Single strain model (fungicide only).

We first consider a single pathogen strain, for the purposes of explanation, before introducing the full model. We assign this pathogen strain a ‘fungicide trait value’ k, which describes its fitness with respect to the fungicide. The trait value k is between 0 and 1, with 0 being fully sensitive (i.e. the pathogen is fully controlled by a fungicide application, at least directly after it is sprayed) and 1 being fully resistant (i.e. completely unaffected by a fungicide application). We use ‘strain k’ as a shorthand for ‘the pathogen strain with trait value k’.

The density of susceptible host at time t is denoted by S(t), and the density of host tissue infected by strain k at time t is denoted by I(k, t). The rate of loss of healthy (susceptible) leaf tissue is given by: (1) where g(t), Γ(t) and β(k, t) are host growth, host senescence and the pathogen infection rate respectively, which are all described below.

The rate of increase of host tissue infected by strain k is given by: (2)

We normalise the tissue quantities, which correspond to leaf area, so that the total density of healthy host tissue (S) and infected host tissue (I) is 1 when host (leaf) growth is complete and before host senescence begins. ‘Disease severity’ (D) is the percentage of leaf tissue/leaf area that is infected by any pathogen strain. Define (3) as the total density of tissue infected by any strain. Although we only have one strain at this stage, we will in general require this integral. Then disease severity is calculated according to: (4)

The pathogen infection rate, β(k, t), depends on the pathogen response to the fungicide. [26, 28, 46] use a fungicide dose response of the form: (5) where ϵ is the proportionate effect of the fungicide on strain k, ω is the asymptote of the dose response curve and θ(k) is the curvature of the dose response curve with respect to the pathogen, which here also depends on the strain k. We choose the asymptote parameter, ω, to equal 1, as was used for high risk fungicide in [28]. This means that the infection rate at theoretical infinite dose is 0.

The relationship between k and θ(k) is that θ(k) = −log(k). Then, since we have assumed the asymptote ω = 1, we find: (6)

This means that the proportionate effect of the fungicide on the infection rate of this pathogen strain is k when the concentration is 1. In general the effect depends on the fungicide concentration, which is assumed to decay exponentially with time t, at rate Λ, after each application as in [26, 28]. The fungicide concentration equation is: (7) where C is initially 0 but instantaneously increases by 1 every time the fungicide is applied. We choose a decay rate of Λ = 0.5 × (6.91 + 11.1) × 10⁻³ = 9.005 × 10⁻³ degree-days^-1, which is the average of the two fungicide decay rates in [24, 28]. The decay rates in [24, 28] were based on the half lives reported in [25] and the relation Λ = −log(0.5)/T_0.5, where T_0.5 is the half life. For ‘azoxystrobin’, the half life times reported in [25] were 6.9, 3.6, 4.8 and 1.2 days for UK trial data from Edinburgh 2002, Inverness 2002 and Terrington 2002 and 2003, respectively. We follow [26, 28] in assuming perfect exponential decay in fungicide concentration. In general the decay rate parameter will vary depending on the fungicide.

The fungicide effect acts as a multiplier on the underpinning infection rate β₀. When the concentration is 0, the effect is 1, so when there is no fungicide present all strains behave identically. When the fungicide is applied the infection rates of strains with lower k values are suppressed more than that of strains with higher k values: (8)

The fungicide can be applied at up to three time points: T₁ (growth stage 32), T₂ (growth stage 39) and T₃ (growth stage 61) [28, 36]. A one treatment programme is applied at T₂, a two treatment programme is applied at T₂ and T₃, a three treatment programme is applied at T₁, T₂ and T₃ (Table 2).

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Table 2. Times that are required in running the model.

