Explain the product of infected and susceptible proportions

$1/[y(1-y)]=1/y+1/(1-y)$. Hence $\ln[y/(1-y)]=kt+C$, since the ratio is positive. Thus $y/(1-y)=Ae^{kt}$ with $A=e^C>0$, giving $y=Ae^{kt}/(1+Ae^{kt})$. The equilibrium endpoints $0,1$ are outside this question’s explicit strict range.

$1/4=A/(1+A)$ gives $A=1/3$, so $y=e^{kt}/(3+e^{kt})=1/(1+3e^{-kt})$. It increases through $(0,1/4)$, approaches $1$ as $t\to\infty$, and its formal past limit is $0$. Since $y^{\prime\prime}=k^2y(1-y)(1-2y)$, it is initially concave up and changes to concave down at $(\ln3/k,1/2)$. The formula-based sketch resolves the early concavity that the rough source sketch does not distinguish.

More infected people, represented by $y$, create more opportunities to pass on infection. More susceptible people, represented by $1-y$, create more opportunities for those contacts to cause new infections. Under homogeneous mixing the rate is proportional to their product $y(1-y)$, with positive proportionality constant $k$.