= Solution
If the seed $v_0$ is positive, infection initially increases precisely when
$$
\boxed{\mathcal R_0>1.}
$$
Indeed, $v'(0)=(\mathcal R_0-1)v_0$. Susceptibility is nonincreasing, so if $\mathcal R_0\leq1$, 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.
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