Here we use Zadoks’ growth scale [47]. The times T₁, T₂ and T₃ are as in [28, 36]. The start time is growth stage 32, corresponding to the emergence of leaf 3 (T₁). The final time t_end (growth stage 75) is interpolated from the value at growth stage 61 (T_GS61 = T₃ = 2066 degree-days) and growth stage 87 (T_GS87 = 2900 degree-days) from [28, 36]. This assumes that the temperature remains approximately constant in the late stages of the season. We also round the values to the nearest integer. Growth stage 75 was chosen for the end time because it was the time in Dataset D which links yield to disease severity (Table 3). Any host or pathogen growth before t_start can be scaled into the amount of inoculum I₀ in the model at t_start; the procedure used to fit the initial inoculum parameter, I₀, is reported in S1 Text.

https://doi.org/10.1371/journal.pcbi.1010969.t002

The host is assumed to grow within the season before a period of senescence. We use the same form for growth and senescence as in [25, 28]. The growth equation is: (9) where A(t) is the total amount of healthy and infected tissue. The value for r is 0.0126 degree-days^-1 as in [24, 28]. We non-dimensionalised so that the maximum leaf area is 1, as in [46]. The initial total quantity of host tissue is H₀ = 0.05/4.1, using the values for initial host area and maximum host area from [24].

The senescence equation, which controls the time-dependent rate at which leaf senescence occurs, is: (10) This is the same function / parameterisation as in [28], and originally derived in [24]. See Table 2.

Model accounting for diversity within the pathogen population (fungicide only).

The population model, which accounts for diversity within the pathogen population (Table 1) addresses a continuous spectrum of pathogen strains with trait values (k) varying from 0 to 1. It generalises the single strain model to an entire population of different pathogen strains with differing initial densities. Healthy host tissue S gets infected by different components of the pathogen population, at different rates when the fungicide is present. We assume that the different strains behave identically in the absence of control (i.e. we neglect modelling fitness costs). Ideally we would incorporate fitness costs into the model, but to do so would require a complete data set describing the infection rates for every different pathogen trait value k (i.e. each pathogen strain).

Pathogen mutation is considered, so a small proportion of pathogen offspring has a different trait value to its parent. We assume mutation events occur with probability p, and we choose a Gaussian mutation kernel with mutation scale σ², following [35]. The model in [35] is of cultivar resistance rather than fungicide resistance, so we assume that mutation for cultivar and fungicide resistant traits acts in the same way. We assume that the rate and scale of mutation are constants, i.e. do not vary through the season and they do not depend on the trait value of the parent. This means that mutation acts in the same way for highly resistant strains as for highly sensitive strains.

For a strain j, we describe the probability P(k, j) of its offspring taking trait value k: (11) Here δ(x) is the Kronecker delta (which is 1 when x = 0 and 0 otherwise). All strains must have trait values within [0, 1], so any offspring that are predicted to take negative values are given trait value 0 and any that are predicted to take values greater than 1 are given trait value 1.

The per capita growth rate of offspring with trait value k is given by the integral over all possible parents with trait value j: (12) The total per capita growth rate of new infection at time t is given by the integral over all new strains: (13) This means fastest growing strains have relatively higher contribution to new infections.

The S equation becomes: (14) The I equation, for each strain with trait value k, becomes: (15)

The total amount of initial inoculum is fixed at the same value each year, although evolutionary changes mean that its strain composition varies over years. This choice means the integral over all pathogen strains equals I₀ × H₀ at the start of each modelled season (t = T₁, Table 2), where H₀ is the initial amount of host tissue.

Incorporating host plant resistance (full model).

A similar modelling approach can be used to explore how the pathogen population adapts to resistant cultivars. We introduce a ‘host trait’ l. The main difference between the host trait and the fungicide trait in the model is that the host protection is assumed to act in the same way throughout the season (no time-variation), whereas fungicide control depends on the fungicide concentration which depends on time. The value of k is as before, whereas the value of l corresponds to the host effect at any time: (16) Higher host trait values give higher infection rates corresponding to increased pathogen fitness (similarly to the fungicide trait values).

Mutation is assumed to act in a similar way to before, with both k and l changing, again on a [0, 1] scale. The per capita growth rate of the pathogen strain with fungicide trait k and host trait l is given by the integral over all possible parent strains with fungicide trait j and host trait m: (17) Host and fungicide mutations are assumed to act independently of each other. This assumption should hold for host/fungicide combinations where different genes control fitness relative to the host/fungicide.

