E-mail: neil.paveley@adas.co.uk

# Predicting effective doses for the joint action of two fungicide applications

Article first published online: 29 OCT 2003

DOI: 10.1046/j.1365-3059.2003.00881.x

Additional Information

#### How to Cite

Paveley, N. D., Thomas, J. M., Vaughan, T. B., Havis, N. D. and Jones, D. R. (2003), Predicting effective doses for the joint action of two fungicide applications. Plant Pathology, 52: 638–647. doi: 10.1046/j.1365-3059.2003.00881.x

#### Publication History

- Issue published online: 29 OCT 2003
- Article first published online: 29 OCT 2003
- Accepted 19 April 2003

- Abstract
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- Cited By

### Keywords:

- dose–response curves;
- effective dose;
- fungicide programmes;
- multiplicative survival model;
*Septoria tritici*;- tebuconazole

### Abstract

A function was derived to predict fungicide efficacy when more than one application of a single active ingredient is made to a crop, given parameters describing the dose–response curves of the component single-spray applications. In the function, a second application is considered to act on that proportion of the total pathogen population which was uncontrollable at the time of the first application (represented by the lower asymptote of the dose–response curve for the first treatment), plus any additional part of the population which survived the first application as a result of a finite dose being applied. Data to estimate the single-spray dose–response curve parameters and validate predictions of two-spray programme efficacy were obtained from separate subsets of treatments in four field experiments. A systemic fungicide spray was applied to wheat at a range of doses, at one or both of two times (*t*_{1} and *t*_{2}), in all dose combinations. Observed values of the area under the disease progress curve (AUDPC) for septoria leaf blotch (*Mycosphaerella graminicola*) were used to construct response surfaces of dose at *t*_{1} by dose at *t*_{2} for each culm leaf layer. Parameters were estimated from single-spray and zero-dose treatment data only. The model predicted a high proportion (*R*^{2} = 71–95%) of the variation in efficacy of the two-spray programmes. AUDPC isobols showed that the dose required at *t*_{2} was inversely related to the dose at *t*_{1}, but the slope of the relationship varied with the relative timings of *t*_{1} and *t*_{2} in relation to culm leaf emergence. Isobols were curved, so the effective dose – the total dose required to achieve a given level of disease suppression – was lower when administered as two applications.

### Introduction

A single fungicide application to wheat provides efficient control of foliar disease on the culm leaf layer which emerges (and is therefore exposed to infection) shortly before application (Paveley *et al*., 2000). Less effective control is obtained on preceding and succeeding layers. Consequently, when the weather is particularly conducive to disease or host resistance is inadequate, a single fungicide application may not provide sufficient control to prevent premature green-area loss from all the upper leaf layers that are important to yield formation. There is little theory to describe or predict the joint effect of two or more fungicide applications within a spray programme. Such theory should support decisions on the appropriate dose (Paveley *et al*., 2001) for the current application, for any given dose applied previously. It might also help to discriminate between circumstances under which the total dose required to achieve a given level of disease control (the effective dose, ED) would be minimized by applying a lower dose more often, or a higher dose less often.

Simple models to describe the joint effect of active ingredients applied simultaneously in a spray mixture are long established (Bliss, 1939; Wadley, 1945; Scardavi, 1966; Colby, 1967; Rummens, 1975), and have been tested against experimental data for control of weeds, invertebrate pests and fungal diseases (Morse, 1978; Kozial & Witkowski, 1982; Gisi *et al*., 1985; Grabski & Gisi, 1987; Streibig & Kudsk, 1993). Kosman & Cohen (1996) extended mixture theory to account for nonsimultaneous action of mixtures in which, for example, one fungicide affects spore germination, and another affects mycelial growth but not spore germination. Hence the components could be considered as acting at different times. This paper explores whether such models might be adapted to predict the efficacy of fungicide spray programmes from the performance of their component applications, where those applications are of the same active ingredient but applied at different times. Two mixture models have been widely reported: the multiplicative survival model (MSM) and the additive dose model (ADM). Each has proponents and critics, but consensus suggests that the ADM and MSM are most appropriate when the components of the mixture have the same or different modes of action, respectively (Morse, 1978).

