*5.3. Numerical Simulations for High-Risk Population*

In this subsection, we focus our attention to the French MSM high-risk population. We remind that this MSM subgroup consists of individuals with multi-partner intercourse, sex relationships while using drugs (chemsex), etc. The HIV risk of infection is thus much higher than the rest of the MSM population. This is one of the reasons why this subgroup is the PrEP treatment target for the French health insurance. The protected compartment in this particular case plays a major role as a reservoir. This was clearly confirmed in our model simulations, as shown in Figure 6, where the number of susceptibles clearly drops as the number of protected individuals increases. Note that the infected population rises, but for the first time, reaches a threshold. This means that barely any new infected cases appear. We remind here that the model is an *SI* and not an *SIR*, that is no recovered individuals can appear since *HIV* is a disease with a lifetime treatment.

**Figure 6.** Plot of the evolution of the different compartments of the high-risk french population along time (over 15 years). The crosses in the last plot represent the real values of the number of PrEP users obtained from Table 4. *ψ* verifies the logistic equation, and f is a Hill function.

In Figure 7, the incidence evolution is plotted with or without PrEP treatment. The reservoir effect is even more explicit there in the sense that, with PrEP, the incidence declines to reach a low plateau, while without, it keeps on growing indefinitely.

Finally, as mentioned before, for the MSM high-risk population, the R<sup>0</sup> in France is higher than 1. Figure 8 reveals that, as the number *ψ* of new PrEP users increases, R<sup>0</sup> not only decreases below one, but also keeps on falling off. Here, the critical value of *<sup>ψ</sup>*, which keeps <sup>R</sup><sup>0</sup> below one, is *<sup>ψ</sup>critical* <sup>=</sup> 0.113 person.months−<sup>1</sup> . According to our data, *ψ* = 0.071 person.months−<sup>1</sup> , as we assume the maximum threshold has been reached. The decay appears, however with a smaller slope, which means, as in the previous subsection, that, after a certain threshold (0.2 here), very large values of *ψ* do not influence the reduction of R<sup>0</sup> much.

**Figure 7.** Plot of the evolution of the incidence with and without PrEP (4) over 15 years among French high-risk MSM.

This reservoir effect is so important that, if for any reason, there is the decision to stop PrEP treatment for this MSM group, the rise in new infected individuals would be inevitable, as shown in Figure 9.

**Figure 8.** Plot of the R<sup>0</sup> as a function of *ψ*.

**Figure 9.** Plot of the susceptibles and infected if PrEP were stopped in 15 years. We considered that all the protected individuals become susceptibles again.

### **6. Conclusions and Discussion**

The aim of this work was to propose a brand-new model for prescribing PrEP for 3 months, with the choice to continue treatment or to stop it at any time after this period. This choice changes many things in the new modeling approach by introducing difference equations with delay and, thus, the possibility of describing the effect of inertia and the reservoir of protected individuals, not only analytically, but also numerically. This approach could also be generalized to other epidemiological models describing the effect of vaccine or booster doses for improved efficacy of protection.

Furthermore, the double insights of the MSM general and high-risk populations seemed very important to us, especially to depict the effect of PrEP treatment on the disease incidence of the disease and the French health insurance policy to target the high-risk subgroup. With our model and simulations, it appeared that these strategies are much more effective than random targeting of any MSM individual.

Our future work, in production, will take into account that the exponential growth of susceptibles is not realistic for a longer period of time, so it would be better to model it as a logistic growth and then numerically simulated on a longer period of time.

The data we worked on were from 2016 to 2019. It would be interesting to analyze the effect of the SARS-CoV-2 lockdown effect on the incidence and PrEP users. It would also be informative to analyze the booster effect of the non lockdown period (mid-2021 and after). For this purpose, it is necessary to wait a few more years to collect a sufficient amount of data.

A future model should also consider *θ* time dependent. Indeed, depending on certain periods (a lockdown, for example, or a seasonal effect), the number of PrEP users willing to take the treatment at the end of their 3-month prescription may vary over time. Some users may also take it for a short period of time (e.g., 2 years on average) and discontinue treatment after choosing a long-term closed relationship.

The addition of a compartment of the infected population under tritherapy should also be more realistic. Indeed, with tritherapy, HIV individuals can no longer be infectious and can be removed from the *I* compartment. The estimate of *β* would then be more accurate and reflect a more realistic situation.

Finally, it would be interesting to add the cost of treatment and the cost-efficient strategy with the threshold of PrEP users in the compartment for optimal disease control and, eventually, its disappearance. For this purpose, our next goal is to compare data and policies across multiple countries (some with full financial support for treatment, some with partial support, and some with no support).

**Author Contributions:** Supervision, M.A., L.P.-M. and J.W.; Writing—original draft, J.M.; Writing review & editing, G.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** G.R. was supported by the Natural Sciences and Engineering Research Council and the York Research Chairs program.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable

**Conflicts of Interest:** The authors declare no conflict of interest.
