The time step for calculating disease progress is one day. Thus, a pathogen cohort is defined as the number germinating each day and subsequently maturing together. However, weather variables are considered on hourly and daily bases. Dew period will be determined on an hourly basis. We calculate hours of dew (HD) and mean temperature (THD) for those hours only, every 24 hours. In other words, first we find which hours have dew or leaf wetness, then we also find mean temperature for those hours. This is used for germination each day. For calculation of degree days for maturation during latent period and daily sporulation, we will just use mean temperature (T) for all 24 hours. Small t is the index for day of simulation.

The purpose of the model is to simulate disease dynamics during early reproductive growth stages of soybean. Depending on when and where the crop is planted in Brazil, this period likely occurs during November through March. Table 1 shows the monthly mean temperatures for these months in Brazil. All means are in the range 21.3-28.3°C. Except for Rio Grande do Sul in southern Brazil, where the mean varied by 3°C over the five months, the mean temperatures at each location varied by<1.5° C demonstrating relatively constant temperatures. Daily fluctuating temperatures will certainly exceed the range 21-28° C, but the model must perform well over this range.

Leaf area development is a function of soybean physiological days, pd, which is linearly related to temperature on a daily basis (Wilkerson et al., 1985).

where Tav is the average daily temperature, and Tmin, Tmax and Topt are the minimum, maximum and optimum temperatures for growth of soybean. The values of Tmin, Topt and Tmax were obtained from Yang et al. (1991) as Tmin = 7° C, Tmax = 45° C and Topt = 30° C.

Cumulative physiological days, Cpd, equals the sum of pd over all days after germination of soybean. Leaf area per plant is calculated from Cpd using one of the following two equations (Yang et al., 1991).

Equation 2a is for medium and long maturing cultivars and equation 2b is for short maturity cultivars. The leaf area index per m² (LAI) is calculated after multiplying the leaf area per plant by the number of plants in a m². We used 30 plants per m², a typical value for Brazil.

In our model, we assume that the disease does not limit development of leaf area. This simplification is reasonable for low levels of disease severity during the early phases of an epidemic when the management of the disease is either successful or not (Scherm et al., 2009). This soybean development function along with the entire model is inadequate for describing epidemics that overwhelm a soybean crop. If the pathogen invades the crop after all leaves have formed in determinate growth soybeans, then the reduction in leaf growth may be less of a concern.

Dew period (HD) was calculated using classification and regression tree well known as CART model (Gleason et al., 1994). The CART model utilizes dew point depression (DPD), hourly air temperature, dew point temperature, wind speed (m/s), and relative humidity (RH) data. The mean temperature during the hours of dew each 24-hour period, THD, was calculated as expected by averaging the 24 or fewer hourly temperatures.

All of the following studies were performed in labs or greenhouses. In Brazil, Alves et al. (2006) found that uredospores germinated at all temperatures ranging from 8 to 30°C and that the optimum temperatures for germination varied from 15 to 25°C. These results agreed with Tan et al. (1996), who showed that the optimum temperature range for uredospore germination in China was 15 to 26°C. Kochman (1979) in Australia found that germination dropped by half between 24.5 and 28.5°C with a maximum proportion of 0.54. According to Singh and Thapliyal (1977) in India, no uredospores germinated at 10 and 35°C. The maximum proportion was 0.33 at 25°C after 24 hours. In a different experiment, they found that proportional germination ranged from 0.05 to 0.18 at constant 30°C to 0.30 when temperatures varied from 20-30°C. As long as temperatures were 20-25°C, germination ranged from 0.45 to 0.61, with germination at higher end as temperature increased. Marchetti et al. (1976) observed a maximum 0.40 proportional germination in darkness for a Taiwanese isolate with the optimal temperatures 20-25°C. They stated that germination declined rapidly between 25 and 28.5°C and ceased between 28.5 and 31°C. In a different experiment with isolates from four countries (not Brazil), Marchetti et al. (1976) observed much higher proportional germination in darkness (0.60-0.80) and a lower range of optimal temperatures. Melching and Bromfield (1975) and Melching et al. (1989) found that temperatures below 9°C or above 28°C did not favor germination of uredospores. Melching and Bromfield also determined that the optimum temperature for germination was from 12 to 21°C, while Melching et al. (1989) found 18-26.5°C optimal. Melching et al. (1988) observed a germination range of 0.41-0.64 at 21°C and discovered that germination rates on agar are much higher than those on leaves after the same period of incubation. Thus, many of the results described above are overestimates. However, after 6 hours, the rates on the two substrates tended to converge to 0.55 proportional germination.

