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Liu Y, Wang Y, Jiang D. Dynamic behaviors of a stochastic virus infection model with Beddington-DeAngelis incidence function, eclipse-stage and Ornstein-Uhlenbeck process. Math Biosci 2024; 369:109154. [PMID: 38295988 DOI: 10.1016/j.mbs.2024.109154] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Revised: 01/13/2024] [Accepted: 01/25/2024] [Indexed: 02/05/2024]
Abstract
In this paper, we present a virus infection model that incorporates eclipse-stage and Beddington-DeAngelis function, along with perturbation in infection rate using logarithmic Ornstein-Uhlenbeck process. Rigorous analysis demonstrates that the stochastic model has a unique global solution. Through construction of appropriate Lyapunov functions and a compact set, combined with the strong law of numbers and Fatou's lemma, we obtain the existence of the stationary distribution under a critical condition, which indicates the long-term persistence of T-cells and virions. Moreover, a precise probability density function is derived around the quasi-equilibrium of the model, and spectral radius analysis is employed to identify critical condition for elimination of the virus. Finally, numerical simulations are presented to validate theoretical results, and the impact of some key parameters such as the speed of reversion, volatility intensity and mean infection rate are investigated.
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Affiliation(s)
- Yuncong Liu
- College of Science, China University of Petroleum (East China), Qingdao, Shandong 266580, China.
| | - Yan Wang
- College of Science, China University of Petroleum (East China), Qingdao, Shandong 266580, China.
| | - Daqing Jiang
- College of Science, China University of Petroleum (East China), Qingdao, Shandong 266580, China.
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2
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Babamohamadi M, Mohammadi N, Faryadi E, Haddadi M, Merati A, Ghobadinezhad F, Amirian R, Izadi Z, Hadjati J. Anti-CTLA-4 nanobody as a promising approach in cancer immunotherapy. Cell Death Dis 2024; 15:17. [PMID: 38191571 PMCID: PMC10774412 DOI: 10.1038/s41419-023-06391-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2023] [Revised: 11/25/2023] [Accepted: 12/13/2023] [Indexed: 01/10/2024]
Abstract
Cancer is one of the most common diseases and causes of death worldwide. Since common treatment approaches do not yield acceptable results in many patients, developing innovative strategies for effective treatment is necessary. Immunotherapy is one of the promising approaches that has been highly regarded for preventing tumor recurrence and new metastases. Meanwhile, inhibiting immune checkpoints is one of the most attractive methods of cancer immunotherapy. Cytotoxic T lymphocyte-associated protein-4 (CTLA-4) is an essential immune molecule that plays a vital role in cell cycle modulation, regulation of T cell proliferation, and cytokine production. This molecule is classically expressed by stimulated T cells. Inhibition of overexpression of immune checkpoints such as CTLA-4 receptors has been confirmed as an effective strategy. In cancer immunotherapy, immune checkpoint-blocking drugs can be enhanced with nanobodies that target immune checkpoint molecules. Nanobodies are derived from the variable domain of heavy antibody chains. These small protein fragments have evolved entirely without a light chain and can be used as a powerful tool in imaging and treating diseases with their unique structure. They have a low molecular weight, which makes them smaller than conventional antibodies while still being able to bind to specific antigens. In addition to low molecular weight, specific binding to targets, resistance to temperature, pH, and enzymes, high ability to penetrate tumor tissues, and low toxicity make nanobodies an ideal approach to overcome the disadvantages of monoclonal antibody-based immunotherapy. In this article, while reviewing the cellular and molecular functions of CTLA-4, the structure and mechanisms of nanobodies' activity, and their delivery methods, we will explain the advantages and challenges of using nanobodies, emphasizing immunotherapy treatments based on anti-CTLA-4 nanobodies.
