SIR model

I've been toying with the following SIR-type model in my head, and I want to write it down. The goal is to encapsulate as much of the standard epidemic structure in some time varying parameters. Let `I_t` be infections at time `t` and `c_t` be new reported cases at time `t`. We do this in discrete time.

Suppose that the infection model is:
```
I_{t+1} = b_t * I_t * e_t
log(b)_{t+1} = log(b)_t + v_t
v_{t+1} = v_t + z_t
z_t ~ N(0, sigma_z)
```

In the standard SIR, we usually have `I(t)' = bS(t)I(t)-gI(t)`, which yields `r(t) = bS(t)-g`. So we can think of `b_t` above as being "like" `(1+r(t))`. So we're saying that `log(r(t))` varies slowly. The specification with the time varying `v` is very similar to putting 2nd-order AR behavior on `log(b)`. Then we model cases as linear in past infections:
```
c_t = P * I + f_t
f_t ~ N(0, sigma_f)
```
Here, I is some vector of past infections (say last 21) and P is a weight vector. So this part looks much like Maria's nowcasting model.

---

1. I suspect that "folding compartments into b_t" means that this structure encapsulates lots of models. I have more to think on how exactly it relates. (multi-strain, SEIR, etc.)
2. Fitting is non-trivial. My goal would be to avoid MC as much as possible. I'm hoping that this can be fit with the unscented Kalman Filter (and that that is relatively easy).
3. I would probably set P deterministically, say with a Gamma distribution. The parameters of that could also be but into the state equation, but that makes avoiding MC harder.