Using epidemiological principles to explain fungicide resistance management strategies: why do mixtures outperform alternations?

Whether fungicide resistance management is optimised by spraying chemicals with different modes of action as a mixture (i.e. simultaneously) or in alternation (i.e. sequentially) has been studied by experimenters and modellers for decades, largely inconclusively. We use previously-parameterised and validated mathematical models of wheat septoria leaf blotch and grapevine powdery mildew to test which strategy provides better resistance management, using the total yield before fungicide-resistance causes disease control to become economically-ineffective (“lifetime yield”) to measure effectiveness. Lifetime yield is optimised by spraying as much low-risk fungicide as is permitted, combined with slightly more high-risk fungicide than needed for acceptable initial disease control, applying these fungicides as a mixture. This is invariant to model parameterisation and structure, as well as the pathosystem in question. However if comparison focuses on other metrics, for example lifetime yield at full label dose, either mixtures or alternation can be optimal. Our work shows how epidemiological principles can explain the evolution of fungicide resistance, and highlights a theoretical framework to address the question of whether mixtures or alternation provide better resistance management. Our work also demonstrates that precisely how spray strategies are compared must be given extremely careful consideration.


INTRODUCTION
in which T is the time of exposure to fungicide. Selection for resistance can therefore be Since each fungicide is sprayed twice as often when part of a mixture, we halve the dose The parameters ω (maximum effect) and θ (curvature) vary between fungicides. For 157 mixtures we assume independent action   1. There is no primary inoculum compartment, and epidemics are instead initiated by 241 a small amount of tissue being set to be latently infected (i.e. exposed) at the start 242 of each season.

243
2. An additional compartment is included in the model, accounting for leaves 244 developing Ontogenic resistance by virtue of age. For powdery mildew we model trifloxystrobin as the high-risk fungicide, and sulphur as 250 the low-risk fungicide, assuming both chemicals combine protectant and eradicant modes 251 of action (Reuveni, 2001). We assume flowering occurs at day 163 of the season 252 (Mammeri et al., 2014), and that spraying is done either side of this, two days before and 253 twelve days after flowering. This is a smaller number of sprays than normally used in 254 French viticulture (Calonnec et al., 2006;Savary et al., 2009), although it is within the 255 range leading to acceptable control (Gadoury et al., 2003).  al., 1984;Calonnec et al., 2004). Quantifying fine details of this would require a more 261 detailed treatment than appropriate here. However, there is a strong positive correlation 262 between leaf infection and berry infection (Calonnec et al., 2006;Delière et al., 2015). We

272
Initial disease control at full doses 273 For septoria and applying full doses of both fungicides, all three strategies lead to 274 adequate control in the first season (yield >95% of the disease-free yield) (Figure 3a).

277
The first-season yield is highest for mixtures because of the concave dose response 278 curve, with diminished returns from increased concentrations. Spraying half the dose 279 twice as often therefore leads to better control (recall full dose corresponds to half dose 280 in both sprays under mixture). The alternation high-low strategy slightly outperforms the low-high strategy in the first season since the high-risk fungicide is assumed more efficacious (maximum effect ωH = 1.0 > 0.48 = ωL). All other things being equal, control 283 is improved by applying the high-risk fungicide earlier, since it then targets the pathogen 284 when its relative growth rate is larger.

285
Evolution of resistance at full doses 286 For all three strategies, there is a sharp breakdown of control after ~15 seasons ( Figure   287 3a). This is driven by a rapid increase in the proportion of the resistant strain, which 288 increases sigmoidally from being practically undetectable (<1%) to near fixation (>99%) although this level of control is not economically-viable. The improved performance of the 294 low-high strategy relative to high-low alternation after resistance has taken over is again 295 due to timing: control is improved by applying the sole effective fungicide earlier.