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Affiliation(s)
- Mehregan Babamohamadi
- Department of Biology, School of Natural Sciences, University of Tabriz, Tabriz, Iran
- Stem Cell and Regenerative Medicine Innovation Center, Tehran University of Medical Sciences, Tehran, Iran
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran
| | - Nastaran Mohammadi
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Student Research Committee, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Department of Immunology, School of Medicine, Kermanshah University of Medical Sciences, Kermanshah, Iran
| | - Elham Faryadi
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Student Research Committee, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Department of Immunology, School of Medicine, Kermanshah University of Medical Sciences, Kermanshah, Iran
| | - Maryam Haddadi
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Student Research Committee, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Department of Immunology, School of Medicine, Kermanshah University of Medical Sciences, Kermanshah, Iran
| | - Amirhossein Merati
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Student Research Committee, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Department of Medical Laboratory Sciences, School of Paramedical, Kermanshah University of Medical Sciences, Kermanshah, Iran
| | - Farbod Ghobadinezhad
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Student Research Committee, Kermanshah University of Medical Sciences, Kermanshah, Iran
| | - Roshanak Amirian
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Student Research Committee, Kermanshah University of Medical Sciences, Kermanshah, Iran
- Pharmaceutical Sciences Research Center, Health Institute, Kermanshah University of Medical Sciences, Kermanshah, Iran
| | - Zhila Izadi
- USERN Office, Kermanshah University of Medical Sciences, Kermanshah, Iran.
- Pharmaceutical Sciences Research Center, Health Institute, Kermanshah University of Medical Sciences, Kermanshah, Iran.
| | - Jamshid Hadjati
- Department of Immunology, School of Medicine, Tehran University of Medical Sciences, Tehran, Iran.
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Han R, Dai B, Chen Y. Chemotaxis-driven stationary and oscillatory patterns in a diffusive HIV-1 model with CTL immune response and general sensitivity. CHAOS (WOODBURY, N.Y.) 2023; 33:073142. [PMID: 37463094 DOI: 10.1063/5.0150072] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/11/2023] [Accepted: 06/30/2023] [Indexed: 07/20/2023]
Abstract
In this paper, a reaction-diffusion-chemotaxis HIV-1 model with a cytotoxic T lymphocyte (CTL) immune response and general sensitivity is investigated. We first prove the global classical solvability and L∞-boundedness for the considered model in a bounded domain with arbitrary spatial dimensions, which extends the previous existing results. Then, we apply the global existence result to the case with a linear proliferation immune response and an incidence rate. We study the spatiotemporal dynamics about the three types of spatially homogeneous steady states: infection-free steady state S0, CTL-inactivated infection steady state S1, and CTL-activated infection steady state S∗. Our analyses indicate that S0 is globally asymptotically stable if the basic reproduction number R0 is less than 1; if R0 is between 1 and a threshold, then S1 is globally asymptotically stable. However, if R0 is larger than the threshold, then the chemoattraction and chemorepulsion can destabilize S∗, and thus, a spatiotemporal pattern forms as the chemotactic sensitivity crosses certain critical values. We obtain two kinds of important patterns, which are induced by chemotaxis: stationary Turing pattern and irregular oscillatory pattern. We also find that different chemotactic response functions can affect system's dynamics. Based on some empirical parameter values, numerical simulations are given to illustrate the effectiveness of the theoretical predications.
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Affiliation(s)
- Renji Han
- School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, People's Republic of China
| | - Binxiang Dai
- School of Mathematics and Statistics, Central South University, Changsha 410083, People's Republic of China
| | - Yuming Chen
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada
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4
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Agusto FB, Numfor E, Srinivasan K, Iboi EA, Fulk A, Saint Onge JM, Peterson AT. Impact of public sentiments on the transmission of COVID-19 across a geographical gradient. PeerJ 2023; 11:e14736. [PMID: 36819996 PMCID: PMC9938658 DOI: 10.7717/peerj.14736] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2021] [Accepted: 12/21/2022] [Indexed: 02/17/2023] Open
Abstract
COVID-19 is a respiratory disease caused by a recently discovered, novel coronavirus, SARS-COV-2. The disease has led to over 81 million confirmed cases of COVID-19, with close to two million deaths. In the current social climate, the risk of COVID-19 infection is driven by individual and public perception of risk and sentiments. A number of factors influences public perception, including an individual's belief system, prior knowledge about a disease and information about a disease. In this article, we develop a model for COVID-19 using a system of ordinary differential equations following the natural history of the infection. The model uniquely incorporates social behavioral aspects such as quarantine and quarantine violation. The model is further driven by people's sentiments (positive and negative) which accounts for the influence of disinformation. People's sentiments were obtained by parsing through and analyzing COVID-19 related tweets from Twitter, a social media platform across six countries. Our results show that our model incorporating public sentiments is able to capture the trend in the trajectory of the epidemic curve of the reported cases. Furthermore, our results show that positive public sentiments reduce disease burden in the community. Our results also show that quarantine violation and early discharge of the infected population amplifies the disease burden on the community. Hence, it is important to account for public sentiment and individual social behavior in epidemic models developed to study diseases like COVID-19.
