Answer
The number of people infected will start to decrease after approximately $9$ days.
Work Step by Step
The number of people infected (in hundreds) after $t$ days since the start of the infection, is given by the function :
$P(t)=\frac{10ln(0.19t+1)}{0.19t+1}$
The number of people infected will start reducing when $\frac{dP}{dt}<0$ since the function $P(t)$ is decreasing as $\frac{dP}{dt}<0$.
i.e. $\frac{10(0.19t+1)\frac{0.19}{(0.19t+1)}-10ln(0.19t+1)0.19}{(0.19t+1)^2}<0$
$=1-\ln(0.19t+1)<0$
$=\ln(0.19t+1)>1$
$=0.19t+1>e$
$=t>\frac{e-1}{0.19}\implies t>9.04 \hspace{0.1cm}\text{days}$
Therefore the number of persons infected will start to reduce after approximately $9 \hspace{0.1cm}\text{days}$.