The ADM assumes a constant ratio between equipotential doses, which implies parallelism of the dose–response curves of the components, after log dose transformation (Kosman & Cohen, 1996). Paveley *et al*. (2000) described fungicide dose–response curves by a parsimonious function which can be expressed as:

*D*_{d}=*D*_{0}[1 −*b*(1 −*e*^{−kdose})](1)

where *D*_{d} and *D*_{0} are disease severity at dose *d* and dose = 0, respectively, *b* the amount of disease that might potentially be controlled with an infinite dose (expressed as a proportion of *D*_{0}), and *k* defines the rate of change of disease severity with dose.

The *k* parameter was shown to be constant across spray timings, whereas *b* varies substantially and predictably according to the timing of application in relation to culm leaf emergence. Hence, if two sprays were applied some days apart, the lower asymptote (*D*_{0} − *b*) would differ between the two applications. Thus the curves for the two component applications cannot be parallel, making use of the ADM inappropriate.

The MSM does not require parallelism, but assumes that the mixture components can be thought of as acting independently and sequentially. Hence active substance B acts with the same efficacy as if it were applied alone, but only on that part of the pathogen population which ‘survives’ the effect of active substance A. This notion fits well with the sequential spray case. The approximation to independence of action of the two components arises in the mixture case by the two modes of action working on different target sites within the pathogen. Where sequential sprays are of the same mode of action, independence of action might still occur, as different subsets of individuals in the pathogen population are likely to be in a susceptible phase of their life cycle at different times. In effect, the second application might be considered as acting on that part of the total pathogen population which was uncontrollable at the time of the first application (represented by the lower asymptote of the dose–response curve from the first application), plus any additional part of the population that survived the first application as a result of a finite dose being applied.

These principles were used to derive a function to predict fungicide efficacy when more than one fungicide application of a single active ingredient is made in a crop, given parameters describing the dose–response curves of the component single-spray applications. Predictive value was tested against experimental data, using septoria leaf blotch (causal organism *Septoria tritici*, anamorph of *Mycosphaerella graminicola*) and winter wheat (*Triticum aestivum*) as the test pathosystem.

### Materials and methods

#### Theory: extension to a two-spray programme

To extend Eqn 1 to describe a two-spray programme, using MSM principles, let spray 1 have parameters *k* and *b*_{1} and be applied at dose *d*_{1}, and let spray 2 have parameters *k* and *b*_{2} and be applied at dose *d*_{2}. The amount of disease, *D*, remaining after both applications is given by:

- (2)

The first and second terms (in square parentheses) represent the proportions of disease remaining after the effects of *d*_{1} and *d*_{2}, respectively. Dose can be measured either in absolute units or as proportions of the recommended dose (the dose recommended for use on a particular crop by the fungicide manufacturer and approved by the pesticide regulatory authority), with estimates for *k* varying accordingly. Proportions of the recommended dose are used here.

Because control might be pest density-dependent, Kosman & Cohen (1996) suggested that where action was nonsimultaneous, the efficacy of the second mixture component should be determined against a pest population of the same density minus the efficacy of the first component. This is impractical in field conditions. Thus it was assumed that the proportional efficacy of the second application was unaffected by the amount of disease remaining after the first treatment.

#### Experimental design and treatments

Four field experiments were conducted during the 1996/97 and 1997/98 seasons on winter wheat cv. Riband, which is susceptible to septoria leaf blotch, at ADAS Rosemaund, Hereford, England. Experiments were established using a plot drill on 10 October 1996 and 4 October 1997 (experiments 1 and 2, respectively) in a three-replicate randomized block design. Plot sizes were 18 × 2 m in 1996 and 24 × 2 m in 1997. At Markle Mains, East Lothian, Scotland, experiments were marked out in existing crops drilled on 8 October 1996 and 27 September 1997 (experiments 3 and 4, respectively) and were arranged in fully randomized and randomized block designs, respectively, with four replicates of each treatment. Plot sizes were 20 × 2 m.

