### Model structure

We model a vaccine trial using an SI deterministic compartmental model; this is a simple epidemic model that has two populations, the Susceptible (S) and the Infected (I).
We start the model with all trial individuals in the S group; over time S individuals move into the I group as they become infected.

More details are on IDM's Github repo here.

Note that this is specfically a vaccine trial model and not a more commonly used epidemic model (e.g. there are no births or deaths, and no recovery); we are not modeling infections from the I to S compartments, but rather only infections from a theoretical, un-modeled, outside (non-trial) population.
We do this with one parameter, `lambda`

, which is the rate that individuals in S get infected (move to I). This model structure also removes the possibility of indirect effects from vaccination.

### Parameters

beta

transmission rate (per contact)

c

exposure rate (serodiscordant sexual contacts per time)

prev

prevalence (prevalence of viremic individuals)

lambda

lambda = beta * c * prev (this parameter defines the rate of infection)

epsilon

per contact vaccine efficacy; vaccine-induced reduction in the risk of HIV infection from a single exposure

The infection rate per time step is a combination of population prevalence `prev`

(of viremic individuals), the exposure rate
(serodiscordant sexual exposure per time) `c`

, and the transmission rate (per exposure) `beta`

.

The per exposure effect of vaccination is `epsilon`

; `epsilon`

is not time-varying (the per-exposure vaccine effect does not decay over time) and assumes a homogeneous effect
(does not vary by viral genotype or individual traits).

We include three subgroups in the heterogeneous exposure population: high, medium, and low exposure.
In reality we never fully know the correct size of HIV risk subgroups (i.e. fraction of the population) or their relative contribution to overall incidence.

The `risk`

parameter (the risk multiplier) is an amalgam of increases in transmission risk that could be due to higher per-contact transmission or infection risk,
higher exposure rate (number of contacts), or higher prevalence of HIV viremia in partners. Individual risk of infection can vary for
these separately or in combination. Note that exposure heterogeneity, i.e. variation among individuals in the risk of infection, is most commonly discussed as variation in the number of HIV exposures.
It can also include variation in the individual per-contact risk of acquisition, perhaps due to co-infections, viral load in the source individual or host genetics. Our `risk`

parameter only multiplies the baseline risk of the medium risk subgroup, and it
possible (in the underlying code) for the low risk subgroup to have zero risk.