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Affiliation(s)
| | - Eric Numfor
- Augusta University, Augusta, Georgia, United States
| | | | | | | | - Jarron M. Saint Onge
- University of Kansas, Lawrence, Kansas, United States,University of Kansas Medical Center, Kansas City, Kansas, United States
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5
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Deng J, Shu H, Wang L, Wang XS. Viral dynamics with immune responses: effects of distributed delays and Filippov antiretroviral therapy. J Math Biol 2023; 86:37. [PMID: 36695964 DOI: 10.1007/s00285-023-01869-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2022] [Revised: 10/17/2022] [Accepted: 01/06/2023] [Indexed: 01/26/2023]
Abstract
In this paper, we propose a general viral infection model to incorporate two infection modes (virus-to-cell mode and cell-to-cell mode), the CTL immune response, and the distributed intracellular delays during the processes of viral infection, viral production, and CTLs recruitment. We investigate the existence, the uniqueness, and the global stability of three equilibria: infection-free equilibrium [Formula: see text], immune-inactivated equilibrium [Formula: see text] and immune-activated equilibrium [Formula: see text], respectively. We prove that the viral dynamics are determined by two threshold parameters: the basic reproduction number for infection [Formula: see text] and the basic reproduction number for immune response [Formula: see text]. We also numerically explore the viral dynamics beyond stability. We use bifurcation diagrams to show that increasing the delay in CTL immune cell recruitment can induce a switch in viral load from a stable constant level to sustained oscillations, and then back to a stable equilibrium. We also compare the contributions of the two infection modes to the total infection level and identify the key parameters that would affect the percentages of virus-to-cell infection and cell-to-cell infection. Finally, we explore how Filippov control can be applied in antiretroviral therapy to reduce the viral loads.
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Affiliation(s)
- Jiawei Deng
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710062, People's Republic of China
| | - Hongying Shu
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710062, People's Republic of China.
| | - Lin Wang
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada
| | - Xiang-Sheng Wang
- Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70503, USA
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6
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Wang Y, Liu J, Zhang X, Heffernan JM. An HIV stochastic model with cell-to-cell infection, B-cell immune response and distributed delay. J Math Biol 2023; 86:35. [PMID: 36695912 DOI: 10.1007/s00285-022-01863-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2022] [Revised: 12/07/2022] [Accepted: 12/15/2022] [Indexed: 01/26/2023]
Abstract
In this study, a delayed HIV stochastic model with virus-to-cell infection, cell-to-cell transmission and B-cell immune response is proposed. We first transform the stochastic differential equation with distributed delay into a high-dimensional degenerate stochastic differential equation, and then theoretically analyze the dynamic behaviour of the degenerate model. The unique global solution of the model is given by rigorous analysis. By formulating suitable Lyapunov functions, the existence of the stationary Markov process is obtained if the stochastic B-cell-activated reproduction number is greater than one. We also use the law of large numbers theorem and the spectral radius analysis method to deduce that the virus can be cleared if the stochastic B-cell-inactivated reproduction number is less than one. Through uncertainty and sensitivity analysis, we obtain key parameters that determine the value of the stochastic B-cell-activated reproduction number. Numerically, we examine that low level noise can maintain the number of the virus and B-cell populations at a certain range, while high level noise is helpful for the elimination of the virus. Furthermore, the effect of the cell-to-cell infection on model behaviour, and the influence of the key parameters on the size of the stochastic B-cell-activated reproduction number are also investigated.
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Affiliation(s)
- Yan Wang
- College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China
| | - Jun Liu
- College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China
| | - Xinhong Zhang
- College of Science, China University of Petroleum (East China), Qingdao, 266580, Shandong, China
| | - Jane M Heffernan
- Modelling Infection and Immunity Lab, Centre for Disease Modelling, Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada.