With the exception of fungicide applications, the experimental areas were treated according to local commercial practice. Fungicide treatments of tebuconazole (as the commercial product Folicur, 250 g active ingredient L^{−1}, Bayer plc, Bury St Edmunds, UK) were applied using a hand-held sprayer in 225–250 L water ha^{−1} at 250 kPa through Lurmark F110-04 nozzles (Lurmark Ltd, Longstanton, Cambridge, UK) at the doses and timings given in Table 1. Doses above the recommended rate were used for experimental purposes, to better estimate dose–response curve lower asymptotes. There was no evidence of phytotoxicity resulting from these high-dose treatments. Tebuconazole was selected as a representative of the widely used ergosterol biosynthesis-inhibiting group, with eradicant and protective activity against *S. tritici*. This group of active ingredients plays a key role in wheat disease control, used in combination with fungicides of other modes of action (Paveley & Clark, 2000). All growth stages (GS) quoted are from the decimal code (Tottman, 1987).

Treatment number | Fungicide dose^{a} applied at | |
---|---|---|

Timing 1 (GS 32)^{b} | Timing 2 (GS 33)^{b} | |

^{a}Tebuconazole as commercial product Folicur (L ha ^{−1}at 250 g ai L^{−1}).^{b}Treatments at timings 1 and 2 were applied when leaf III or II, respectively, was the youngest emerged leaf with ligule visible on more than 50% of shoots. Timing 1 dates were 1 May, 29 April, 6 May and 5 May in experiments 1–4, respectively; timing 2 dates were 15, 13, 20 and 19 May, respectively.
| ||

1 | 0·0 (untreated) | 0·0 (untreated) |

2 | 0·0 | 0·25 |

3 | 0·0 | 0·50 |

4 | 0·0 | 1·00 |

5 | 0·0 | 2·00 |

6 | 0·25 | 0·0 |

7 | 0·25 | 0·25 |

8 | 0·25 | 0·50 |

9 | 0·25 | 1·00 |

10 | 0·25 | 2·00 |

11 | 0·50 | 0·0 |

12 | 0·50 | 0·25 |

13 | 0·50 | 0·50 |

14 | 0·50 | 1·00 |

15 | 0·50 | 2·00 |

16 | 1·00 | 0·0 |

17 | 1·00 | 0·25 |

18 | 1·00 | 0·50 |

19 | 1·00 | 1·00 |

20 | 1·00 | 2·00 |

21 | 2·00 | 0·0 |

22 | 2·00 | 0·25 |

23 | 2·00 | 0·50 |

24 | 2·00 | 1·00 |

25 | 2·00 | 2·00 |

First treatments (at GS 32) were applied 1–3 days before full emergence of leaf III (where leaf layers are counted downwards from the flag leaf, leaf I). The date of full emergence was defined as the first assessment on which the leaf ligule was visible on at least 50% of shoots. Second treatments were applied when leaf II was fully emerged (at GS 33). The precise timing of fungicide sprays to target particular leaf layers was determined by dissecting five randomly selected shoots from across the experiment every 3 or 4 days from GS31. Leaves II and III were targeted as there was a greater likelihood of obtaining high disease severities to discriminate treatment effects, than on leaf I.

#### Disease assessments

Disease severity was measured as the percentage of leaf area affected by septoria leaf blotch in all plots at weekly intervals from the time of the first fungicide treatment until crop senescence. On each occasion, 10 randomly selected shoots per plot were assessed using disease assessment keys (Anonymous, 1972). At each assessment date prior to GS 39, shoots were dissected to allow disease scores to be recorded under the positions that they would eventually occupy on the mature plant. In experiments 1 and 2 at ADAS Rosemaund, every leaf layer with green leaf area remaining was assessed, and data for leaves I–III are presented. In experiments 3 and 4 at Markle Mains, only leaf II was assessed.

#### Analysis

The severity of the leaf blotch epidemic was expressed as the area under the disease progress curve (AUDPC) from sequential assessments, to avoid confounding the effects of treatment and assessment timing. AUDPC values (% days) were calculated from the severity data by numerical integration using the trapezoidal rule, and provided a measure of the damage caused by the disease, integrated over the life of the leaf.

