Supplementary Materials from Trade-off between disease resistance and crop yield: a landscape-scale mathematical modelling perspective

The deployment of crop varieties that are partially resistant to plant pathogens is an important method of disease control. However, a trade-off may occur between the benefits of planting the resistant variety and a yield penalty, whereby the standard susceptible variety outyields the resistant one in the absence of disease. This presents a dilemma: deploying the resistant variety is advisable only if the disease occurs and is sufficient for the resistant variety to outyield the infected standard variety. Additionally, planting the resistant variety carries with it a further advantage in that the resistant variety reduces the probability of disease invading. Therefore, viewed from the perspective of a grower community, there is likely to be an optimal trade-off and thus an optimal cropping density for the resistant variety. We introduce a simple stochastic, epidemiological model to investigate the trade-off and the consequences for crop yield. Focusing on <i>SIR</i> epidemic dynamics, we use the final size equation to calculate the surviving host population in order to analyse the yield, an approach suitable for rapid epidemics in agricultural crops. We identify a single compound parameter, which we call the efficacy of resistance and which incorporates the changes in susceptibility, infectivity and durability of the resistant variety. We use the compound parameter to inform policy plots that can be used to identify the optimal strategy for given parameter values when an outbreak is certain. When the outbreak is uncertain, we show that for some parameter values planting the resistant variety is optimal even when it would not be during the outbreak. This is because the resistant variety reduces the probability of an outbreak occurring.