For developing the conceptual and mathematical structure of the model, we conducted a systematic literature search (Okoli and Schabram, 2010; Biesbroek et al., 2013; Candel, 2014) to collect data on the biology, ecology, and epidemiology of Colletotrichum spp. from academic articles, articles in the press, and conference papers. Articles were searched by combining the following keywords: (i) Colletotrichum; (ii) conidia OR spore OR acervuli OR primary inoculum OR production OR development OR overwinter OR dissemination OR release OR dispersion OR deposition OR infection OR penetration OR germination OR incubation OR lesion development OR lesion onset OR latency OR latent period, and (iii) temperature OR wetness OR relative humidity OR rain OR environment. Search strings are shown in Table 1 of the Supplementary material . The literature search was carried out in 2022 in three bibliographical databases: Scopus (https://www.scopus.com/ accessed on February 25), Web of Science (https://www.webofscience.com/ accessed on February 28), and CAB Abstracts (https://www.cabdirect.org/cabdirect/search/ accessed on March 1). Each article found was first reviewed on the basis of the information in the title and abstract; articles of potential interest were then read in full. Additional articles were retrieved from the “References” section of the found articles; these articles were also reviewed.

Selected papers were used to extract information on the influence of environmental conditions on the considered biological processes. Data on the pathogen or the disease were obtained directly from the text, tables, or graphs in the papers; the GetData Graph Digitizer 2.24 (http://getdata-graph-digitizer.com accessed on 5 May 2021) was used to obtain precise data from graphs.

Although different life styles have been described for Colletotrichum spp. (i.e., necrotrophic, biotrophic, hemibiotrophic, quiescent, and endophytic life styles), the life cycles of these species have some common attributes (Peres et al., 2005; De Silva et al., 2017), which include both sexual and asexual reproduction. Because the perithecial stage is rarely observed in the field (Dowling et al., 2020), the asexual conidia are the most important stage for disease development; our model, therefore, focuses on the asexual life cycle. A general, asexual life cycle for Colletotrichum spp. is shown in Figure 1 .

Colletotrichum spp. overwinter as mycelium or sclerotia in crop debris (Farley, 1976; Casela and Frederiksen, 1993; Yoshida et al., 2002; Conner et al., 2019), fruit mummies (Moral and Trapero, 2012; Samuelian et al., 2012; McKay et al., 2014), and buds (Verma et al., 2006; Stensvand et al., 2017; Everett et al., 2018). In some hosts like strawberry and Citrus spp., the pathogen survives as melanized appressoria on asymptomatic leaves (Denham and Waller, 1981; Leandro et al., 2001). The pathogen can also be introduced in the field with infected seeds (Thomas and Sweetingham, 2004) or transplants (Howard et al., 1992); infected seeds and transplants, however, are not considered as primary inoculum sources in our model, because we assume that farmers use pathogen-free propagating material.

Primary inoculum consists of fresh conidia produced in acervuli under favorable conditions of temperature (10 to 30°C for most of the clades) and moisture (e.g., wetness periods longer than 6 h) on overwintered sources for the entire cropping season (Moral and Trapero, 2012; Stensvand et al., 2017; Everett et al., 2018). Conidia are splash-dispersed by rain (Denham and Waller, 1981; Chakraborty and Billard, 1995; Madden et al., 1996; Madden, 1997; Guyot et al., 2005; Buirchell et al., 2006), are deposited on plant surfaces, and germinate; the germ tubes then produce appressoria that in turn produce infection pegs that penetrate the host cuticle and epidermal cell walls (Perfect et al., 1999). Penetration through stomata or wounds by germ tubes without the formation of appressoria is rare (Zulfiqar et al., 1996). The infection of epidermal and mesophyll cells occurs after a symptomless biotrophic phase and results in a necrotic phase in which host cells are killed and secondary hyphae grow intra- and inter-cellularly (O'Connell et al., 2012). In some cases, the pathogen has a quiescent phase (Peres et al., 2005; Prusky et al., 2013), and the switch to the necrotrophic phase occurs only when host tissue and environmental conditions become conducive to the continuation of the infection cycle. Except for those few species that exist entirely as endophytes, the majority of Colletotrichum species develop a necrotrophic stage at some point in their life cycle (Prusky et al., 2013).