##### AUDPC dose–response surfaces

The analysis proceeded in two distinct steps: parameter estimation for Eqn 2, then the test of the equation with independent data. To test the extent to which Eqn 2 predicted the efficacy of two-spray treatments (closed triangles in Figs 1–2), response surfaces were derived, using parameters describing the component single-spray dose–response curves (estimated from observations of zero-dose and single-spray treatments; closed circles in Figs 1–2). For each leaf layer in each experiment, the parameters of Eqn 2, namely *D*_{0} (AUDPC at dose = zero), *b*_{1} and *b*_{2} (proportions of disease that might potentially be controlled with an infinite dose of fungicide applied at *t*_{1} or *t*_{2}, respectively), and *k* (rate of change of AUDPC with dose) were estimated using FITNONLINEAR in genstat (Payne *et al*., 1993) using only the AUDPC values from plots making up the dose–response curves for single sprays (those that were untreated or received only a single spray treatment at either *t*_{1} or *t*_{2}). Fitted dose–response surfaces were plotted with dose applied at *t*_{1} (*d*_{1}), dose applied at *t*_{2} (*d*_{2}), and AUDPC in the *x*, *y* and *z* dimensions, respectively. Two sets of percentage fit (*R*^{2}) values were calculated. The first () described the variance in the single-spray dose–response curves accounted for by the fitted function. The second () quantified the proportion of the variation in the efficacy of two-spray treatments predicted by Eqn 2, using parameters from the fitted single-spray dose–response curves.

### Results

Septoria leaf blotch was severe on leaves I, II and III in all experiments, except for moderate severity on leaf I in experiments 1 and 2. Other foliar diseases remained at zero or low levels, and would not have interfered with leaf blotch epidemic development.

#### AUDPC dose–response surfaces

The 3D response surfaces fitted to the single-spray and untreated points are shown in Figs 1 and 2 for experiments 1–4, with first dose, second dose and AUDPC on the *x*, *y* and *z* axes, respectively, and parameter estimates (with their units and standard errors) in Table 2. Data points are shown attached to their *t*_{1} and *t*_{2} dose points on the 3D surface by vertical lines.

D_{0} (% days) | b_{1} | b_{2} | k (dose^{−1})^{a} | i^{b} (dose)^{a} | R^{2}_{test} (%)^{c} | R^{2}_{test} (%)^{d} | |
---|---|---|---|---|---|---|---|

(dimensionless) | |||||||

- a
Expressed as proportions of the recommended dose. ^{b}Amount by which the dose at one time should exceed the dose at the other time in order to minimize the total dose applied. Derived from fitted parameters via Eqn 6; see text. ^{}^{c}*R*^{2}for percentage fit of AUDPC values observed from the zero-dose (untreated) and single-spray treatments to values calculated by Eqn 2 (see text) after parameter estimation (calibration) using those values.^{}^{d}*R*^{2}for percentage fit of AUDPC values observed from two-spray treatments to values predicted by Eqn 2, given parameters estimated using independent observed AUDPC data from zero-dose (untreated) and single-spray treatments only.^{}nd, not determined; na, not applicable.
| |||||||

