Model description

[paigemiller]

adding description of planned repo contents:

• next generation function to calculate R0 for SIR model with 4 groups
• SIR model assumptions
• desired outputs: infections and severe infections

[afmagee42]

Tagging @jrcpulliam who seemed to have ideas on how to get at the questions of interest without final sizes

[jrcpulliam]

Yes, this is because I have a different idea about what the outputs of interest are. Namely, I think what we care about are:

• the effective reproduction number (and possibly the associated probability of a large outbreak); you get this from the dominant eigenvalue of the NGM for the vaccination scenario
• the distribution of infections (and associated severe outcomes - in this case, death); you get this from the dominant eigenvector of the NGM for the vaccination scenario

[afmagee42]

I know this is inherited from other docs, but I am now realizing I am confused by the combination of "vaccination is all or nothing" with "VE is a parameter" (per below).

[jrcpulliam]

"All or nothing" is jargon and essentially means that the assumption is that each vaccinated individual's immunity is determined by a coin flip with a probability of being immune equal to the VE. If immune, they're completely protected (all); otherwise, they're just like someone who has never been infected or vaccinated (nothing). This is in contract to what's usually called "leaky" vaccination, which conceptually puts a coin flip on each exposure event.

[afmagee42]

Suggested change

	* Recovery rate
	* Recovery rate (assumed to be the same across all groups)

(Sorry, I like large dots on all my "is" and thick crosses on my "t"s)

[jrcpulliam]

You don't need a recovery rate input at all.... just parameterize using within and between group reproduction numbers. (sorry, I know I'm a broken record!)

[afmagee42]

If I'm understanding this right, we're saying something like

$d I_k / d t = \beta S_k \sum_i \zeta_i I_i - \dots$ where $\beta$ is shared and $\zeta_i$ is either this "high" or "low" value?

Could we maybe call $\beta$ the "reference" transmission constant or something?

[jrcpulliam]

Again, think instead of in terms of basic reproduction numbers between all your groups. In a naive population, how many core group members would a core group member infect on average, how many children would they infect, how many travelers, how many genpop? Then the same if your initial case were in each of the other groups.

Note that the beauty of a widget is that you don't need to know these numbers precisely. You can make some informed guesses to provide as defaults and then let the users adjust those numbers if they think they're different.

[swo]

and note that $R_{ij}$ is also a sum over modes of transmission: one sex act is worth a lot more than 1 conversation

[afmagee42]

Is there a framing where this could be called a per-group probability of severe infection? Or the per-group expected proportions of infections which are severe? As-is, I had to try a few times to parse this sentence.

[jrcpulliam]

Yes, I would go with "per-group probability of severe infection"

[afmagee42]

Same @jrcpulliam tag as above

[jrcpulliam]

See suggestions on outputs in comments on line 28 above.

[afmagee42]

My "request changes" is mostly so that we don't go any further without resolving this "final size" thing, everything else is more suggestions on clarity for dummies (like me) than hard requirement.

[jrcpulliam]

Yes, this is because I have a different idea about what the outputs of interest are. Namely, I think what we care about are:

• the effective reproduction number (and possibly the associated probability of a large outbreak); you get this from the dominant eigenvalue of the NGM for the vaccination scenario
• the distribution of infections (and associated severe outcomes - in this case, death); you get this from the dominant eigenvector of the NGM for the vaccination scenario

[jrcpulliam]

Nitpicky, but in the NGM framework, you'd multiply the proportion of infections in each group by the group-specific probability of your outcome of interest. You could then multiply that by the total number of infections that you want to use as your reference point.

[jrcpulliam]

You don't need betas, you need within and between group reproduction numbers; these are the entries in the NGM. No need to go through a beta / gamma relationship - that just makes things harder to parameterize and think about.

[jrcpulliam]

Again, think instead of in terms of basic reproduction numbers between all your groups. In a naive population, how many core group members would a core group member infect on average, how many children would they infect, how many travelers, how many genpop? Then the same if your initial case were in each of the other groups.