Acervuli finally develop within lesions and produce masses of conidia that serve as secondary inoculum for the repetition of infection cycles and the further spread of the disease (Peres et al., 2005; De Silva et al., 2017).

Information from the literature was organized in a relational diagram representing the epidemic structure as developed by Zadoks (1971) and by using the systems analysis syntax (Leffelaar and Ferrari, 1989; Rossi et al., 2010; Rossi et al., 2015) ( Figure 2 and Table 1 ). In this structure, the crop is composed of a large but finite number of sites that have equal dimensions and equal probability of becoming infected; a site is defined as a fraction of the host tissues where an infection may occur and where a lesion may develop (Zadoks, 1971; Savary et al., 2015). During the epidemic, a site goes through the following stages: healthy, infected without visible lesions (latent infection), infected with visible lesions (i.e., disease symptoms or signs), infected with fertile (i.e., sporulating) lesions, and finally, infected with no-longer sporulating lesions (or removed lesions). In the model, the site stages are state variables (boxes); the flow from one stage to the next (solid arrows) is regulated by rates (valves), which are in turn influenced by external and auxiliary variables (short segments and circles, respectively). The external variables include weather variables such as air temperature (T, in °C), wetness duration (WD, in h), and rainfall (P, in mm). Mathematical equations link (through dotted arrows) the external variables to rates, and were developed by using the data extracted from the literature; different parameterization was used for the different phylogenetic clades of Colletotrichum (Salotti et al., 2022).

For equation development and parameterization, the original data for each response (independent) variable collected in different experiments were rescaled between 0 and 1 by dividing each value by the maximum value obtained in each experiment. This was necessary because the experiments often used different measurement units (e.g., conidia production has been expressed as the average number of conidia per lesion, the number of conidia per cm² of a lesion, or the number of conidia per Petri dish in laboratory experiments, depending on the paper), fungal species or strains, host plants, etc. As an example of rescaling, Wilson et al. (1990) reported that 4 and 24 h of continuous wetness during infection on strawberry fruits by C. acutatum gave a disease incidence of 2 and 100%, respectively; therefore, the rescaled data were x_4h = 2/100 = 0.2, and x_24h = 100/100 = 1.

Rescaled data were regressed against the influencing environmental factors (the independent variables). Different nonlinear regression models were fit, and the goodness-of-fit was compared based on the Akaike’s Information Criterion (AIC); the model providing the smallest AIC value was considered the most correct (Brunham and Anderson, 2002). Equation parameters were estimated using the function nls of the “stats” package of R software (Team, R Core. R: A Language and Environment for Statistical Computing. 2019; available at https://www.r-project.org/). For parameter estimation, the data obtained from different experiments were considered to be replicates; the data concerning the different species belonging to the same clade were also considered to be replicates.

The goodness-of-fit of equations to original (rescaled) data was evaluated based on the adjusted R², the concordance correlation coefficient (CCC), the root mean square error (RMSE), and the coefficient of residual mass (CRM) (Nash and Sutcliffe, 1970; Lin, 1989). The adjusted R² was estimated by conducting a linear regression between the observed values and the model predicted values; the linear regression was conducted with the lm function of the R “stats” package (Wickham, 2019). CCC is an indication of the difference between the best fitting line and the perfect agreement line (if CCC = 1, the agreement is perfect) (Lin, 1989). CCC was obtained using the CCC function of the R “DescTools” package (Signorell, 2020). RMSE, which represents the average distance of real data from the fitted line (Nash and Sutcliffe, 1970), was obtained using the rmse function of the R “modeler” package (Wickham, 2019). CRM was calculated as described in Manstretta and Rossi (2015); CRM is a measure of the tendency of the equation to overestimate or underestimate the observed values (a negative CRM indicates a tendency of the model toward overestimation; Nash and Sutcliffe, 1970).

The model was developed and run in Excel^® (Microsoft 365^®).

