Example 1, a monocyclic epidemic: Flax wilt is caused by the fungus Fusarium oxysporum f. sp. lini. Chlamydospores of the fungus persist for several years in the soil, and when flax is planted in an infested field, the young plants are infected through the roots. An extensive soil survey was made of a heavily infested field, and it was found to contain an average of 57 colony-forming units per gram of soil. When a susceptible flax cultivar was planted in this field, the percent of plants showing wilt symptoms increased with time as follows:

Days After % Plants Planting Infected 10 18 20 56 30 82 40 91 50 96 60 98

In a plot of disease progress, note how the percent infection asymptotically approaches 100 percent.

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Flax wilt disease progress

To estimate the product, QR, we first have to convert percent infection to the proportion, x, and then using the transformation appropriate for the monocyclic model, calculate ln(1/(1-x)).

t x ln(1/(1-x)) 10 .18 0.198 20 .56 0.821 30 .82 1.71 40 .91 2.41 50 .96 3.22 60 .98 3.91

From a plot of ln(1/(1-x)) versus t, we can fit a straight line to the data points using least squares regression.

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Flax wilt, multiple hit transformation

The slope of the line estimated by the regression equation is 0.076, which is the value of QR. Therefore,

       R = 0.076/57 = 0.0013/CFU/Day.

Example 2, a polycyclic epidemic: Halo blight of beans is caused by the bacterium Pseudomonas syringae pv. phaseolicola. The major source of initial inoculum is infected seeds that when planted give rise to plants with lesions on the primary leaves. Bacteria produced in these lesions are splash dispersed to adjacent healthy plants. New lesions can themselves produce secondary inoculum within about 4-5 days. Under conditions moderately favorable for disease development, the following observations were made of disease progress:

Days After % Plants Planting Infected 10 1 20 4 30 15 40 31 50 65 60 88 70 94

Disease progress shows the sigmoid-shaped curve characteristic of a polycyclic epidemic.

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Halo blight disease progress

As with the previous epidemic, we have to convert the percents to proportions (x), but this time the transformation that we use is ln(x/(1-x)).

t x ln(x/(1-x)) 10 .01 -4.60 20 .04 -3.18 30 .15 -1.73 40 .31 -0.80 50 .65 0.62 60 .88 1.99 70 .94 2.75

By plotting ln(x/(1-x)), sometimes called the logits of x, versus t, we can fit a straight line to the data points with least squares regression.

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Halo blight, logistic transformation

The regression gives us a slope of 0.124/day, which is our estimate of the apparent infection rate, r.

Obviously the more data points that we have, particularly if they are relatively evenly distributed on both sides of 50% infection, the better estimate we will have of the apparent infection rate. However, it is possible to make a rough estimate of the apparent infection rate with just two data points. Let us suppose for a moment that instead of observations every ten days during the epidemic, we only made two observations, one early (day 10) and one late (day 70). How might we estimate the apparent infection rate? In this case we would be using only the first and the final points in the plot of the transformed data above and calculating the slope as the rise over the run:

      r  =  (ln(0.94/(1.0-0.94)) - ln(0.01/1.0-0.01))) / (70 - 10)

         =  (2.75 + 4.60) / 60

         =  0.123/day