The total per capita growth rate of infection is given by: (18)

The model equations become: (19) (20)

Notes on model implementation.

When solving the model, we discretise the integral in Eqs 14 and 19. We use at least 100 values for k (and l if using the full model) in each model run. The ‘discretisation number’ is specified in the parameter values in each figure caption, denoted by n_k for the number of k values (and n_l for the number of l values if applicable). We typically chose larger values for n_k when using the fungicide only model, since the full model was slower for large values of n_k and n_l (because the discretisation was then over a grid of n_k × n_l values rather than a one dimensional spectrum of n_k values).

The fungicide concentration equation (Eq 7) can be solved for each spray programme and written explicitly. For example, the concentration of a 3 spray programme is given by: (21) where (22)

The form is similar for the 1 and 2 spray programmes, except the term is omitted for the 1 and 2 spray programmes and the is omitted for the 1 spray programme. In our implementation of the model we use this form rather than solving the differential equation each time the model is run.

Initial inoculum.

We fitted the initial amount of inoculum at the start of each season using Dataset A (see Table 3 and also S1 Text). We assumed this initial level was constant over seasons, with differences in disease pressure from season to season captured entirely by the infection rate (β₀), which varies over seasons in our model. The fitted value of I₀ was used for all model runs unless stated otherwise. Dataset A was collected in Denmark from seven locations across three years. The trials reported in Dataset A were conducted by Aarhus University and the knowledge centre for agriculture, SEGES.

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Table 3. Datasets used in model fitting.

https://doi.org/10.1371/journal.pcbi.1010969.t003

Infection rate.

To find the distribution of infection rates β₀, we used data from cultivar trials which were run from 1996 until 2017 (Dataset B). The trials from Dataset B were conducted by the knowledge centre for agriculture, SEGES and the Danish Institute of Technology. The data set includes data about disease severity at growth stage 75 on untreated plots, depending on location and cultivar.

We filtered the raw trial data to consider plots that were untreated by fungicide and excluding those with cultivar mixtures. This is because we were interested in the level of disease in the absence of control measures. We also filtered to only include high pressure locations (higher than 6 on a 1–10 scale), because we were most interested in testing disease management strategies in the regions where disease is the most damaging. Effective disease management is even more important in these locations than in lower disease pressure locations. Note that due to environmental variation, the high disease pressure locations contained a wide range of disease severities from very low values in some years/locations to very high values in others.

We grouped the data by location, cultivar and year, before finding the mean severity score for each location/cultivar/year combination. For each year/location, we found the worst performing cultivar and assumed that this mean severity is representative of the average severity in the absence of control (i.e. that the pathogen has reached peak fitness for that cultivar). We chose ‘Mariboss’ as the disease-resistant cultivar to analyse. It was one of only 4 cultivars which appeared for 10 or more years in the dataset, and it offered a relatively good level of control initially. We filtered the dataset to only include location/year combinations where ‘Mariboss’ was grown. All data for the worst performing cultivar (or cultivars if multiple cultivars had an equally poor average) in each year and location that ‘Mariboss’ was found were used as the values for severity in the absence of control.

We found that a truncated exponential was a good fit to these severity data. The probability density function of the truncated exponential is: (23) Fitting the truncated exponential to the data allowed us to sample many values for final disease severities in the absence of control, which we used to generate infection rate (β₀) values. The fitted value of λ is in Table 4. This was done by numerically solving the ODEs using the value for I₀ as found above. For each severity, we used a numerical optimiser (‘minimize’ from the open-source Python package ‘scipy.optimise’) to find the corresponding value of β₀. Since this fit was for the absence of control, it meant the entire population behaved like a single pathogen strain with infection rate β₀. This meant that the output had no k or l dependence, so the initial host and fungicide distributions and the mutation parameters had no effect on the output. Therefore we only needed the host growth/senescence parameters already described and the value for I₀ already found.

