*5.1. Choice of the Parameters*

In Table 1, we give the parameters used in Model (1), as well as their meaning and values for the general and high-risk MSM community in France estimated from [2], assuming that 60% of the the population considered in the dataset of the study are MSM.

PrEP treatment in France started in 2016, and since the SARS-CoV-2 epidemic drastically modified sexual habits (through successive lockdowns), we considered the population data until 2019 only. Then, after, we simulated our model based on the estimated parameter values for the following 15 years. Note here that we did not take the SARS-CoV-2 nor the monkey pox epidemic into account for three reasons: first, because these periods are a bias for our model and need to be carefully adjusted with updated parameter values; second, to the best of our knowledge, the official French data related to these past two years have not been released yet; finally, we preferred keeping our model as simple as possible for this present work. A more complex approach will be the object of a future work.


**Table 1.** Parameters used in Model (1) with values applied to the French population estimated from [2].

In [2], information between 2016 and 2019 was collected every 6 months. We remind here that, at the end of a 3-month period, the patient decides whether to give up the treatment or to continue. This is the reason why our simulations below were computed on a monthly basis.

Initial conditions for functions *t* → *S*(*t*), *t* → *I*(*t*) and *t* → *u*(*t*) were chosen to be January, the 1st of 2016, according to [2].

Because of the delay equation for *<sup>u</sup>*, the function *<sup>u</sup>*init : [−3, 0] <sup>→</sup> <sup>R</sup>+, *<sup>t</sup>* → *<sup>u</sup>*init(*t*) was adjusted with a cubic spline interpolation of 60% of each total number of the last column in Table 2. Note that *u*init is the same for both populations' (general MSM and its high-risk subset) simulations because we assumed here that most of the French PrEP users belong to the high-risk MSM group.

Regarding the susceptible: based on an AVAC study (https://www.avac.org/sites /default/files/u3/MSM\_in\_Europe\_Euro\_Rave.pdf, page 5) (accessed on 13 September 2022), 4–10% of French males declare belonging to the MSM group. Assuming that France has about 65,000,000 inhabitants, it seems then reasonable to assume that *S*(0) = 2,600,000.

Furthermore, on the official website of UNAIDS (https://www.unaids.org/en/region scountries/countries/france) (accessed on 13 September 2022), we obtained that 170,000 individuals were infected with HIV in France in 2016. We assumed that a majority belong to the MSM group, but not the entire number (indeed, a large portion of new cases was reported to be foreign heterosexuals, among other subcases). This is why we took *I*(0) = 90,000. It is important to remind here that this value may not be the exact one, since the number of HIV infections detected is always below the real number of HIV infections (HIV testing is not compulsory, and thus, several individuals may carry the virus for years without knowing it).

**Table 2.** Summary of initial conditions for susceptibles and infected among general MSM and MSM high-risk populations.


Finally, when simulating the MSM high-risk population, the estimated susceptible number would be between 3000 and 50,000 depending on the literature. Thus, we took *S*(0) = 40,000 according to [2]. According to French data (https://vih.org/20190328/stabili te-des-chiffres-du-vih-en-france/) (accessed on 13 September 2022), French infections have been stable since 2010. On average, there are 6500 contaminations per year, and infections within the MSM population represent 43%. Then, we chose *I*(0) = 9000.

According to the official French data (https://www.insee.fr/fr/statistiques/2383440) (accessed on 13 September 2022), *μ* = 0.000758333 individuals.months−<sup>1</sup> at the country scale. We chose this value for the French MSM removal rate (death, as well as removed from sexual life). For the high-risk MSM, *μ* is considered much larger because a high-risk sexual life style does not last longer than the average MSM one. Thus, *μ* = 0.0076 individuals.months−<sup>1</sup> in this subgroup.

The 2016–2019 semester official French datasets are given in Table 3.


**Table 3.** Total number of PrEP users in France since 2016, given by semester (see Table 3 in [2] ).

Thanks to Tables 2 and 3, we are able to assess the values of functions *θ* and *ψ* for the years 2016–2019. Indeed, for each semester, we obtained a different *θ* by using the following formula coming from its definition:

$$\theta(\text{semester}) = \frac{\text{Number of renewal treatment of the current semester}}{\text{Total of PrEP users of the previous semester}}.$$

We assumed that this value is the same for every month of the semester. Then, we chose *ψ*, per month, as follows:

$$
\psi(\text{months}) = \frac{\text{Number of individuals who begin the treatment}}{S(0) \ast 6}
$$

where *S*(0) is the initial condition for the susceptible compartment *S*. These values are given in Table 4.

**Table 4.** Values of parameters *ψ* and *θ* per month according to each semester, computed according to Tables 2 and 3.


We selected *θ* = 0.83 as the average value of all the previous *θ* from Table 4.

As mentioned in Section 2, we assumed that *ψ* is the solution of a logistic equation given by:

$$\frac{d\psi}{dt} = r\psi \left(1 - \frac{\Psi}{K}\right)\prime$$

where *K* is the carrying capacity and *r* exponential growth. We remind that the explicit form of *ψ* is then written by the following expression:

$$\psi(t) = \frac{K}{1 + \left(\frac{K}{\Psi0} - 1\right) \exp(-rt)},$$

where *ψ*<sup>0</sup> is the initial condition depending on the population type. Using the data for *ψ* given in Table 4, we summarize our results in Table 5.

**Table 5.** Values of parameters K and r depending on the population.


We assumed a Hill function for the dynamics of the function *f* :

$$f(S) = S\_{\rm sat} \frac{S^n}{\gamma^n + S^n},$$

with *Ssat* the saturation of the Hill function, *γ* the abscissa of the inflection point, and *n* the intensity of the slope. We summarize the values of these parameters in Table 6.

**Table 6.** Values of parameters *Ssat*, *γ*, and *n* depending on the general French MSM and the high-risk French MSM.


One of our goals was to estimate the HIV epidemic R<sup>0</sup> for French MSM. We used the package in the R® language untitled R0 (https://www.rdocumentation.org/packages/R0 /versions/1.2-6) (accessed on 13 September 2022) and, precisely, the function *est.R0.SB*, which estimates R<sup>0</sup> using a Bayesian approach following the idea developed in [13]. Thanks to our data of new HIV infections ([14,15]), we obtained R<sup>0</sup> = 0.93 for the normal MSM French population.

The two remaining parameters, *β* and *σ*, were estimated to keep R<sup>0</sup> equal to 0.93. We decided to choose *σ* = 3000; in other words, each month, 3000 individuals might join the susceptible compartment (by reaching the sexual consent age or deciding to join the MSM group). Thus, *<sup>β</sup>* = 1.821 × <sup>10</sup>−10(individuals.months)−<sup>1</sup> was estimated for the French general MSM group.

On the other hand, according to the official French data (https://solidarites-sante.go uv.fr/IMG/pdf/argumentaire\_depistage\_vih\_HAS\_2009-2.pdf) (accessed on 13 September 2022), the basic reproduction number related to the high-risk MSM ranges between 1.27 to 1.44. We took R<sup>0</sup> = 1.31 and estimated *<sup>β</sup>* = 2.85 × <sup>10</sup>−7(individuals.months)−1, with *μ* = 0.0076 months−<sup>1</sup> and *σ* = 562 given by Table 1.
