The model here is that an observation comes from an equal mixture of two normal distributions, which is not at all the same as saying that the value of an observation is the sum of two normally-distributed quantities. The pdf for the mixture distribution is just 1/2 the pdf for one of the normals plus 1/2 the pdf for the other normal.

]]>How did you calculate the pdf of the sum of these two normal distribution. By looking at your R code, why don’t you calculate by actually treat it like a single normal distribution since the sum of two normal should still be a normal. ]]>

That’s the sort of question that the later parts of the Kissler, et al. paper try to answer. I’ll be discussing those parts in a future post. I haven’t finished examining what they did, so I can’t say exactly what I’ll conclude about that.

Leaving aside the particulars of this paper, though, I think it’s reasonable to see COVID-19 staying around as a seasonal (in temperate regions) disease as one possible scenario. If so, it might also evolve to become serious (though of course in many people it is already not serious). I think it’s pretty hard to make definite predictions.

]]>It’s not like the literature hasn’t discussed these:

Russell Timothy W , Hellewell Joel , Jarvis Christopher I , van Zandvoort Kevin , Abbott Sam , Ratnayake Ruwan , CMMID COVID-19 working group , Flasche Stefan, Eggo Rosalind M , Edmunds W John , Kucharski Adam J . “Estimating the infection and case fatality ratio for coronavirus disease (COVID-19) using age-adjusted

data from the outbreak on the Diamond Princess cruise ship”, February 2020. Euro Surveill. 2020;25(12):pii=2000256. https://doi.org/10.2807/1560-7917. ES.2020.25.12.2000256.

T. Jombart, *et al*, “Inferring the number of COVID-19 cases from recently reported deaths” [version 1; peer review: 2 approved], Wellcome Open Research 2020, 5:78 Last updated: 26 MAY 2020.

T. W. Russell, *et al*, “Using a delay-adjusted case fatality

ratio to estimate under-reporting”, https://fondazionecerm.it/wp-content/uploads/2020/03/Using-a-delay-adjusted-case-fatality-ratio-to-estimate-under-reporting-_-CMMID-Repository.pdf.

Given that infections are far from Poisson events, being overdispersed because of the superspreader phenomenon, it seems a good deal more investigation of those long tails would be warranted. After all, the nice thing about Poisson statistics is that they imply a certain stability and predictability in outcome. Forcing a Poisson model on top of an actually Negative Binomial model with a big variance means the of the Poisson is going to be exaggerated. Sure, it looks like the Poisson is exaggerating. But, in fact, there’s bigger latent risk: Can’t know how the big tail events are going to behave.

Indeed, if there’s anything specific to be criticized about Imperial College is that they did not acknowledge this feature of epidemics in their analysis. The superspreader phenomenon has been known for a while, since 2000 at least. See

J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp & W. M. Getz, “Superspreading and the effect of individual variation on disease emergence”, *Nature*,438(17), November 2005, doi:10.1038/nature04153.

And see its references.

]]>https://www.medrxiv.org/content/10.1101/2020.05.22.20110403v1

]]>I agree that the impetus to reach for a closed form density, like the Weibull, seems to be more a compulsion than well grounded.

]]>I think you’ve reversed “ILI” and “non-ILI” in your second paragraph. The seasonality of non-ILI visits seems to be tied to calendar events, such as the start of school. The ILI seasonality seems to be tied to virus dynamics, which has a general seasonality, but with peaks that aren’t precisely timed by the calendar (varying by several weeks, perhaps because of the weather that year, or whatever). In both cases, there could be a bit of the other kind of seasonality, but putting in both kinds for both ILI and non-ILI visits seems too complicated.

The ability of sums of sin and cos to produce any phase is important to getting the calendar sort of seasonality to have peaks in the right places, but doesn’t help if the peaks move around from year to year. One could build a model with an AR(2) component (which can exhibit oscillations of around some period, but not precisely fixed), but that seems too elaborate in this context.

]]>where is the observed count or index at time point , is the periodic, is some zero mean noise, and is a deterministic function of which describes the upramp/downramp of LI visits, probably a ratio of sums of exponentials. Over the course of a season the net contribution from is zero.

On the second point, okay, but per “Unlike for non-ILI visits, the seasonality for ILI visits is not firmly fixed to the calendar — the time of the peak varies in different years by several weeks — so a Fourier representation of seasonality that is common to all years is not appropriate”, mightn’t there be departures season by season even for ILI? Or does the uniform distribution for phase take care of that? And if it doesn’t, why doesn’t it take care of it for non-ILI?

]]>Allowing a linear combination of sin and cos doesn’t impose phase restrictions, since you can get a sine wave with any phase that way. In fact, even in a Bayesian context, this scheme works fine, since independent Gaussian priors for the regression coefficients on the sin and cos components results in a uniform distribution for the phase of the combined wave.

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