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Table 4. Parameter values used in the model (to 3 significant figures).

The parameter values were found in the model fitting process using the data described in Table 3. The sampled infection rates are generated from a truncated exponential distribution with maximum 100 and parameter λ (see Eq 23).

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Mutation.

The mutation kernel has two parameters, one for the proportion of offspring which have a different trait value and one for the scale of mutation, which affects how different these trait values are (Eq 11). We assume that the proportion and scales of mutation are the same for fungicide and host traits. We had no data to justify an alternative choice, but reassuringly find via a sensitivity scan that changing the mutation proportion and then re-fitting the model has no effect on the model results, and neither does excluding mutation altogether (S3 Text).

The value for the mutation proportion we use is based on estimates for the total number of spores generated in a growing season and the number of spores per season carrying adaptive mutations for fungicides and resistant cultivars [48]. See S2 Text.

To identify an appropriate value for the mutation scale, we first find an upper bound for it. In general, the loss in control efficacy is caused by a combination of selection on strains that are already present in the population as well as on strains that arise due to mutation. This means we can find an upper bound from the value for mutation scale for which the loss in efficacy could be attributed to mutation alone with no initial standing variation (S2 Text). We then arbitrarily choose 10% of this value to the be the value for the mutation scale used in the model, but we test this assumption in S3 Text and find that the model output is very similar for a wide range of choices of mutation scale.

Fungicide trait distribution.

We fitted an initial distribution for the fungicide trait to reflect the fact there is some initial variation in pathogen fitness relative to the fungicide. To find this distribution we used information about control efficacy derived from disease severity data (Dataset C) from UK disease trials from 2001–2018 [14, 49]. The level of initial control in the data is linked to the mean of the pathogen distribution, because the average pathogen trait value determines how effective control is. The rate of decline in control in the data is linked to the variance of the pathogen distribution, because more variance in the pathogen population leads to a faster breakdown in control. [49] fitted curves to these data, of the form (24) to 5 different dosages D, where S_rel is related to control by control = 100 − 100S_rel, so that control takes values between 0% and 100%. They reported the mean value and standard error for the two parameters R_D and κ in each year, but did not report the actual control values (nor the distribution of control values). The parameter R_D relates to the maximum reduction in severity at theoretical infinite dose and κ is a curvature parameter.

We used a t-distribution [50] with 3 degrees of freedom to drive a procedure based on the reported standard errors to sample 500 values per year for R_D and κ. We then used these values to generate 500 control values each year for use when fitting the model (which helps give an idea of the uncertainty in the measurements and therefore a better measure of model performance). We used a t-distribution because the sample sizes were small and the true standard deviation of the parameters was unknown. We filtered out any generated control values less than 0 and greater than 100 and resampled to replace them until we had 500 values in every year. The bias introduced by this filtering process was small, changing the mean value by 0.83% on average and 2.3% at most across the 18 years in the dataset. Using the uncertainty rather than just a single fitted value per year allowed a clearer assessment of how the model performed on the training and the test set given the variability in the data, and meant that years with lower uncertainty would have a larger influence on the optimal model parameters found in the model fitting process.

We use a gamma distribution to characterise the initial distribution of fungicide curvature parameter, θ = −log(k). The curvature parameter θ takes values in (0, ∞). The gamma distribution is a continuous, two parameter distribution with support on (0, ∞). We use the shape and rate parameterisation. It provides a flexible distribution with support on the appropriate range taken by θ. This allows us to compare the mean of the fitted distribution for θ with those found in the literature in [26, 28, 46]. We found the optimal values for the two-parameter gamma distribution (i.e. the values that minimised the sum of squared distance between the data and the model output) using the open-source Python package Optuna (v2.10.0) to find the optimal values, sampling our search-space using their implementation of a so-called ‘tree-structured Parzen estimator’ (see [51]), which is more efficient than random sampling since it uses information from previous samples (i.e. a Bayesian approach) to reduce the time spent evaluating ‘bad’ hyperparameters.

