Classic approaches to modeling biological invasions predict a "traveling wave" of constant velocity determined by the invading organism's reproductive capacity, generation time, and dispersal ability. Traveling wave models may not apply, however, for organisms that exhibit long-distance dispersal. Here we use simple empirical relationships for accelerating waves, based on inverse power law dispersal, and apply them to diseases caused by pathogens that are wind dispersed or vectored by birds: the within-season spread of a plant disease at spatial scales of <100 m in experimental plots, historical plant disease epidemics at the continental scale, the unexpectedly rapid spread of West Nile virus across North America, and the transcontinental spread of avian influenza strain H5N1 in Eurasia and Africa. In all cases, the position of the epidemic front advanced exponentially with time, and epidemic velocity increased linearly with distance; regression slopes varied over a relatively narrow range among data sets. Estimates of the inverse power law exponent for dispersal that would be required to attain the rates of disease spread observed in the field also varied relatively little (1.74-2.36), despite more than a fivefold range of spatial scale among the data sets.

Field data on disease gradients are essential for understanding the spread of plant diseases. In particular, dispersal far from an inoculum source can drive the behavior of an expanding focal epidemic. In this study, primary disease gradients of wheat stripe rust, caused by the aerially dispersed fungal pathogen Puccinia striiformis, were measured in Madras and Hermiston, OR, in the spring of 2002 and 2003. Plots were 6.1 m wide by 128 to 171 m long, and inoculated with urediniospores in an area of 1.52 by 1.52 m. Gradients were measured as far as 79.2 m downwind and 12.2 m upwind of the focus. Four gradient models-the power law, the modified power law, the exponential model, and the Lambert's general model-were fit to the data. Five of eight gradients were better fit by the power law, modified power law, and Lambert model than by the exponential, revealing the non-exponentially bound nature of the gradient tails. The other three data sets, which comprised fewer data points, were fit equally well by all the models. By truncating the largest data sets (maximum distances 79.2, 48.8, and 30.5 m) to within 30.5, 18.3, and 6.1 m of the focus, it was shown how the relative suitability of dispersal models can be obscured when data are available only at a short distance from the focus. The truncated data sets were also used to examine the danger associated with extrapolating gradients to distances beyond available data. The power law and modified power law predicted dispersal at large distances well relative to the Lambert and exponential models, which consistently and sometimes severely underestimated dispersal at large distances.

Abstract. Understanding landscape effects on disease spread can contribute to the prediction and control of epidemic invasions. We conducted large-scale field experiments with wheat stripe rust, which is caused by a wind-dispersed rust fungus. Three landscape heterogeneity variables were altered: host frequency (mixtures of susceptible and resistant plants), host patch size (different plot sizes), and size of initial disease focus (attained by artificial inoculation). Assessments of disease prevalence at different distances from the disease foci were used to quantify effects of landscape variables. We expected that a low frequency of susceptible hosts, small host patch sizes, and small initial disease foci would reduce secondary inoculum levels and thus suppress disease spread. Low host frequency and small initial disease foci greatly reduced epidemic spread. We did not detect an effect of host patch size on disease spread, though artificial inoculations did not allow us to measure the potential for small patches to escape infection under conditions of random deposition of initial inoculum. Our results suggest that, for diseases epidemiologically similar to wheat stripe rust, epidemic invasions may be suppressed by decreasing host frequency, limiting the size of initial outbreak foci, and applying control measures soon after epidemic establishment.

Disease spread has traditionally been described as a traveling wave of constant velocity. However, aerially dispersed pathogens capable of long-distance dispersal often have dispersal gradients with extended tails that could result in acceleration of the epidemic front. We evaluated empirical data with a simple model of disease spread that incorporates logistic growth in time with an inverse power function for dispersal. The scale invariance of the power law dispersal function implies its applicability at any spatial scale; indeed, the model successfully described epidemics ranging over six orders of magnitude, from experimental field plots to continental-scale epidemics of both plant and animal diseases. The distance traveled by epidemic fronts approximately doubled per unit time, velocity increased linearly with distance (slope ~(1/2)), and the exponent of the inverse power law was approximately 2. We found that it also may be possible to scale epidemics to account for initial outbreak focus size and the frequency of susceptible hosts. These relationships improve understanding of the geographic spread of emerging diseases, and facilitate the development of methods for predicting and preventing epidemics of plants, animals, and humans caused by pathogens that are capable of long-distance dispersal.

The velocity of expansion of focal epidemics was studied using an updated version of the simulation model EPIMUL, with model parameters relevant to wheat stripe rust. The modified power law, the exponential model, and Lambert's general model were fit to primary disease gradient data from an artificially initiated field epidemic of stripe rust and employed to describe dispersal in simulations. The exponential model, which fit the field data poorly (R (2) = 0.728 to 0.776), yielded an epidemic that expanded as a traveling wave (i.e., at a constant velocity), after an initial buildup period. Both the modified power law and the Lambert model fit the field data well (R(2) = 0.962 to 0.988) and resulted in dispersive epidemic waves (velocities increased over time for the entire course of the epidemic). The field epidemic also expanded as a dispersive wave. Using parameters based on the field epidemic and modified power law dispersal as a baseline, life cycle components of the pathogen (lesion growth rate, latent period, infectious period, and multiplication rate) and dispersal gradient steepness were varied within biologically reasonable ranges for this disease to test their effect on dispersive wave epidemics. All components but the infectious period had a strong influence on epidemic velocity, but none changed the general pattern of velocity increasing over time.