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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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2
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Xu J. Dynamic analysis of a cytokine-enhanced viral infection model with infection age. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:8666-8684. [PMID: 37161216 DOI: 10.3934/mbe.2023380] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
Recent studies reveal that pyroptosis is associated with the release of inflammatory cytokines which can attract more target cells to be infected. In this paper, a novel age-structured virus infection model incorporating cytokine-enhanced infection is investigated. The asymptotic smoothness of the semiflow is studied. With the help of characteristic equations and Lyapunov functionals, we have proved that both the local and global stabilities of the equilibria are completely determined by the threshold $ \mathcal{R}_0 $. The result shows that cytokine-enhanced viral infection also contributes to the basic reproduction number $ \mathcal{R}_0 $, implying that it may not be enough to eliminate the infection by decreasing the basic reproduction number of the model without considering the cytokine-enhanced viral infection mode. Numerical simulations are carried out to illustrate the theoretical results.
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Affiliation(s)
- Jinhu Xu
- School of Sciences, Xi'an University of Technology, Xi'an 710048, China
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3
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Yosyingyong P, Viriyapong R. Global dynamics of multiple delays within-host model for a hepatitis B virus infection of hepatocytes with immune response and drug therapy. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:7349-7386. [PMID: 37161155 DOI: 10.3934/mbe.2023319] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
In this paper, a mathematical model describing the hepatitis B virus (HBV) infection of hepatocytes with the intracellular HBV-DNA containing capsids, cytotoxic T-lymphocyte (CTL), antibodies including drug therapy (blocking new infection and inhibiting viral production) with two-time delays is studied. It incorporates the delay in the productively infected hepatocytes and the delay in an antigenic stimulation generating CTL. We verify the positivity and boundedness of solutions and determine the basic reproduction number. The local and global stability of three equilibrium points (infection-free, immune-free, and immune-activated) are investigated. Finally, the numerical simulations are established to show the role of these therapies in reducing viral replication and HBV infection. Our results show that the treatment by blocking new infection gives more significant results than the treatment by inhibiting viral production for infected hepatocytes. Further, both delays affect the number of infections and duration i.e. the longer the delay, the more severe the HBV infection.
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Affiliation(s)
- Pensiri Yosyingyong
- Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand
| | - Ratchada Viriyapong
- Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand
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4
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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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5
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Deng J, Jiang P, Shu H. Viral infection dynamics with mitosis, intracellular delays and immune response. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:2937-2963. [PMID: 36899566 DOI: 10.3934/mbe.2023139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
In this paper, we propose a delayed viral infection model with mitosis of uninfected target cells, two infection modes (virus-to-cell transmission and cell-to-cell transmission), and immune response. The model involves intracellular delays during the processes of viral infection, viral production, and CTLs recruitment. We verify that the threshold dynamics are determined by the basic reproduction number $ R_0 $ for infection and the basic reproduction number $ R_{IM} $ for immune response. The model dynamics become very rich when $ R_{IM} > 1 $. In this case, we use the CTLs recruitment delay $ \tau_3 $ as the bifurcation parameter to obtain stability switches on the positive equilibrium and global Hopf bifurcation diagrams for the model system. This allows us to show that $ \tau_3 $ can lead to multiple stability switches, the coexistence of multiple stable periodic solutions, and even chaos. A brief simulation of two-parameter bifurcation analysis indicates that both the CTLs recruitment delay $ \tau_3 $ and the mitosis rate $ r $ have a strong impact on the viral dynamics, but they do behave differently.
