For some inexplicable reason, I have recently been interested in epidemiology. One of the classical and simplistic models in epidemiology is the SIR model given by the following system of first-order autonomous nonlinear ordinary differential equations in the real variables $s,i,r$ satisfying $s,i,r\geq 0$ and $s+i+r=1$: $$ \begin{align} s' &= -\beta i s\\ i' &= \beta i s - \gamma i\\ r' &= \gamma i \end{align} \tag{$*$} $$ (where prime denotes derivative w.r.t. time, $s,i,r$ represent the proportion of “susceptible”, “infected” and “recovered” individuals, $\beta$ is the kinetic constant of infectiousness and $\gamma$ that of recovery; the “replication number” here is $\kappa := \beta/\gamma$). It is easy to see that for every initial value $(s_0,i_0,r_0)$ at $t=0$ the system has a unique $\mathscr{C}^\infty$ (even real-analytic) solution (say $r_0=0$ to simplify, which we can enforce by dividing by $s_0+i_0$); it does not appear to be solvable in exact form in function of time, but since $s'/r' = -\kappa\cdot s$ and $i'/r' = \kappa\cdot s - 1$ (where $\kappa := \beta/\gamma$) it is possible to express $s$ and $i$ as functions of $r$, viz., $s = s_0\,\exp(-\kappa\cdot r)$ and of course $i = 1-s-r$. This makes it possible to express, e.g., the values of $s,i,r$ at peak epidemic (when $i'=0$): namely, $s = 1/\kappa$ and $r = \log(\kappa)/\kappa$); or when $t\to+\infty$: namely, $s = -W(-\kappa\,\exp(-\kappa))/\kappa$ where $W$ is (the appropriate branch of) Lambert's transcendental W function; this is upon taking $i_0$ infinitesimal and $r_0=0$.

Now one of the many ways in which this SIR model is simplistic is that it assumes that recovery follows an exponential process (hence the $-\gamma i$ term in $i'$) with characteristic time $1/\gamma$. A slightly more realistic hypothesis is recovery in constant time $T$. This leads to the following delay-difference equation: $$ \begin{align} s' &= -\beta i s\\ i' &= \beta i s - \beta (i s)_T\\ r' &= \beta (i s)_T \end{align} \tag{$\dagger$} $$ where $(i s)_T = i_T\cdot s_T$ and $f_T(t) = f(t-T)$. When I speak of a “$\mathscr{C}^\infty$ solution” of ($\dagger$) on $[0;+\infty[$ I mean one where the functions $s,i,r$ are $\mathscr{C}^\infty$ and satisfy the equations whenever they make sense (so the second and third are only imposed for $t\geq T$), although one could also look for solutions on $\mathbb{R}$.

I am interested in how ($\dagger$) behaves with respect to ($*$). Specifically,

Does ($\dagger$) admit a $\mathscr{C}^\infty$ (or better, real-analytic) solution on $[0;+\infty[$ for every initial value $(s_0,i_0,r_0)$ at $t=0$? Or maybe even on $\mathbb{R}$? Is this solution unique? (Note that we could try to specify a solution by giving initial values on $[0;T[$ and working in pieces, but this does

*not*answer my question as it would not, in general, glue at multiples of $T$ to give a $\mathscr{C}^\infty$ solution.)Can we express this solution in closed form or, at least, express $s$ and $i$ in function of $r$?

Qualitatively, how do the (at least continuous!) solutions of ($\dagger$) differ from those of ($*$), and specifically:

How do the behaviors compare when for $t\to 0$?

How do the values compare around peak epidemic $i'=0$?

How do the limits when $t\to+\infty$ compare?