The default untreated disease severity reported by [49] was 37%, corresponding to the top 10% of disease severities measured in untreated plots, averaged across all years of trials. [49] did not report a separate value for each year nor standard error terms, so we used this single value for the disease severity in the absence of control in each year. The cultivars used in the experiment were selected for their susceptibility to septoria [49], so we assume that they offer negligable levels of control. We used an infection rate (β) value corresponding to the 37% value for every year.

We split our data into a training and a test set, using 2/3 of the data by year for the training set (the training set was from 2001–2012, and the test set from 2013–2018).

Host trait distribution.

In each year and location (Dataset C), we found the level of control offered by the disease-resistant cultivar ‘Mariboss’ relative to the worst performing cultivar, using the following formula: (25) where S_M is the mean severity for ‘Mariboss’ and S_W is the mean severity for the worst cultivar in that particular year y and location L, where severity is measured on leaf 2.

We use a beta distribution to describe the initial distribution of host trait values l. Beta distributions are continuous, two parameter distributions defined on the range (0, 1). They take a wide variety of shapes and commonly appear in population genetics contexts [52], and have support on the appropriate parameter range for l: (0, 1). The host trait value l is defined on this range and so the beta distribution was an appropriate choice (in contrast to the fungicide trait parameter, for which a positive value was the only constraint since we fit a distribution of curvature values θ rather than trait values k).

Again we use Optuna to optimise the values of the two distribution parameters, allowing us to find the pair of values for these parameters which gave the best fit between the model output and the data. This was characterised in terms of a loss function (Eq 27), which was the squared distance between the model and the data, weighted so that the control values that were informed by more data points carried greater importance to the model fit.

Let M_y be the control prediction from the model with no fungicide applications in year y, i.e. the control value obtained from comparing the disease severity (S_N) with no control (so β(k, l, t) = β₀ for all k, l, t), and the disease severity (S_C) with cultivar control but not fungicide control (so β(k, l, t) = β₀l for all k, l, t). Then M_y is given by: (26) analogously to how C_y,L was calculated (Eq 25).

The cultivar control observed in the data in year y and location L is C_y,L. We have multiple locations in each year, up to N_y in year y. Then let n_y,L denote the minimum of the number of trials from the worst performing cultivar and of ‘Mariboss’ in the each year/location combination. (27) Again we separate the data into a training and test set with a 2/3 split by year (the training set was 2007–2012, and the test set was 2013–2016).

Yield.

To analyse the relationship between disease severity and yield, we used trial data from Soenderborg (Dataset D), a high disease pressure region of southern Denmark. The data were collected in 2019, and reports disease severity on leaf 2 and yield for a variety of cultivars, including ‘Sheriff’, ‘Kalmar’ and ‘Informer’, meaning that our yield relationship is effectively averaged across different cultivars. Although we expect that yield will vary depending on the cultivar, the dataset was too small to further restrict it to a single choice of cultivar. The dataset corresponds to trials from Soenderborg, but we assume that the response of wheat yields to a particular level of disease is the same regardless of location. Rather than arbitrarily choose a functional form to model the yield-severity relationship, we used a generalised additive model (GAM) to find a smooth curve relating yield and severity as determined by the data. We used the open-source Python package ‘pyGAM’ to fit a monotonically decreasing GAM with 5 splines. The choice of 5 splines was arbitrary, but chosen so that there were enough splines to fit the data well but not so many that the model was overfitting. See S1 Data Data for a CSV file relating disease severity values to yield values.

Comparing numbers of fungicide applications.

We explore the effect of the number of fungicide applications per year on the pathogen population, disease severity and yield. The results shown in Figs 3 and 4 do not include disease-resistant cultivar control and consider the effect of fungicide applications only. Fig 3 shows the average result across an ensemble of runs, whereas Fig 4 shows the variability in outcomes from the ensemble.

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Fig 3. More fungicide applications per year leads to faster selection for resistant strains and to lower yield in the long term.