297
Although the timing of the sharp increase in the frequency of fungicide-resistant pathogen 298 is similar for all three strategies, it occurs earliest for mixtures then for alternation high-299 low then for low-high (Figures 3a and b). This is precisely the order of the efficacy of the 300 strategies for disease control in the first season. Applying fungicides as a mixture leads 301 to slightly more effective disease control, butin part as a consequence of thisexerts 302 a stronger selective pressure. Considered over the effective lifetime, the alternation low-303 high strategy therefore has the highest lifetime yield (Figure 3c). At full dose, however, 304 differences between the strategies are relatively minor. Patterns in lifetime yield are more complex (Figure 4b). There is a region of dose-space 319 (hatch-shaded grey) within which no strategy leads to sufficient control even before 320 resistance has spread. This outcome is associated with low doses of both chemicals, 321 although even at full dose of low-risk, some high-risk is required (recall the high-risk 322 chemical is the more efficacious The largest lifetime yield over all strategies and pairs of doses is marked with the red 330 arrow on Figure 4b. It corresponds to spraying a mixture of a full dose of low-risk with a 331 dose of high-risk slightly larger than that required for economically-acceptable yield in the 332 first season. As the low-risk fungicide exerts no selection, it is unsurprising that the 333 maximal permissible amount of low-risk should be optimal, since this allows the smallest 334 amount of high-risk to be applied while maintaining acceptable disease control. However,

335
it is less obvious why the optimal strategy should be to apply the high-risk fungicide as 336 part of a mixture, and why this particular dose of high-risk (i.e. just above the amount 337 required to ensure effective control) is required.

339
We therefore examine responses to the dose of high-risk (CH) with the low-risk fixed at

348
To understand optimum performance in more detail, we compare the selection ratios at

357
The optimal dose of the high-risk fungicide (CH) is slightly higher than the minimum CH 358 ensuring acceptable control in the first season. This is because the effective lifetime is life. This is difficult to see in Figure 4e, but the "horizontal" parts of the response are not 365 quite horizontal. The red arrow on Figure 4b is therefore above the boundary between the 366 grey and dark-green regions.

367
Balancing selection and control 368 We further dissect the trade-off between selection and control by considering equal doses changing parameter values can cause this not to be optimal. We then consider the 397 response of lifetime yield to changing CH.

399
As an example, we examine in some detail the effect of the infection rate (β) (Figure 6a).

400
If β is made significantly larger than the default, all three strategies fail to give sufficient 401 control at any CH (dark-grey hatching). If β is made sufficiently smaller, then control can 402 be maintained indefinitely through the low-risk fungicide alone (light-grey hatching). Both 403 cases are unrealistic. Within the realistic range of values of β, exactly the same pattern is 404 seen as before. At low CH, no spray strategy can provide effective control. At slightly 405 higher CH mixtures perform best. At the highest CH alternation performs better, although 406 for large infection rates this might require CH > 1.0 (i.e. a dose above the permissible 407 maximum label dose). As β is increased, the threshold CH at which mixtures first become 408 effective shifts upwards, as more fungicide is required to provide acceptable disease 409 control.

411
The optimal dose is always lower for mixtures than either of the alternations (Figure 6b; 412 the saw-tooth pattern is because the effective lifetime is discrete). The corresponding 413 optimal lifetime yield is always larger under mixture (Figure 6c), andfor all strategies -414 corresponds to selecting CH close to the threshold required for effective first season 415 control ( Figure 6d). For all values of the infection rate, β, the optimal strategy is therefore again to spray a little more high-risk fungicide than required for effective control in the first 417 season, and to do so under mixture.

419
The pattern is consistent for all parameters tested in our sensitivity analysis ( Figure 7).

420
For parameters which cause disease to spread faster as they are increased, more high- To facilitate inter-model comparison, we return to comparing strategies in dose-space.

431
The simplest modelwith both pathogen strains growing exponentiallyis similar to 432 models used in the early fungicide resistance modelling literature (Delp, 1980;Kable & 433 Jeffery, 1980;Skylakakis, 1981). Indeed, if we additionally assume fungicides do not 434 decay, an analytical prediction of which strategy leads to better resistance management 435 at a given pair of doses can be generated (Methods S4). The other models are too 436 complicated for mathematical analysis, although the same pattern is seen for selection in 437 dose-space in every model ( Figure 8). The only real differences between models are the 438 slightly larger regions within which alternation provide better resistance management 439 when models include a latent period. When a latent period is not included, the high-risk 440 fungicide loses its eradicant mode of action, and so becomes generally less efficacious.