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7
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Allali K, Harroudi S, Torres DFM. Optimal control of an HIV model with a trilinear antibody growth function. DISCRETE & CONTINUOUS DYNAMICAL SYSTEMS - S 2022. [DOI: 10.3934/dcdss.2021148] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
<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>
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8
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Getz M, Wang Y, An G, Asthana M, Becker A, Cockrell C, Collier N, Craig M, Davis CL, Faeder JR, Ford Versypt AN, Mapder T, Gianlupi JF, Glazier JA, Hamis S, Heiland R, Hillen T, Hou D, Islam MA, Jenner AL, Kurtoglu F, Larkin CI, Liu B, Macfarlane F, Maygrundter P, Morel PA, Narayanan A, Ozik J, Pienaar E, Rangamani P, Saglam AS, Shoemaker JE, Smith AM, Weaver JJA, Macklin P. Iterative community-driven development of a SARS-CoV-2 tissue simulator. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2021:2020.04.02.019075. [PMID: 32511322 PMCID: PMC7239052 DOI: 10.1101/2020.04.02.019075] [Citation(s) in RCA: 23] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/08/2023]
Abstract
The 2019 novel coronavirus, SARS-CoV-2, is a pathogen of critical significance to international public health. Knowledge of the interplay between molecular-scale virus-receptor interactions, single-cell viral replication, intracellular-scale viral transport, and emergent tissue-scale viral propagation is limited. Moreover, little is known about immune system-virus-tissue interactions and how these can result in low-level (asymptomatic) infections in some cases and acute respiratory distress syndrome (ARDS) in others, particularly with respect to presentation in different age groups or pre-existing inflammatory risk factors. Given the nonlinear interactions within and among each of these processes, multiscale simulation models can shed light on the emergent dynamics that lead to divergent outcomes, identify actionable "choke points" for pharmacologic interventions, screen potential therapies, and identify potential biomarkers that differentiate patient outcomes. Given the complexity of the problem and the acute need for an actionable model to guide therapy discovery and optimization, we introduce and iteratively refine a prototype of a multiscale model of SARS-CoV-2 dynamics in lung tissue. The first prototype model was built and shared internationally as open source code and an online interactive model in under 12 hours, and community domain expertise is driving regular refinements. In a sustained community effort, this consortium is integrating data and expertise across virology, immunology, mathematical biology, quantitative systems physiology, cloud and high performance computing, and other domains to accelerate our response to this critical threat to international health. More broadly, this effort is creating a reusable, modular framework for studying viral replication and immune response in tissues, which can also potentially be adapted to related problems in immunology and immunotherapy.
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9
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Guo T, Qiu Z, Shen M, Rong L. Dynamics of a new HIV model with the activation status of infected cells. J Math Biol 2021; 82:51. [PMID: 33860365 PMCID: PMC8049625 DOI: 10.1007/s00285-021-01604-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2020] [Revised: 03/28/2021] [Accepted: 04/03/2021] [Indexed: 11/06/2022]
Abstract
The activation status can dictate the fate of an HIV-infected CD4+ T cell. Infected cells with a low level of activation remain latent and do not produce virus, while cells with a higher level of activation are more productive and thus likely to transfer more virions to uninfected cells during cell-to-cell transmission. How the activation status of infected cells affects HIV dynamics under antiretroviral therapy remains unclear. We develop a new mathematical model that structures the population of infected cells continuously according to their activation status. The effectiveness of antiretroviral drugs in blocking cell-to-cell viral transmission decreases as the level of activation of infected cells increases because the more virions are transferred from infected to uninfected cells during cell-to-cell transmission, the less effectively the treatment is able to inhibit the transmission. The basic reproduction number \documentclass[12pt]{minimal}
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\begin{document}$$R_{0}$$\end{document}R0 of the model is shown to determine the existence and stability of the equilibria. Using the principal spectral theory and comparison principle, we show that the infection-free equilibrium is locally and globally asymptotically stable when \documentclass[12pt]{minimal}
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\begin{document}$$R_{0}$$\end{document}R0 is less than one. By constructing Lyapunov functional, we prove that the infected equilibrium is globally asymptotically stable when \documentclass[12pt]{minimal}
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\begin{document}$$R_{0}$$\end{document}R0 is greater than one. Numerical investigation shows that even when treatment can completely block cell-free virus infection, virus can still persist due to cell-to-cell transmission. The random switch between infected cells with different activation levels can also contribute to the replenishment of the latent reservoir, which is considered as a major barrier to viral eradication. This study provides a new modeling framework to study the observations, such as the low viral load persistence, extremely slow decay of latently infected cells and transient viral load measurements above the detection limit, in HIV-infected patients during suppressive antiretroviral therapy.