We evaluated the ability of the model to describe real epidemics by using 17 disease progress curves (or epidemics) obtained from field data and retrieved from the literature. Details on Colletotrichum species and clades, primary inoculum origin (natural or artificial), hosts, and experimental sites used for model validation are summarized in Table 2 .

In the validation, we assumed that each epidemic is triggered by a potential quantity of inoculum sources (represented by the parameter k in equation [1]), which are responsible for the production of primary conidia. The values of k used to initialize the model are reported in Table 2 . Because there was no information in the literature for an exact calculation of the parameter k for each epidemic, the values in Table 2 were estimated empirically, i.e., we selected the value of k that resulted in the closest agreement between predicted and observed disease severities by running the model with varying values of k (in the rage 0 to 1).

For epidemics resulting from natural inoculum, the model was operated starting from January 1. For epidemics triggered by the experimental placement of diseased plants (see epidemics MA-14 and MA-15 in Table 2 ) or diseased fruit (see epidemics OH-90 and OH-91 in Table 2 ) in the crop or near the crop, the model was operated starting from the day in which the diseased plants or fruit were placed in the field. For MA-14 and MA-15, the inoculum was introduced by the sowing of Colletotrichum-infested white bean seeds (along with non-infested seeds), which developed into plants that produced the primary conidia; the model was operated from June 2 at MA-14 and from June 9 at MA-1. For OH-90 and OH-91, the inoculum was provided by Colletotrichum-diseased strawberry fruits that experimenters placed within the crop, and the model was operated from July 16 at OH-90 and from August 7 at OH-91.

Predicted disease severities (i.e., sum of visible, infectious, and removed sites) and observed disease severities were compared, and CCC, RMSE, and CRM (Nash and Sutcliffe, 1970; Lin, 1989), which were used to evaluate the goodness-of-fit of predicted vs. observed data, were calculated by using R software as described before. The different sources of predictive errors in the model were identified by calculating Theil’s partial inequality (Theil’s U statistic) coefficients (Smith and Rose, 1995; Fall et al., 2016); among these coefficients, Ubias indicates the mean difference between observed and predicted disease; Uslope indicates the deviations from the 1:1 line; and Uerror indicates the unexplained variance.

The relational diagram of the processes leading to epidemics of Colletotrichum spp. is shown in Figure 2 ; abbreviations are explained in Table 1 . The model is based on host sites, each of which belongs to one of the following mutually exclusive conditions: (i) healthy (H); (ii) latent (L), i.e., without disease symptoms; (iii) visible (V), i.e., with visible lesions; (iv) infectious (I), i.e., with visible lesions producing spores; and (iv) removed (R), i.e., older and non-sporulating lesions removed from the system. The model does not incorporate host growth and senescence and does not account for lesion expansion.

Sites become infected (i.e., they change from healthy sites into latent sites) at an infection rate (INFR), which depends on the abundance of conidia (CON), which can be both primary (CON’) and secondary (CON”) conidia, and on their rain-dependent dispersal rate (DISR). Primary conidia (produced on k overwintered inoculum sources at a sporulation rate of SPOR’) and secondary conidia (produced on lesions on the host plant at a sporulation rate of SPOR”) accumulate in CON’ and CON”, respectively; SPOR’ depends on temperature and wetness duration, and SPOR” depends only on temperature because the moisture required for sporulation is provided by the host plant. Infection rate also depends on a correction factor (CF) for diseased sites (i.e., stages L, V, I, and R), a multiplication factor (Rc) representing the proportion of daughter lesions generated per mother lesion that depends on temperature (RcT) and wetness duration (RcWD). Infection occurs during the period of host susceptibility (HS), which is regulated by the plant growth stage and varies with the host. Latent sites become visible at the end of an incubation period (IP), and visible sites become infectious at the end of a latency period (LP), both of which depend on temperature; the progress of IP and LP depends on an incubation (INCR) and latency (LATR) rate, respectively. Infectious sites produce secondary conidia (CON”) during the infectious period (iP) and finally become removed at a removal rate (REMR) ( Figure 2 , Table 1 ).

At the beginning of model calculations, H = 1 (i.e., the whole crop is healthy), and the model represents the flow from one state to the following one as a proportion of the H, i.e., on a 0 to 1 scale. The model works at hourly time steps.

