Draft:AUDPC & AUDPS

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Area Under the Disease Progress Curve (AUDPC) and the related Area Under the Disease Progress Stairs (AUDPS) are quantitative summary statistics used in plant pathology and epidemiology to combine both the duration and the intensity of a plant-disease epidemic into a single numerical value. The single value index thus allows for a simple comparison of cultivars (host genotypes) or treatments instead of using an entire time series.

Even though AUDPC and AUDPS were developed for quantification of plant disease progress, the same principle as used in AUDPC (trapezoidal integration) or AUDPS (staircase integration) approach has since been adapted, frequently under a new acronym, to summarize other biological progressive quantities that are assessed repeatedly over time; for example decline in postharvest quality, progress in plant growth and development, changes in chlorophyll content, increase in seed germination, and progress in insect infestation as detailed in a later section.

Background

The majority of plant pathogens that cause major foliar diseases are polycyclic; i.e. the pathogen propagates and produces new inoculum to reinfect the host repeatedly over a growing season. Therefore, disease severity (or disease incidence) typically rises from no disease at healthy plants to a maximum disease by the end of the season, frequently resembling a sigmoid (S-shaped) disease progress curve. To combine evaluations of disease progress across all time points when assessments were taken, researchers have developed methods that can summarize the entire disease progress into a single value. The first such approach to combine the area under the disease progress curve using the trapezoidal rule was proposed by van der Plank in the chapter "Sanitation with Special Reference to Wheat Stem Rust" of his book Plant Diseases: Epidemics and Control. The author argued that crop injury from a disease could be modeled using the total amount of disease and its duration.

Probably one of the earliest and most cited empirical applications of this method was performed by Shaner and Finney in 1977 to study resistance to powdery mildew in Knox wheat. They calculated the Area Under the Disease-progress Curve (which they termed ADPC) using the same approach as suggested by van der Plank:

${\text{ADPC}}=\sum _{i=1}^{n-1}\left[{\frac {Y_{i+1}+Y_{i}}{2}}\right]\left[X_{i+1}-X_{i}\right]$

where $Y_{i}$ is mildew severity at the ith observation, $X_{i}$ is the time (in days) of the ith observation, and n is the total number of observations. The authors reported that this statistic had a lower error variance than statistics derived from a logit transformation of severity readings, and concluded it was a superior, more practical measurement of partial (slow-mildewing) resistance in a plant-breeding context.

Area Under the Disease Progress Curve (AUDPC) – trapezoidal rule approach

Assuming a series of n disease assessments, where $y_{i}$ is the disease severity or incidence (often expressed as a diseased plant area, percentage or proportion, or visual disease rating score) recorded at time $t_{i}$, AUDPC is calculated by the trapezoidal (midpoint) rule. Using this rule, each pair of consecutive observations is treated as defining a trapezoid, and the areas of the trapezoids are summed.

${\text{AUDPC}}=\sum _{i=1}^{n-1}{\frac {y_{i}+y_{i+1}}{2}}\left(t_{i+1}-t_{i}\right)$

Area Under the Disease Progress Stairs (AUDPS) – staircase (step) rule approach

Rearranging the AUDPC expression shows that the area can equivalently be computed by multiplying each individual assessment by an associated weight and summing the results. The weight associated with each assessment is the time from the midpoint of the preceding interval to the midpoint of the following interval. However, since the first and last assessments do not have an interval on one side, their relative weights are extrapolated in one direction only. For example, if disease score assessments are made every ten days, each interior observation is effectively weighted by 10 (five days extrapolated in each direction), while the first and last observations are weighted by only 5. Thus this approach systematically undervalues the contribution of the first and last assessments relative to the rest of the series, though for the majority of experimental situations there is no obvious reason for the endpoints to matter less.

To correct this boundary effect, Simko and Piepho (2012) proposed the Area Under the Disease Progress Stairs (AUDPS), which extrapolates the weight of the first and last observations in the missing direction as well. The authors suggested using half of the average interval length between all observations as supplemental weight for the two observations located at each end. By applying this approach, every assessment, including the two endpoints, receives an equal effective weight when they are evenly spaced. Graphically, this corresponds to the area under a step (staircase) function fitted to the assessments.