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Affiliation(s)
- Ting Guo
- School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China.,Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA
| | - Zhipeng Qiu
- Center for Basic Teaching and Experiment, Nanjing University of Science and Technology Jiangyin Campus, Jiangyin, 214443, China
| | - Mingwang Shen
- Department of Epidemiology and Biostatistics, School of Public Health, Xi'an Jiaotong University Health Science Center, Xi'an, 710061, China
| | - Libin Rong
- Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA.
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10
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Hamza A, Ahmad I, Uneeb M. Fuzzy logic and Lyapunov-based non-linear controllers for HCV infection. IET Syst Biol 2021; 15:53-71. [PMID: 33780147 PMCID: PMC8675797 DOI: 10.1049/syb2.12014] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2020] [Revised: 12/15/2020] [Indexed: 11/19/2022] Open
Abstract
Hepatitis C is the liver disease caused by the Hepatitis C virus (HCV) which can lead to serious health problems such as liver cancer. In this research work, the non‐linear model of HCV having three state variables (uninfected hepatocytes, infected hepatocytes and virions) and two control inputs has been taken into account, and four non‐linear controllers namely non‐linear PID controller, Lyapunov Redesign controller, Synergetic controller and Fuzzy Logic‐Based controller have been proposed to control HCV infection inside the human body. The controllers have been designed for the anti‐viral therapy in order to control the amount of uninfected hepatocytes to the desired safe limit and to track the amount of infected hepatocytes and virions to their reference value which is zero. One control input is the Pegylated interferon (peg‐IFN‐α) which acts in reducing the infected hepatocytes and the other input is ribavirin which blocks the production of virions. By doing so, the uninfected hepatocytes increase and achieve the required safe limit. Lyapunov stability analysis has been used to prove the stability of the whole system. The comparative analysis of the proposed nonlinear controllers using MATLAB/Simulink have been done with each other and with linear PID. These results depict that the infected hepatocytes and virions are reduced to the desired level, enhancing the rate of sustained virologic response (SVR) and reducing the treatment period as compared with previous strategies introduced in the literature.
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Affiliation(s)
- Ali Hamza
- Department of Electrical Engineering, School of Electrical Engineering and Computer Science, National University of Sciences and Technology (NUST), Islamabad, Pakistan
| | - Iftikhar Ahmad
- Department of Electrical Engineering, School of Electrical Engineering and Computer Science, National University of Sciences and Technology (NUST), Islamabad, Pakistan
| | - Muhammad Uneeb
- Department of Electrical Engineering, School of Electrical Engineering and Computer Science, National University of Sciences and Technology (NUST), Islamabad, Pakistan
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11
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Backward bifurcation in within-host HIV models. Math Biosci 2021; 335:108569. [PMID: 33636199 DOI: 10.1016/j.mbs.2021.108569] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2020] [Revised: 02/16/2021] [Accepted: 02/16/2021] [Indexed: 11/21/2022]
Abstract
The activation and proliferation of naive CD4 T cells produce helper T cells, and increase the susceptible population in the presence of HIV. This may cause backward bifurcation. To verify this, we construct a simple within-host HIV model that includes the key variables, namely healthy naive CD4 T cells, helper T cells, infected CD4 T cells and virus. When the viral basic reproduction number R0 is less than unity, we show theoretically and numerically that bistability for RC<R0<1 can be caused by a backward bifurcation due to a new susceptible population produced by activation of healthy naive CD4 T cells that become helper T cells. An extended model including the CTL dynamics may also show this backward bifurcation. In the case that the homeostatic source of healthy naive CD4 T cells is large, RC is approximately the threshold for HIV to persist independent of initial conditions. The backward bifurcation may still occur even when we consider latent infections of naive CD4 T cells. Thus to control the spread of within-host HIV, it may be necessary for treatment to reduce the reproduction number below RC.
