# American Institute of Mathematical Sciences

February  2018, 11(1): 119-141. doi: 10.3934/dcdss.2018008

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

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

* Corresponding author: Cristiana J. Silva

Received  September 2016 Revised  February 2017 Published  January 2018

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.

Citation: Cristiana J. Silva, Delfim F. M. Torres. Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 119-141. doi: 10.3934/dcdss.2018008
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##### References:
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
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
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 $(\psi, \theta) = (0, 0)$
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 $\psi = 0.9$
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
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 $\psi = 0.9$
(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 $\psi = 0.61$
Extremals of the optimal control problem (17)–(20) with $\theta = 0.001$
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
 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
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]
 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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