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.

Controversy has long existed over whether plant disease epidemics spread with constant or with increasing velocity. We conducted largescale field experiments with wheat stripe rust at Madras and Hermiston, Oregon, where natural stripe rust epidemics were rare, to test these competing models. Data from three location-years were available for analysis. A susceptible winter wheat cultivar was planted in pure stand and also in a 1:4 or 1:1 mixture with a cultivar immune to the stripe rust race utilized in the experiments. Plots were 6.1 m wide and varied from 73 to 171 m in length. A 1.5 by 1.5-m focus was inoculated in either the center (2001) or upwind of the center (2002 and 2003) of each plot. Disease severity was evaluated weekly throughout the epidemics in each plot at the same points along a transect running upwind and downwind from the focus. Velocity of spread was calculated from the severity data and regressed separately on time and on distance from the focus. In all location-years and treatments, and at all levels of disease severity, velocity consistently increased linearly with distance, at an average rate of 0.59 m/week per m, and exponentially with time. Further, across epidemics there was a significant positive relationship between the apparent infection rate, r, and the rate of velocity increase in both space and time. These findings have important implications for plant diseases with a focal or partially focal character, and in particular for the effectiveness of ratereducing disease management strategies at different spatial scales.

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.

Hosts of soybean rust (Phakopsora pachyrhizi) are sensitive to low temperatures, limiting this obligate parasite in the United States to overwintering sites in a restricted area along the Gulf Coast. This temperature sensitivity of soybean rust hosts allowed us to study spatial spread of epidemic invasions over similar territory for seven sequential years, 2005–2011. The epidemic front expanded slowly from early April through July, with the majority of expansion occurring from August through November. There was a 7.4-fold range of final epidemic extent (0.4 to 3.0 million km2) from the year of smallest final disease extent (2011) to that of the largest (2007). The final epidemic area of each year was regressed against epidemic areas recorded at one-week intervals to determine the association of final epidemic extent with current epidemic extent. Coefficients of determination for these regressions varied between 0.44 to 0.62 during April and May. The correlation coefficients varied between 0.70 and 0.96 from early June through October, and then increased monotonically to 1.0 by year's end. Thus, the spatial extent of disease when the epidemics began rapid expansion may have been a crucial contributor to subsequent spread of soybean rust. Our analyses used presence/absence data at the county level to evaluate the spread of the epidemic front only; the subsequent local intensification of disease could be strongly influenced by other factors, including weather.