Median values for disease severity and yield from each year of the 10,000 simulations (A, B). Initially three fungicide applications per year gives the lowest severity and highest yield. Panels C-F relate to the ‘mean distribution’ across the 10,000 ensembles, and the mean value of this mean distribution in C. Due to more rapid selection for resistant strains with more fungicide applications (C-F), the one fungicide application per year strategy outperforms the two and three applications per year strategies in terms of both severity and yield by year 7. Here n_k = 500.

https://doi.org/10.1371/journal.pcbi.1010969.g003

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Fig 4. The optimal disease management strategy depends on the relevant timeframe.

In the left, middle and right columns we compare the performance of zero, one and three fungicide applications annually with the performance of two applications annually. Zero applications (no control) leads to higher severity (A) and lower yield (B) than two applications, although the difference reduces over time due to selection for resistant strains (C). Initially one application leads to increased severity (D) and decreased yield (E), but by year 7 one application is better than two due to reduced selection for resistant strains (F). Conversely, three fungicide applications initially leads to lower severity (G) and higher yield (H), but by year 5 this benefit is lost due to increased resistance (I). The black dots (panels D,E,G,H) denote when the median line crosses 0, i.e. when severity/yield changes so that 1 spray outperforms 2 / 2 sprays outperform 3. Here n_k = 400.

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Initially, a greater number of applications offers greater control than fewer (Fig 3A). The figure shows the median value for disease severity and yield in each year over an ensemble of 10,000 runs, each one using randomly sampled values (from our fitted distribution) for the infection rate β₀. It shows the mean density of the distribution at each discretised value for k, resulting in an ‘average’ distribution across the ensemble (Fig 3C, 3D, 3E and 3F). The strategies with more fungicide applications lead to faster selection for higher trait values, leading to a gradual shift in distribution (Fig 3D, 3E and 3F). This faster shift in distribution leads to the order in terms of disease severity and yield reversing such that 1 fungicide application per year outperforms 2 and 3 applications per year (Fig 3A and 3B). The higher the number of applications, the more rapidly the distributions shift, which can be shown most clearly by looking at the mean of the distributions (Fig 3C).

Initially the control offered by just one application is strong. This means that the improvement in severity and yield offered by further applications is marginal, with two and three applications performing comparably initially. Their performance remains similar in subsequent years due to a trade off—more applications means greater amount of time with fungicide control, but also faster selection for resistance. This trade off leads to two and three strategies behaving similarly, whereas by year 7 the reduced selection from the one fungicide application strategy outweighs the reduced control caused by fewer applications, and after year 7 the single fungicide application programme performs optimally for yield.

In the early years almost all of the pathogen density is found close to k = 0, i.e. highly sensitive strains. Due to strong selection (and also mutation) the distributions broaden over time (Fig 3D, 3E and 3F). Eventually, for 2 or 3 sprays, the distributions narrow around the highly resistant strains with high k values (Fig 3E and 3F). Given more years, this narrowing also occurs for a 1 spray programme. These distribution changes are caused by strong selection for highly resistant strains which are at very low density (or arise due to mutation). As these strains outcompete more sensitive strains, we find a range of pathogen strains coexisting briefly, but eventually the sensitive strains are so strongly suppressed that (given enough time/enough sprays) the population comprises almost exclusively of highly resistant strains (e.g. Fig 3F). Note that the distributions approximately match for the same number of sprays. For example, the 1 spray programme distribution looks very similar after 6 years as a 2 spray programme after 3 years, at which point both populations have been subjected to 6 fungicide applications (Fig 3D and 3E).

Ranges of outcomes under different numbers of fungicide applications.

To further explore the relative performance of differing numbers of fungicide applications, we compare the full distribution of outcomes under zero, one and three applications per year to two applications (Fig 4). We use 1,000 simulations with infection rates sampled randomly from our fitted distribution from Fig 2B. The same set of sampled infection rates were used for each application strategy.

Zero applications per year (no treatment) gives higher severity and lower yield (Fig 4A and 4B) than two applications per year, with particularly extreme differences in the early years whilst the fungicide still offers good control in the two application programme. Within 15 years the fungicide loses efficacy due to the shift in pathogen distribution (Fig 4C), meaning that the control offered is reduced.