Experiment 1 | |||||||

Leaf I | 357 | 0·21 | 0·84 | 3·55 | 0·8 | 93·2 | 88·8 |

(29) | (0·09) | (0·07) | (1·07) | (nd) | (na) | (na) | |

Leaf II | 965 | 0·38 | 0·97 | 3·90 | 1·0 | 80·9 | 94·9 |

(147) | (0·14) | (0·12) | (1·83) | (nd) | (na) | (na) | |

Leaf III | 1494 | 0·85 | 0·82 | 1·71 | −0·1 | 82·9 | 92·5 |

(162) | (0·12) | (0·12) | (0·68) | (nd) | (na) | (na) | |

Experiment 2 | |||||||

Leaf I | 490 | 0·67 | 0·74 | 3·15 | 0·1 | 81·2 | 71·3 |

(51) | (0·08) | (0·08) | (1·07) | (nd) | (na) | (na) | |

Leaf II | 917 | 0·71 | 0·74 | 3·41 | 0·0 | 92·6 | 92·0 |

(59) | (0·05) | (0·05) | (0·69) | (nd) | (na) | (na) | |

Leaf III | 1082 | 0·81 | 0·71 | 2·82 | −0·2 | 90·1 | 93·0 |

(87) | (0·06) | (0·06) | (0·70) | (nd) | (na) | (na) | |

Experiment 3 | |||||||

Leaf II | 1023 | 0·32 | 0·81 | 2·61 | 0·9 | 81·6 | 86·0 |

(119) | (0·12) | (0·11) | (1·19) | (nd) | (na) | (na) | |

Experiment 4 | |||||||

Leaf II | 1369 | 0·15 | 0·78 | 1·32 | 2·3 | 98·8 | 92·6 |

(31) | (0·03) | (0·05) | (0·20) | (nd) | (na) | (na) |

##### Parameter estimation

Equation 2 provided a good description of the variation in AUDPC with single fungicide sprays of varying dose (including zero) and timing. Percentage variance accounted for () by the fits to the untreated and single-spray AUDPC values ranged from 80·9 to 98·8% between sites and leaf layers (Table 2). Tests for correlations between parameters suggested a negative association between *D*_{0} and *k* (correlation coefficient = −0·75) and a weak negative association (−0·46) between *b*_{1} and *b*_{2}.

##### Test of predictive precision

Prediction of the variation in AUDPC with two-spray treatments of varying dose and timing, using dose–response curve parameters estimated from the single-spray treatments, was good. Fits of the predicted values to observed data from the two-spray treatments gave values for percentage variance accounted for () ranging from 71·3 to 94·9% (Table 2), and *F* probability values were always *P* < 0·001. Plots of residuals provided no consistent evidence of bias, so unexplained variation represented random error in the data.

##### Interpretation of response surfaces

The response surfaces (Figs 1 and 2) show AUDPC decreasing with increasing dose in both *x* and *y* dimensions, with each additional increment of dose having a decreasing effect. However, the extent to which a unit dose decreased disease severity was determined by the *b* parameter. On leaf II in all experiments *b*_{2} was greater than *b*_{1} (albeit not always significantly so, according to the standard errors), indicating the greater efficacy of the *t*_{2} applications, which were administered closer to the time when the leaf was fully emerged. The *t*_{1} dose was less influential in reducing disease severity. In experiments 1 and 2, where leaves I and III were also assessed, the pattern of effects seen on leaf I was similar to that observed on leaf II, with greater efficacy per unit dose from the *t*_{2} timing. Values for *b*_{1} and *b*_{2} were not significantly different on leaf III, but in both experiments the estimate for the former exceeded the latter. Differences between *b*_{1} and *b*_{2}, and changes in their relative values between leaf layers, were less marked in experiment 2 than in experiment 1. Nevertheless, on all leaf layers in all experiments, good efficacy (indicated by large *b* values) was obtained from the treatment timing nearest to full leaf emergence.

The *k* parameter was positively related to the degree of curvature of the dose–response. Standard errors indicated that estimated values for *k* varied little, with few significant differences within experiments and no consistent trends across leaf layers (Table 2). Minimum and maximum values across the experiments were, respectively, 1·32 on leaf II in experiment 4, and 3·9 on leaf II in experiment 1. In the latter case the dose–response curvature was such that AUDPC values were close to their asymptote values at dose = 1, whereas in the former case AUDPC values were still decreasing beyond dose = 1.

#### Effective dose isobols

Contour diagrams corresponding with Figs 1 and 2 are shown in Figs 3 and 4, respectively (axes are arranged to allow direct comparison between the two sets of figures). The lines of equal effect or isobols (Loewe & Muischnek, 1926; Tammes, 1964) represent the effective total dose (ED_{tot}) at each timing required to achieve a given percentage control from 10% (ED_{tot}10) to 90% (ED_{tot}90), i.e. for an ED_{tot} of *x*%, AUDPC = *D*_{0}(1 − *x*/100). Hence the isobols quantify how the dose at *t*_{2} required to achieve a given efficacy relates to the dose at *t*_{1}. In the UK, the recommended dose is the maximum that can be applied. Thus where an isobol crosses an axis at a value below or equal to unity, that level of disease control is achievable with a single application of a legal dose at the time represented by that axis. However, that level of control may be obtainable with a lower total dose, or higher levels of control may be attainable by applying two sprays. To test this, a set of parallel lines were considered, each running diagonally across the isobol diagram and joining points of equal total dose (for example, the line AB for leaf III in Fig. 3e). Point C, at which a line of equal total dose meets an isobol, defines the point at which the total dose required to achieve a given level of control is minimized.

