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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Karam Allali ◽  
Sanaa Harroudi ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Nonlinear Differential Equations ◽  
Adaptive Immune Response ◽  
Growth Function ◽  
Optimal Control Strategy ◽  
Hiv Model ◽  
Adaptive Immune ◽  
Immunodeficiency Virus ◽  
Infected Cells ◽  
Disease Free

<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>

Nonlinear incidence Human immunodeficiency virus infection model with optimal control

2020 ◽  
Vol 34 (11) ◽  
pp. 2050100
Author(s):  
David Yaro ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Optimal Control ◽  
Human Immunodeficiency Virus ◽  
Evolutionary Dynamics ◽  
Control Strategies ◽  
Cellular Level ◽  
Infection Model ◽  
Hiv Model ◽  
Immunodeficiency Virus ◽  
Infected Cells ◽  
Horizontal Spread

Mathematical modeling plays a crucial role in understanding the dynamics of Human immunodeficiency virus (HIV) disease. Most models deal with the vertical and horizontal spread of disease, but few studies have focused on the evolutionary dynamics of HIV at the cellular level. In this paper, we present an HIV model to analyze the dynamics of HIV infection at the cellular level to produce more natural results. We present a detailed stability analysis of disease-free and viral-persistence equilibrium in the system. In addition, sensitivity analysis and optimal control strategies are used to analyze the role of antiretroviral drug therapy and dietary supplements in controlling the concentration of infected cells and viruses.

scholarly journals Mathematical Modeling of Transmission Dynamics and Optimal Control of Vaccination and Treatment for Hepatitis B Virus

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Ali Vahidian Kamyad ◽  
Reza Akbari ◽  
Ali Akbar Heydari ◽  
Aghileh Heydari
Keyword(s):  
Optimal Control ◽  
Hepatitis B Virus ◽  
Hepatitis B ◽  
Public Health Problem ◽  
Hbv Infection ◽  
Optimal Control Strategy ◽  
B Virus ◽  
Existence And Stability ◽  
Steady State Solutions ◽  
Disease Free

Hepatitis B virus (HBV) infection is a worldwide public health problem. In this paper, we study the dynamics of hepatitis B virus (HBV) infection which can be controlled by vaccination as well as treatment. Initially we consider constant controls for both vaccination and treatment. In the constant controls case, by determining the basic reproduction number, we study the existence and stability of the disease-free and endemic steady-state solutions of the model. Next, we take the controls as time and formulate the appropriate optimal control problem and obtain the optimal control strategy to minimize both the number of infectious humans and the associated costs. Finally at the end numerical simulation results show that optimal combination of vaccination and treatment is the most effective way to control hepatitis B virus infection.

scholarly journals MATHEMATICAL ANALYSIS OF THE ROLE OF DETECTION RATE IN THE DYNAMICAL SPREAD OF HIV-TB CO-INFECTION

2016 ◽  
Vol 11 (10) ◽  
pp. 5715-5741
Author(s):  
SUNDAY OLUMUYIWA Adewale ◽  
I.A Olopade ◽  
I.T Mohammed ◽  
S.O Ajao ◽  
O.T Oyedemi
Keyword(s):  
Detection Rate ◽  
Equilibrium Points ◽  
Population Level ◽  
Transformation Method ◽  
Infection Model ◽  
Hiv Model ◽  
Public Health Burden ◽  
Immunodeficiency Virus ◽  
Effect Of Treatment ◽  
Disease Free

Human Immunodeficiency Virus (HIV) co-existing with Tuberculosis (TB) in individuals remains a major global health challenges, with an estimated 1.4 million patients worldwide. These two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. We formulated new fifteen (15) compartmental models to gain more insight into the effect of treatment and detection of infected undetected individuals on the dynamical spread of HIV- TB co-infection. Sub models of HIV and TB only were considered first, followed by the full HIV-TB co-infection model. Existence and uniqueness of HIV and TB only model were analyzed quantitatively, and we shown that HIV model only and TB only model have solutions, moreover, the solutions are unique. Stability of HIV model only, TB model only and full model of HIV-TB co-infection were analyzed for the existence of the disease free and endemic equilibrium points. Basic reproduction number () was analyzed, using next generation matrix method (NGM), and it has been shown that the disease free equilibrium point is locally asymptotically stable whenever  and unstable whenever this threshold exceeds unity. i.e., Numerical simulation was carried out by maple software using differential transformation method, to show the effect of  treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Significantly, all the results obtained from this research show the importance of treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Detection rate of infected undetected individuals reduce the spread of HIV-TB co-infections.

