 Solution

Write $N=S+I$. Inthe 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. Inthe 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 equationThe disease therefore changes how the population is divided between healthy and diseased plants but has 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.