[Vp-integration-subgroup] Another case study of model integration

[William Waites]

Hi Robin!

> Thanks for this. All the generation time distributions (i.e. expected infectiousness as a function of the time since infection) that we have been considering are normalised so that they integrate to 1.

This is very interesting. The shape of all of the distributions for generation that you find (particularly striking in Fig 4a of your paper) have the characteristic shape of the curves for log viral load, and if we suppose that chance of developing symptoms is proportional to this quantity we indeed get a cumulative fraction of infectious individuals that is lagged, as would be expected. See, for example,

 

modulo perhaps some stretching. The vertical axis for n is arbitrary units where n is allowed to vary between 0 and 2 proportional to log virus population within the host.

Coupling this to a transmission model at the population level is straightforward. If we also suppose that infectiousness is also proportional to log viral load (traditionally the quantity denoted β) then we can get to generation time distributed as you find it. The point is, we can get at this mechanism from an underlying model of adaptive immune response, as opposed to the more phenomenological approach of estimating the chance that someone is infectious or has symptoms as a function of time. It is good that this latter approach agrees, but this same result seems to follow from some more basic (not to say simpler, just more fundamental) considerations. 

The implications for combining models with these two approaches are important. It is possible (though a bit subtle for various reasons) to derive generation time distributions from immune models like this. It is easier and more “correct” in a certain sense to couple them directly to transmission by giving an account of the actual mechanism of transmission (giving a bit of virus to another person). It seems to me that something important might get lost by going via a population-level distribution derived from a process that happens within individuals and exhibits quite a lot of heterogeneity between them. This particular jumping between scales needs to be looked at closely and properly understood — it’s not going to be as simple as just agreeing to normalise in a certain way.

Cheers,
-w

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