One application per year initially gives higher severity and lower yield (Fig 4D and 4E) than two application per year, but by year 7 the median yield is higher and severity is lower due to two applications leading to higher levels of resistance (Fig 4F), as indicated by the summarised results in Fig 3.

Three applications per year initially reduces severity and increases yield relative to two (Fig 4G and 4H). However, by year 5 the increased resistance in the pathogen population (Fig 4I) leads to two applications outperforming three. Interestingly the yield difference in high disease pressure years (95th percentile) in years 5 and 6 are more extreme than the differences in the early years. That means that the more rapid increase in resistance from using 3 sprays can cause bigger yield losses than the improvement in control when the population is highly sensitive. The differences in performance are less exaggerated for the three versus two comparison than for the one versus two or zero versus two comparisons. This is because the difference in control (at least initially) between two and three sprays is smaller than between one and two, which is smaller than between zero and two.

Note that the variation in trait mean is smaller than in yield and severity (Fig 4). Selection increases as disease pressure (and correspondingly severity) increases. However, this increase is much smaller than the difference obtained by extra sprays (e.g. in Fig 4F and 4I the difference from 0 is much bigger than the width of the uncertainty bands). Interestingly in the one versus two and two versus three spray comparison, the difference in trait mean reduces eventually (Fig 4F and 4I). This is because the pathogen distributions eventually approach a trait mean of 1 for all of the strategies, excluding the zero spray strategy (see Fig 3D, 3E and 3F).

Introducing disease-resistant cultivars.

Fig 5 shows the results of introducing disease-resistant cultivar control, comparing the results of a two-spray programme with and without host-plant protection from the disease-resistant cultivar ‘Mariboss’. The model fit for ‘Mariboss’ is shown in Fig 5A, corresponding to the model run using the initial trait parameters. The optimal trait parameters are the ones which best matched the data showing decline in cultivar control in the absence of fungicide treatment. The disease-resistant cultivar helps reduce disease severity and increase yield (Fig 5B and 5C). The use of the resistant cultivar reduces the disease pressure, which protects the fungicide, as shown by the difference in fungicide trait mean between the strategies (Fig 5D). This protection offered by the host to the fungicide means that although the difference in yield and severity in year 1 is small, by year 7 the improvement in yield is much larger, particularly in high disease pressure years. This is because the control offered by the fungicide alone is excellent initially, so both strategies offer excellent control and the difference in yield is minimal. However, in later years the benefit of choosing a resistant cultivar becomes significant—greater than the initial benefit of using 2 sprays rather than 1 or 3 rather than 2 in Fig 4E and 4H.

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Fig 5. Disease-resistant cultivars reduce severity and protect the fungicide.

The optimal host trait initial distribution parameters were fitted using data from 2007 to 2012 (solid line, A). We used data from 2013 to 2016 as a test set (dotted line). When comparing a two application fungicide programme with and without the disease-resistant cultivar, the disease-resistant cultivar reduces the severity (B) and increases yield (C); in each case the response shown is the difference between two spray fungicide programmes with and without the resistant cultivar. Further, the fungicide develops resistance more slowly due to reduced disease pressure when the resistant cultivar is used (D). This means the benefit of using the resistant cultivar becomes greater as time goes on, and is particularly noticeable in high disease pressure years with yield improvements of up to 1.39 tons/ha in the 95th percentile in year 7. Here n_l, n_k = 100.

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Varying host control and regular replacement of disease-resistant cultivars.

To further explore the extent to which a disease-resistant cultivar protects the fungicide (and vice versa), we ran a parameter scan testing the performance of different hypothetical cultivars. We were interested to determine how changes in cultivar efficacy would impact the rate of fungicide breakdown. We allowed the initial mean of the host trait distribution to vary whilst keeping the shape parameter of the distribution fixed at the value from the model fitting. Some examples of the resulting distributions are shown in S4 Text. For simplicity, we used the median value β₀ = β_0,M (Table 4) in all examples, i.e. the underlying simulations did not account for environmental stochasticity.

