Host-pathogen time series data in wildlife support a transmission function between density and frequency dependence

Abstract

A key aim in epidemiology is to understand how pathogens spread within their host populations. Central to this is an elucidation of a pathogen's transmission dynamics. Mathematical models have generally assumed that either contact rate between hosts is linearly related to host density (density-dependent) or that contact rate is independent of density (frequency-dependent), but attempts to confirm either these or alternative transmission functions have been rare. Here, we fit infection equations to 6 years of data on cowpox virus infection (a zoonotic pathogen) for 4 natural populations to investigate which of these transmission functions is best supported by the data. We utilize a simple reformulation of the traditional transmission equations that greatly aids the estimation of the relationship between density and host contact rate. Our results provide support for an infection rate that is a saturating function of host density. Moreover, we find strong support for seasonality in both the transmission coefficient and the relationship between host contact rate and host density, probably reflecting seasonal variations in social behavior and/or host susceptibility to infection. We find, too, that the identification of an appropriate loss term is a key component in inferring the transmission mechanism. Our study illustrates how time series data of the host-pathogen dynamics, especially of the number of susceptible individuals, can greatly facilitate the fitting of mechanistic disease models.

Fig. 1.

Time series data of cowpox infection status in one natural population of field voles. We show total population size [N(t), red] and the population sizes of susceptible [S(t), blue] and infected [I(t), black] individuals; for brevity, we have omitted the data on recovered individuals. We overlay the mean and 95% credibility intervals of estimated I(t) for our best-fit model (Eq. 5) by using the mean best-fit parameters (q = 0.62, β = 0.18, M = 7.1, Δ = 0.69, A = 0.67, gray). The qualitative patterns are very similar for all four populations studied (see SI Text, Fig. S1).

Fig. 2.

The shape and effects of nonlinear contact rate–abundance relationships. (A) Example realizations of the relationship between contact rate and population size for different values of q in Eq. 2, and the mean and 95% credibility intervals supported by our best model (Eq. 5). The lines have all been scaled to have the same mean value for β_qD = κυ/A, which was set equal to the mean value from our best-fit model. (B) Qualitative demonstration of the potential importance of q to the dynamics predicted by our previously published vole-disease model (24). N(T) is the total population size at time T, the start of the reproductive season. See SI Text and Fig. S5 for full details of the model.

Fig. 3.

Parameter estimation results for the best-fit model of cowpox infection in field vole populations (Eq. 5). (A) Probability density function for estimated q (see Methods). See Fig. S2 for probability distributions of the other parameters. (B) Example realization of the best-fit seasonal loss rate (M = 7.1, Δ = 0.69, A = 0.67), with the time series data of infected population density overlaid [taking mean of log₁₀(I(t)) of the four sites and for the four complete years of time series data]. (C) Example results of making the parameters q and β functions of time when fitting Eq. 5 to the data, alongside the mean annual variation in log₁₀(total vole density), averaged across all sites.