Comparison of AUDPC and AUDPS calculation methods, with weight-zone widths

Assuming a series of n assessments at times $t_{1},\dots ,t_{n}$, with $D=t_{n}-t_{1}$, the general AUDPS formula is:

${\text{AUDPS}}=\left[y_{1}\left({\frac {t_{2}-t_{1}}{2}}+{\frac {D}{2(n-1)}}\right)\right]+\left[\sum _{i=2}^{n-1}y_{i}\cdot {\frac {t_{i+1}-t_{i-1}}{2}}\right]+\left[y_{n}\left({\frac {t_{n}-t_{n-1}}{2}}+{\frac {D}{2(n-1)}}\right)\right]$

The authors algebraically proved that this expression, which can be used for both evenly and unevenly spaced assessment times, simplifies to the AUDPC plus a correction term that provides an equivalent full weight to the first and last observations:

${\text{AUDPS}}={\text{AUDPC}}+{\frac {y_{1}+y_{n}}{2}}\cdot {\frac {D}{n-1}}$

AUDPS approach has since been implemented in software, including the open-source R package epifitter.

When all assessments are equally spaced, AUDPS formula further simplifies to a multiple of the arithmetic mean of the assessments, ${\bar {y}}$:

${\text{AUDPS}}={\bar {y}}\times {\frac {Dn}{n-1}}$

and if assessments are made at every time unit (so that $D=n-1$), the formula simplifies further still to ${\text{AUDPS}}={\bar {y}}\times n$.

Testing the AUDPC and AUDPS formulas' precision across 50 trials spanning several crops and diseases, Simko and Piepho (2012) found that the standardized form of AUDPS (sAUDPS, see below) achieved lower root-mean-square error (RMSE) than the standardized form of AUDPC in 46 of the 50 trials, a lower coefficient of variation (CV) in 42 of 50 trials, and lower Euclidean distance (ED) between relative optimal and relative actual weights in 49 of 50 trials. F-value from one-way analysis of variance (ANOVA) was higher (more significant) for sAUDPS in 35 trials while for sAUDPC in 15 of the tested trials. They also showed theoretically that AUDPS is expected to be more precise than AUDPC whenever the first and the last assessments are, on average, no more variable than the assessments made in the middle of the series. The authors argue that this situation is more common in practice than the opposite, since replicate variance in a trial is often largest during the middle stages of disease increase.

Illustrative example

AUDPC and AUDPS calculations based on three evaluations of disease severity — 40%, 80%, and 90% — recorded 7 days apart (on days 14, 21, and 28):

${\text{AUDPC}}={\frac {40+80}{2}}(7)+{\frac {80+90}{2}}(7)=420+595=1015$

${\text{AUDPS}}=1015+{\frac {40+90}{2}}\cdot {\frac {28-14}{2}}=1015+455=1470$

This example matches the result given in the documentation of the agricolae R package's audps() function.

When assessments are made at every consecutive time unit (so that all intervals equal 1), the formulas simplify further: AUDPS is the arithmetic mean of the assessments multiplied by the number of assessments, n, while AUDPC is the sum of all the assessments minus half of the first and the last assessment. For example, four assessments taken on consecutive days with scores 1, 2, 3, and 10 give AUDPS = mean(1,2,3,10) × 4 × (3/3) = 16, since here D = n − 1 = 3, and AUDPC = (1+2+3+10) − (1+10)/2 = 10.5.

