Standardises generation of infections

[seabbs]

This PR standardises the generation of infections and the use of covariates (currently GP and BP only). Work in progress.

[sbfnk]

Not having looked in detail can‘t this be simplified further by using the calculate_Rt function and separating out setting of the GP prior (whether on Rt or infections)?

[sbfnk]

I.e., can't we write the model as in #211 (comment)_ where R_t, I_t and r are all parameters? The only difference between the models is then in the prior choice, e.g. R_t could be a GP prior or an uninformed prior (i.e., a backcalculation model in which case it will just be calculated from uncertain I_t as in the first equation in the comment referenced), and this could be done in separate functions (and what's currently generate infections becomes calculate_Rt). Or am I missing something?

If this can be done, it would have the nice effects of, e.g., being able to use breakpoints/covariates on Rt in a model that has a GP prior on I_t and uninformed prior on R_t.

[sbfnk]

Looking again most of this is there already: all it would need is for all models to have R_t, r_t and I_t as parameters, and all models to use generate_infections_with_infectiousness (which I think should just be called renewal_model) whereas generate_infections_with_infectiousness would be re-named to reflect that all it does is set a prior on infections (which is all that determines infections if the priors on R_t and r_t are unconstrained - though in practice one might want to bound them an turn into uninformative proper priors even when not using a GP, e.g. uniform with some bound, that way their posteriors become proper).

Wouldn't change anything in practice but reduce code complexity and remove duplicate code (e.g. calculate_Rt vs generate_infections, where the former would no longer be needed if R_t was always a parameter).

[seabbs]

Hmm so in my mind it was a choice (i.e infections directly, infections via renewal equation, infections via growth. It is not totally clear to me that you could do all of them at once as the renewal equation is iterative (in that it is forwards whereas when you calculate Rt from it it is backwards) and stan will try and fit all the parameters - will stare at the code a bit.

[sbfnk]

Hmmm, yeah so it might depend on whether Stan interprets an equals sign as a statement of equality or an assignment. If there's no way to have it interpret as equality than my suggestion won't work.

[seabbs]

So my understanding of what it is doing is assignment - that could be incorrect however.

What I think would work would be something like this (where the infection and growth model are currently folded into a single function):

# set up covariates for each model
R_cov = update_covariate(...)
r_cov = update_covariate(...)
inf_cov = update_covariate(...)

# initialise unobserved infections using constant growth model
uobs_infs = generate_seed(...)
# generate infections using renewal 
infections = renewal_model(R_cov, uobs_infs,...)
infections = growth_model(infections, r_cov, ...)
infections = infection_model(constant = infections, inf_cov, ...)

# later as quantities
R = calculate_rt(infections, ...)
r = calculate_growth(infections, ...)

Where each later model using infections from the previous model as an input. That would let you have covariates/interventions on Rt and a GP on infections for example. This would add quite a bit of weight + code for a use case I am not entirely clear about.

The alternative would be the parallel approach (i.e they are options). So:

# set up covariate
cov = update_covariate(...)

# initialise unobserved infections using constant growth model
uobs_infs = generate_seed(...)
if (model == 1) {
  infections = renewal_model(cov, uobs_infs,...)
}else if (model ==2) {
  infections = growth_model( cov, uobs_infs ...)
}else if (model == 3) {
  infections = infection_model(constant, cov, uobs_infs, ...)
}

# later as quantities
R = calculate_rt(infections, ...)
r = calculate_growth(infections, ...)

[sbfnk]

Closing in favour of #307