Modeling and optimal control of HIV/AIDS prevention through PrEP

Cristiana J. Silva; Delfim F. M. Torres

doi:10.3934/dcdss.2018008

Article Contents

2018, Volume 11, Issue 1: 119-141. Doi: 10.3934/dcdss.2018008

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Modeling and optimal control of HIV/AIDS prevention through PrEP

Cristiana J. Silva^, and
Delfim F. M. Torres

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

* Corresponding author: Cristiana J. Silva
* Corresponding author: Cristiana J. Silva

Received: August 31, 2016

Revised: January 31, 2017

Published: February 2018

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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.

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Mathematics Subject Classification: Primary: 34C60, 92D30; Secondary: 34D23, 49K15.

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Figure 1. Model (1) fitting the total population of Cape Verde between 1987 and 2014 [25,42]. The $l_2$ norm of the difference between the real total population of Cape Verde and our prediction gives an error of $1.9\%$ of individuals per year with respect to the total population of Cape Verde in 2014

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Figure 2. Model (1) fitting the data of cumulative cases of HIV and AIDS infection in Cape Verde between 1987 and 2014 [25]. The $l_2$ norm of the difference between the real data and the cumulative cases of infection by HIV/AIDS given by model (1) gives, in both cases, an error of $0.03\%$ of individuals per year with respect to the total population of Cape Verde in 2014

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Figure 3. Top left: cumulative HIV and AIDS cases. Top right: pre-AIDS HIV infected individuals $I$. Bottom left: HIV-infected individuals under ART treatment $C$. Bottom right: HIV-infected individuals with AIDS symptoms $A$. Expression "with PrEP" refers to the case $(\psi, \theta) = (0.1, 0.001)$ and "no PrEP" refers to the case $(\psi, \theta) = (0, 0)$

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Figure 4. Top left: Individuals under PrEP, $E$. Top right: pre-AIDS HIV infected individuals, $I$. Bottom left: HIV-infected individuals under ART treatment, $C$. Bottom right: HIV-infected individuals with AIDS symptoms, $A$. The continuous line is the solution of model (10) for $\psi = 0.1$, the dashed line "$-\, -$" is the solution of the optimal control problem with no mixed state control constraint and "$\cdot \, -$" is the solution of model (10) for $\psi = 0.9$

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Figure 5. Solutions of the optimal control problem with no mixed state control constraint. (a) Optimal control. (b) Total number of individuals that take PrEP at each instant of time

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Figure 6. Top left: Individuals under PrEP, $E$. Top right: pre-AIDS HIV infected individuals, $I$. Bottom left: HIV-infected individuals under ART treatment, $C$. Bottom right: HIV-infected individuals with AIDS symptoms, $A$. The continuous line is the solution of model (10) for $\psi = 0.1$, the dashed line "$-\, -$" is the solution of the optimal control problem with the mixed state control constraint (21) and "$\cdot \, -$" is the solution of model (10) for $\psi = 0.9$

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Figure 7. (a) Optimal control $\tilde{u}$ considering the mixed state control constraint (21). (b) Total number of individuals under PrEP at each instant of time for $t \in [0, 25]$ associated with the optimal control $\tilde{u}$. (c) Total number of individuals under PrEP at each instant of time for $t \in [0, 25]$ associated with $\psi = 0.61$

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Figure 8. Extremals of the optimal control problem (17)–(20) with $\theta = 0.001$

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Table 1. Cumulative cases of infection by HIV/AIDS and total population in Cape Verde in the period 1987–2014 [25,42]

Year	1987	1988	1989	1990	1991	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010	2011	2012	2013	2014
HIV/AIDS	61	107	160	211	244	303	337	358	395	432	471	560	660	779	913	1064	1233	1493	1716	2015	2334	2610	2929	3340	3739	4090	4537	4946
Population	323972	328861	334473	341256	349326	358473	368423	378763	389156	399508	409805	419884	429576	438737	447357	455396	462675	468985	474224	478265	481278	483824	486673	490379	495159	500870	507258	513906

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Table 2. Parameters of the HIV/AIDS model (1) for Cape Verde

Symbol	Description	Value	Reference
$N(0)$	Initial population	$323 972$	[38]
$\Lambda$	Recruitment rate	$13045$	[38]
$\mu$	Natural death rate	$1/69.54$	[38]
$\beta$	HIV transmission rate	$0.752$	Estimated
$\eta_C$	Modification parameter	$0.015$, $0.04$	Assumed
$\eta_A$	Modification parameter	$1.3$, $1.35$	Assumed
$\phi$	HIV treatment rate for $I$ individuals	$1$	[30]
$\rho$	Default treatment rate for $I$ individuals	$0.1 $	[30]
$\alpha$	AIDS treatment rate	$0.33 $	[30]
$\omega$	Default treatment rate for $C$ individuals	$0.09$	[30]
$d$	AIDS induced death rate	$1$	[39]

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