The model was validated for the acutatum, dematium, gloeosporioides, graminicola, and orbiculare clades. The model’s ability to predict disease development throughout the season was evaluated for 17 epidemics ( Table 2 ) recorded between 1980 and 2019 in Italy, the USA, Canada, and Japan on six hosts, i.e., olive, strawberry, mulberry, grape, bluegrass, and dry bean. In this manuscript, only a few epidemics are described in detail; details on the remaining epidemics are provided in the Supplementary material .

For the acutatum clade, model validation was performed for 4 epidemics on drupes recorded in naturally infested olive orchards in South Italy, and 2 epidemics in artificially infested strawberry fields in Ohio, USA; in the latter case, researchers provided the inoculum by placing affected fruit between the strawberry rows. Epidemics on olive developed between August and December, with final disease incidence on drupes ranging from 7% to 26% ( Table 5 ); CCC between predicted and observed disease ranged from 0.789 to 0.953, and RMSE ranged from 0.022 to 0.047. Epidemics in the two strawberry fields occurred in July to August in OH-90, and in August to September in OH-91, with final disease incidences of 28% and 45%, respectively; in the comparison of predicted and observed disease, CCC = 0.874 and 0.898, and RMSE = 0.055 and 0.061, respectively ( Table 5 ). Across all 6 epidemics of the acutatum clade, CCC = 0.895, RMSE = 0.048, and CRM = −0.099 (the latter indicated a slight tendency of the model toward overestimation). An example of model output for acutatum clade is shown in Figure 3 for IT-17A. In the olive orchard, flowering (BBCH 61, i.e., the beginning of host susceptibility) began on May 15, and harvest occurred at the end of December. From mid-May to the end of December, the average daily temperature was 20.3°C (min = 4.5°C, max = 31.1°C), with an average RH = 71%, a total of 365 mm of rain on 69 rainy days, and a total of 1617 h of leaf wetness ( Figure 3A ). Rains were frequent and intense between September and November, with prolonged wetness periods that led to the prediction of repeated infection periods ( Figure 3B ). Disease outbreak occurred on August 17 (disease incidence 2%) and was followed by a regular disease increase that resulted in a final disease incidence of 18%, which was correctly predicted by the model ( Figure 3B ). For IT-17A, CCC = 0.953, RMSE = 0.022, and CRM = 0.017 (the latter indicated no substantial underestimation).

For the dematium clade, 3 disease progress curves on mulberry in Japan were used to validate the model ( Table 2 ). Epidemics developed on mulberry leaves between August and November, with final disease incidence ranging from 13% to 37%. In the comparison of observed and predicted disease progress curves, CCC ranged from 0.862 to 0.991 and RMSE ranged from 0.014 to 0.055; across all three epidemics, CCC = 0.95, RMSE = 0.036, and CRM = −0.023 (the latter indicated a slight tendency toward overestimation ( Table 5 ). An example of model output for the dematium clade is shown in Figure 4 for JA-94. Disease assessment began in June after the summer-pruning of mulberry trees, and ended in November. During this period, the average daily temperature was 21.9°C (min = 7.0°C, max = 30.1°C), with an average RH = 82%, a total of 625.5 mm of rain on 61 rainy days, and a total of 1688 h of leaf wetness ( Figure 4A ). Regularly distributed rainfall ensured that the epidemic would progress during the season, starting from the time of disease onset in August until November when a sharp increase in disease led to a 37% disease incidence ( Figure 4B ). The model correctly predicted this dynamic; across all three epidemics, CCC = 0.991, RMSE = 0.02, and CRM = 0.078.

