I am confused about the notation used in the SIR model.
Most of the works found online present the equations of the compartments as following:
$s_{t+1}=s_t-αs_ti_t$
$i_{t+1}=i_t+αs_ti_t-γi_t$
$r_{t+1}=γi_t$
But theoretically these are functions of time so shouldn't they be represented as:
$s(t+1)=s(t)-α·s(t)·i(t)$
$i(t+1)=i(t)+α·s(t)·i(t)-γ·i(t)$
$r(t+1)= r(t)+γi(t)$?
Is it because they are recursive formulas referring to the position in time so the subscript form should be used instead? But then, when I am referring exclusively to the function of susceptibles, infected, or recovered at a particular time, is the $S(t)$, $I(t)$ and $R(t)$ notation correct or should I still employ $S_t$, $I_t$ and $R_t$?
In some pieces of work online these equations are differentiated as $\frac{ds(t)}{dt}$, $\frac{di(t)}{dt}$ and $\frac{dr(t)}{dt}$ but in that case shouldn't the derivatives be $s'(t)$, $i'(t)$ and $r'(t)$? What would be the correct notation for the first derivatives if $s_t$, $i_t$ and $r_t$ where the original functions?
It's a discrete-time model, where $t$ is an integer, and then it's very common to write $s_t$ instead of $s(t)$ (both are fine).
If you see these equations with derivatives, then you are looking at a continuous-time model instead, and in that case the notation is almost always $s(t)$, except that often one write just $s$ for simplicity. The derivative can be written as $s'$ or $\dot s$ or $ds/dt$, just choose your favourite...