower than that over other dogs. In the ith shelter we estimate the arrival rate µ i as the median number of dogs arriving per day to that shelter over the duration of the observation period. The departure rate in shelter i is estimated by taking the inverse of the median length of stay for dogs in that shelter over the duration of the observation period. Equilibrium shelter population size for shelter i is then given as N i = µ i /δ i . We reconstructed the shelter's actual population size over time by subtracting cumulative departures from cumulative arrivals, and visually checked the stationarity of each shelter's actual population size against the equilibrium value given by N i . Eight shelters recorded whether outcomes were transfers to other shelters. In each of these eight shelters we calculated the proportion of all outcomes that were transfers. The average of this value across shelters is τ = 0.1 (0.1 s.d). When estimating intake, output, and transfer we excluded dogs whose length of stay was greater than 40 days, as these represented rare atypical cases.
To parameterize the metapopulation model we need the arrival rate of dogs to each shelter from each other shelter µ ij . We also need the per-capita rate at which shelter i transfers dogs to shelter j, δ ij . Let µ ii represent arrivals from outside the metapopulation and let δ ii represent the percapita rate of outcomes that are not transfers to another shelter (e.g. adoption, return to owner, euthanasia).
Here is how we go from the demographic data we have to the metapopulation parameters we need. The mean per capita departure rate for the shelter associated with a randomly chosen dog (which we have) satisfies where there are M shelters in the metapopulation.
A proportion τ of these outcomes are transfers to other shelters, and we have an estimate of τ . Assume that shelters accept dogs transferred from other shelters in the metapopulation proportional to the size of the recipient shelter, so that large shelters accept more transfers than small ones. For shelter i, define the proportion of the metapopulation outside of shelter i that resides in shelter j as which we can also calculate from the demographic data we have.
Assuming the metapopulation is closed, so that no dogs are transferred to shelters not in the metapopulation, we get δ ij as Substituting into equation 38 yields, Now let µ ij represent the number of dogs that arrive to shelter i from j per day. Then We assume that during transfers dogs are selected at random from the shelter without regard to their disease state. Thus the probability that a susceptible dog is transferred is S i /N i and so on for the other classes.
Finally, we choose µ ii to balance the relation where the left-hand side is given by data.
In contrast to µ ij , the flow represented by µ ii is assumed to consist of entirely susceptible dogs, because the prevalence of CIV among dogs not in animal shelters is very low.
Our metapopulation approach ignores shelter proximity when calculating transfer probabilities, because we have no data on how transfer rates vary with geographic distance. We therefor use the mean transfer rate τ as an appropriately simple first approximation for transfer dynamics among shelters. The relative low mean transfer probability of τ = 0.1 suggests that transfer of dogs among shelters may be infrequent enough to prevent panmixia. That is, on a continuum from completely isolated to completely intermixed, dog populations in animal shelters in the US are closer to being solitary than to being totally connected. This is broadly consistent with finding of geographic segregation in the phylogenetic data.