Preparing to Fit a Segmented Model

The segmented model you plan to fit has two parts: a logistic growth model and an asymptotic regression model.

Choosing starting values for the logistic model

The logistic model has the form:

Generally, b₁>0, b₂>0, and b₃>0. This model has an "S" shaped curve.

b₁ represents the upper asymptote for viral growth. Looking at the chart, even the largest values fall short of 0.65, so that's a reasonable starting value.
b₂ is the ratio between the value of y when x=0 and the upper asymptote. A reasonable starting value is the ratio of b₁ to the minimum value of y. Looking at the chart, say that's about 0.65/0.13=5.
b₃ can be roughly initially estimated by the slope between two "well separated" points on the plot. Looking at the chart there are a few points about x=3, y=0.12, and about x=19, y=0.60. The slope between these points is (0.60-0.12)/(19-3)=0.03, thus a rough initial estimate for b₃ is 0.03.

Choosing starting values for the asymptotic regression model

The asymptotic regression model has the form:

When a₁>0, a₂>0, and a₃<0, this model initially decreases quickly with increasing values of x, but then it slows and finally tapers off just above the value a₁.

a₁ represents the lower asymptote for the proportion of infected e-mails. The lowest value this can be is 0, so that's a reasonable starting value.
a₂ is the difference between the value of y when x=20 and the lower asymptote. A reasonable starting value is the maximum value of y minus a₁. Looking at the chart, say that's about 0.6-0.0=0.6.
a₃ can be roughly initially estimated by the slope between two "well separated" points on the plot. Looking at the chart there are points about x=20, y=0.6, and about x=40, y=0.1. The slope between these points is (0.6-0.1)/(20-40)=-0.025, thus a rough initial estimate for a₃ is -0.025.