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Affiliation(s)
- Jiawei Deng
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710062, China
| | - Ping Jiang
- School of Management, Shanghai University of International Business and Economics, Shanghai 201620, China
| | - Hongying Shu
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710062, China
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6
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Elaiw AM, Shflot AS, Hobiny AD. Stability analysis of general delayed HTLV-I dynamics model with mitosis and CTL immunity. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:12693-12729. [PMID: 36654018 DOI: 10.3934/mbe.2022593] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
This paper formulates and analyzes a general delayed mathematical model which describe the within-host dynamics of Human T-cell lymphotropic virus class I (HTLV-I) under the effect Cytotoxic T Lymphocyte (CTL) immunity. The models consist of four components: uninfected CD$ 4^{+} $T cells, latently infected cells, actively infected cells and CTLs. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearance rates for all types of cells. The incidence rate of infection is also modeled by a general nonlinear function. These general functions are assumed to be satisfy some suitable conditions. To account for series of events in the infection process and activation of latently infected cells, we introduce two intracellular distributed-time delays into the models: (ⅰ) delay in the formation of latently infected cells, (ⅱ) delay in the activation of latently infected cells. We determine a bounded domain for the system's solutions. We calculate two threshold numbers, the basic reproductive number $ R_{0} $ and the CTL immunity stimulation number $ R_{1} $. We determine the conditions for the existence and global stability of the equilibrium points. We study the global stability of all equilibrium points using Lyapunov method. We prove the following: (a) if $ R_{0}\leq 1 $, then the infection-free equilibrium point is globally asymptotically stable (GAS), (b) if $ R_{1}\leq 1 < R_{0} $, then the infected equilibrium point without CTL immunity is GAS, (c) if $ R_{1} > 1 $, then the infected equilibrium point with CTL immunity is GAS. We present numerical simulations for the system by choosing special shapes of the general functions. The effects of proliferation of CTLs and time delay on the HTLV-I progression is investigated. We noted that the CTL immunity does not play the role in clearing the HTLV-I from the body, but it has an important role in controlling and suppressing the viral infection. On the other hand, we observed that, increasing the time delay intervals can have similar influences as drug therapies in removing viruses from the body. This gives some impression to develop two types of treatments, the first type aims to extend the intracellular delay periods, while the second type aims to activate and stimulate the CTL immune response.
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Affiliation(s)
- A M Elaiw
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
- Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut Branch, Assiut, Egypt
| | - A S Shflot
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
- Department of Mathematics, Faculty of Science, King Khalid University, P. O. Box 960, Abha 61421, Saudi Arabia
| | - A D Hobiny
- Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
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Huynh PK, Setty A, Phan H, Le TQ. Probabilistic domain-knowledge modeling of disorder pathogenesis for dynamics forecasting of acute onset. Artif Intell Med 2021; 115:102056. [PMID: 34001316 PMCID: PMC8493977 DOI: 10.1016/j.artmed.2021.102056] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2020] [Revised: 03/01/2021] [Accepted: 03/22/2021] [Indexed: 11/18/2022]
Abstract
Disease pathogenesis, a type of domain knowledge about biological mechanisms leading to diseases, has not been adequately encoded in machine-learning-based medical diagnostic models because of the inter-patient variabilities and complex dependencies of the underlying pathogenetic mechanisms. We propose 1) a novel pathogenesis probabilistic graphical model (PPGM) to quantify the dynamics underpinning patient-specific data and pathogenetic domain knowledge, 2) a Bayesian-based inference paradigm to answer the medical queries and forecast acute onsets. The PPGM model consists of two components: a Bayesian network of patient attributes and a temporal model of pathogenetic mechanisms. The model structure was reconstructed from expert knowledge elicitation, and its parameters were estimated using Variational Expectation-Maximization algorithms. We benchmarked our model with two well-established hidden Markov models (HMMs) - Input-output HMM (IO-HMM) and Switching Auto-Regressive HMM (SAR-HMM) - to evaluate the computational costs, forecasting performance, and execution time. Two case studies on Obstructive Sleep Apnea (OSA) and Paroxysmal Atrial Fibrillation (PAF) were used to validate the model. While the performance of the parameter learning step was equivalent to those of IO-HMM and SAR-HMM models, our model forecasting ability was outperforming those two models. The merits of the PPGM model are its representation capability to capture the dynamics of pathogenesis and perform medical inferences and its interpretability for physicians. The model has been used to perform medical queries and forecast the acute onset of OSA and PAF. Additional applications of the model include prognostic healthcare and preventive personalized treatments.
