# Forecasting the Effect of Pre-Exposure Prophylaxis (PrEP) on HIV Propagation with a System of Differential–Difference Equations with Delay

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## Abstract

**:**

## 1. Introduction

## 2. Introducing Our Model

## 3. Mathematical Analysis

#### 3.1. Well-Posedness of the Model

**Proposition**

**1.**

**Proof.**

#### 3.2. Steady-States and Basic Reproduction Number

**Lemma**

**1.**

**Proof.**

**Proposition**

**2.**

- If ${\mathcal{R}}_{0}\left(0\right)<1$, then we have ${\mathcal{R}}_{0}\left(\tau \right)<1$, for all $\tau \ge 0$.
- If ${\mathcal{R}}_{0}(\infty )>1$, then we have ${\mathcal{R}}_{0}\left(\tau \right)>1$, for all $\tau \ge 0$.
- If ${\mathcal{R}}_{0}\left(0\right)>1$ and ${\mathcal{R}}_{0}(\infty )<1$, then there exists a unique $\overline{\tau}>0$ such that ${\mathcal{R}}_{0}\left(\tau \right)>1$ for $0\le \tau <\overline{\tau}$ and ${\mathcal{R}}_{0}\left(\tau \right)<1$ for $\tau >\overline{\tau}$, with ${\mathcal{R}}_{0}\left(\overline{\tau}\right)=1$.

**Property**

**1.**

**Proof.**

**Corollary**

**1.**

- (i)
- Suppose that there exists ${\tau}_{0}\ge 0$ such that ${\mathcal{R}}_{0}\left({\tau}_{0}\right)>1$. Then, the disease-free equilibrium (9) is unstable for $\tau ={\tau}_{0}$.
- (ii)
- Suppose that there exists ${\tau}_{1}\ge 0$ such that ${\mathcal{R}}_{0}\left({\tau}_{1}\right)<1$ and the disease-free equilibrium (9) is locally asymptotically stable for $\tau ={\tau}_{1}$. Then, it is locally asymptotically stable for all $\tau \ge {\tau}_{1}$.
- (iii)
- Suppose that ${\mathcal{R}}_{0}\left(0\right)<1$. Then, the disease-free equilibrium (9) is locally asymptotically stable for all $\tau \ge 0$.

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

## 4. Global Stability Analysis

#### 4.1. Global Asymptotic Stability of the Disease-Free Steady-State

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 4.2. Global Asymptotic Stability of the Endemic Equilibrium

**Theorem**

**2.**

**Proof.**

## 5. Applications to French Clinical Data

#### 5.1. Choice of the Parameters

^{®}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 ${\mathcal{R}}_{0}$ using a Bayesian approach following the idea developed in [13]. Thanks to our data of new HIV infections ([14,15]), we obtained ${\mathcal{R}}_{0}=0.93$ for the normal MSM French population.

#### 5.2. Numerical Simulations for the General French MSM

#### 5.3. Numerical Simulations for High-Risk Population

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the compartmental model (1). Continuous arrows represent movements between compartments. Dashed ones represents the transmission of the infection.

**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.

**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. $\psi $ verifies the logistic equation, and f is a Hill function.

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

**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.

Symbols | Signification | Value for General MSM | Value for High-Risk MSM | Unity |
---|---|---|---|---|

$\theta $ | Probability to keep the PrEP treatment | $0.83$ | $0.83$ | |

$\sigma $ | Recruitment | $3000$ | 562 | indiv.months${}^{-1}$ |

$\mu $ | Removal rate from the compartments | $0.000758333$ | $0.0076$ | indiv.months${}^{-1}$ |

$\beta $ | HIV transmission rate per infected individual | $1.821\times {10}^{-10}$ | $2.85\times {10}^{-7}$ | (indiv.months)${}^{-1}$ |

$\tau $ | Duration of the period of PrEP taking | 3 | 3 | months |

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

Initial Conditions | Value for MSM | Value for High-Risk MSM |
---|---|---|

$S\left(0\right)$ | 2,600,000 | 40,000 |

$I\left(0\right)$ | 90,000 | 9000 |

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

Semester | Initiation of PrEP | Renewal of the Treatment | Total PrEP Users |
---|---|---|---|

S1—2016 | 1166 | ### | 1166 |

S2—2016 | 1826 | 911 | 2737 |

S1—2017 | 2193 | 2273 | 4466 |

S2—2017 | 2564 | 3807 | 6371 |

S1—2018 | 3138 | 5413 | 8551 |

S2—2018 | 4488 | 7647 | 12,135 |

S1—2019 | 5103 | 10,398 | 15,501 |

Semester | Values of $\mathit{\psi}$ for General MSM | Values of $\mathit{\psi}$ for High-Risk MSMS | Values of $\mathit{\theta}$ |
---|---|---|---|

S1—2016 | $0.0000747$ | $0.0048583$ | ### |

S2—2016 | $0.000117$ | $0.007608$ | $0.7813$ |

S1—2017 | $0.00014$ | $0.00913$ | $0.8305$ |

S2—2017 | $0.000164$ | $0.0106$ | $0.8159$ |

S1—2018 | $0.000201$ | $0.0130$ | $0.8496$ |

S2—2018 | $0.00029$ | $0.0187$ | $0.8943$ |

S1—2019 | $0.00033$ | $0.0216$ | $0.8569$ |

Parameters | Values for General MSM | Values for High-Risk MSM |
---|---|---|

K | $0.0007$ | $0.072$ |

r | $0.0000222$ | $0.000026$ |

**Table 6.**Values of parameters ${S}_{sat}$, $\gamma $, and n depending on the general French MSM and the high-risk French MSM.

Parameters | Values for MSM | Values for High-Risk MSM |
---|---|---|

${S}_{sat}$ | $5\times {10}^{6}$ | 230,000 |

$\gamma $ | 120 | 50 |

n | 2 | $1.56$ |

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**MDPI and ACS Style**

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

**AMA Style**

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(21):4093.
https://doi.org/10.3390/math10214093

**Chicago/Turabian Style**

Adimy, Mostafa, Julien Molina, Laurent Pujo-Menjouet, Grégoire Ranson, and Jianhong Wu.
2022. "Forecasting the Effect of Pre-Exposure Prophylaxis (PrEP) on HIV Propagation with a System of Differential–Difference Equations with Delay" *Mathematics* 10, no. 21: 4093.
https://doi.org/10.3390/math10214093