Note that the beauty of a widget is that you don't need to know these numbers precisely. You can make some informed guesses to provide as defaults and then let the users adjust those numbers if they think they're different.

[jrcpulliam]

You don't need a recovery rate input at all.... just parameterize using within and between group reproduction numbers. (sorry, I know I'm a broken record!)

[jrcpulliam]

Yes, I would go with "per-group probability of severe infection"

[jrcpulliam]

See suggestions on outputs in comments on line 28 above.

[jrcpulliam]

I'm sure there's some fun math in this paper but am skeptical that you actually need final sizes. If you think final sizes are relevant, simple ODEs are (in my opinion) the wrong way to go... But the question here is about preventing epidemic growth, so we're interested in the exponential growth phase of the epidemic, which is where simple ODEs shine!

And the NGM approach is equivalent to the non-dimensionalized multi-type SIR. In the simplest version, you assume that every type has the same recovery rate. You then rescale time by the mean duration of infectiousness (1 / recovery rate), which turns all your transmission coefficients into type-specific basic reproduction numbers, and rescale your compartments by your population size.

[jrcpulliam]

I've always found this paper to be basically impenetrable... But this reminds me of the request from @afmagee42 to provide some references for this stuff... I'm guessing it's covered pretty well in Diekmann & Heesterbeek 2020, but Carl took our copy into the office so I can't check right now. Will pick his brain and get back to you on this!

[jrcpulliam]

FWIW, my introduction to matrix models came from Caswell's Matrix Population Models, which might be a useful reference as well, though he doesn't go into the subtleties of how these models relate to continuous time formulations, and I don't think he has any examples for infectious disease systems (or at least, he didn't in the older edition that I'm familiar with!)

[jrcpulliam]

@jasonasher: did you have a reference you could share for the probability of a large outbreak calculations? I haven't done this since grad school and don't remember there being a good reference for it at the time...

[jrcpulliam]

This looks like a more accessible reference that the original 1990 paper (but would still not recommend diving into it until after the response needs are met): Diekmann, Heesterbeek, and Roberts 2009

Van den Driessche & Watmough 2002 is what I've typically used for teaching, but as noted in the above, the procedure it describes yields a different matrix (with the same dominant eigenvalue), which can be confusing.

[swo]

Generally this is a good start!

I think we could simplify a bunch of the intermediate people; I think you can jump right to "the matrix has elements $R_{ij}$"

I also think the first output from this widget could be the very, very simple matrix multiplication that says "give these population sizes, and this NGM, this is the number of next-generations (and other outcomes) you expect per group"

[swo]

Suggested change

	This repo contains code to apply the next-generation method of Diekman et al. (1990) to calculate R0 for an SIR model with 4 risk groups and flexible inputs for varying vaccine allocation to the 4 groups.
	This repo contains code to apply the next-generation method of [Diekman et al. (1990)](https://doi.org/10.1007/BF00178324) to calculate R0 for an SIR model with 4 risk groups and flexible inputs for varying vaccine allocation to the 4 groups.

Assuming this is the right paper? On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

[jrcpulliam]

Recommended reference is Diekmann & Heesterbeek 2020. I haven't been able to find an online copy that's accessible, but I'll ask Sarah what options are available from the CDC library. I can also try reaching out to Hans and seeing if he's willing to share a PDF with just the relevant pages. The book is much more easier to follow than the original paper.

[jrcpulliam]

See also the Interlude in Diekmann et al 2009, which is a good description of the way I actually think about these things in practice

[swo]

My understanding is that the dominant eigenvalue is $R_0$?

[jrcpulliam]

The dominant eigenvalue is $R_0$ when the population is fully susceptible and $R_e$ under vaccination. The entries in the NGM are group-specific $R_0$'s and $R_e$'s respectively