Use individual level shedding data to estimate G, phi, + hinge function parameters

[kaitejohnson]

Problem

We are currently using a hinge function on the log scale for fecal shedding kinetics, which is based in part on nasal shedding kinetics and in part on data from fecal shedding studies. However, there is individual fecal shedding kinetics data, and it would be of interest to the broader wastewater community to use the individual shedding data to identify the parameters and form of a shedding kinetics function

Context

Need to use the data from Arts et al and loaded in this Rmd to estimate the following parameters:
V_peak : relative peak of viral load (when viral shedding is normalized) in log10 scale

t_peak: timing of peak viral load relative to start of infection

waning_rate: rate of waning on the log10 scale

G: mean number of viral genomes shed during the course of an infection (integral over the viral kinetics)

phi: individual level dispersion in G

We will basically use pieces of the stan code from the wastewater model, but only the relevant ones. We will also pass in the incubation period (from Park et al). The idea for this is we could make a notebook that loads in the data, sets the priors and known distributions, and describes how to go about jointly estimating these parameters, in a similar way to how has estimated time from symptom onset to hospitalization. Then we would use this separate analysis as priors for these values in the larger WW model.

Basic assumptions:

likehood:
$$V_{obs}(t) \sim NegBinom(V(t), \phi_{measurement})$$

Data generation (something like this…):

$$V(t-t_{sym}) = G_i*S(t-t_{sym})$$

$$t_{sym} \sim Weibull(\alpha, \beta) $$

$$S(t) = H(V_{peak}, t_{peak}, w)$$

$$G_i \sim NegBinom(G_{mean}, \phi_{ind})$$

Where $H$ is the normalized hinge function, exponentiated.
Of note, this might be a fun task for someone getting familiar with stan (which is still me, but also doesn’t have to be me!)

[seabbs]

Some variation of the model in here might help with this (note very preliminary package): https://github.com/seabbs/epict/