Reason I: We have more than one partnership type

There is one straightforward reason to model the sex act, and make the process of infection transmission contingent upon that sex act: the presence of >1 partnership types. Therefore, it makes sense to have a dichotomous condition that is TRUE if one (or more) of those sex acts occurs on any one day, and compute the probability of infection contingent on each of those sex acts. each

Reason II: Distributing 2.4 sex acts / week does not capture the intended process

In the current formulation, the empirical value of the numebr of sex acts (2.4 sex acts/week) is distributed over the 7 day period. While this does not make conceptual sense, it would have been okay if mathematically the transmission probability over a week did not change by distributing the number of weekly sex acts uniformly.

We start by defining constants

   n_sex_p_week <- 2.4 ## number of sex acts/week
   beta <- seq(0, 1, 1e-5) ## probability of transmission per sex act

and the linear and binomial formulations

   lin_scaling <- beta*n_sex_p_week ## linear model for probability of transm./week 
   binom_scaling <- 1-(1-beta)^n_sex_p_week ## linear model for probability of transm./week

Now, let’s compare the linear and binomial models for transmission probabilities over a week:

For even moderately high beta (>0.15), the binomial models predict lower probabilites of transmission per week than the linear model.

If we do distribute 2.4 sex acts uniformly across each day of the week, then the daily and weekly transmission probabilities will be given by

   transm_p_one_day <- 1-(1-beta)^(2.4/7)
   transm_p_one_week <- transm_p_one_day*7

Whereas, with a binomial transmission model, the weekly transmission probability is as defined above, by binom_scaling.

So now, we compare, * linear scaling, * binomial scaling with 2.4 sex acts / week distributed over 7 days * binomial scaling with 2.4 sex acts/week respectively, below:

While all three formulations seem to be roughtly equal for beta (i.e. transmission probability/sex act) < 0.10, for higher values, they diverge.

This example provides evidence that we need to have a decision about a sex act on any given day, and compute if an infection event occurs on that day, rather than including sex acts and infectivities at the same time.