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Affiliation(s)
- Phat K Huynh
- Department of Industrial and Manufacturing Engineering, North Dakota State University at Fargo, ND, USA
| | | | - Hao Phan
- Pham Ngoc Thach University of Medicine at Ho Chi Minh City, Viet Nam
| | - Trung Q Le
- Department of Industrial and Manufacturing Engineering, North Dakota State University at Fargo, ND, USA; Department of Biomedical Engineering, North Dakota State University at Fargo, ND, USA.
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8
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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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9
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Han B, Jiang D, Hayat T, Alsaedi A, Ahmad B. Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation. CHAOS, SOLITONS, AND FRACTALS 2020; 140:110238. [PMID: 32868968 PMCID: PMC7449666 DOI: 10.1016/j.chaos.2020.110238] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2020] [Revised: 08/06/2020] [Accepted: 08/22/2020] [Indexed: 05/29/2023]
Abstract
Focusing on deterministic AIDS model proposed by Hyman (2000) and the detailed data from the World Health Organization (WHO), there are three stages of AIDS process which are described as Acute infection period, Asymptomatic phase and AIDS stage. Our paper is therefore concerned with a stochastic staged progression AIDS model with staged treatment. In view of the complexity of random disturbances, we reasonably take second-order perturbation into consideration for realistic sense. By means of our creative transformation technique and stochastic Lyapunov method, a critical value R 0 H > 1 is firstly obtained for the existence and uniqueness of ergodic stationary distribution to the stochastic system. Not only does it respectively reveal the corresponding dynamical effects of the linear and second-order perturbations to the model, but the unified form of second-order and linear fluctuations is derived. Next, some sufficient conditions about extinction of stochastic system are established in view of the basic reproduction number R 0 . Finally, some examples and numerical simulations are introduced to illustrate our analytical results. In addition, some advantages of our new method and theory are highlighted by comparison with other existing results at the end of this paper.
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Affiliation(s)
- Bingtao Han
- College of Science, China University of Petroleum (East China), Qingdao 266580, PR China
| | - Daqing Jiang
- College of Science, China University of Petroleum (East China), Qingdao 266580, PR China
- Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
| | - Tasawar Hayat
- Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
- Department of Mathematics, Quaid-i-Azam University 45320, Isamabad 44000, Pakistan
| | - Ahmed Alsaedi
- Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
| | - Bashir Ahmad
- Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
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10
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Geng Y, Xu J. Stability and bifurcation analysis for a delayed viral infection model with full logistic proliferation. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500333] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we study a delayed viral infection model with cellular infection and full logistic proliferations for both healthy and infected cells. The global asymptotic stabilities of the equilibria are studied by constructing Lyapunov functionals. Moreover, we investigated the existence of Hopf bifurcation at the infected equilibrium by regarding the possible combination of the two delays as bifurcation parameters. The results show that time delays may destabilize the infected equilibrium and lead to Hopf bifurcation. Finally, numerical simulations are carried out to illustrate the main results and explore the dynamics including Hopf bifurcation and stability switches.
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Affiliation(s)
- Yan Geng
- School of Science, Xi’an Polytechnic University, Xi’an 710048, Shaanxi, P. R. China
| | - Jinhu Xu
- School of Sciences, Xi’an University of Technology, Xi’an 710049, Shaanxi, P. R. China
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11
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Mathematical Analysis and Clinical Implications of an HIV Model with Adaptive Immunity. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2019; 2019:7673212. [PMID: 31827588 PMCID: PMC6885180 DOI: 10.1155/2019/7673212] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/30/2018] [Accepted: 11/01/2019] [Indexed: 11/17/2022]
Abstract
In this paper, a mathematical model describing the human immunodeficiency virus (HIV) pathogenesis with adaptive immune response is presented and studied. The mathematical model includes six nonlinear differential equations describing the interaction between the uninfected cells, the exposed cells, the actively infected cells, the free viruses, and the adaptive immune response. The considered adaptive immunity will be represented by cytotoxic T-lymphocytes cells (CTLs) and antibodies. First, the global stability of the disease-free steady state and the endemic steady states is established depending on the basic reproduction number R 0, the CTL immune response reproduction number R 1 z , the antibody immune response reproduction number R 1 w , the antibody immune competition reproduction number R 2 w , and the CTL immune response competition reproduction number R 3 z . On the other hand, different numerical simulations are performed in order to confirm numerically the stability for each steady state. Moreover, a comparison with some clinical data is conducted and analyzed. Finally, a sensitivity analysis for R 0 is performed in order to check the impact of different input parameters.
