Using digitized images of the three-dimensional, branching structures for root systems of bean seedlings, together with analytical and numerical methods that map a common susceptible-infected- recovered ('SIR') epidemiological model onto the bond percolation problem, we show how the spatially correlated branching structures of plant roots affect transmission efficiencies, and hence the invasion criterion, for a soil-borne pathogen as it spreads through ensembles of morphologically complex hosts. We conclude that the inherent heterogeneities in transmissibilities arising from correlations in the degrees of overlap between neighbouring plants render a population of root systems less susceptible to epidemic invasion than a corresponding homogeneous system. Several components of morphological complexity are analysed that contribute to disorder and heterogeneities in the transmissibility of infection. Anisotropy in root shape is shown to increase resilience to epidemic invasion, while increasing the degree of branching enhances the spread of epidemics in the population of roots. Some extension of the methods for other epidemiological systems are discussed.