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Examples of growth models for plant disease progress

In plant pathology we are often interested in studying disease progress over time, where time (*t*) is modeled as a continuous variable rather than as a discrete variable. Many different population growth models have been used for modeling disease progress curves (Gilligan 1990; Madden *et al.* 2007). Five common growth curve models are discussed below along with their assumptions. For the illustration of some of these models in R, we refer to Table 1 from Gottwald *et al*. (1989) which compared different models for disease progress, based on nonlinear regression analysis. In this table, the initial disease incidence for orange and grapefruit is *y*_{0} = 1/(13*45) = 0.0017 and for swingle is *y*_{0} = 1/(15*67) = 0.001.

Model | Differential Equation Form | Integrated Form | Linearized Form |
---|---|---|---|

Exponential | logy= logy + _{0}rt |
y = y_{0}exp(rt) |
logy = logy_{0} + rt |

Monomolecular | ln{1/(1-y)} = ln{1/(1- y_{0})} + rt |
y = 1-(1-y_{0})exp(-rt) |
ln{1/(1-y)} = ln{1/(1- y_{0})} + rt |

Logistic | y = 1/[1 + {-lny_{0}/(1- y_{0}) + rt}] |
y = 1/[1 + exp{-lny_{0}/(1- y_{0}) + rt}] |
ln(y/(1-y) = ln{y_{0}/(1- y_{0}) + rt} |

Gompertz | dy/dt = ry ln(1/y) = ry(-lny) |
y = exp(lny_{0}exp(-rt)) |
-ln(-lny) = -ln(-lny_{0}) + rt |

Weibull | dy/dt = c/b{(t-a)/b}^{(c-1)exp[-{(t-a)/b}c]} |
y = 1-exp[-{(t-a)/b}^c] |
ln[ln{1/(1-y)}] = -clnb + ln(t-a) |

The exponential model assumes that the absolute rate of disease increase (*dy/dt*) is proportional to the disease intensity (*y*). The following R code creates a function, `plotexp`

, that plots an exponential relationship between disease incidence and time. After you create the `plotexp`

function, you can apply it using different values of the parameters to better understand how the parameters affect the shape of the curve. In the illustration, the parameter *y*_{0} is set to 0.0017, the parameter *r* is set to 0.01579, and the maximum time for the illustration is set to 100. Try changing those values when applying the `plotexp`

function.

## Exponential Model Example plotexp <- function(y0,r,maxt){ curve( y0*exp(r*x), from=0, to=maxt, xlab='Time', ylab='Disease Incidence', col='mediumblue') } plotexp(0.0017, 0.01579, 100)

The monomolecular model assumes a carrying capacity of one, that is, the maximum level of disease is one, so disease severity or incidence is measured as a proportion. Diseased plant tissue may only lie between zero (healthy) and one (complete disease). It also assumes the absolute rate of change is proportional to the healthy tissue i.e., (1-*y*). After creating the `plotmono`

function and trying the example set of parameter values, try replacing the parameter values with others to see how the shape of the relationship changes.

## Monomolecular Model Example plotmono <- function(y0,r,maxt){ curve( 1-(1-y0)*exp(-r*x), from=0, to=maxt, xlab='Time', ylab='Disease Incidence', col='mediumblue') } plotmono(0.0017, 0.00242, 2000)

The logistic model assumes that the absolute rate of change in disease level depends on both healthy tissue (*y*) and diseased tissue (*1-y*) present at the time. The curve is perfectly symmetric with an inflection point at *t* = 1/*r*ln *y*_{0}/(1- *y*_{0}) when *y* = 1/2. That is, *dy/dt* increases up until *y* = 1/2 and decreases thereafter. After creating the `plotlog`

function, try applying it with different parameter values.

## Logistic Model Example plotlog <- function(y0,r,maxt){ curve( 1/(1+(1-y0)/y0*exp(-r*x)), from=0, to=maxt, xlab='Time', ylab='Disease Incidence', col='mediumblue' ) } plotlog(0.001, 0.01636, 1000)

The Gompertz model assumes that the absolute rate of change depends on *y* and ln(1/*y*) and is very similar to the logistic model. However, the Gompertz model is more asymmetric, with an inflection point attained at 0.37(1/*e*) instead. After creating the `plotgomp`

function and comparing it to the other models before, try changing the parameter values to see the effect on the shape of the curve.

## Gompertz Model Example plotgomp <- function(y0,r,maxt){ curve( exp(log(y0)*exp(-r*x)), from=0, to=maxt, xlab='Time', ylab='Disease Incidence', col='mediumblue' ) } plotgomp(0.0017,0.02922, 250)

The Weibull model includes a larger number of parameters, and so can describe more complicated curves. The parameters are:

*a*- units of time, indicating the time of disease onset,*b*- the scale parameter,inversely related to the rate of disease increase,*c*- the unitless shape parameter that controls the skewness of the curve.

If *c* = 1, this model is identical to the monomolecular model with the rate parameter *r* = 1/*b* and the initial disease level *y*_{0} = 1-exp(*a/b*). With suitable values of parameters 1, 2, and 3, other models can be approximated by the Weibull model.

## Weibull Model Example plotweib <- function(a,b,c,maxt){ curve( 1-exp(-((x-a)/b)^c), from=0, to=maxt, xlab='Time', ylab='Disease Incidence', col='mediumblue' ) } plotweib(1, 331.10, 10.04, 500)

Note that for some parameter combinations, these different models may produce very similar curves. For this reason, when the fit of these curves is being compared for a real data set, more than one of the models may give a good fit. In that case, it may be best to go with the simplest model. On the other hand, if many different data sets are being compared, a more complicated model such as the Weibull may make it easier to obtain reasonable fits to all the data sets being considered. For an in-depth comparison of the different uses of growth models in plant pathology we refer the reader to Gilligan (1990) and Madden *et al.* (2007).