The minimum total dose points were calculated for each isobol by calculating the point at which the tangent of the isobol was −1. Equation 2 was rearranged to give the contour curve for dose *d*_{2} as a function of dose *d*_{1}, at a given disease level *D*_{a}, by making *d*_{2} the subject:

- (3)

To obtain the values of *d*_{2} and *d*_{1} for a given tangent, Eqn 3 was differentiated with respect to *d*_{1} and the resulting expression made equal to the required tangent value – in this case, to unity. The differential of Eqn 3 with respect to *d*_{1} is:

- (4)

The expression has a negative sign as the tangents of the isobols represented by Eqn 3 are negative, because *d*_{2} decreases as *d*_{1} increases. By setting Eqn 4 equal to −1 so that, in effect, the numerator is equal to the denominator, the resulting expression can be solved for the specific dose () on the isobol which fulfils the condition, giving the quadratic:

- (5)

The positive sign before the square root was found to give the correct value of for further analysis. This value of , when put into Eqn 3, gave the corresponding value, , for the minimum total dose point on the isobol.

#### Line of minimum total dose

The form of Eqn 2 dictates that, for all experiments and leaf layers, the minimum total dose points must lie in a straight line (for example, DE for leaf III in Fig. 3e) where , about which the isobols, and hence the surfaces that they represent, are symmetrical. The constant represents the intercept (*i*) on the *y* axis: the amount by which the dose at one time should exceed the dose at the other in order to minimize the total dose (*d*_{tot}) applied.

Simplifying the expression for *d*_{2} (Eqn 3) minus the expression for *d*_{1} (Eqn 5) provided the intercept (*i*) (Table 2) as a function of *b*_{1}, *b*_{2} and *k*, so:

- (6)

The size of the constant *i*, that defines the position of the line of minimum total dose, depends on the relationship between amounts of controllable disease *b _{1}* and

*b*

_{2}. When two spray timings are of approximately equal efficacy, so

*b*

_{1}≈

*b*

_{2}, the response surface is symmetrical about a line close to

*d*

_{2}=

*d*

_{1}(Fig. 3e, leaf III; Fig. 3b, leaf I and II), so

*i*≈ 0. In contrast, where

*d*

_{1}and

*d*

_{2}differ in efficacy, so

*b*

_{1}is < or >

*b*

_{2}, respectively (Fig. 3a,c and Fig. 4a,b), then

*i*has a substantial positive or negative value. The form of Eqn 6 dictates that for any given difference between

*b*

_{1}and

*b*

_{2},

*i*will increase as

*k*tends towards zero.

#### Minimum total dose–response curve

A vertical section through the response surface along the line of minimum total dose (line DE in Fig. 3e) would show the relationship between *d*_{tot} and *D*_{a} when the amount of *d*_{tot} applied at *t*_{1} and *t*_{2} was optimized. The function for this minimum total dose–response curve can be derived because, along the line of minimum total dose, .

Substituting these functions for *d*_{1} and *d*_{2} in Eqn 2 gives:

- (7)

By defining each of the component doses as a function of *d*_{tot} and the intercept, in effect Eqn 2 for the response surface is constrained only to calculate *D*_{a} along the line of minimum total dose.

Similarly, along the line of minimum total dose , hence combining Eqn 5 for with Eqn 6 for *i* provides the function for the minimum total dose () required to achieve any given disease level, *D*_{a}:

- (8)

In countries where a maximum legal dose per application is defined by regulatory authorities (one dose unit), only those parts of the isobol diagram (and hence those parts of the line of minimum total dose) that are enclosed by the square labelled F in Fig. 3a are of practical interest. Higher doses are of experimental value to estimate the shape of the response surface more precisely.