scholarly journals Optimal Control of an HIV Model with Gene Therapy and Latency Reversing Agents

2021 ◽  
Vol 26 (4) ◽  
pp. 77
Author(s):  
Zachary Abernathy ◽  
Kristen Abernathy ◽  
Andrew Grant ◽  
Paul Hazelton
Keyword(s):  
Gene Therapy ◽  
Optimal Control ◽  
Combination Treatment ◽  
Latent Reservoir ◽  
Sweep Method ◽  
Hiv Model ◽  
Future Work ◽  
Disease Free ◽  
Forward Backward Sweep Method ◽  
Reversing Agents

In this paper, we study the dynamics of HIV under gene therapy and latency reversing agents. While previous works modeled either the use of gene therapy or latency reversing agents, we consider the effects of a combination treatment strategy. For constant treatment controls, we establish global stability of the disease-free equilibrium and endemic equilibrium based on the value of R0. We then consider time-dependent controls and formulate an associated optimal control problem that emphasizes reduction of the latent reservoir. Characterizations for the optimal control profiles are found using Pontryagin’s Maximum Principle. We perform numerical simulations of the optimal control model using the fourth-order Runge–Kutta forward-backward sweep method. We find that a combination treatment of gene therapy with latency reversing agents provides better remission times than gene therapy alone. We conclude with a discussion of our findings and future work.

Global dynamics in an age-structured HIV model with humoral immunity

2021 ◽  
pp. 2150047
Author(s):  
Zhongzhong Xie ◽  
Xiuxiang Liu
Keyword(s):  
Differential Equations ◽  
Humoral Immunity ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Hiv Model ◽  
Infection Age ◽  
Number Formula ◽  
Infected Cells ◽  
Age Structured ◽  
Disease Free

In this paper, we formulate an age-structured HIV model, in which the influence of humoral immunity and the infection age of the infected cells are considered. The model is governed by three ordinary differential equations and two first-ordered partial differential equations and admits three equilibria: disease-free, immune-inactivated and immune-activated equilibria. We introduce two important thresholds: the basic reproduction number [Formula: see text] and immune-activated reproduction number [Formula: see text] and further show the global stability of above three equilibria in terms of [Formula: see text] and [Formula: see text], respectively. The numerical simulations are presented to illustrate our results.

scholarly journals An Efficient Therapy Strategy under a Novel HIV Model

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Chunming Zhang ◽  
Xiaofan Yang ◽  
Wanping Liu ◽  
Lu-Xing Yang
Keyword(s):  
Optimal Control ◽  
Human Immunodeficiency Virus ◽  
Immune System ◽  
Virus Spread ◽  
Hiv Model ◽  
Hiv Virus ◽  
Therapy Strategy ◽  
Spread Model ◽  
Immunodeficiency Virus ◽  
The Cost

By incorporating the chemotherapy into a previous model describing the interaction of the immune system with the human immunodeficiency virus (HIV), this paper proposes a novel HIV virus spread model with control variables. Our goal is to maximize the number of healthy cells and, meanwhile, to minimize the cost of chemotherapy. In this context, the existence of an optimal control is proved. Experimental results show that, under this model, the spread of HIV virus can be controlled effectively.

