# American Institute of Mathematical Sciences

March  2022, 15(3): 639-654. doi: 10.3934/dcdss.2021156

## Stability and optimal control of a delayed HIV/AIDS-PrEP model

 Center for Research & Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal

Received  February 2020 Revised  March 2021 Published  March 2022 Early access  December 2021

In this paper, we propose a time-delayed HIV/AIDS-PrEP model which takes into account the delay on pre-exposure prophylaxis (PrEP) distribution and adherence by uninfected persons that are in high risk of HIV infection, and analyze the impact of this delay on the number of individuals with HIV infection. We prove the existence and stability of two equilibrium points, for any positive time delay. After, an optimal control problem with state and control delays is proposed and analyzed, where the aim is to find the optimal strategy for PrEP implementation that minimizes the number of individuals with HIV infection, with minimal costs. Different scenarios are studied, for which the solutions derived from the Minimum Principle for Multiple Delayed Optimal Control Problems change depending on the values of the time delays and the weights constants associated with the number of HIV infected individuals and PrEP. We observe that changes on the weights constants can lead to a passage from bang-singular-bang to bang-bang extremal controls.

Citation: Cristiana J. Silva. Stability and optimal control of a delayed HIV/AIDS-PrEP model. Discrete & Continuous Dynamical Systems - S, 2022, 15 (3) : 639-654. doi: 10.3934/dcdss.2021156
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##### References:
Partial derivative of $R_0(\tau)$ with respect to $\tau$ (with logarithmic scale in the $Y$ axis): (a) $\beta = 0.0752$ corresponding to $R_0(\tau) < 1$; (b) $\beta = 0.752$ corresponding to $R_0(\tau) > 1$
Influence of the time delay on $R_0(\tau)$, for $\tau \in [0, 50]$ and $\beta \in [0.16, 1.72]$
Stability of the disease free equilibrium $\Sigma_0$ for $d = 1$, $\Lambda = 10724$, $\beta = 0.0752$ and the other parameter values from Table 2, with $t \in [0, 1000]$ and $\tau \in \{0, 10,100 \}$
Stability of the endemic equilibrium $\Sigma_+$ for $d = 1$, $\Lambda = 10724$, $\beta = 0.752$ and the other parameter values from Table 2, with $t \in [0, 1000]$ and $\tau \in \{0, 10,100 \}$
Extremal solutions $\tilde{S}$, $\tilde{I}$ and $\tilde{C}$ associated to the extremal control $\tilde{u}$, given by (18)
Extremal solutions $\tilde{A}$, $\tilde{E}$ and extremal control $\tilde{u}$
Extremal control $\tilde{u}$ and associated state trajectory $\tilde{I}$, for different weight constants: $w_1 = w_2 = 1$; $w_1 = 1$, $w_2 = 50$; $w_1 = 5$, $w_2 = 1$
Description of the parameters of the HIV/AIDS-PrEP model (2)
 Symbol Description $\Lambda$ Recruitment rate $\mu$ Natural death rate $\lambda$ Infection rate for $S$ individuals $\beta$ Transmission coefficient for HIV transmission $\eta_C$ Modification parameter $\eta_A$ Modification parameter $\phi$ HIV treatment rate for $I$ individuals $\rho$ Default treatment rate for $I$ individuals $\alpha$ AIDS treatment rate $\omega$ Default treatment rate for $C$ individuals $d$ AIDS induced death rate $\psi$ Proportion of susceptible individuals that takes PrEP $\theta$ Proportion of susceptible individuals who default PrEP
 Symbol Description $\Lambda$ Recruitment rate $\mu$ Natural death rate $\lambda$ Infection rate for $S$ individuals $\beta$ Transmission coefficient for HIV transmission $\eta_C$ Modification parameter $\eta_A$ Modification parameter $\phi$ HIV treatment rate for $I$ individuals $\rho$ Default treatment rate for $I$ individuals $\alpha$ AIDS treatment rate $\omega$ Default treatment rate for $C$ individuals $d$ AIDS induced death rate $\psi$ Proportion of susceptible individuals that takes PrEP $\theta$ Proportion of susceptible individuals who default PrEP
Parameters values of models (3) and (11), taken from [24]
 Symbol Value Symbol Value $\mu$ $1/69.54$ $\rho$ $0.1$ $\Lambda$ $10724$ $\alpha$ $0.33$ $\beta$ $0.582$ $\omega$ $0.09$ $\eta_C$ $0.04$ $d$ $0$ $\eta_A$ $1.35$ $\psi$ $0.1$ $\phi$ $1$ $\theta$ $0.01$
 Symbol Value Symbol Value $\mu$ $1/69.54$ $\rho$ $0.1$ $\Lambda$ $10724$ $\alpha$ $0.33$ $\beta$ $0.582$ $\omega$ $0.09$ $\eta_C$ $0.04$ $d$ $0$ $\eta_A$ $1.35$ $\psi$ $0.1$ $\phi$ $1$ $\theta$ $0.01$
Cost functional and switching times for different weight constant values
 Weight constant values Cost functional $J(\tilde{u})$ Switching time $t_1$ Switching time $t_2$ $w_1 = w_2 = 1$ $J(\tilde{u}) \simeq 1818.64$ $t_1 \simeq 13.30$ $t_2 \simeq 18.02$ $w_1 = 1, \, w_2 = 50$ $J(\tilde{u}) \simeq 2125.28$ $t_1 \simeq 2.93$ $t_2 \simeq 11.28$ $w_1 = 5, \, w_2 = 1$ $J(\tilde{u}) \simeq 9020.09$ $t_1 \simeq 19.10$
 Weight constant values Cost functional $J(\tilde{u})$ Switching time $t_1$ Switching time $t_2$ $w_1 = w_2 = 1$ $J(\tilde{u}) \simeq 1818.64$ $t_1 \simeq 13.30$ $t_2 \simeq 18.02$ $w_1 = 1, \, w_2 = 50$ $J(\tilde{u}) \simeq 2125.28$ $t_1 \simeq 2.93$ $t_2 \simeq 11.28$ $w_1 = 5, \, w_2 = 1$ $J(\tilde{u}) \simeq 9020.09$ $t_1 \simeq 19.10$
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