Plant diseases pose a significant threat to food security, and fungal plant pathogens can be particularly damaging due to enormous evolutionary potential. Two major control strategies for fungal pathogens include chemical control with fungicides and use of disease-resistant cultivars. Both strategies are threatened by pathogen evolution leading to fungicide resistance and loss of cultivar efficacy (‘cultivar breakdown’). Both fungicide resistance and cultivar breakdown can be characterised as qualitative or quantitative, depending on the genetic basis for resistance/breakdown. Mathematical models have an important role in understanding fungicide resistance and cultivar breakdown, and modelling studies offer numerous advantageous over experimental studies.

We present a model of qualitative fungicide resistance and one of quantitative fungicide resistance and cultivar breakdown. The model of qualitative resistance extends a model from the literature, while the quantitative resistance model is entirely novel. Both models are parameterised to address Septoria, the most prevalent disease of wheat. Although qualitative resistance has been widely studied, many single-site fungicides (i.e. those challenged by qualitative resistance) face widespread resistance, meaning that quantitative resistance is increasingly important. However, the mechanisms underlying quantitative resistance/breakdown are more complex and suitable data for model fitting is harder to source. We present the first model of quantitative resistance to be fitted to field data for both fungicide resistance and cultivar breakdown.

Fungicide mixtures can help delay resistance. Although many studies focus on mixtures of fungicides to which resistance is qualitative, the optimal strategy is not characterised if the initial resistance frequencies (RFs) to two fungicides in a mixture differ. Past work showed that equal selection for single resistant strains in the first year was the optimal strategy when initial RFs were equal but did not consider when the initial RFs differ. We show that this strategy is often sub-optimal if the initial RFs differ and present an alternative strategy based on equalising RFs in the breakdown year. We test the robustness of this strategy to changes in parameters controlling fungicide efficacy and pathogen epidemiology, including between-season pathogen sexual reproduction. Previous modelling studies of Septoria neglected pathogen sexual reproduction, for simplicity, despite evidence that sexual reproduction is an important part of Septoria’s life cycle.

We use the quantitative resistance model to determine how the number of fungicide applications per year affects resistance development, disease severity and yield. We consider how the optimal strategy varies depending on the time-frame of interest, before exploring how cultivar and fungicide control can be optimally combined to control the pathogen whilst minimising degradation of host and fungicide. We explore the effect of fungicide dose and compare the recommendations from the quantitative resistance model to those from the qualitative resistance model.

Most fungicide resistance studies consider strategies which are fixed in time – i.e. rely on the same tactic used every year until failure. We present a flexible approach to optimise time-variable strategies, based on dynamic programming. This allows us to address an otherwise computationally infeasible problem to find the optimal strategy when time-variable strategies are permitted. We compare the improvement offered by the optimal time-variable strategy to the optimal fixed-time strategy.