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I wouldn't normally consider the Stats SE to be the best place for this question, but neither the Medical Sciences SE nor the Biology SE seems to focus on macroscopic modelling, while this SE seems to contain SIR modelling questions. Please let me know if there is a more appropriate forum.

I'm new to the modelling of disease spread, but not new to difference equations or rate equations. I am following the Model Properties section of a model for nearcasting disease proliferation, trying to square it off with the basic SIR-based models at Wikipedia. In the latter $\gamma$ is the number of people who become immunized or die per unit time. None of the models based on the SIR model make simplifying assumptions to eliminate $\gamma$.

However, the only way that I can get the difference equations in the nearcasting model is to assume +1 person becomes immune per unit time, i.e., $\gamma=1$. So far, I am unable to find the rationale for this in nearcasting. Can anyone point to an online explanation, or explain this themselves?

Example

$N$ = Population size

$S$ = Number of susceptible people in population (time varying)

$I$ = Number of infectious people in population (time varying)

$R$ = Number of immune or dead people in population (time varying)

$\beta$ = number of people contacted by any one person per unit time

$\gamma$ = probablity per unit time of an infectious person recovering or dying, i.e., becoming no longer infectious nor susceptible

Basic reproduction number::
$R_0 = \beta / \gamma$

Effective reproduction number (time varying):
$ R_e = \dfrac{S}{N} R_0 = \dfrac{S}{N} \dfrac{\beta}{\gamma} $

From the SIR model:
$ \dfrac{dI}{dt} = \dfrac{\beta I S}{N} - \gamma I $

Discretizing with forward differencing:
$ \dfrac{I_{t+1}-I_t}{\Delta t} = \beta I_t S / N - \gamma I_t $

With some manipulation (see Annex below):
$ I_{t+1} = I_t [ 1 + \Delta t \gamma ( R_e -1 ) ] $

The challenge is identifying the approximations that yield the nearcasting model's $I_{t+1}=R_e(t)I_t$.

Since this is a nearcasting model, assume that the author is focusing on initial behaviour, i.e., small $t$, such that $I \ll N$ and $S \simeq N^-$. Hence, $R_e \simeq R_0^- = ( \beta / \gamma )^- $.

Let's also assume that the author is only interested in epidemics, so:
$ R_e \simeq R_0^- = ( \beta / \gamma )^- > 1 $ .

With this, $I_{t+1}$ simplifies to:
$ I_{t+1} \simeq I_t [ 1 + \Delta t \gamma R_e ] \equiv I_t [ 1 + \Gamma R_e] $ ,

In the above, I replaced $ \Delta t \gamma \leftarrow \Gamma$, representing the probability of an infectious individual becoming immune or dead within the time step, as induced by the rate $\gamma$. Whatever the time step, the time rates $\gamma$ and $\beta$ get replaced by probabilities and numbers of people, respectively.

To make $I_{t+1}$ match the nearcasting model, we have to further assume that $\Gamma R_e \gg 1$ so that:
$ I_{t+1} \simeq I_t [ \Gamma R_e ] $ ,

and further assume that $\Gamma = 1$ so that:
$ I_{t+1} \simeq I_t R_e $ .

That last equation is how the nearcasting model represents the time stepping behaviour of $I_t$.

I'm having a hard time justifying the last assumption -- unless $\Delta t$ is chosen such that $ \Gamma \equiv \Delta t \gamma \equiv 1 $. If this is indeed the case, then the assumption $ \Gamma R_e \gg 1 $ (needed to reach the nearcasting equation) becomes $R_e\gg 1$. In fact, if $\Delta t \gamma \equiv 1$, we don't seem to need any approximations to get the nearcasting formula:

$ I_{t+1} = I_t [ 1 + \Delta t \gamma ( R_e - 1 ) ] = I_t [ 1 + R_e - 1 ] = R_e I_t $ ,

as required by the nearcasting model.

Is choosing $\Delta t$ so that $\Delta t \gamma \equiv 1$ a common practice in nearcasting? If so, that's a likely candidate for the answer.

