IHR(t) should be on the time of infection, not time of admissions

[kaitejohnson]

Problem

As written

$$H(t) = \omega(t) p\mathrm{hosp}(t) \sum{\tau = 0}^{T_d} d(\tau) I(t-\tau)$$

and as implemented, the time-varying IHR applies is multiplied by the incident admissions. But biologically, we want to associate the probability of admissions with a time of infection, thus the implementation and write up should read:

$$H(t) = \omega(t) \sum_{\tau = 0}^{Td} d(\tau) I(t-\tau) p\mathrm{hosp}(t - \tau)$$

Thanks for the flag @damonbayer @dylanhmorris

[seabbs]

Just noting that for a late of factors this is true but it may not be true for things like hospital pressures and changing attitudes to being ill

[gvegayon]

@kaitejohnson, would you like me to work on this one?

[kaitejohnson]

I don't think this is highest priority. I added a bunch in the R sprint milestone.

[damonbayer]

Since this is still somewhat up for discussion (based on @seabbs's comment), wanted to note that while $p\mathrm{hosp}(t)$ is only used for the observation period, the $p\mathrm{hosp}(t)$ that is generated is actually long enough to cover the unobserved and observed period. The portion from the unobserved is never used, so there is no point to generating it.