Without loss of generality we assume that $\min \left(a_{1}, a_{2}, \ldots, a_{2019}\right)=a_{1}$. Note that if $a, b, c$ $(b \neq c)$ are positive, then $\frac{a}{|b-c|}>\min \left(\frac{a}{b}, \frac{a}{c}\right)$. Hence $$S=\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\cdots+\frac{a_{2019}}{\left|a_{1}-a_{2}\right|}>0+\min \left(\frac{a_{2}}{a_{3}}, \frac{a_{2}}{a_{4}}\right)+\cdots+\min \left(\frac{a_{2017}}{a_{2018}}, \frac{a_{2017}}{a_{2019}}\right)+\frac{a_{2018}}{a_{2019}}+\frac{a_{2019}}{a_{2}}=T.$$ Take $i_{0}=2$ and for each $\ell \geqslant 0$ let $i_{\ell+1}=i_{\ell}+1$ if $a_{i_{\ell}+1}>a_{i_{\ell}+2}$ and $i_{\ell+1}=i_{\ell}+2$ otherwise. There is an integral $k$ such that $i_{k}<2018$ and $i_{k+1} \geqslant 2018$. Then $$T \geqslant \frac{a_{2}}{a_{i_{1}}}+\frac{a_{i_{1}}}{a_{i_{2}}}+\cdots+\frac{a_{i_{k}}}{a_{i_{k+1}}}+\frac{a_{2018}}{a_{2019}}+\frac{a_{2019}}{a_{2}}=A.$$ We have $1 \leqslant i_{\ell+1}-i_{\ell} \leqslant 2$, therefore $i_{k+1} \in\{2018,2019\}$. Since $$2018 \leqslant i_{k+1}=i_{0}+\left(i_{1}-i_{0}\right)+\cdots+\left(i_{k+1}-i_{k}\right) \leqslant 2(k+2),$$ it follows that $k \geqslant 1007$. Consider two cases. (i) $k=1007$. Then in the inequality we have equalities everywhere, in particular $i_{k+1}=2018$. Applying AM-GM inequality for $k+3$ numbers to $A$ we obtain $A \geqslant k+3 \geqslant 1010$. (ii) $k \geqslant 1008$. If $i_{k+1}=2018$ then we get $A \geqslant k+3 \geqslant 1011$ by the same argument as in the case (i). If $i_{k+1}=2019$ then applying AM-GM inequality to $k+2$ summands in $A$ (that is, to all the summands except $\frac{a_{2018}}{a_{2019}})$ we get $A \geqslant k+2 \geqslant 1010$. So we have $S>T \geqslant A \geqslant 1010$. For $a_{1}=1+\varepsilon, a_{2}=\varepsilon, a_{3}=1+2 \varepsilon, a_{4}=2 \varepsilon, \ldots, a_{2016}=1008 \varepsilon, a_{2017}=1+1009 \varepsilon, a_{2018}=\varepsilon^{2}, a_{2019}=1$ we obtain $S=1009+1008 \varepsilon+\frac{1008 \varepsilon}{1+1009 \varepsilon-\varepsilon^{2}}+\frac{1+1009 \varepsilon}{1-\varepsilon^{2}}$. Then $\lim _{\varepsilon \rightarrow 0} S=1010$, which means that the constant 1010 cannot be increased.