We derived our function for calculating proportional germination from two functions of temperature developed by Bonde et al. (2007) based on observations of a Brazilian isolate. (They did not observe significant differences across the three isolates). They observed an optimum temperature of 22-23°C and their functions have optima at 23-24°C. We simply took the average of the five parameters from their two functions with the same structure and created

where Y is proportion of spore germination as a function of temperature, THD is the mean temperature during the period of germination each day. Bonde et al. (2007) defined Tmin as 9.57 and Tmax as 29.54. The maximum proportion calculated with our version is 0.54.

We considered successful germination would always lead to infection on leaves of susceptible soybean plants. Bonde et al. (2007) also measured infection and determined the minimum and maximum temperatures for infection to be on the average 10.2°C and 29.8°C, which are essentially the same as those for germination.

For temperatures 20-24°C, rust lesions only develop after a minimum 6-hour dew period on leaves (Kitani, 1960; Melching et al., 1989). Marchetti et al. (1976) found the same threshold at temperatures 15-25°C and much more germination after 10 hours of dew. Therefore, germination was estimated as a function of both temperature and dew hours with

where G is the proportion of uredospores on leaves that germinate, and HD is the hours of dew or leaf wetness duration on soybean leaves each day. This function permits higher germination rates for temperatures below 20 °C than those observed by Marchetti et al. (1976) ( Figure 1 ) because of dew period/temperature interactions. Parameter G is multiplied by the number of uredospores in the field about to germinate. If uredospores do not germinate within 24 hours in the model, they are no longer viable.

We define latent period (LP) as the time between host infection and onset of pathogen sporulation. The latent period determines the maturation of latent lesions and the interval between successive generations of pathogens. For our model, we used a temperature-based function to estimate the latent period. Table 2 shows the latent period in days observed in a variety of studies. Melching et al. (1979) studied isolates from around the world. We display the observations for older plants (37-42-days-old soybean) made by Melching et al. (1989).

A temperature of 6°C has been used as a base for calculating degree days (Danelli and Reis, 2016). We therefore calculated degree days (DD) by subtracting 6°C from the mean daily temperature in each study ( Table 2 ). The Danelli and Reis (2016); Marchetti et al. (1975); Melching et al. (1979) and Melching et al. (1988) results are in the range 150-185 DD for LP but this may only be valid for mean daily temperatures of 15-30°C (or shorter range). The warmest temperatures used in a study ( Table 2 ) were the 22-32°C range of Kochman (1979). The extremely high calculated DD of 231 suggests that maturation slows above 27-30°C; subtracting 5°C per day (32-27) gives 176 DD. There is significant variability no matter how LP is evaluated. Therefore, we chose to use the most recent and most relevant results (Danelli and Reis, 2016) for Brazil. The calculated mean of the five values is 156.8 DD, but we simplified this to 155 DD for the model. Once the cumulative DD exceeds 155 for each daily cohort of pathogen, the cohort will mature into the infectious period and start sporulating.

The infectious period (IP) is defined as the period during which cohorts of lesions produce uredospores under favorable environmental conditions. Marchetti et al. (1975) observed a 3-week infectious period. The observations of Melching et al. (1979) and Yeh et al. (1982) support an average IP of 5 weeks. For our model, we used a 35-day IP from first sporulation to end of sporulation. Each day, a cohort of lesions will shift into the next of 35 age classes during the IP. Maturation is calculated first for oldest cohort then proceeds in declining order to youngest cohort. After day 35, the cohort will become senescent (dead).

Lesion size and number determine the level of disease severity. Melching et al. (1979) studied four Asian and Australian isolates. They inoculated susceptible soybean plants when the fourth trifoliate was fully expanded. Two weeks after inoculation, the mean lesion areas were 0.61-0.77 mm² for three isolates and 0.3-0.43 mm² for the Australian isolate. Seven weeks after inoculation, the mean lesion areas for the four isolates were 1.76-1.81, 1.70-1.89, 1.75-2.01, and 1.43-1.45 mm². Hernández (2004) described lesions as 1-2 mm in diameter, which suggests that circular lesions range in size from 0.79 mm² to 3.14 mm². Alves et al. (2006) observed similar lesion sizes at different combinations of leaf wetness duration and temperature. We used a lesion size of 2 mm² in the model, which is the size also chosen by (Yang et al., 1991).

