With the standard article
class, with no option, the text width is set to 345pt. On the other hand, your longest equations are 195.14384pt and 205.44582pt wide respectively.
This makes for slightly more than 400pt, so there's no chance to set the two blocks side by side unless you increase the text width; in order to ensure space for the mid rule, along with some padding, no less than 420pt are needed.
Here's with 420pt (wider is obviously possible). The showframe
option is just to show the margins of the text block.
\documentclass{article}
\usepackage[textwidth=420pt,showframe]{geometry}
\usepackage{amsmath}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
\begin{document}
\[
\begin{aligned}[t]
\frac{\diff R1(t)}{\diff I(t)}
& = \frac{\gamma \bar{k} I(t) S(t) + \alpha \bar{k} S(t) R2(t) + \delta S(t)}
{- \bar{k} I(t)S(t)} \\
& = - \gamma - \alpha \frac{R2(t)}{I(t)} - \frac{\delta}{\bar{k} I(t)}
\end{aligned}\hspace{1000pt minus 1fil}\vrule\hspace{1000pt minus 1fil}
\begin{aligned}[t]
\frac{\diff R1(t)}{\diff I(t)}
& = \frac{5.3 I(t) S(t) + 4.5 S(t) R2(t) + 0.5 S(t)}{-10 I(t) S(t)\vphantom{\bar{k}}} \\
& = - 0.53 - 0.45 \frac{R2(t)}{I(t)\vphantom{\bar{k}}} - \frac{0.05}{I(t)\vphantom{\bar{k}}}
\end{aligned}
\]
\end{document}
Note the phantoms to make the lines the same height and depth.

Alternatively, a single align*
:
\documentclass{article}
\usepackage{amsmath}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
\begin{document}
\begin{align*}
\frac{\diff R1(t)}{\diff I(t)}
& = \frac{\gamma \bar{k} I(t) S(t) + \alpha \bar{k} S(t) R2(t) + \delta S(t)}
{- \bar{k} I(t)S(t)} \\
& = - \gamma - \alpha \frac{R2(t)}{I(t)} - \frac{\delta}{\bar{k} I(t)} \\[2ex]
& = \frac{5.3 I(t) S(t) + 4.5 S(t) R2(t) + 0.5 S(t)}{-10 I(t) S(t)} \\
& = - 0.53 - 0.45 \frac{R2(t)}{I(t)} - \frac{0.05}{I(t)}
\end{align*}
\end{document}

\end{minipage}
gives a good breakpoint, and the additional space needed by the\hfill\vline\hfill
takes the total length over the line width. Add%
after the\end{minipage}
and decrease the width of the minipages to0.48\textwidth
. That should be enough to fix it. – barbara beeton Aug 19 '19 at 2:00\noindent
instruction before the first minipage. – Mico Aug 19 '19 at 10:40