For very short timescales (5 years), more fungicide applications gives improved yield (Fig 6A). However over longer timescales more applications leads to reduced yield for all but the strongest of our theoretical host cultivars (Fig 6B). This is because selection for fungicide resistance is stronger with more applications, although for particularly strong host protection higher numbers of fungicide applications are preferable due to increased mutual protection between host and fungicide. It should be noted that these particularly strong hosts have vastly stronger protection than the fitted cultivar ‘Mariboss’ (dotted vertical line).

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Fig 6. More effective cultivars protect fungicides more strongly and more frequent replacement leads to effective control for longer.

In panels A-E we vary the initial host trait mean whilst keeping the shape parameter fixed. The vertical dotted lines show the fitted trait mean value for ‘Mariboss’. Lower host trait means than this value represent hypothetical cultivars which are even more disease-resistant. Over a short timescale (5 years) more applications is preferable than only one (A). However, for longer timescales and initial trait values close to that of ‘Mariboss’ (dotted line) one application outperforms two and three. For a given number of fungicide applications per year, the stronger the host protection (lower trait values), the slower the fungicide resistance develops (C). For a given initial host strength, higher numbers of fungicide applications protects the host more effectively (D). One application keeps the yield above a threshold of 10.1 tons/ha for the longest for all initial host trait values greater than 0.56 (E). Each year value is an integer but we offset each line vertically by a small amount to show the ordering. If cultivars are replaced regularly then the yield remains above the threshold for much longer (F). In this panel the cultivars are assumed to have identical performance to ‘Mariboss’. Here n_l, n_k = 100.

https://doi.org/10.1371/journal.pcbi.1010969.g006

Strategies with more fungicide applications lead to faster selection for resistant strains and correspondingly a higher fungicide trait mean in year 10 (Fig 6C). Stronger host protection (lower initial host trait mean) leads to lower values of the fungicide trait mean due to the protection offered by the cultivar to the fungicide via reducing disease pressure. Similarly, the host trait mean after 10 years of the integrated management strategy is reduced when more fungicide applications are used (Fig 6D), since more intensive fungicide application programmes offer stronger protection to the host. This difference is less exaggerated for cultivars with efficacy comparable to ‘Mariboss’ than for hypothetical ones with extremely low initial trait means.

As an overall summary of the durability of the combined programme, we compared the number of years for which yield remains above a threshold of 10.1 tons/ha. This threshold value was chosen as a value slightly higher than the yield achieved with a resistant cultivar but no fungicide input (9.98 tons/ha). One application outperforms two and three for all host trait values above 0.56 (Fig 6E), although note that this result depends on the choice of threshold (for very high yield thresholds one fungicide application is not sufficient to achieve a high enough yield even in year one).

We ran a second scan where we assume that the cultivar is replaced at regular intervals, from every year to every N years (Fig 6F). In practice this would correspond to wheat breeders releasing new varieties to market. Each time the current cultivar is replaced with another cultivar with efficacy matching that of ‘Mariboss’, but to which the pathogen population has not yet adapted to cultivar resistance. This prolongs the effectiveness of the host protection and increases the protective effect offered by the host to the fungicide. For very regular replacement (replacement frequency 1 or 2), the number of years above the threshold is much higher than without replacement. However, it should be noted that the cost of producing new disease-resistant cultivars can be very large (and take many years), so very regular replacement may be unrealistic in practice. The jagged nature of the lines is caused by the hard 10.1 threshold—for example, there is a sharp jump between 10 years and 11 years for 3 sprays, because without any replacement the strategy would fail in 10 years but if replacement happens in the final (10th) year then the sudden improvement in cultivar control leads to several more years of acceptable yields. With a replacement frequency of 5 years we get replacement in the 5th and 10th years before failure, but with a replacement frequency of 6 years the strategy fails in the 11th year just before the second replacement, so the outcome is significantly worse.