Sporulation is mainly determined by air temperature. Bonde et al. (2012) discovered that after germination, relative humidity had no effect on number of lesions or uredospores. Under a greenhouse temperature regime of 20-29°C, Melching et al. (1979) calculated 2,028, 3,768, 6,600, and 6,268 uredospores per lesion for isolates from four different countries. Thus, the mean uredospore production during the lifespan of an infectious lesion on a soybean leaf was 4,666 uredospores per lesion. Production peaked about 10 days into infectious period and declined afterwards.

Bonde et al. (2012) made observations over the first 3 weeks of the IP to determine the number of uredospores produced per lesion as a function of constant temperature. The following pairs of temperature and uredospores per lesion represent their data: (17 °C, 750), (21 °C, 4,050), (25 °C, 5,500), (29 °C, 3,300), and (33 °C, 1,400). No uredospores were produced at 37°C. The mean number of uredospores at the middle three temperatures is 4,283, which is very close to the mean from the Melching et al. (1979) study. We fit a function to the data of Bonde et al. (2012) with zero at 37°C ( Figure 1 ). In the model, the total number of uredospores potentially produced by infectious lesions is calculated as

where T is mean daily temperature. The R^2 = 0.97. The value of Z is circa 0 at temperatures 16 and 34°C. No sporulation occurs beyond those limits.

Based on the triangular shape of production over the infectious age classes seen in figures of Yeh et al. (1982) and Melching et al. (1979), the proportion of uredospores produced in age class j of the IP (with peak at age 10 days) is

Essentially, K distributes the total in Z over the entire IP. Then the number of uredospores produced by each cohort per 24-hour period per unit ground area, P_t is

where j is age class in IP and I is the number of lesions in a cohort. P_t is calculated before maturing cohorts to next age class.

Uredospores cannot adhere to leaves during the first 30 minutes after landing on the surface (Dufault et al., 2010) and there is a chance that some proportion of uredospores may be removed from the leaf surface (Andrade et al., 2009). Although uredospore removal from soybean leaves depends on several factors including canopy location and duration of rainfall, Dufault et al. (2010) found that, for rain intensities of 45-85 mm, the percentage of uredospores removed from leaves ranged from 23% to 82% as the duration increased from 1 to 30 minutes. However, rainfall of 85 mm in 30 minutes seems extremely high, so we focused on 45 mm as a threshold for designating an hour as contributing to spore removal. Furthermore, because it is difficult for a model with a time horizon of a few months and a time step of 1 day to account for these minutes of rainfall, we chose to model this effect as 1 - exp(-0.1 x HR), where the proportion of uredospores removed is a function of the number of hours in a 24 hour period (HR) in which rainfall exceeds 45 mm in each hour. This implies that an hour with rainfall exceeding 45 mm results in 10% loss of the uredospores and 12 hours with rainfall exceeding 45 mm in each hour results in 70% loss of the uredospores.

The number of uredospores adhering to soybean leaves on day t, N_t is

where H is the number of uredospores immigrating into field, P is the number produced in the field, and E is the proportion emigrating from the field due to wind. Zidek (2007) reported that about 10% of uredospores may escape a closed canopy due to wind and turbulence. Thus, E=0.1 in the model. CC is the proportion of uredospores captured by leaf area, which increases as LAI increases

where 0.75 is light-extinction coefficient for 30 reproductive-stage plants per m² (Flénet et al., 1996) and D is disease severity (proportion of leaf area covered with infectious and senesced lesions).

The number of lesions in cohort of LP defined by day of germination t, L_t, is

where N is number of uredospores adhering to soybean leaves and G is proportion germinating.

The proportion of leaf area (undersides only) covered by live or dead lesions is defined as disease severity, D, and calculated per m² each day as

where I_tot(t) is the sum of I(j,t) in a m² over all age classes j, S is the total number of senesced (dead) lesions in a m², and 2 is the lesion size in mm². The factor 10⁶ converts leaf area to number of mm² patches. Although, our focus is on early stages of an epidemic when disease severity is low, the model does include the following statements to maintain logic as severity gets close to 1. If D_t > 1 in Equation 11, then D_t = 1 and I_{1, t} = [1-D_t-1] x LAI x 0.5 x 10⁶. This last statement adjusts the incoming age 1 infectious lesions to the maximum permitted given the amount of leaf area remaining without lesions.

