**1. Introduction**

Since its onset in the early 1980s and its clear identification in 1983, Human Immunodeficiency Virus (HIV) and then Acquired Immune Deficiency Syndrome (AIDS) have still comprised one of the most deadly active worldwide epidemics. In the 2019 UN-AIDS study, about 38 million persons lived with HIV, 1.7 million became infected, and 690,000 died of AIDS-related diseases (UNAIDS, Global HIV & AIDS statistics—2020 fact sheet: https://www.unaids.org/en/resources/fact-sheet) (accessed on 13 September 2022), becoming one of the most serious public health challenges.

It is well known now that this infection evolves in three stages: first a short acute phase where flu-like symptoms appear, followed by a symptom-free chronic phase that lasts between 10 and 15 years. It eventually ends up with AIDS when the virus has killed enough TCD4, leading to the failure of the immune system (https://www.hiv.gov/hiv-ba sics/overview/about-hiv-and-aids/symptoms-of-hiv) (accessed on 13 September 2022).

Note here that the latency stage appears as one of the crucial problems. Indeed, the dormant period of the virus drastically delays HIV/AIDS diagnosis if not detected and plays a major role in the epidemic's spread (see Figure 1 in [1]).

Despite extensive investigations, there is still no existing therapy that helps the organism to fully get rid of the virus.

**Citation:** Adimy, M.; Molina, J.; Pujo-Menjouet, L.; Ranson, G.; Wu, J. Forecasting the Effect of Pre-Exposure Prophylaxis (PrEP)on HIV Propagation with a System of Differential–Difference Equations with Delay. *Mathematics* **2022**, *10*, 4093. https://doi.org/10.3390/ math10214093

Academic Editor: Alexandra Kashchenko

Received: 15 September 2022 Accepted: 26 October 2022 Published: 2 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

However, since 1996, AntiRetroviral Therapy (ART), consisting of a combination of three (or more) antiretroviral agents taken daily, has appeared to reduce significantly the presence of the virus under a detectable threshold. Currently, not only efficient at avoiding the fatality of the disease, ART allows also HIV-infected individuals stop spreading the disease. In 2019, 25.4 million of them successfully accessed antiretroviral therapy, (UNAIDS, Global HIV & AIDS statistics—2020 fact sheet: https://www.unaids.org/en/resources/fact-sheet (accessed on 13 September 2022); World Health Organization (WHO)—HIV/AIDS: https: //www.who.int/news-room/fact-sheets/detail/hiv-aids (accessed on 13 September 2022).

In the late 2010s, Pre-Exposure Prophylaxis (PrEP) was introduced as a new way of preventing HIV transmission. This successful treatment addresses HIV-free individuals showing a high risk of becoming infected. PrEP consists of taking a combination of two antiretrovirals in two different protocols: either on a daily basis (continuous treatment) or on demand (discrete treatment), which is just at least 2 h before and 2 days after sexual intercourse. Every 3 months, continuous PrEP users need to have a global Sexually Transmitted Infections (STIs) screening in order to obtain a prescription for another 3 months (World Health Organization (WHO)—HIV/AIDS: https://www.who.int/news-room/fact-sheets/detail/hiv-aids; https://www.who.int/team s/global-hiv-hepatitis-and-stis-programmes/hiv/prevention/pre-exposure-prophylaxis) (accessed on 13 September 2022). Theoretically, they can stop anytime, but they are practically more likely to choose at the end of each quarter. We remind here that PrEP is preventive and does not cure HIV. However, it effectively reduces HIV transmission (World Health Organization (WHO)—HIV/AIDS: https://www.who.int/news-room/fact-sheets/detail/hiv-aids) (accessed on 13 September 2022).

Our objective here was to design a new model to mathematically forecast the influence of PrEP users among high-risk profiles on the epidemic. The application of our study focused only on the French population, where PrEP has been used since 2016, and is currently followed by an average of 20,000 individuals [2].

In the past few decades, many epidemiological models have been used to describe HIV's dynamics. Some of them used the standard Susceptible (*S*)–Infected (*I*)–Infected under ART (*C*) with low viral charge–Infected with AIDS symptoms (*A*) (*SICA*) model (see [3] for instance), and for a good review, we suggest [4,5].

It has only been in the past few years that new models including a PrEP user compartment appeared and started to receive our attention. A good example is the preliminary work [6], where the authors suggested a vaccination approach, which could be seen as a precursor to PrEP. Their numerical simulations showed that the epidemic spread could be controlled thanks to vaccination, even if the basic reproduction number remains above 1. In [7,8], an explicit PrEP compartment was included in the *SICA* model. The parameters in [8] were adjusted with clinical data and PrEP's effectiveness mathematically proven. In [9], the authors divided the PrEP compartment according to the adherence of users to the treatment. They showed that, with at least 70% of PrEP users in the male homosexual population, the HIV epidemic could be effectively controlled. Finally, in [10], PrEP was combined with screening and the result was compared to Portuguese data showing that these two processes appeared necessary to control the HIV epidemic.

In our paper, we introduce a new approach with a Susceptible–Infected–Protected (*SIP*) model. Inspired by [8,11], it includes an age structure on the PrEP (protected) compartment, corresponding to the time spent after the onset of the new 3-month treatment period. Once the term is reached, the user chooses either to keep taking PrEP or to stop it, becoming susceptible again. On the other hand, individuals of the *S* compartment may decide to start or resume this preventative process and, hence, reach the *P* population.

Our goal was to propose a more accurate model than that of [8] that still possible to handle analytically by considering a more complex susceptible interaction *F*(*t*, *S*) depending both on time *t* (through a function *ψ*) and the total *S* population (through a function *f*), which can be seen as a political or economic decision made depending on the size of the HIV-exposed population(see Section 2 for details). After standard modifications, we prove in Section 2 that our new model can be written as a nonlinear differential–difference

system with discrete delay. As presented below in Sections 3 and 4, the stability analysis is then possible when the *ψ* term is constant. However, since it may take a while to convince a population set to start PrEP, it appears natural to us that *ψ* may follow a logistic equation. The model becomes then too challenging to be investigated in an analytical way, and thus, numerical simulations take over this work in Section 5. Thanks to this new assumption, we show eventually that it is possible to fit the data with realistic values. Before this, we proceed to its well-posedness and stability study and investigate the constant and nonconstant joining PrEP rate subcases in Section 3 and 4. Then, we compare our theoretical work to official clinical French data in order to understand the role of PrEP's dynamics on the HIV/AIDS epidemic in Section 5. With some numerical simulations, we observe that, by choosing a logistic equation for the nonlinear joining PrEP rate and a Hill function for the dynamics of *f*(*S*), we perfectly fit our data.
