Focus expansion and stability of the spread parameter estimate of the power law model for dispersal gradients

Peter S Ojiambo; David H Gent; Lucky K Mehra; David Christie; Roger Magarey

doi:10.7717/peerj.3465

Focus expansion and stability of the spread parameter estimate of the power law model for dispersal gradients

PeerJ. 2017 Jun 20:5:e3465. doi: 10.7717/peerj.3465. eCollection 2017.

Authors

Peter S Ojiambo¹, David H Gent², Lucky K Mehra¹, David Christie³, Roger Magarey³

Affiliations

¹ Center for Integrated Fungal Reserach, Department of Entomology and Plant Pathology, North Carolina State University, Raleigh, NC, USA.
² U.S. Department of Agriculture-Agricultural Research Service, Forage Seed and Cereal Reserach Unit, Department of Botany and Plant Pathology, Oregon State University, Corvallis, OR, USA.
³ Center for Integrated Pest Management, North Carolina State University, Raleigh, NC, USA.

Abstract

Empirical and mechanistic modeling indicate that pathogens transmitted via aerially dispersed inoculum follow a power law, resulting in dispersive epidemic waves. The spread parameter (b) of the power law model, which is an indicator of the distance of the epidemic wave front from an initial focus per unit time, has been found to be approximately 2 for several animal and plant diseases over a wide range of spatial scales under conditions favorable for disease spread. Although disease spread and epidemic expansion can be influenced by several factors, the stability of the parameter b over multiple epidemic years has not been determined. Additionally, the size of the initial epidemic area is expected to be strongly related to the final epidemic extent for epidemics, but the stability of this relationship is also not well established. Here, empirical data of cucurbit downy mildew epidemics collected from 2008 to 2014 were analyzed using a spatio-temporal model of disease spread that incorporates logistic growth in time with a power law function for dispersal. Final epidemic extent ranged from 4.16 ×10⁸ km² in 2012 to 6.44 ×10⁸ km² in 2009. Current epidemic extent became significantly associated (P < 0.0332; 0.56 < R² < 0.99) with final epidemic area beginning near the end of April, with the association increasing monotonically to 1.0 by the end of the epidemic season in July. The position of the epidemic wave-front became exponentially more distant with time, and epidemic velocity increased linearly with distance. Slopes from the temporal and spatial regression models varied with about a 2.5-fold range across epidemic years. Estimates of b varied substantially ranging from 1.51 to 4.16 across epidemic years. We observed a significant b ×time (or distance) interaction (P < 0.05) for epidemic years where data were well described by the power law model. These results suggest that the spread parameter b may not be stable over multiple epidemic years. However, b ≈ 2 may be considered the lower limit of the distance traveled by epidemic wave-fronts for aerially transmitted pathogens that follow a power law dispersal function.

Keywords: Dispersive waves; Epidemic spread; Inverse power law; Long-distance dispersal; Parameter stability.

Grants and funding

This work was supported by Hatch Funds from the North Carolina Agriculture Experiment Station for Project NC02432, U.S. Department of Agriculture CRIS project 5358-21000-046-00D, and National Institutes of Food and Agriculture grant number 2016-68004-24931. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.