Annex

$ \dfrac{I_{t+1}-I_t}{\Delta t} = \beta I_t S / N - \gamma I_t $

$ I_{t+1} - I_t = \Delta t \beta I_t \dfrac{S}{N} - \Delta t \gamma I_t $

$ I_{t+1} = \left[ 1 + \Delta t \left( \beta \dfrac{S}{N} - \gamma \right) \right] I_t = I_t \left[ 1 + \Delta t \gamma \left( \dfrac{\beta}{\gamma} \dfrac{S}{N} - 1 \right) \right] $

$ I_{t+1} = I_t [ 1 + \Delta t \gamma ( R_e -1 ) ] $

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  • $\begingroup$ Why is there a link back to this question at the top of the question? $\endgroup$
    – Glen_b
    Apr 13 '20 at 4:31
  • $\begingroup$ Sorry, that's just the way I email myself my own postings, along with the link. Will fix it. P.S. I suspect that my suspicions are right that the author chose $\Delta t = 1 / \gamma$. I have to sit down and work it through, but his paper says that $I = R_0^t$, which probably mutually implies $\Delta t = 1 / \gamma$ with some approximations that are appropriate for early in an epidemic. $\endgroup$ Apr 13 '20 at 5:57
  • $\begingroup$ Would clicking "follow" on your own questions be sufficient? $\endgroup$
    – Glen_b
    Apr 13 '20 at 6:39
  • $\begingroup$ I think that would just send me a notification of responses. I want to keep the original draft in addition to the link. I can forward either to any point of contact for tracking. It would be active tracking, meaning that one would have to follow the link rather than receiving notifications. The presence of the link in the actual post was an error, as my "procedure" is only to include the link in my copies. Sometimes, though, I add to the post, and I do that a power editor like vim. I accidentally end up coping the link along with the rest of the text. Rarely, though. $\endgroup$ Apr 13 '20 at 20:07
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I believe that condition $\Delta t \gamma = 1$ is not an approximation, but rather, a deliberate choice of simulation parameter $\Delta t = 1 / \gamma$. This is from the paper's Model development section: "$t$ may be defined as $I=R_0^t$". It's a rather loose specification, and I believe that what is meant is that $I=R_0^{t/\Delta t}$, with the approximations describe in the posted question. Specifically, for small $t$ , $I \ll N$ , $S \simeq N^-$ , and $R_e \simeq R_0^- = ( \beta / \gamma )^- $. The result in the posted question

$ I_{t+1} = R_e I_t $

becomes

$ I_{t+1} \simeq R_0 I_t $

It is then clear how the model yields $I = R_0^t$.

It occurred to me that the article does not distinguish between a time $t$ and integer enumeration of time step, $t/\Delta t$. Rather, it uses $t$ for the latter. This is why $I = R_0^t$ instead of $I = R_0^{t / \Delta t}$.

Since this is my own oversight, I might delete this question.

AFTERNOTE caveat There seems to be a discrepancy between $\Delta t = 1 / \gamma$ and the paper's term "serial interval" (SI). SI is the time from a individual being infected to the time when they pass that on to someone else. From the simple SIR model, the infectious period is $1/\gamma$. The reason why the number of people infected by one individual is $R_0=\beta/\gamma$ is because the individual contacts people at a rate of $\beta$ throughout the duration of $1/\gamma$. So the average time from the when the individual got infected to when they infect a contact would be approximately half of $1/\gamma$, i.e., SI=$1/2\gamma$

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  • $\begingroup$ Please don't delete. I think if you were confused others will be as well, and your presentation of both the question and discussion in an answer are likely to be helpful. $\endgroup$
    – Glen_b
    Apr 13 '20 at 22:43
  • $\begingroup$ OK. Thanks for the feedback. $\endgroup$ Apr 13 '20 at 22:46
  • $\begingroup$ @user2153235 the average time between infections is $1/\beta$ $\endgroup$ Apr 21 '20 at 7:07
  • $\begingroup$ @Sextus Empiricus: Thanks. That's a bit different from average time between infection of an individual and the time he/she infects someone else. $\endgroup$ Apr 21 '20 at 11:58
  • $\begingroup$ @user2153235 the infection rate is $\beta$, the time between events (any events whether is time between two infections or the time between infections of the individual and time they infect someone else, is irrelevant) is $1/\beta$. $\endgroup$ Apr 21 '20 at 12:36

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