I refer to J.H. Jones' Notes on R0.
The basic SIR model - as described in Jones' Notes - considers three factors that make up the reproduction number:
$\tau$ = the transmissibility (i.e., probability of infection given contact between a susceptible and infected individual)
$\overline{c}$ = the average rate of contact between susceptible and infected individuals
$d$ = the duration of infectiousness
The (basic) reproduction number then is
$$R_0 = \tau \cdot \overline{c} \cdot d$$
The duration of infectiousness enters the basic SIR model as the so-called removal rate $\nu$ which is nothing but the inverse of the duration of infectiousness: $\nu = 1/d$:
$\frac{ds}{dt} = -\beta s i$
$\frac{di}{dt} = \beta s i - \nu i$
$\frac{dr}{dt} = \nu i$
with
$s$ = the fraction of susceptible persons
$i$ = the fraction of infected persons
$r$ = the fraction of removed persons (recovered or died)
$\beta = \tau \cdot \overline{c} = R_0/d$ = effective contact rate or infection rate
My question concerns the way that $d$ enters the SIR model, because I find it not so plausible:
- to consider all persons that are infected today and take a fraction $\nu$ of them that will have recovered tomorrow.
Wouldn't it be much more plausible
- to consider all persons that became infected $d$ days ago and let these be recovered tomorrow?
The latter approach would be especially valid when the death rate can be neglected, i.e. when "removed $\approx$ recovered".
My impression is that most papers using a variant of the basic SIR model let enter the duration of infectiousness in the first way - leading to significantly different predictions than in the second case.
I implemented both and this is the difference (only due to the different ways that $d$ enters the progression formula, i.e. the values of $\beta$ and $d$ are fixed):
(In case that you wonder why the curves oscillate: I have modelled some sort of acquired immunity with finite duration of only some month - but in the very same way in both cases.)