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12
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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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13
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Modeling Inhibitory Effect on the Growth of Uninfected T Cells Caused by Infected T Cells: Stability and Hopf Bifurcation. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2018; 2018:3176893. [PMID: 31191708 PMCID: PMC6525856 DOI: 10.1155/2018/3176893] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/01/2018] [Accepted: 04/30/2018] [Indexed: 01/02/2023]
Abstract
We consider a class of viral infection dynamic models with inhibitory effect on the growth of uninfected T cells caused by infected T cells and logistic target cell growth. The basic reproduction number R0 is derived. It is shown that the uninfected equilibrium is globally asymptotically stable if R0 < 1. Sufficient conditions for the existence of Hopf bifurcation at the infected equilibrium are investigated by analyzing the distribution of eigenvalues. Furthermore, the properties of Hopf bifurcation are determined by the normal form theory and the center manifold. Numerical simulations are carried out to support the theoretical analysis.
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14
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Jiang C, Wang K, Song L. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2018; 14:1233-1246. [PMID: 29161858 DOI: 10.3934/mbe.2017063] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number R0. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.
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Affiliation(s)
- Cuicui Jiang
- Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China
| | - Kaifa Wang
- Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China
| | - Lijuan Song
- Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China
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15
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Jenner A, Yun CO, Yoon A, Kim PS, Coster ACF. Modelling heterogeneity in viral-tumour dynamics: The effects of gene-attenuation on viral characteristics. J Theor Biol 2018; 454:41-52. [PMID: 29857083 DOI: 10.1016/j.jtbi.2018.05.030] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2017] [Revised: 05/25/2018] [Accepted: 05/28/2018] [Indexed: 01/16/2023]
Abstract
The use of viruses as a cancer treatment is becoming increasingly more robust; however, there is still a long way to go before a completely successful treatment is formulated. One major challenge in the field is to select which virus, out of a burgeoning number of oncolytic viruses and engineered derivatives, can maximise both treatment spread and anticancer cytotoxicity. To assist in solving this problem, an in-depth understanding of the virus-tumour interaction is crucial. In this article, we present a novel integro-differential system with distributed delays embodying the dynamics of an oncolytic adenovirus with a fixed population of tumour cells in vitro, allowing for heterogeneity to exist in the virus and cell populations. The parameters of the model are optimised in a hierarchical manner, the purpose of which is not to obtain a perfect representation of the data. Instead, we place our parameter values in the correct region of the parameter space. Due to the sparse nature of the data it is not possible to obtain the parameter values with any certainty, but rather we demonstrate the suitability of the model. Using our model we quantify how modifications to the viral genome alter the viral characteristics, specifically how the attenuation of the E1B 19 and E1B 55 gene affect the system performance, and identify the dominant processes altered by the mutations. From our analysis, we conclude that the deletion of the E1B 55 gene significantly reduces the replication rate of the virus in comparison to the deletion of the E1B 19 gene. We also found that the deletion of both the E1B 19 and E1B 55 genes resulted in a long delay in the average replication start time of the virus. This leads us to propose the use of E1B 19 gene-attenuated adenovirus for cancer therapy, as opposed to E1B 55 gene-attenuated adenoviruses.
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Affiliation(s)
- Adrianne Jenner
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia
| | - Chae-Ok Yun
- Department of Bioengineering, College of Engineering, Hanyang University, Seoul, South Korea
| | - Arum Yoon
- Department of Bioengineering, College of Engineering, Hanyang University, Seoul, South Korea
| | - Peter S Kim
- School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia
| | - Adelle C F Coster
- School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia.
