= Solution
Write $N=S+I$. In
$$
\dot S=N-NS-\beta IS,
$$
the term $N$ represents production of healthy new plants by both healthy and diseased plants at unit per-capita rate. The term $-NS$ is density-dependent loss of healthy plants through competition, and $-\beta IS$ is loss through <mass-action infection>. In
$$
\dot I=-NI+\beta IS,
$$
the term $-NI$ is the same density-dependent loss for diseased plants, while $+\beta IS$ is the matching gain when healthy plants become diseased. Thus $\beta$ is the transmission-rate coefficient per healthy-diseased encounter.
Adding the equations cancels the infection terms and gives the <logistic differential equation>
$$
\boxed{\dot N=N(1-N).}
$$
The disease therefore changes how the population is divided between healthy and diseased plants but has \b[no effect on the total-population dynamics]: infected plants reproduce at the same rate, all offspring enter the healthy class, and infection causes no additional death in this model.
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