441
As with dose and fungicide parameter values, less effect from the high-risk fungicide then 442 favours mixture.

444
In the models that include host-limited infection, and thus the loss of host tissue to 445 disease, we also investigate how predictions of lifetime yield are affected by model 446 structure ( Figure 9). Predictions vary between models, which is perhaps unsurprising 447 given the additional complexity underlying the yield metric. However, although the 448 patterns vary, the characteristic pattern in dose-space is conserved. At low CH both 449 strategies fail to give acceptable yield, at slightly higher doses mixtures out-perform 450 alternation, and at the highest doses alternation out-performs mixture. Exactly as before, 451 the optimal strategy is therefore again to spray a little more high-risk fungicide than is 452 required for effective control in the first season, and to do so under mixture (the bright 453 green saw-tooth lines on Figure 9).  long-term yield is Alternation Low-High.

468
However, when normalising strategies by the level of initial control (Figures 10c and 10d We considered resistance management of a fungicide at high-risk of resistance,   (Figures 8 and 9) and pathosystem (Figures 10a and b). An underlying driver of 505 the variation in performance at different doses iswhen normalising by the applied doses

506
that different strategies lead to varying levels of disease control in the absence of resistance (Figures 4c and 5a). An alternate normalisationas in our third comparisonaccounts for this by selecting combinations of pairs of fungicide doses for each strategy 509 that lead to identical initial levels of control (Figure 5f and 5g). The effective lifetime and 510 so lifetime yield for any given level of disease control then depends strongly on the 511 amount of high-risk fungicide that is sprayed (Figure 5h). Since the low-risk chemical 512 exerts no selection in our model, it is optimal to include as much low-risk fungicide as 513 possible in any spray programme, and to combine this with as little high-risk fungicide as 514 provides effective control (Figures 4b and 10b). Arguably this is unsurprising (Shaw, good evidence that applying fungicides as mixtures will be the best resistance 551 management strategy in a range of situations. Furthermore, for a set of 1000 parameter 552 sets in which each parameter was sampled uniformly at random from the ranges shown in Figure 7, no case was found where mixture did not provide an overall better lifetime yield than alternation. (Methods S5).

556
The majority of our results are explained by dose-splitting and suppression by the mixing 557 partner, both of which are simply explained by the governing principles. However, the fact 558 that the two alternation strategies do not perform identicallyand that these two 559 strategies differ only in the order in which high-risk and low-risk chemicals are applied -

560
shows that timing of fungicide application can also be important. These can likely also be 561 explained by the governing principles, but with greater difficulty due to the non-trivial 562 interactions between the end of the season, the growth rate of the pathogen at any given 563 time, and the critical period for yield formation. We have therefore not pursued these 564 differences here.

566
While we have identified how the optimal strategy and combination of doses could be principles, our work has developed a firm base to which these complexities can be added.

586
Our future work will do this, albeit with the expectation that mixtures will very often be the 587 better strategy.

861
Methods S1 Model of septoria leaf blotch on winter wheat

863
As described in the main text, the model of septoria leaf blotch on winter wheat is a semi-

876
We denote the total upper leaf LAI as A, with This LAI grows at rate g, which is monomolecular after the emergence of the first leaf 880 tracked, and in which disease has no effect on growth The system of differential equations describing the system is then

1005
The ratio of these two quantities quantifies whether mixture (ratio greater than 1) or 1006 alternation (ratio less than 1) provides better resistance management  (ωH) has no effect on relative strategy performance; neither does the underlying 1079 pathogen growth rate (β). The values of the other fungicide parametersi.e. the 1080 maximum effect of low-risk fungicide (ωL), and the curvature parameters of both 1081 fungicides (θH and θL)therefore control the shape of the response. The first row (a-1082 d) has ωL = 0.3; the second row (e-h) has ωL = 0.5; and the third row (i-l) has ωL = 1083 0.7. The first column (a, e, i) has θH = 3 and θL = 2, the second (b, f, j) has θH = 7 and 1084 θL = 2, the third (c, g, k) has θH = 3 and θL = 6, and the fourth (d, h, l) has θH = 7 and 1085 θL = 6.