**5. Applications to French Clinical Data**

In order to apply our model to real data, we chose the French MSM population, and later, we focused more specifically on the high-risk individuals within this group. We define here the high-risk MSM population as multi-partner gay men, as well as sexual intercourse while using drugs, usually called chemsex, or new generations not inclined to use any protection anymore (mainly condoms). Of course, this reduces our study to a small part of the population, but, because of their having the highest rate of infectivity, they belong to the target of PrEP treatments at the infectious disease center in France.

Furthermore , for a more realistic approach, *ψ* was considered time-dependent, and *ψ* is defined by the logistic equation for a more accurate data fit (details and parameters in the first subsection below). The PrEP users were collected between January 2016 (starting year of PrEP in France) and June 2019 (before the SARS-CoV-2 outbreak) (see [2]).

Section 5 is structured into three subsection: First, we introduce the data and the methods to approximate the parameters. Then, we propose a set of simulations adapted to the general French MSM population. The last subsection is focused on the so-called high-risk individuals as described previously.

It is important at this stage to remind that HIV is not curable and that PrEP does not have a direct influence on the decreasing number of the infected compartment over time, which is mainly due to natural death (with a life expectancy for treated persons as high as non-infected ones) or, more rarely now, thus neglected here for the French group, untreated AIDS individuals. However, PrEP policy in the population may have an important impact on the infectivity rate. Since PrEP's efficiency can be measured in a long-term range and using incidence, we simulated over 15 years (that is, 180 months) for a forecast of this prevention strategy over time.