Estimating AUDPC from limited assessments – logistic growth model

Because AUDPC is normally calculated from many repeated assessments, its use can be costly in time, labor, and field space, particularly in breeding trials that screen large numbers of cultivars, lines, or treatments. Jeger and Viljanen-Rollinson (2001) showed that when a disease progress curve follows a logistic growth model, AUDPC can be estimated accurately from as few as two assessments — one taken early and one taken late in the epidemic — rather than calculated from a full time series. If $y_{0}$ and $y_{T}$ are the disease severities recorded at an early time (normalized to $t=0$) and a later time $T$, the apparent infection rate $r$ can be estimated as

$r={\frac {\ln \left({\dfrac {y_{T}}{1-y_{T}}}\right)-\ln \left({\dfrac {y_{0}}{1-y_{0}}}\right)}{T}}$

and AUDPC is then estimated as

${\text{AUDPC}}=T+{\frac {\ln(y_{0}/y_{T})}{r}}$

The authors tested this approach on field trials of stripe rust (Puccinia striiformis f. sp. tritici) on ten wheat cultivars. They found excellent agreement between AUDPC values calculated from complete disease-progress curves (seven or eight assessments per season) and those estimated from only the first and last assessments (Spearman rank correlations ranged from 0.95 to 0.99 across trials). The authors cautioned, however, that the method assumes the underlying epidemic growth is well-described by a logistic curve, that the assessment period is comparable across the units being compared, and that disease increases monotonically without interruption. When these assumptions do not hold, calculating AUDPC from all repeated assessment remains preferable.

Standardized and relative forms of AUDPC and AUDPS

Because raw AUDPC and AUDPS values vary depending on both the length of the assessment period and the used scale, they are difficult to compare across experiments with different duration and/or scales. Two normalized forms are commonly used to address this challenge.

Standardized AUDPC/AUDPS (sAUDPC, sAUDPS) are calculated by dividing the area by the time span the formula covers, expressing the result back in the original severity units and allowing comparison of experiments with different numbers or spacing of assessment dates. Because AUDPC and AUDPS formally cover different time spans ($D$ for AUDPC, and the slightly extended $Dn/(n-1)$ for AUDPS), they are standardized differently:

${\text{sAUDPC}}={\frac {\text{AUDPC}}{D}}\qquad {\text{sAUDPS}}={\frac {{\text{AUDPS}}\times (n-1)}{D\times n}}$

When assessments are equally spaced, sAUDPS reduces exactly to the arithmetic mean of all the assessments, in the original measurement units, whereas sAUDPC remains affected by the reduced weight given to the first and last observations.

Relative AUDPC/AUDPS (rAUDPC, rAUDPS) parameters are calculated by dividing the area by the maximum theoretically possible area; i.e. the hypothetical area if the maximum possible severity or incidence is recorded at every assessment $y_{\max }$. This dimensionless value can be compared across experimental results that were obtained using different disease scales or durations. This approach for AUDPC was introduced by Fry (1978) in a study of general resistance to potato late blight: and was extended to AUDPS by Simko and Piepho:

${\text{rAUDPC}}={\frac {\text{AUDPC}}{y_{\max }\,D}}\qquad {\text{rAUDPS}}={\frac {{\text{AUDPS}}\times (n-1)}{D\times n\times y_{\max }}}={\frac {\text{sAUDPS}}{y_{\max }}}$

When the disease-rating scale has a nonzero minimum possible value $y_{\min }$ (for example, a 1-to-9 ordinal scale), rAUDPS should in principle be adjusted accordingly, although in practice this adjustment is rarely needed for disease-severity assessments because such scales are uncommon and negative severities do not occur:

${\text{rAUDPS}}={\frac {{\text{sAUDPS}}-y_{\min }}{y_{\max }-y_{\min }}}$

AUDPC and AUDPS software implementations

AUDPC and AUDPS calculations were implemented in several statistical software and scripts used in plant pathology and crop science:

• R: the agricolae package provides audpc() and audps() functions, including standardized and relative forms; the epifitter package provides AUDPC() and AUDPS() functions alongside tools for fitting and comparing population-dynamics models (exponential, monomolecular, logistic, and Gompertz) to disease-progress data.
• SAS: Simko and Piepho published SAS code for calculating AUDPS and sAUDPS directly from a table of assessment times and severity scores.
• Spreadsheets: IdeTo, an Excel- and Google Sheets-based calculator released in 2021, computes AUDPC, AUDPS, and their standardized and relative forms for up to 200 individuals across 200 timepoints, along with descriptive statistics, correlation analysis, and plots. The spreadsheet calculator was developed to make both metrics accessible to researchers without a statistical-programming background.
• Python: AUDPC can be computed directly with general-purpose numerical integration functions such as numpy.trapz/numpy.trapezoid or scipy.integrate.trapezoid.

