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Case Study #1: A simple generic infection model for foliar fungal plant pathogens

Magarey, R. D., Sutton , T. B., and Thayer, C. L. 2005. A simple generic infection model for foliar fungal plant pathogens. Phytopathology 95: 92-100.

Many models have been developed for forecasting the probability of infection for particular plant diseases, but there is also the need for generic models that can be applied when new and little-studied pathogens are introduced. Magarey et al. (2005) developed a generic infection model based on temperature and wetness duration. The model is generic in the sense that it was developed to describe any pathosystem when appropriate parameters are supplied.

The principle behind this model is what Magarey et al. (2005) defined as the Critical Disease Threshold (CDT): 20% disease incidence or 5% disease severity under conditions of non-limiting inoculum concentration. Furthermore, the model incorporates a measure for the Wetness Duration Requirement (WDR) function, denoted as W(T), which is the number of hours to reach the CDT at a temperature T. Upper and lower bounds of W(T) are fixed at Wmax and Wmin, respectively. Other parameters in the model are Tmin, Topt and Tmax, which denote the minimum, optimum and maximum temperature (°C) for infection, respectively.

Magarey et al. (2005) used the following model for predicting the probability of infection based on temperature (Yin et al. 1995):

This model is based on the Beta function, commonly used as a probability function in statistics to describe skewed distributions, and was originally introduced to describe the effect of temperature on the rate of crop development (Yin et al. 1995). Since it describes a probability, it has values between 0 and 1. So, f(T) = 0 at T = Tmin and T = Tmax and f(T) = 1 at T = Topt. The Beta function is flexible enough to describe many curves, namely, exponential, positive linear, parabolic and negative responses across a range of temperatures (Garrett et al. 2007).

Using the parameter values from Table 2 in Magarey et al. (2005), an illustration of the temperature response function for Venturia inaequalis, causal agent of apple scab, can be generated in R using the following code:

fT <- function(Tmin,Topt,Tmax,Wmin,Wmax){
  T = seq(0,35,1)
  fT = (Tmax-T)/(Tmax-Topt)*
      ((T-Tmin)/(Topt-Tmin))^((Topt-Tmin)/(Tmax-Topt))
  new.fT = ifelse(fT<0, 0, fT)
  WT = Wmin/new.fT
  new.WT = ifelse(WT<Wmax, WT, Wmax)
  output=plot(T, new.fT, type='b', pch=20, lty=1, col='black',
              xlab='Temperature',
              ylab='f(T) = Probability of Infection')
  return(output)
} 
fT(1,20,35,6,40.5)

The resulting graph looks like

Click to enlarge.

The complete generic infection model takes into account f(T) and wetness duration and is defined as:

Magarey et al. (2005) estimated the values of the five parameters Wmin, Wmax, Tmin, Topt and Tmax from published literature or best estimates based on current knowledge of pathosystems. Daily or hourly temperature data were used and T was the daily (or hourly) mean temperature.

Using the parameter values in Table 2 of Magarey et al. (2005), an illustration of the generic infection model for Venturia inaequalis can be generated in R using the following code:

fT <- function(Tmin,Topt,Tmax,Wmin,Wmax){
  T = seq(0,35,1)
  fT = (Tmax-T)/(Tmax-Topt)*
      ((T-Tmin)/(Topt-Tmin))^((Topt-Tmin)/(Tmax-Topt))
  new.fT = ifelse(fT<0, 0, fT)
  WT = Wmin/new.fT
  new.WT = ifelse(WT<Wmax, WT, Wmax)
  output = plot(T, new.WT, type='b', pch=20, lty=1, col='black',
                xlab='Temperature',
                ylab='f(T)')
  return(output)
} 
fT(1,20,35,6,40.5)

With the resulting plot

Click to enlarge.

Suggested Exercise

Compare different risk curves for the following pathosystems and corresponding temperature and wetness data. You will need to set up a vector of this information to use with the two functions. Also, using plotting techniques described throughout the ecology and epidemiology in R series, try plotting different curves together to compare responses for different pathosystems to a range of environments.

Due to differences among pathogens or their hosts, additional methods for calculations and disease assessments were used, some of which are listed below:

The authors have compared original observations and model predictions for 52 different pathogens using Pearson’s correlation coefficient (r) and Root Mean Square (RMS). In order to take into account the influence of Wmin in RMS, they have also used Standardized Root Means Square (SRMS) = RMS/Wmin. The average and median values of r were found to be 0.83 and 0.94 respectively. Likewise, mean values of RMS and SRMS were 4.8h and 0.6 respectively.

This model is useful because it does not require extensive amounts of biological data. It can incorporate subjective estimates of Tmin, Topt, Tmax, Wmin and Wmax, which can be obtained from published literature for many pathosystems. Moreover, all five parameters have a clear biological meaning (Yan and Hunt 1999). Another good point is that it uses the temperature response approach, which is widely applicable. However, the model does not include the interaction between temperature and wetness, so will be less useful for pathosystems for which this interaction is important. The validation provided by the authors also testifies to its successful prediction in most cases. The model can certainly be adapted to exotic pathogens because of its generic nature and is being used by the USDA Animal Plant Health Inspection Service for several applications (R.D. Magarey, personal communication).

Next, Development and evaluation of a disease prediction model