# The role of duration of infectiousness in SIR models

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.)

• This question reminds me of the equations for radioactive decay (see hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli2.html). Over there, we assume that a constant fraction of particles decay per unit time. A particular particle can have a lifetime varying from zero to infinity, but the average lifetime of all the particles is simply the reciprocal of the decay constant. Perhaps something similar applies here? (Think of infected persons as particles, removal as radioactive decay, and $\nu$ as the decay constant.) If that is so, the first of the two approaches you mentioned is correct. Commented May 17, 2020 at 10:07
• I guess the main difference is that radioactive decay follows a Poisson distribution, but recovering of a disease does not: I assume it is somehow Gauss distributed around a characteristic maximum. Commented May 17, 2020 at 11:20
• While the number of decays in a given time period for a sample follows a Poisson distribution, the particle lifetimes do not. In fact they have an exponential distribution. Also, how can recovery times for a disease have a normal distribution? A normal distribution stretches infinitely in both directions, whereas recovery time can never be negative. Commented May 17, 2020 at 14:47
• Thank you, you are right in both cases. Nevertheless the distribution of recovering times might be "almost Gaussian" - because nobody recovers immediately after having been infected. So maybe THIS distribution is Poisson - as opposed to exponential? Commented May 17, 2020 at 18:03

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 $$ν$$ of them that will have recovered tomorrow.

Well, it is in fact not very 'realistic' as you point out, but in the assumptions of the model, we see that the population has no structure (it is well-mixed, constant) and there are no birth-death events. So, in this case, it is not so problematic to take $$v$$ as a constant across the entire simulation, because what you are trying to calculate is the rate at which the three sub-fractions of $$N$$ ($$s$$, $$i$$, and $$r$$) change, not really which individuals are moving from one class to another (which you cannot really know anyways, if you talk about fractions of $$N$$).

Wouldn't it be much more plausible

• to consider all persons that became infected $$d$$ days ago and let these be recovered tomorrow?

So, considering my previous comment, it would not make much sense really to take a time-delay form on $$d$$, because you cannot really know which individuals got infected at any given point, you can only talk about fractions of $$N$$ (there is no population structure, as they say in the model formulation). So, the fact that your formulation seems to have a slower dynamics might not be very informative, because what you did is simply applying $$d$$ to a fraction-of-the-fraction of the population classes, so it makes sense mathematically that it runs slower, but as per the model formulation it does not make much sense, unless you have population structure defined from the beginning (which in this case is not), and unless you can explicitely know individual-per individual class transitions. In fact, I believe that taking the fraction-of-the-fraction would lead to an artifactual 'undercounting' of these individuals that actually needed to be in the $$i$$ class (and overcounting of the other classes).

• I don't see it problematic to take $\nu$ as a constant. I don't understand what you mean with "constant across the entire simulation". Furthermore, there is no single rate at which $s$, $i$, and $r$ change but two: $\beta$ and $\nu$. And I don't understand your which-individuals-argument. All I want to say: When "duration of infectiousness" does mean what is seems to be supposed to mean, it makes sense to remove all those newly infected $d$ days ago as recovered tomorrow, and not a fraction $\nu = 1/d$ of those that are infected today. Commented May 16, 2020 at 21:33
• The persons infected today consist of some newly infected yesterday, some newly infected the day before yesterday, ..., and some newly infected $d$ days ago. And only the latter will have recovered tomorrow (in the mean). Commented May 16, 2020 at 21:36
• You want to treat the model as it had population structure, but it doesn't. That's the main problem as I see it. Commented May 16, 2020 at 21:45
• So, it's not a problem with the model but how you use it. Commented May 16, 2020 at 21:50
• Sorry, but I still don't understand your argument. And I still don't understand what's wrong with my argument. The only structure I am supposing is a "timely" structure: that it does matter when a person got infected. Commented May 17, 2020 at 21:06

I am not sure if I quite understand your question, but I think your problem is here: removal (and your d) is a rate (time/removal). It does not matter what time you choose; a day, a week, a year, as long as you adjust your c (which is /time) to same timescale. In other words, if you wish to use d over several days, you need to calculate your contacts over several days, and changing only one will erroneously lead to different results.

• I'm afraid this misses the point. Commented May 16, 2020 at 11:44