Exponential growth model for initial infections

[gvegayon]

Goal

Implement the initialization of the infections using an exponential growth model as in the wastewater model (see here):

This process is initialized by estimating an initial exponential growth[^EpiNow2] of infections for 50 days prior to the calibration start time $t_0$:

$$ I(t) = I_0\exp(rt) $$

where $I_0$ is the initial per capita infection incident infections and $r$ is the exponential growth rate.

Context

During discussions of feature prioritization. This particular component was not listed.

Required features

The feature should be implemented as a RandomVariable object part of the latent submodule.

Specifications

• Mathematical description of the component.
• Unit test evaluating the feature.
• Integration test with the basic model and hospital admissions model.
• Integration as part of a tutorial.

These specs may be moved to a different issue involving other components of the replication of the hospital admissions model in the wastewater project.

Out of scope

• TBD

[gvegayon]

Hey @damonbayer, as we discussed, this could be implemeted via a latent process either extending the current implementation of the infections module or adding a new one called something like latent.InfectionsWithExpGrowth(RandomVariable).

[damonbayer]

From my understanding, there are three key times related to this implementation:

• $t_0$: the time where the first latent infection(s) occur
• $t_\text{seeded}$: the time at which there is a sufficient infection history such that the renewal process can begin.
• $t_\text{obs}$: the time at which the first observation occurs.

There is no reason that $t_\text{obs}$ couldn't come before $t_\text{seeded}$, but in the current hospitalization model tutorial (example-with-datasets), we have $t_0 < t_\text{seeded} < t_\text{obs}$.

This issue relates to generating $I(t)$ for $t \in [t_0, t_{\text{seeded}}]$. In the current model, we have $I(t) = I_0$ for $t \in [t_0, t_{\text{seeded}}]$ and we only save $I(t)$ for $t \geq t_{\text{seeded}}$.

When we have variation in $I(t)$ for $t \in [t_0, t_{\text{seeded}}]$, I assume we will want to keep track of all of these values. Is that correct?

@gvegayon @dylanhmorris

[gvegayon]

When we have variation in [math] for [math], I assume we will want to keep track of all of these values. Is that correct?

Yes. Since the only sampled quantities in the equation are $I_0$ and $r$, the rest of the sequence is fully deterministic. You can use the numpyro.deterministic function to tag that and ask the model to keep track of it, e.g.,

multisignal-epi-inference/model/src/pyrenew/process/rtrandomwalk.py

Line 123 in a7d6d2d

Rt = npro.deterministic("Rt", self.Rt_transform.inverse(Rt_trans_ts))

That said, I am not sure I'm following why we need $t_{seeded}$ and not only $t_0$ and $t_{obs}$. What am I missing?

[damonbayer]

$t_\text{seeded} \neq t_\text{obs}$ corresponds to the idea of padding in the current hospitalization model tutorial (example-with-datasets).

In that model, we have $I(t) = I_0$ for $t \in [t_0, t_{\text{seeded}}]$. Then $I(t)$ follows the renewal process for $t > t_\text{seeded}$. The observations don't start until some time later, $t_\text{obs} = t_\text{seeded} +$ padding

You could also imagine a scenario where $t_\text{obs} < t_\text{seeded}$ which would be like negative padding. This points to the idea that the $I(t)$ vector should contain $I(t)$ for $t > t_0$. In the current implementation it only contains $I(t)$ for $t > t_\text{seeded}$.

[damonbayer]

So, I am thinking latent/i0.py should be made more generic to contain the concept of $I(t)$ for $t \in [t_0, t_{\text{seeded}}]$. Then that would be used with sample_infections_rt to get $I(t)$ for $t > t_0$, which forms the basis for the observations.

[dylanhmorris]

I agree with @damonbayer; my only minor quibble is notational. Proposed default: let $t_0 = 0$ denote the first timepoint at which $I(t)$ is generated by the renewal process according to some $\mathcal{R}(t_0)$ (versus "seeded").

i.o.w. seeding times are by default negative: $t_\mathrm{firstseed} < t_0 = 0$