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12
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Li B, Jiao F. A delayed HIV-1 model with cell-to-cell spread and virus waning. JOURNAL OF BIOLOGICAL DYNAMICS 2020; 14:802-825. [PMID: 33084532 DOI: 10.1080/17513758.2020.1836272] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2020] [Accepted: 09/29/2020] [Indexed: 06/11/2023]
Abstract
In this paper, we propose and analyse a delayed HIV-1 model with both viral and cellular transmissions and virus waning. We obtain the threshold dynamics of the proposed model, characterized by the basic reproduction number R0 . If R0<1 , the infection-free steady state is globally asymptotically stable; whereas if R0>1 , the system is uniformly persistent. When the delays are positive, we show that the intracellular delays in both viral and cellular infections may lead to stability switches of the infected steady state. Both analytical and numerical results indicate that if the effect of cell-to-cell transmission is ignored, then the risk of HIV-1 infection will be underestimated. Moreover, the viral load of model without virus waning is higher than the one of model with virus waning. These results highlight the important role of two ways of viral transmission and virus waning on HIV-1 infection.
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Affiliation(s)
- Bing Li
- School of Mathematical Science, Harbin Normal University, Harbin, People's Republic of China
| | - Feng Jiao
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
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13
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Wang Y, Lu M, Jiang D. Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2020; 18:274-299. [PMID: 33525091 DOI: 10.3934/mbe.2021014] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
In this paper, an HIV infection model with latent infection, Beddington-DeAngelis infection function, B-cell immune response and four time delays is formulated. The well-posedness of the model solution is rigorously derived, and the basic reproduction number $\mathcal{R}_0$ and the B-cell immune response reproduction number $\mathcal{R}_1$ are also obtained. By analyzing the modulus of the characteristic equation and constructing suitable Lyapunov functions, we establish the global asymptotic stability of the uninfected and the B-cell-inactivated equilibria for the four time delays, respectively. Hopf bifurcation occurs at the B-cell-activated equilibrium when the model includes the immune delay, and the B-cell-activated equilibrium is globally asymptotically stable if the model does not include it. Numerical simulations indicate that the increase of the latency delay, the cell infection delay and the virus maturation delay can cause the B-cell-activated equilibrium stabilize, while the increase of the immune delay can cause it destabilize.
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Affiliation(s)
- Yan Wang
- College of Science, China University of Petroleum (East China), Qingdao 266580, China
| | - Minmin Lu
- College of Science, China University of Petroleum (East China), Qingdao 266580, China
| | - Daqing Jiang
- College of Science, China University of Petroleum (East China), Qingdao 266580, China
- Key Laboratory of Unconventional Oil & Gas Development, China University of Petroleum (East China), Qingdao 266580, China
- Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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14
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Yang J, Bi S. Stability and Hopf bifurcation of a delayed virus infection model with latently infected cells and Beddington–DeAngelis incidence. INT J BIOMATH 2020. [DOI: 10.1142/s179352452050045x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, the dynamical behaviors for a five-dimensional virus infection model with Latently Infected Cells and Beddington–DeAngelis incidence are investigated. In the model, four delays which denote the latently infected delay, the intracellular delay, virus production period and CTL response delay are considered. We define the basic reproductive number and the CTL immune reproductive number. By using Lyapunov functionals, LaSalle’s invariance principle and linearization method, the threshold conditions on the stability of each equilibrium are established. It is proved that when the basic reproductive number is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable; when the CTL immune reproductive number is less than or equal to unity and the basic reproductive number is greater than unity, the immune-free infection equilibrium is globally asymptotically stable; when the CTL immune reproductive number is greater than unity and immune response delay is equal to zero, the immune infection equilibrium is globally asymptotically stable. The results show that immune response delay may destabilize the steady state of infection and lead to Hopf bifurcation. The existence of the Hopf bifurcation is discussed by using immune response delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results.