Both agricolae and IdeTo provide descriptive statistics and correlation analysis alongside the area calculations, while epifitter provides a growth-model fitting feature.

Applications of AUDPC and AUDPS in phytopathology

AUDPC (and increasingly also AUDPS) are used throughout plant pathology, plant breeding, and related fields including:

• Comparing the level of quantitative (partial) disease resistance among crop cultivars or breeding lines, as in the original slow-mildewing wheat study and numerous subsequent trials
• Genome-wide association and quantitative trait locus mapping studies, where AUDPC or AUDPS is used as the phenotype summarizing disease resistance across a growing season
• Evaluating and comparing the efficacy of fungicides or other disease-management treatments over time
• Modeling the relationship between disease intensity and crop yield loss, an approach that traces back to van der Plank's original proposal that yield injury is proportional to the area under the disease progress curve

AUDPC, AUDPS and related metrics used in other fields

The trapezoidal rule method of calculating AUDPC and recently also the staircase rule weighting of AUDPS has been adapted under new names to summarize other quantities that are assessed repeatedly over time, in fields well outside plant disease progress evaluation:

• Plant physiology (chlorophyll content): in a study of winter wheat, Byamukama et al. (2012) calculated an Area Under the SPAD Progress Curve (AUSPC) from repeated chlorophyll-meter (SPAD) readings to summarize the decline in leaf chlorophyll content caused by viral infection.
• Postharvest quality decline: Simko, Hayes, and Kramer (2012) introduced the Area Under the Decay (or Deterioration) Progress Stairs (AUDePS) to combine repeated visual decay ratings of fresh-cut lettuce stored in modified-atmosphere packaging into a single index, using the AUDPS formula to combine evaluations of a physiological deterioration process. Similarly, Grzyb et al. (2026) used AUDPC approach to compute an area under the quality-degradation curve for five separate postharvest quality parameters (weight loss, cap opening, color change, browning index, and decay incidence) of essential-oil-fumigated mushrooms. They applied the resulting values to identify the most effective fumigation treatment.
• Plant and bacterial growth traits: Liu et al. (2019) defined an Area Under the Growth Progress Curve (AUGPC) using the identical trapezoidal formula given by Madden, Hughes, and van den Bosch (2007) to summarize the season-long height, percentage ground coverage, and visual-quality ratings of prairie grasses and sedums grown in different substrates on an experimental green roof.. Corbin-Augusti et al. (2026) applied AUGPC to combine values of optical density measurement taken at a wavelength of 600 nm (OD600) to compare growth of a bacterium in growth media.
• Seed germination dynamics: Marcinkowska et al. (2026) applied a cumulative measure that integrates both the speed and extent of germination termed Area Under the germination Curve (AUC). The authors used AUC data to compare temperature-dependent germination of herbicide-resistant and susceptible biotypes of two grass weeds.
• Insect infestation: in a study of banana germplasm resistance to banana bunchy top virus and its aphid vector, Ngatat et al. (2022) calculated an Area Under the Infestation Progress Curve (AUIPC) for aphid population buildup, directly alongside a conventional AUDPC for the viral disease symptoms.

These adaptations illustrate that the underlying idea of combining a repeatedly measured quantity into a single cumulative value via trapezoidal or staircase integration is treated in the literature as a general-purpose method rather than one specific to plant disease, even though it originated, and remains most widely used, in that context.

See also

• Area under the curve (pharmacokinetics)
• Disease progress curve
• Plant pathology
• Trapezoidal rule

External links

• American Phytopathological Society: Calculating AUDPC
• Simko & Piepho (2012), "The Area Under the Disease Progress Stairs"
• Jeger & Viljanen-Rollinson (2001), "Estimating AUDPC from Limited Assessments"

Category:Plant pathology Category:Epidemiology