Model validation for the gloeosporioides clade was performed for 3 epidemics on grapevines in North Carolina, USA ( Table 2 ). These epidemics developed between June and September, with final disease incidence on berries ranging from 24% to 54%. A small average distance of real data from the fitted line was observed for the three epidemics, with RMSE ranging from 0.012 to 0.101. Concordance between observed and predicted values for NC-80 and NC-82 gave CCC = 0.843 and 0.993, respectively. The NC-81 epidemic is reported in Figure 5 . Berries were susceptible between June 9 (BBCH 71, i.e., fruit set) and October 1 (BBCH 89, i.e., berries ripe for harvest). During this period, the average daily temperature was 24.6°C (min = 14.5°C, max = 29.8°C), with an average RH = 77%, a total of 216.1 mm of rain on 31 rainy days, and a total of 746 h of leaf wetness ( Figure 5A ). Disease incidence on berries was assessed on September 17 and 24, and on October 1, with a final disease incidence of 24%. Contrary to observations, the model anticipated by some days the observed increase in disease at the end of the season ( Figure 5B ), and CCC = 0.497, probably because of an imprecise estimation of the incubation length in relation to the berry growth stage at the time of infection. In the comparison of predicted and observed disease progress curves for the 3 epidemics of the gloeosporioides clade, CCC = 0.910, RMSE = 0.068, and CRM = −0.109 (the latter indicated a slight tendency of the model toward overestimation).

Model validation for the graminicola clade was performed for 3 bluegrass epidemics in New Jersey, USA, in 2009 and 2010, and Michigan, USA, in 1982 ( Table 2 ). The epidemics were observed between May and August, with final disease severity ranging from 16% to 41%. In the comparison of model output and observed data, CCC ranged from 0.947 to 0.997, RMSE ranged from 0.006 to 0.037 ( Table 5 ), and CRM ranged from −0.15 to 0.19 (the latter indicated a slight over- or underestimation of observed disease progress curves, depending on the epidemic). Across all three epidemics of the graminicola clade, CCC = 0.973, RMSE = 0.026, and CRM = 0.018 (the latter indicated a slight tendency of the model toward underestimation). An example of model output for the graminicola clade is shown in Figure 6 . At MI-82, the bluegrass field was inspected at 10-day intervals for anthracnose symptoms from the beginning of May to the end of July. During this period, the average daily temperature was 18.9°C (min = 10.6°C, max = 26.7°C), with an average RH = 75%, a total of 202.6 mm of rain on 31 rainy days, and a total of 651 h of leaf wetness ( Figure 6A ). The disease outbreak at MI-82 occurred on July 7 (disease severity 3%), and the disease increased regularly to a final disease incidence of 25%, which was correctly predicted by the model ( Figure 6B ), with CCC = 0.997, RMSE = 0.006, and CRM = 0.050 (the latter indicated a slight tendency of the model toward underestimation).

To validate the model for the orbiculare clade, 2 epidemics that occurred on white bean in Manitoba, Canada were used ( Table 2 ). Crops were scouted for disease from the end of June, when artificial inoculum was introduced into the field, to mid-August. At MA-14, the disease was recorded only in traces (<1%), so that proper model evaluation was impossible. At MA-15 ( Figure 7 ), the average daily temperature was 18.9°C (min = 10.6°C, max = 26.7°C), with an average RH = 75%, a total of 202.6 mm of rain on 31 rainy days, and a total of 651 h of leaf wetness ( Figure 7A ). Regularly distributed rainfalls and prolonged wet periods ensured a progressive development of the epidemic from the first third of July until mid-August, with a final disease severity of 20%. The model slightly overestimated the disease dynamics in the second half of July ( Figure 7B ), with CCC = 0.896, RMSE = 0.035, and CRM = 0.19 (the latter indicated a tendency of the model toward underestimation). Across both orbiculare clade epidemics, CCC = 0.931, RMSE = 0.027, and CRM = 0.158 (the latter indicated a tendency of the model toward underestimation).

In an overall comparison of predicted versus observed values for the 17 epidemics listed in Table 5 , CCC = 0.928 and RMSE = 0.044; the similarity between the observed data and the fitted line indicated that the model accurately represented the mechanisms leading to Colletotrichum epidemics on different host plants. The model, however, showed a slight tendency toward overestimation (CRM = −0.052) when evaluated for the epidemics on different host plants. The variance explained by the relationship of observed versus predicted data was R² = 0.866; based on Theils’ statistic, the deviation from the unexplained variance was Uerror = 86%, whereas the percentage of error associated with model bias and deviation from the 1:1 line was Ubias = 2% and Uslope = 12%, respectively.