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16
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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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17
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Global properties and bifurcation analysis of an HIV-1 infection model with two target cells. ACTA ACUST UNITED AC 2017. [DOI: 10.1007/s40314-017-0523-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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18
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Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase. APPLIED SCIENCES-BASEL 2017. [DOI: 10.3390/app7080861] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/04/2023]
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19
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Ciupe SM, Heffernan JM. In-host modeling. Infect Dis Model 2017; 2:188-202. [PMID: 29928736 PMCID: PMC6001971 DOI: 10.1016/j.idm.2017.04.002] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2017] [Revised: 04/24/2017] [Accepted: 04/26/2017] [Indexed: 01/14/2023] Open
Abstract
Understanding the mechanisms governing host-pathogen kinetics is important and can guide human interventions. In-host mathematical models, together with biological data, have been used in this endeavor. In this review, we present basic models used to describe acute and chronic pathogenic infections. We highlight the power of model predictions, the role of drug therapy, and advantage of considering the dynamics of immune responses. We also present the limitations of these models due in part to the trade-off between the complexity of the model and their predictive power, and the challenges a modeler faces in determining the appropriate formulation for a given problem.
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Affiliation(s)
- Stanca M. Ciupe
- Department of Mathematics, Virginia Tech, Blacksburg, VA, USA
| | - Jane M. Heffernan
- Centre for Disease Modelling, Department of Mathematics & Statistics, York University, Toronto, ON, Canada
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20
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Alshorman A, Wang X, Joseph Meyer M, Rong L. Analysis of HIV models with two time delays. JOURNAL OF BIOLOGICAL DYNAMICS 2017; 11:40-64. [PMID: 26889761 DOI: 10.1080/17513758.2016.1148202] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Time delays can affect the dynamics of HIV infection predicted by mathematical models. In this paper, we studied two mathematical models each with two time delays. In the first model with HIV latency, one delay is the time between viral entry into a cell and the establishment of HIV latency, and the other delay is the time between cell infection and viral production. We defined the basic reproductive number and showed the local and global stability of the steady states. Numerical simulations were performed to evaluate the influence of time delays on the dynamics. In the second model with HIV immune response, one delay is the time between cell infection and viral production, and the other delay is the time needed for the adaptive immune response to emerge to control viral replication. With two positive delays, we obtained the stability crossing curves for the model, which were shown to be a series of open-ended curves.
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Affiliation(s)
- Areej Alshorman
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Xia Wang
- b College of Mathematics and Information Science , Xinyang Normal University , Xinyang , People's Republic of China
| | - M Joseph Meyer
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Libin Rong
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
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21
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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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22
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Abstract
Infection of human immunodeficiency virus (HIV) is determined through the decay of healthy CD4+ T-cells in a well-mixed compartment, such as a bloodstream. A mathematical model is considered to illustrate the effects of combined drug therapy, i.e. reverse transcription plus protease inhibitor, on viral growth and T-cell population dynamics. This model is used to explain the existence and stability of infected and uninfected steady states in HIV growth. An analytical technique, called variational iteration method (VIM), is used to solve the mathematical model. This method is modified to obtain the rapidly convergent successive approximations of the exact solution. These approximations are obtained without any restrictions or the transformations that may change the physical behavior of the problem. Numerical simulations are computed and exhibited to illustrate the effects of proposed drug therapy on the growth or decay of infection.
