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’ &= […]

- Tags 'viz', “infected” and “recovered” individuals, \exp(-\kappa))/\kappa$ where $W$ is (the appropriate branch of) Lambert's transcendental W function; this is upon taking $i_0$ infinitesimal, \exp(-\kappa\cdot r)$ and of course $i = 1-s-r$. This makes it possible to express, $\beta$ is the kinetic constant of infectiousness and $\gamma$ that of recovery; the “replication number” here is $\kappa := \beta/\gamma$)., $s = -W(-\kappa\, $s = 1/\kappa$ and $r = \log(\kappa)/\kappa$); or when $t\to+\infty$: namely, $s = s_0\, although one could also look for solutions on $\mathbb{R}$. I am interested in how ($\dagger$) behaves with respect to ($*$). Specifically, 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, at least, 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, but this does not answer my question as it would not, Does ($\dagger$) admit a $\mathscr{C}^\infty$ (or better, e.g., express $s$ and $i$ in function of $r$? Qualitatively, For some inexplicable reason, glue at multiples of $T$ to give a $\mathscr{C}^\infty$ solution.) Can we express this solution in closed form or, how do the (at least continuous!) solutions of ($\dagger$) differ from those of ($*$), i, I have recently been interested in epidemiology. One of the classical and simplistic models in epidemiology is the SIR model given by the fol, i_0, in general, r_0)$ at $t=0$ the system has a unique $\mathscr{C}^\infty$ (even real-analytic) solution (say $r_0=0$ to simplify, 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 value, 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, r$ are $\mathscr{C}^\infty$ and satisfy the equations whenever they make sense (so the second and third are only imposed for $t\geq T$), r$ at peak epidemic (when $i'=0$): namely, r$ represent the proportion of “susceptible”, r$ satisfying $s, real-analytic) solution on $[0;+\infty[$ for every initial value $(s_0, s, the values of $s, 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