*5.2. Numerical Simulations for the General French MSM*

In Figure 2, protected (*P*) in green (graph on the right) keeps track of the exact data (crosses) for the first semesters, then reaches a threshold due to the French policy of regulating PrEP users. Indeed, the treatment is fully taken care of by the health insurance in France. Thus, a threshold needs to be set up (estimated at 60,000 MSM individuals here). With this threshold, it seems already that, within 15 years, the number of infected drops and keeps on decreasing. The number of susceptibles (blue curve on the left of the graph) keeps on growing with a lower slope between 50 and 100 months due to the increase of the protected and then the threshold. However, even with the increase of the susceptibles, the infected remain low. The protected compartment plays the role of a reservoir to prevent the HIV epidemic from growing back.

**Figure 2.** Plot of the evolution of the French MSM population (4) along the time (over 15 years). The crosses in the last plot represent the real values of PrEP users taken from Table 3. Function *ψ* verifies the logistic equation, and *f* is a Hill function.

Just as a comparison point, we simulated our model with a function *f* defined as identity (and not a Hill function anymore) (see Figure 3). We easily observed that the graph of the protected population does not fit the data in the first months with the same realistic parameter sets. Then, it seems that *f* needs to be more complex than identity for the case of French MSM, and the Hill function seems to be validated here.

In Figure 4, we plot the incidence with or without PrEP treatment. It seems obvious that without the PrEP reservoir (in blue), the incidence keeps on growing with no chance for the HIV epidemic to be controlled. On the other hand, with the PrEP compartment (in red), it is possible to keep the incidence at a low level (and even to decrease it significantly for the first years). The slight increase at the end (after Month 125) is due to the threshold of the number of PrEP users imposed in our model and likely used by the French health insurance policy.

**Figure 3.** Plot of the evolution of the different compartments of the model (4) along time (over 15 years). The crosses in the last plot represent the real values of PrEP users taken from Table 3. *ψ* verifies the logistic equation, and f is identity.

**Figure 4.** Plot of the evolution of the incidence with PrEP (in red) and without (in blue) from the system (4) over 15 years among the general French MSM.

In Figure 5, we depict the effect of the variation of R<sup>0</sup> as a function of *ψ*. It appears clearly here that, as *ψ* increases (that is, the number of PrEP users rises), R<sup>0</sup> declines drastically. This is, however, not linear, and a plateau may be reached after a certain threshold (above 0.2), which indicates that the flow of new PrEP users does not need to expand drastically. Indeed, even at *ψ* larger than 0.125, we obtain R<sup>0</sup> already below 0.3, which is quite satisfactory. Augmenting *ψ* would decrease R0, but not as fast as the first values of *ψ*.

**Figure 5.** Plot of the R<sup>0</sup> as a function of *ψ*.
