# Modeling and optimal control of HIV/AIDS prevention through PrEP

## TL;DR

This paper develops a new epidemiological model for HIV/AIDS incorporating PrEP, proves its mathematical properties, calibrates it with real data from Cape Verde, and determines optimal PrEP strategies to reduce HIV transmission effectively.

## Contribution

It introduces a novel HIV/AIDS model with PrEP, proves its stability, calibrates it with real data, and derives optimal control strategies considering cost and coverage constraints.

## Key findings

- PrEP significantly reduces HIV transmission in simulations.
- The model accurately predicts HIV/AIDS cases in Cape Verde.
- Optimal strategies balance PrEP costs and epidemic control.

## Abstract

Pre-exposure prophylaxis (PrEP) consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals and has recently demonstrated to be highly efficacious for HIV prevention. We propose a new epidemiological model for HIV/AIDS transmission including PrEP. Existence, uniqueness and global stability of the disease free and endemic equilibriums are proved. The model with no PrEP is calibrated with the cumulative cases of infection by HIV and AIDS reported in Cape Verde from 1987 to 2014, showing that it predicts well such reality. An optimal control problem with a mixed state control constraint is then proposed and analyzed, where the control function represents the PrEP strategy and the mixed constraint models the fact that, due to PrEP costs, epidemic context and program coverage, the number of individuals under PrEP is limited at each instant of time. The objective is to determine the PrEP strategy that satisfies the mixed state control constraint and minimizes the number of individuals with pre-AIDS HIV-infection as well as the costs associated with PrEP. The optimal control problem is studied analytically. Through numerical simulations, we demonstrate that PrEP reduces HIV transmission significantly.