Formal Methods for Control Synthesis: An Optimization Perspective

2019 ◽  
Vol 2 (1) ◽  
pp. 115-140 ◽  
Author(s):  
Calin Belta ◽  
Sadra Sadraddini
Keyword(s):  
Optimal Control ◽  
Temporal Logic ◽  
Nonlinear Differential Equations ◽  
Fundamental Property ◽  
Control Synthesis ◽  
Future Research ◽  
Optimal Control Strategy ◽  
Formal Synthesis ◽  
Finite State ◽  
Model Computer

In control theory, complicated dynamics such as systems of (nonlinear) differential equations are controlled mostly to achieve stability. This fundamental property, which can be with respect to a desired operating point or a prescribed trajectory, is often linked with optimality, which requires minimizing a certain cost along the trajectories of a stable system. In formal verification (model checking), simple systems, such as finite-state transition graphs that model computer programs or digital circuits, are checked against rich specifications given as formulas of temporal logics. The formal synthesis problem, in which the goal is to synthesize or control a finite system from a temporal logic specification, has recently received increased interest. In this article, we review some recent results on the connection between optimal control and formal synthesis. Specifically, we focus on the following problem: Given a cost and a correctness temporal logic specification for a dynamical system, generate an optimal control strategy that satisfies the specification. We first provide a short overview of automata-based methods, in which the dynamics of the system are mapped to a finite abstraction that is then controlled using an automaton corresponding to the specification. We then provide a detailed overview of a class of methods that rely on mapping the specification and the dynamics to constraints of an optimization problem. We discuss advantages and limitations of these two types of approaches and suggest directions for future research.

scholarly journals Mathematical Modeling Identifies the Role of Adaptive Immunity as a Key Controller of Respiratory Syncytial Virus (RSV) Titer in Cotton Rats

2018 ◽  
Author(s):  
Darren Wethington ◽  
Olivia Harder ◽  
Karthik Uppulury ◽  
William C. L. Stewart ◽  
Phylip Chen ◽  
...  
Keyword(s):  
Immune Response ◽  
Adaptive Immune Response ◽  
Viral Titer ◽  
Vaccine Candidates ◽  
Adaptive Immune ◽  
Rsv Infection ◽  
Cotton Rats ◽  
Infected Cells ◽  
Syncytial Virus

AbstractRespiratory syncytial virus (RSV) is a common virus that can have varying effects ranging from mild cold-like symptoms to mortality depending on the age and immune status of the individual. We combined mathematical modeling using ordinary differential equations (ODEs) with measurement of RSV infection kinetics in primary well differentiated human airway epithelial (HAE) cultures in vitro and in immunocompetent and immunosuppressed cotton rats to glean mechanistic details that underlie RSV infection kinetics in the lung. Quantitative analysis of viral titer kinetics in our mathematical model showed that the elimination of infected cells by the adaptive immune response generates unique RSV titer kinetic features including a faster time scale of viral titer clearance than viral production, and a monotonic decrease in the peak RSV titer with decreasing inoculum dose. Parameter estimation in the ODE model using a non-linear mixed effects approach revealed a very low rate (average single cell lifetime > 10 days) of cell lysis by RSV before the adaptive immune response is initiated. Our model predicted negligible changes in the RSV titer kinetics on earlier days (< 5 d.p.i) but a slower decay in RSV titer in immunosuppressed cotton rats compared to that in non-suppressed cotton rats at later days (>5 d.p.i) in silico. These predictions were in excellent agreement with the experimental results. Our combined approach quantified the importance of the adaptive immune response in suppressing RSV infection in cotton rats, which could be useful in testing RSV vaccine candidates.ImportanceA major difficulty in developing vaccines against RSV infection is our rudimentary understanding of the mechanisms that underlie RSV infection. We addressed this challenge by developing a mechanistic computational model with predictive powers for describing RSV infection kinetics in cotton rats. The model was constructed synergistically with in vitro and in vivo measurements. The combined framework determined an important role for CD8+ T cells responses in reducing RSV titers in cotton rats. The framework can be used to design future experiments to elucidate mechanisms underlying RSV infection and test outcomes for potential vaccine candidates. In addition, estimation of the model parameters provides quantitative values for parameters of biological and clinical interest such as the replication rate of RSV, the death rate of infected cells, and the average number of new infections initiated by a single infected cell.

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