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Affiliation(s)
| | - M. D. Sharma
- Department of Mathematics, Kurukshetra University, India
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23
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Sun Q, Min L, Kuang Y. Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model. IET Syst Biol 2015; 9:95-103. [PMID: 26021330 DOI: 10.1049/iet-syb.2014.0046] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022] Open
Abstract
This study proposes a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. This model has an infection-free equilibrium point and an endemic infection equilibrium point. Using Lyapunov functions and LaSalle's invariance principle shows that if the model's basic reproductive number R0 < 1, the infection-free equilibrium point is globally asymptotically stable, otherwise the endemic infection equilibrium point is globally asymptotically stable. It is shown that a forward bifurcation will occur when R0 = 1. The basic reproductive number R0 of the modified model is independent of plasma total CD4⁺ T cell counts and thus the modified model is more reasonable than the original model proposed by Buonomo and Vargas-De-León. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of two group patients' anti-HIV infection treatments. The simulation results have shown that the first 4 weeks' treatments made the two group patients' R'0 < 1, respectively. After the period, drug resistance made the two group patients' R'0 > 1. The results explain why the two group patients' mean CD4⁺ T cell counts raised and mean HIV RNA levels declined in the first period, but contrary in the following weeks.
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Affiliation(s)
- Qilin Sun
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, People's Republic of China
| | - Lequan Min
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, People's Republic of China.
| | - Yang Kuang
- School of Mathematical and Statistical Sciences, Arizona State University, Temp AZ 85287-1804, USA
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24
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Zhou J, Han L, Liu S. Kernel-Based Profile Estimation for Ordinary Differential Equations with Partially Measured State Variables. COMMUN STAT-THEOR M 2015. [DOI: 10.1080/03610926.2013.851224] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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25
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Prakash M, Balasubramaniam P. Stability and multi-parametric Hopf bifurcation analyses of viral infection model with time delay. INT J BIOMATH 2015. [DOI: 10.1142/s179352451550059x] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state. Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E* and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.
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Affiliation(s)
- M. Prakash
- Department of Mathematics, Gandhigram Rural Institute, Deemed University, Gandhigram 624 302, Tamil Nadu, India
| | - P. Balasubramaniam
- Department of Mathematics, Gandhigram Rural Institute, Deemed University, Gandhigram 624 302, Tamil Nadu, India
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26
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Yang Y, Zou L, Ruan S. Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions. Math Biosci 2015; 270:183-91. [PMID: 25998145 DOI: 10.1016/j.mbs.2015.05.001] [Citation(s) in RCA: 87] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2015] [Revised: 04/29/2015] [Accepted: 05/06/2015] [Indexed: 10/23/2022]
Abstract
A within-host viral infection model with both virus-to-cell and cell-to-cell transmissions and three distributed delays is investigated, in which the first distributed delay describes the intracellular latency for the virus-to-cell infection, the second delay represents the intracellular latency for the cell-to-cell infection, and the third delay describes the time period that viruses penetrated into cells and infected cells release new virions. The global stability analysis of the model is carried out in terms of the basic reproduction number R0. If R0≤1, the infection-free (semi-trivial) equilibrium is the unique equilibrium and is globally stable; if R0>1, the chronic infection (positive) equilibrium exists and is globally stable under certain assumptions. Examples and numerical simulations for several special cases are presented, including various within-host dynamics models with discrete or distributed delays that have been well-studied in the literature. It is found that the global stability of the chronic infection equilibrium might change in some special cases when the assumptions do not hold. The results show that the model can be applied to describe the within-host dynamics of HBV, HIV, or HTLV-1 infection.
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Affiliation(s)
- Yu Yang
- School of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, P. R. China
| | - Lan Zou
- Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China
| | - Shigui Ruan
- Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA.
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27
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Sun Q, Min L. Dynamics analysis and simulation of a modified HIV infection model with a saturated infection rate. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2014; 2014:145162. [PMID: 24829609 PMCID: PMC3981026 DOI: 10.1155/2014/145162] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/17/2013] [Accepted: 01/28/2014] [Indexed: 11/17/2022]
Abstract
This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive number R 0 of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; if R 0 of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients' anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients' HIV RNA levels. It can be assumed that if an HIV infected individual's basic virus reproductive number R 0 < 1 then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual's R 0 < 1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual's HIV RNA load to be of unpredictable level, the time that the patient's HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.