A field experiment was conducted in the 2019/2020 growing season at four locations in Brazil ( Table 3 ). BMX-Potência, an ASR sensitive cultivar, was sowed with spacing of 0.5 m between rows and 15 plants per m of row. The experiment consisted of an early and late sowing date (at three locations) to cover low and high ASR spore inoculum concentration during the growing of soybeans. Date of disease outbreak, disease progress and final disease severity were observed. The progress in ASR disease severity was measured at 7 days interval starting from the beginning of the flowering stage (R1) of soybeans.

Weather variables such as rainfall, temperature, relative humidity and wind speed were obtained from weather stations installed in the field experiment locations. Dew period or leaf wetness duration was calculated using classification and regression tree well known as CART model (Gleason et al., 1994; Kim et al., 2005).

The model was simulated to produce the progression of ASR in a soybean field on a daily basis after arrival of the first uredospores. Initial outbreak of disease in the fields were observed 77 to 82 days after planting (DAP) for the early sowed experiments and 52 to 63 days after sowing for the late sowed experiments. We subtracted the typical 9-day LP from date of first disease observation to initialize the model with uredospore arrival. For instance, if the first disease was detected 77 days after sowing, the first uredospores were introduced 68 days after sowing. The number of uredospores initially arriving in the modeled soybean field is 3000 to 4000 per day per m² over a 10-day period. Thus, equal numbers of uredospores arrive in the field on days 68-77. The initial number was chosen to produce a close fit of modeled severity to the initial observation of disease severity.

Model performance was evaluated using the disease progress data obtained from the field experiments. Model simulated severity curves were compared against the ASR severity observed between 50 and 98 DAP at weekly intervals. The correlation between the model simulated and observed ASR severity was evaluated by calculating a coefficient of determination (R²).

Model sensitivity analysis was carried out for the following input parameters: the number of days when uredospores were detected in the soybean field (days with spores), a factor for proportion of uredospores captured by leaf area (canopy cover factor or light-extinction coefficient), wind factor determining escape of uredospores away from field, and rainfall factor removing uredospores from leaves ( Table 4 ). The model was simulated under the conditions observed at each of the four locations for our field studies ( Table 3 ). We used the model sensitivity analysis to explore the response of model output to modest changes in the model input values and to understand the sensitivity of results to uncertainties in the input values of the model.

Key weather conditions that drive ASR infection and development are presented in Figure 2 . The minimum temperature varies 18 to 19°C and the maximum temperature varies between 28 to 32°C during the growing season of soybeans across the four locations in Brazil. The mean temperature for the locations is in the range of 20.5 to 23.8°C. Based on the weather for the experimental sites, the LP lasted 9 days over most of the observed periods with few LP >9. The average daily leaf wetness duration was 15 to 22 hours. Only 6 days exhibited less than 6 hours leaf wetness duration at Chapadão do Sul (MS), 4 days at Indianapólis (MG) and 1 day at Mogi-Mirim (SP). Once uredospores invaded the soybean plots, the local microclimate was conducive for ASR germination and development to happen.

First symptom of ASR was observed 77 to 84 DAP in early planted experiments and 52 to 63 DAP in late planted experiments. Final ASR severity varied between 31 to 62% in early planted and 37% to 100% in late planted soybeans.

As displayed in Figure 3 , the model results generally matched the daily dynamics of ASR progress in the field trials. The predictions reproduced the observed ASR severity well with R² value of 0.84 and root mean square error (RMSE) of 4.71% ( Figure 4 ). The model tends to underestimate some observed severities at low severity levels, however the amount of underestimate varied by trial location and planting period. Additional sensitivity analysis was performed to further investigate the influence of input parameters on the disease severity predictions.

Changes in simulated ASR severity on day 70 after sowing due to changes in the values of four model input parameters are presented in Figure 5 . The results were slightly sensitive to canopy cover factor (light-extinction coefficient). Since ASR tends to infect soybean during its reproductive stage when the canopy is dense, assuming a moderate canopy cover factor (0.75) seems reasonable, but the influence of soybean cultivar should be considered in future work. The results were insensitive to the wind factor and the rain factor.

The results were sensitive to days with initial spores invading the soybean field. Reducing the duration of the initial spore arrival from the standard 10 days to 1-2 days caused an important reduction in the severity, but simulated results for durations of 5-20 days were similar ( Figure 5 ). For instance, the severity at Chapadão do Sul simulated with an invasion period of 10 days of spore was 12.3% and the severity simulated with spore arrival of only 1 day was 6.3% ( Figure 5A ). These results indicated that spore traps or other observations should measure not only the first day of arrival but the subsequent days. Additional invasion by uredospores after the first week will be less important to the epidemic already started by the earliest detected spores.