#### Relative efficacy of single- and two-spray programmes

Equation 8 allows comparisons between the dose required to achieve a given level of efficacy from single- or two-spray programmes. Maximum efficacy obtainable with legal doses can also be quantified by comparing the points at which isobols intercept the axes within the range 0–1 dose units, against the point at which the line of minimum total dose crosses the highest ED isobol before exceeding dose = 1 at either spray timing. Three contrasting examples illustrate the pattern of effects.

In the first example (leaf II in experiment 1), *t*_{2} was close to the optimum spray timing, with high efficacy denoted by *b*_{2} approximating to 1 and an ED90 of 0·68. In contrast, *t*_{1} was too early, with low efficacy denoted by a low *b*_{1} value and effective doses above ED30 exceeding 1. As a result, the line of minimum total dose was substantially displaced from the origin (large positive intercept, *i*) so that two-spray programmes were always less efficient than a single spray at *t*_{2}. In the second case (leaf III in experiment 1), *t*_{1} and *t*_{2} were equally displaced either side of the optimum timing, resulting in equivalent efficacy, so *b*_{1} ≈ *b*_{2} and *i* ≈ 0. ED60 was the highest legal effective dose at both timings, with *d*_{1} = 0·72 and *d*_{2} = 0·78. The ED_{tot}60 for a two-spray programme along the line of minimum total dose was 0·68. Eighty per cent efficacy could be obtained with a total dose below the maximum permitted, as ED_{tot}80 = 1·27. In the third example (leaf III in experiment 2), *t*_{1} was closer to the optimum timing than *t*_{2}. ED70 and ED60 were the highest permitted effective doses for *t*_{1} and *t*_{2}, respectively, at 0·70 and 0·65. The ED_{tot}70 for a two-spray programme was 0·63 and ED_{tot}90 = 1·58.

#### Theory: extension to a multiple spray programme

Equation 2 and its derivatives could be expanded to predict the combined efficacy of multiple sprays of doses *d*_{1}, *d*_{2}…* d*_{n} at times *t*_{1}, *t*_{1}…* t*_{n}, by adding additional terms. Each application would be considered to act on that proportion of the population which survived the combined effect of all previous applications. Similarly, the theory could be expanded to allow the effects of changing the timing of the component sprays to be investigated, because variation in *b* with spray time was found to be described well by the normal function (Paveley *et al*., 2000), which could therefore be substituted for *b* in Eqn 2, thus:

- (9)

AUDPC is minimized when *t* = µ (mean of normal distribution) and increases at a rate dependent on σ (standard deviation of normal function) as *t* increases or decreases from µ. θ is the maximum of the normal term (expressed here as a proportion of *D*_{0}) when *t* = µ, hence θ is inversely proportional to the minimum value of AUDPC. Validation of Eqn 9 requires experimental data from a wider range of spray timings than those tested here, in order to obtain reliable estimates of σ, µ and θ.

### Discussion

This paper demonstrates theoretically and experimentally that it is possible to predict the efficacy of two-spray programmes from the performance of their component single sprays. The assumption underlying use of the MSM model – that the subsets of the population on which the two sprays were acting should be substantially distinct – was acceptable for the pathosystem and fungicide treatments tested. If the theory is extended to other systems, limitations on the proximity of adjacent applications in the programme will need to be respected. Provided the treatment interval is greater than, or a substantial proportion of, one latent period (as in the data presented here), then adjacent applications act on largely distinct generations, and the model should provide reliable predictions of programme efficacy.

There now follows an assessment of the objectives described in the introduction: (i) to support decisions on the appropriate dose for the current application for any given dose applied previously; and (ii) to identify circumstances under which the total effective dose would be minimized by applying a lower dose more often, or a higher dose less often.

