 Solution

If the seed $v_0$ is positive, infection initially increases precisely whenIndeed, $v^{\prime }(0)=({\mathcal{R}}_0-1)v_0$. Susceptibility is nonincreasing, so if ${\mathcal{R}}_0\le 1$, the infectious count never starts an increasing phase. When ${\mathcal{R}}_0>1$, it grows until $u=1/{\mathcal{R}}_0$, the SIR infection peak, and then declines.

The basic reproduction number compares transmission and removal: one infectious individual generates infections at initial rate $\beta S_0$ and remains infectious for mean time $1/\gamma$. Their product $\beta S_0/\gamma$ is the expected number of secondary cases at that susceptible level. Above one, an infectious generation can replace itself with more cases. An exactly zero infectious seed remains zero regardless of this threshold.