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Affiliation(s)
- Qilin Sun
- School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China
| | - Lequan Min
- School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
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28
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Jiang C, Wang W. Complete classification of global dynamics of a virus model with
immune responses. ACTA ACUST UNITED AC 2014. [DOI: 10.3934/dcdsb.2014.19.1087] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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29
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Xiao Y, Miao H, Tang S, Wu H. Modeling antiretroviral drug responses for HIV-1 infected patients using differential equation models. Adv Drug Deliv Rev 2013; 65:940-53. [PMID: 23603208 PMCID: PMC4017332 DOI: 10.1016/j.addr.2013.04.005] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/26/2012] [Revised: 03/29/2013] [Accepted: 04/10/2013] [Indexed: 12/22/2022]
Abstract
We review mathematical modeling and related statistical issues of HIV dynamics primarily in response to antiretroviral drug therapy in this article. We start from a basic model of virus infection and then review a number of more advanced models with consideration of pharmacokinetic factors, adherence and drug resistance. Specifically, we illustrate how mathematical models can be developed and parameterized to understand the effects of long-term treatment and different treatment strategies on disease progression. In addition, we discuss a variety of parameter estimation methods for differential equation models that are applicable to either within- or between-host viral dynamics.
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Affiliation(s)
- Yanni Xiao
- School of Mathematics & Statistics, Xi’an Jiaotong University, Shaanxi, China
| | - Hongyu Miao
- School of Medicine and Dentistry, University of Rochester, New York, USA
| | - Sanyi Tang
- School of Mathematics & Information Sciences, Shaanxi Normal University, Shaanxi, China
| | - Hulin Wu
- School of Medicine and Dentistry, University of Rochester, New York, USA
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30
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ROY PRITIKUMAR, CHOWDHURY SONIA, LI XUEZHI. SATURATION EFFECTS FOR CTL MEDIATED CONTROL OF HIV-1 INFECTION: A MATHEMATICAL STUDY. INT J BIOMATH 2013. [DOI: 10.1142/s1793524513500137] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The relations between the Human Immunodeficiency Virus-1 (HIV-1) and the human immune system are astonishingly multifaceted, where the critical role for cytotoxic T lymphocytes (CTLs) in the suppression of viral replication in HIV-1 infected individuals cannot be ignored. In this research paper, we have proposed a mathematical model incorporating half saturation constant through the CTL mediated killing process and also in that sense, it has been infiltrated in the generation process of CTL through infected cells. To make the model more realistic, a time lag is introduced in the generation term of CTL population. Also an optimal control theory paradigm is used in our mathematical model to suppress the viral production. From our entire analysis, we have found threshold condition of half saturation constant and treatment schedule so that we can handle the situation of Acquired Immunodeficiency Syndrome (AIDS) patients in a better way. Our analysis reveals that, if the half saturation constant is around 47 mm-3in the saturation process and the drug therapy is to be used around 76 days, then we can get adequate results for better treatment of a HIV-1 patient. Based on numerical results, we observed that in a highly unstable situation, administration of chemotherapy at a high dose can stabilize the system.
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Affiliation(s)
- PRITI KUMAR ROY
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - SONIA CHOWDHURY
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - XUE-ZHI LI
- Department of Mathematics, Xinyang Normal University, Xinyang 464000, P. R. China
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31
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BAIRAGI N, ADAK D. HOW SELF-PROLIFERATION OF CD4+T CELLS AFFECT THE HIV DYNAMICS IN AN IN-HOST TARGET-CELL LIMITED HIV MODEL WITH SATURATION INFECTION RATE: A QUASI-STEADY-STATE APPROXIMATION ANALYSIS. INT J BIOMATH 2013. [DOI: 10.1142/s1793524513500046] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this study, we consider two target-cell limited models with saturation type infection rate and intracellular delay: one without self-proliferation and the other with self-proliferation of activated CD4+T cells. We discuss about the local and global behavior of both the systems in presence and absence of intracellular delay. It is shown that the endemic equilibrium of a target-cell limited model would be unstable in presence and absence of intracellular delay only when self-proliferation of activated CD4+T cell is considered. Otherwise, all positive solutions converge to the endemic equilibrium or disease-free equilibrium depending on whether the basic reproduction ratio is greater than or less than unity. Our study suggests that amplitude of oscillation is negatively correlated with the constant input rate of CD4+T cell when intracellular delay is absent or low. However, they are positively correlated if the delay is too high. Amplitude of oscillation, on the other hand, is always positively correlated with the proliferation rate of CD4+T cell for all delay. Our mathematical and simulation analysis also suggest that there are many potential contributors who are responsible for the variation of CD4+T cells and virus particles in the blood plasma of HIV patients.
