Modeling and optimal control of HIV/AIDS prevention through PrEP

Optimal Control and Bifurcation Analysis of HIV Model

In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R ₀ is computed using the next-generation matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R ₀ ≤ 1, the disease-free equilibrium is stable both locally and globally whereas if R ₀ > 1, based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R ₀ = 1, the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin's maximum principle is applied to form an optimality system. Further, forward fourth-order Runge-Kutta's method is applied to obtain the solution of state variables whereas Runge-Kutta's fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population's dynamic behavior.

The effect of PrEP uptake and adherence on the spread of HIV in the presence of casual and long-term partnerships

A model with both casual and long-term partnerships is considered with respect to the impact of a pre-exposure prophylaxis (PrEP) on the spread of HIV. We consider the effect of the effectiveness of PrEP, the rate that susceptible individuals choose to take PrEP, and compliance with the daily dose of the pre-exposure prophylaxis. The rate of infection in long-term partnerships is computed using a linearized expected value as a means for including the nonlocal effects of long-term partnerships while maintaining computational feasibility. The reproduction numbers for models with casual partnerships, long-term partnerships, and a combination of both are analytically computed and global stability of both disease-free and endemic equilibria is shown. Sensitivity and PRCC analysis results suggest that increasing the compliance among the current PrEP users is a more effective strategy in the fight against the HIV epidemic than increased coverage with poor compliance. Furthermore, an analysis of the reproduction number shows that models with either casual or monogamous long-term partnerships can reach the desired $ R_0 < 1 $ threshold for high enough levels of compliance and uptake, however, a model with both casual and monogamous long-term partnerships will require additional interventions. Methods highlighted in this manuscript are applicable to other incurable diseases or diseases with imperfect vaccines effected by long-term partnerships.

Optimal control of an HIV model with a trilinear antibody growth function

<p style='text-indent:20px;'>We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.</p>

Model-free based control of a HIV/AIDS prevention model

Controlling an epidemiological model is often performed using optimal control theory techniques for which the solution depends on the equations of the controlled system, objective functional and possible state and/or control constraints. In this paper, we propose a model-free control approach based on an algorithm that operates in 'real-time' and drives the state solution according to a direct feedback on the state solution that is aimed to be minimized, and without knowing explicitly the equations of the controlled system. We consider a concrete epidemic problem of minimizing the number of HIV infected individuals, through the preventive measure pre-exposure prophylaxis (PrEP) given to susceptible individuals. The solutions must satisfy control and mixed state-control constraints that represent the limitations on PrEP implementation. Our model-free based control algorithm allows to close the loop between the number of infected individuals with HIV and the supply of PrEP medication 'in real time', in such a manner that the number of infected individuals is asymptotically reduced and the number of individuals under PrEP medication remains below a fixed constant value. We prove the efficiency of our approach and compare the model-free control solutions with the ones obtained using a classical optimal control approach via Pontryagin maximum principle. The performed numerical simulations allow us to conclude that the model-free based control strategy highlights new and interesting performances compared with the classical optimal control approach.

Optimal control strategies for the transmission risk of COVID-19

In this paper, we apply optimal control theory to a novel coronavirus (COVID-19) transmission model given by a system of non-linear ordinary differential equations. Optimal control strategies are obtained by minimizing the number of exposed and infected population considering the cost of implementation. The existence of optimal controls and characterization is established using Pontryagin's Maximum Principle. An expression for the basic reproduction number is derived in terms of control variables. Then the sensitivity of basic reproduction number with respect to model parameters is also analysed. Numerical simulation results demonstrated good agreement with our analytical results. Finally, the findings of this study shows that comprehensive impacts of prevention, intensive medical care and surface disinfection strategies outperform in reducing the disease epidemic with optimum implementation cost.

Analysis and optimal control of an HIV model based on CD4 count

A non-linear mechanistic model for the transmission dynamics of HIV/AIDS is developed and analyzed. The model classified the infected individuals based on their CD4 count level. Furthermore, education campaign, voluntary testing and counseling and treatment are considered as intervention strategies for controlling the disease. The analysis of the model reveals that imperfect public enlightenment campaign can induce backward bifurcation. It has been shown that when public enlightenment campaign is [Formula: see text] effective, the disease free equilibrium is globally asymptotically stable for [Formula: see text], whereas for [Formula: see text] the global stability of the endemic equilibrium is proved only in a special case. Time dependent controls of the intervention strategies mentioned above are incorporated into the model and the optimal control strategies with minimal implementation cost are identified. In addition, cost effectiveness analysis in the form of incremental cost effectiveness ratio is carried-out to identify the most cost effective strategies. The results suggest that out of the three non dominated strategies, the strategy of educating the newly entrants only or combination of newly entrants and susceptible individuals is very cost effective using per capita GDP of Nigeria as at 2018. However, the choice of which strategy to implement depends on budgetary allocation and resource availability.

Numerical Optimal Control of HIV Transmission in Octave/MATLAB

We provide easy and readable GNU Octave/MATLAB code for the simulation of mathematical models described by ordinary differential equations and for the solution of optimal control problems through Pontryagin’s maximum principle. For that, we consider a normalized HIV/AIDS transmission dynamics model based on the one proposed in our recent contribution (Silva, C.J.; Torres, D.F.M. A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde. Ecol. Complex. 2017, 30, 70–75), given by a system of four ordinary differential equations. An HIV initial value problem is solved numerically using the ode45 GNU Octave function and three standard methods implemented by us in Octave/MATLAB: Euler method and second-order and fourth-order Runge–Kutta methods. Afterwards, a control function is introduced into the normalized HIV model and an optimal control problem is formulated, where the goal is to find the optimal HIV prevention strategy that maximizes the fraction of uninfected HIV individuals with the least HIV new infections and cost associated with the control measures. The optimal control problem is characterized analytically using the Pontryagin Maximum Principle, and the extremals are computed numerically by implementing a forward-backward fourth-order Runge–Kutta method. Complete algorithms, for both uncontrolled initial value and optimal control problems, developed under the free GNU Octave software and compatible with MATLAB are provided along the article.

Strategies in regulating glioblastoma signaling pathways and anti-invasion therapy

Glioblastoma multiforme is one of the most invasive type of glial tumors, which rapidly grows and commonly spreads into nearby brain tissue. It is a devastating brain cancer that often results in death within approximately 12 to 15 months after diagnosis. In this work, optimal control theory was applied to regulate intracellular signaling pathways of miR-451–AMPK–mTOR–cell cycle dynamics via glucose and drug intravenous administration infusions. Glucose level is controlled to activate miR-451 in the up-stream pathway of the model. A potential drug blocking the inhibitory pathway of mTOR by AMPK complex is incorporated to explore regulation of the down-stream pathway to the cell cycle. Both miR-451 and mTOR levels are up-regulated inducing cell proliferation and reducing invasion in the neighboring tissues. Concomitant and alternating glucose and drug infusions are explored under various circumstances to predict best clinical outcomes with least administration costs.