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Affiliation(s)
- Junxian Yang
- School of Science, Anhui Agricultural University, Hefei 230036, P. R. China
| | - Shoudong Bi
- School of Science, Anhui Agricultural University, Hefei 230036, P. R. China
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15
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Zeinali S, Shahrokhi M. Adaptive Control Strategy for Treatment of Hepatitis C Infection. Ind Eng Chem Res 2019. [DOI: 10.1021/acs.iecr.9b02988] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Affiliation(s)
- Sahar Zeinali
- Chemical and Petroleum Engineering Department, Sharif University of Technology, Tehran 11155-9465, Iran
| | - Mohammad Shahrokhi
- Chemical and Petroleum Engineering Department, Sharif University of Technology, Tehran 11155-9465, Iran
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16
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Kamboj D, Sharma MD. Multidrug Therapy for HIV Infection: Dynamics of Immune System. Acta Biotheor 2019; 67:129-147. [PMID: 30515609 DOI: 10.1007/s10441-018-9340-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2017] [Accepted: 11/26/2018] [Indexed: 11/29/2022]
Abstract
A mathematical model of the dynamics of the immune system is considered to illustrate the effect of its response to HIV infection, i.e. on viral growth and on T-cell dynamics. The specific immune response is measured by the levels of cytotoxic lymphocytes in a human body. The existence and stability analyses are performed for infected steady state and uninfected steady state. In order to keep infection under control, roles of drug therapies are analyzed in the presence of efficient immune response. Numerical simulations are computed and exhibited to illustrate the support of the immune system to drug therapies, so as to ensure the decay of infection and to maintain the level of healthy cells.
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Affiliation(s)
| | - M D Sharma
- Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India
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17
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Miao H, Teng Z, Abdurahman X. Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington-DeAngelis incidence and three delays. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:146-170. [PMID: 29198164 DOI: 10.1080/17513758.2017.1408861] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2016] [Accepted: 11/17/2017] [Indexed: 06/07/2023]
Abstract
In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington-DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results.
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Affiliation(s)
- Hui Miao
- a School of Applied Mathematics , Shanxi University of Finance and Economics , Taiyuan , People's Republic of China
| | - Zhidong Teng
- b College of Mathematics and System Sciences , Xinjiang University , Urumqi , People's Republic of China
| | - Xamxinur Abdurahman
- b College of Mathematics and System Sciences , Xinjiang University , Urumqi , People's Republic of China
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18
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Sahani SK. A delayed HIV infection model with apoptosis and viral loss. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:1012-1034. [PMID: 30462570 DOI: 10.1080/17513758.2018.1547427] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2017] [Accepted: 11/06/2018] [Indexed: 06/09/2023]
Abstract
In this paper, a delayed human immunodeficiency virus (HIV) model with apoptosis of cells has been studied. Both immunological and intracellular delay have been incorporated to make the model more relevant. Firstly, the model has been investigated using local stability analysis. Next, the global stability analysis of steady states has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. The transition of the system from order to chaos has been explored, and the analytical results have been verified by numerical simulations. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been studied in the model numerically.
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Affiliation(s)
- Saroj Kumar Sahani
- a Faculty of Mathematics & Computer Science , South Asian University , New Delhi , India
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19
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Modeling the Adaptive Immunity and Both Modes of Transmission in HIV Infection. COMPUTATION 2018. [DOI: 10.3390/computation6020037] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
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20
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Analysis and Optimal Control of an Intracellular Delayed HIV Model with CTL Immune Response. MATHEMATICS IN COMPUTER SCIENCE 2018. [DOI: 10.1007/s11786-018-0333-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
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21
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Miao H, Teng Z, Abdurahman X, Li Z. Global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response. COMPUTATIONAL AND APPLIED MATHEMATICS 2018. [PMCID: PMC7149116 DOI: 10.1007/s40314-017-0543-9] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
In this paper, the dynamical behaviors for a five-dimensional virus infection model with diffusion and two delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and a general incidence function are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization methods, the threshold conditions on the global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and antibody and CTL responses, respectively, are established if the space is assumed as homogeneous. When the space is inhomogeneous, the effects of diffusion, intracellular delay and production delay are obtained by the numerical simulations.