#### Effective doses at different application timings

Effective doses at the second application were inversely related to the dose at the first application, but the slope of the relationship (represented by the isobols) varied according to the relative efficacy of the two applications. Hence (i) where two timings are equally effective, equal doses should be applied at each time; (ii) if one time is more effective, more of the total dose should be applied at that timing; and (iii) in extreme cases, where *i* is large, all of the dose should be applied at the more effective timing, as the limited isobol curvature within the range of legal doses does not compensate for the poor efficacy of the less effective timing. The decision on the amount of total dose that should be applied at each timing is simplified because *i* is constant, regardless of the appropriate total dose. In practice, fungicide dose decisions are subject to considerable uncertainty. If, in retrospect, the dose applied at the first timing is considered to be excessive or insufficient, the isobol diagrams show that there is scope to compensate by adjusting the dose of the second application in the opposite direction. However, the total effective dose will be higher than if optimal doses are applied at each timing. Spray timing may be less critical when conditions are not conducive to spore transfer and infection. In experiment 1, full emergence of each of leaves III, II and I was followed by substantial rainfall events, whereas emergence of the upper leaves in experiment 2 coincided with a predominantly dry period. Differences in efficacy of the two spray timings were less marked in experiment 2 than in experiment 1.

#### Minimizing total effective dose

In the pathosystem tested, the total effective dose required was reduced when administered as two applications. For a single spray, the optimum application timing is a compromise between decreasing effects on epidemic onset and increasing effects on the relative epidemic growth rate, as the spray is delayed beyond full leaf emergence (Paveley *et al*., 2000). The isobol curvature reported here may arise because two well timed spray applications allow delayed epidemic onset from the first application to be combined with reduced growth rate from the second. At any point in time, there is a proportion of the pathogen population that is not in a fungicide-susceptible phase of its life cycle, as those individuals have yet to arrive on the target leaf or have already expressed symptoms. That proportion of the population is represented by the lower asymptote of the dose–response curve for a fungicide application at that time. Increasing the dose of a single application has only a diminishing effect on the susceptible part of the population, whereas two applications reduce the lower asymptote by catching a higher proportion of the pathogen population in a susceptible state. The beneficial effect of two spray applications on the total effective dose was greatest when there was high isobol curvature within the legal range of doses. This occurred when efficacy at the two timings was similar (small *i* values), and at high ED and *k* values. Hence spray programmes of fungicides with high dose–response curvature can achieve high efficacy at low total dose. However, this beneficial effect does not comply with the strict definition of ‘synergy’ between the two applications, as the observed efficacy for two-spray treatments agreed with the calculated expected efficacy, according to the specified theory of nonsynergistic action (Morse, 1978; Kosman & Cohen, 1996).

Although a lower total dose was usually required to achieve a given level of control when split between two applications, the isobols were seldom sufficiently curved for the financial saving from the lower dose to compensate for the labour and machinery costs of the additional spray (Nix, 2001). Also, repeated applications may increase the risk of fungicide resistance (Metcalfe *et al*., 2000). However, with a susceptible cultivar, favourable weather and high pathogen inoculum concentrations, two sprays were required to achieve acceptable disease control across the upper three leaves, important to resource capture. Where two or more applications are justified on these grounds, isobol curvature should be accounted for when determining the appropriate dose, otherwise an excessive amount may be applied. Equation 9 allows theoretical exploration of the optimum separation of the component sprays in a fungicide programme and the number of applications needed. These should be determined jointly by the period over which a single spray can achieve effective control (quantified by σ), and the difference between optimum application timings (µ) for different parts of the crop canopy. In wheat, the upper culm leaf layers intercept most of the incident photosynthetically active radiation during the yield-forming period. These leaves emerge and expand within a short period (Sylvester-Bradley *et al*., 1997). The long latent period of *Mycosphaerella graminicola* contributes to σ values that are sufficiently large in relation to the differences in µ between upper leaf layers, that two well timed sprays (Hardwick *et al*., 2001) can provide close to optimal control of leaf blotch on those leaf layers critical to resource capture. However, in contrasting pathosystems where canopy expansion is prolonged and the latent period short, closely spaced multiple treatments may be required to obtain effective control.

### Acknowledgements

Funding of this work by the Home-Grown Cereals Authority is gratefully acknowledged. Thanks are due to host farmers for provision of experimental sites, many colleagues for conducting the experiments and statistical advice, and anonymous referees for constructive criticism of the manuscript. The contributions of Simon Oxley of SAC, and Alice Milne and Eric Audsley of Silsoe Research Institute are acknowledged.

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