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Affiliation(s)
- N. BAIRAGI
- Center for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
| | - D. ADAK
- Center for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
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32
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Wang Y, Zhou Y, Brauer F, Heffernan JM. Viral dynamics model with CTL immune response incorporating antiretroviral therapy. J Math Biol 2012; 67:901-34. [DOI: 10.1007/s00285-012-0580-3] [Citation(s) in RCA: 59] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2011] [Revised: 06/28/2012] [Indexed: 01/01/2023]
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33
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WANG SHIFEI, ZHOU YICANG. GLOBAL DYNAMICS OF AN IN-HOST HIV-1 INFECTION MODEL WITH THE LONG-LIVED INFECTED CELLS AND FOUR INTRACELLULAR DELAYS. INT J BIOMATH 2012. [DOI: 10.1142/s1793524512500581] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we investigate global dynamics for an in-host HIV-1 infection model with the long-lived infected cells and four intracellular delays. Our model admits two possible equilibria, an uninfected equilibrium and infected equilibrium depending on the basic reproduction number. We derive that the global dynamics are completely determined by the values of the basic reproduction number: if the basic reproduction number is less than one, the uninfected equilibrium is globally asymptotically stable, and the virus is cleared; if the basic reproduction number is larger than one, then the infection persists, and the infected equilibrium is globally asymptotically stable.
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Affiliation(s)
- SHIFEI WANG
- Department of Applied Mathematics, Xi'an JiaoTong University, Xi'An, 710049, P. R. China
- Department of Physics and Mathematics, Changzhou University, Chang'zhou, 213016, P. R. China
| | - YICANG ZHOU
- Department of Applied Mathematics, Xi'an JiaoTong University, Xi'An, 710049, P. R. China
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34
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A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data. Math Biosci 2011; 235:98-109. [PMID: 22108296 DOI: 10.1016/j.mbs.2011.11.002] [Citation(s) in RCA: 116] [Impact Index Per Article: 8.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2011] [Revised: 10/31/2011] [Accepted: 11/04/2011] [Indexed: 11/22/2022]
Abstract
Mathematical models have made considerable contributions to our understanding of HIV dynamics. Introducing time delays to HIV models usually brings challenges to both mathematical analysis of the models and comparison of model predictions with patient data. In this paper, we incorporate two delays, one the time needed for infected cells to produce virions after viral entry and the other the time needed for the adaptive immune response to emerge to control viral replication, into an HIV-1 model. We begin model analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free and infected steady states, and uniform persistence of the system. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of the steady states. We also fit the model including two delays to viral load data from 10 patients during primary HIV-1 infection and estimate parameter values. Although the delay model provides better fits to patient data (achieving a smaller error between data and modeling prediction) than the one without delays, we could not determine which one is better from the statistical standpoint. This highlights the need of more data sets for model verification and selection when we incorporate time delays into mathematical models to study virus dynamics.
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35
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Li MY, Shu H. Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis. J Math Biol 2011; 64:1005-20. [PMID: 21671033 DOI: 10.1007/s00285-011-0436-2] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2011] [Revised: 05/15/2011] [Indexed: 10/18/2022]
Abstract
To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.
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Affiliation(s)
- Michael Y Li
- Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People's Republic of China.
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36
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Global Dynamics of an In-host Viral Model with Intracellular Delay. Bull Math Biol 2010; 72:1492-505. [DOI: 10.1007/s11538-010-9503-x] [Citation(s) in RCA: 106] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2009] [Accepted: 01/07/2010] [Indexed: 10/20/2022]
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