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Affiliation(s)
- Hui Miao
- School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006 People’s Republic of China
| | - Zhidong Teng
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Xamxinur Abdurahman
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Zhiming Li
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 People’s Republic of China
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22
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Elaiw AM, Ghaleb SA, Hobiny A. Effect of Time Delay and Antibodies on HCV Dynamics with Cure Rate and Two Routes of Infection. ACTA ACUST UNITED AC 2018. [DOI: 10.4236/jamp.2018.65096] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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23
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Wen B, Wang J, Teng Z. A discrete-time analog for coupled within-host and between-host dynamics in environmentally driven infectious disease. ADVANCES IN DIFFERENCE EQUATIONS 2018; 2018:69. [PMID: 32226450 PMCID: PMC7100524 DOI: 10.1186/s13662-018-1522-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2017] [Accepted: 02/09/2018] [Indexed: 05/03/2023]
Abstract
In this paper, we establish a discrete-time analog for coupled within-host and between-host systems for an environmentally driven infectious disease with fast and slow two time scales by using the non-standard finite difference scheme. The system is divided into a fast time system and a slow time system by using the idea of limit equations. For the fast system, the positivity and boundedness of the solutions, the basic reproduction number and the existence for infection-free and unique virus infectious equilibria are obtained, and the threshold conditions on the local stability of equilibria are established. In the slow system, except for the positivity and boundedness of the solutions, the existence for disease-free, unique endemic and two endemic equilibria are obtained, and the sufficient conditions on the local stability for disease-free and unique endemic equilibria are established. To return to the coupling system, the local stability for the virus- and disease-free equilibrium, and virus infectious but disease-free equilibrium is established. The numerical examples show that an endemic equilibrium is locally asymptotically stable and the other one is unstable when there are two endemic equilibria.
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Affiliation(s)
- Buyu Wen
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, People’s Republic of China
| | - Jianpeng Wang
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, People’s Republic of China
| | - Zhidong Teng
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, People’s Republic of China
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Global Stability of Delayed Viral Infection Models with Nonlinear Antibody and CTL Immune Responses and General Incidence Rate. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2017; 2016:3903726. [PMID: 28115980 PMCID: PMC5198507 DOI: 10.1155/2016/3903726] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/12/2016] [Accepted: 11/22/2016] [Indexed: 11/18/2022]
Abstract
The dynamical behaviors for a five-dimensional viral infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses, and nonlinear incidence rate are investigated. The threshold values for viral infection, antibody response, CTL immune response, CTL immune competition, and antibody competition, respectively, are established. Under certain assumptions, the threshold value conditions on the global stability of the infection-free, immune-free, antibody response, CTL immune response, and interior equilibria are proved by using the Lyapunov functionals method, respectively. Immune delay as a bifurcation parameter is further investigated. The numerical simulations are performed in order to illustrate the dynamical behavior of the model.
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25
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Global stability for a class of HIV infection models with cure of infected cells in eclipse stage and CTL immune response. ACTA ACUST UNITED AC 2016. [DOI: 10.1007/s40435-016-0268-4] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/07/2023]
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26
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Wang J, Teng Z, Miao H. Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response. ADVANCES IN DIFFERENCE EQUATIONS 2016; 2016:143. [PMID: 32226448 PMCID: PMC7099752 DOI: 10.1186/s13662-016-0862-y] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/25/2016] [Accepted: 05/10/2016] [Indexed: 06/10/2023]
Abstract
In this paper, a discrete-time analog of a viral infection model with nonlinear incidence and CTL immune response is established by using the Micken non-standard finite difference scheme. The two basic reproduction numbers R 0 and R 1 are defined. The basic properties on the positivity and boundedness of solutions and the existence of the virus-free, the no-immune, and the infected equilibria are established. By using the Lyapunov functions and linearization methods, the global stability of the equilibria for the model is established. That is, when R 0 ≤ 1 then the virus-free equilibrium is globally asymptotically stable, and under the additional assumption ( A 4 ) when R 0 > 1 and R 1 ≤ 1 then the no-immune equilibrium is globally asymptotically stable and when R 0 > 1 and R 1 > 1 then the infected equilibrium is globally asymptotically stable. Furthermore, the numerical simulations show that even if assumption ( A 4 ) does not hold, the no-immune equilibrium and the infected equilibrium also may be globally asymptotically stable.
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Affiliation(s)
- Jianpeng Wang
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Zhidong Teng
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046 People’s Republic of China
| | - Hui Miao
- College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046 People’s Republic of China
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27
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Li B, Chen Y, Lu X, Liu S. A delayed HIV-1 model with virus waning term. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2016; 13:135-157. [PMID: 26776264 DOI: 10.3934/mbe.2016.13.135] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD8+ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number R0. If R0 < 1, then the infection-free steady state is globally asymptotically stable and the infection is cleared from the T-cell population; whereas if R0 > 1, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when R0 > 1 is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage, the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.
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Affiliation(s)
- Bing Li
- Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China.
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