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Guo K, Guo S. Lyapunov functionals for a general time-delayed virus dynamic model with different CTL responses. CHAOS (WOODBURY, N.Y.) 2024; 34:053138. [PMID: 38809906 DOI: 10.1063/5.0204169] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2024] [Accepted: 04/29/2024] [Indexed: 05/31/2024]
Abstract
A time-delayed virus dynamic model is proposed with general monotonic incidence, different nonlinear CTL (cytotoxic T lymphocyte) responses [CTL elimination function pyg1(z) and CTL stimulation function cyg2(z)], and immune impairment. Indeed, the different CTL responses pose challenges in obtaining the dissipativeness of the model. By constructing appropriate Lyapunov functionals with some detailed analysis techniques, the global stability results of all equilibria of the model are obtained. By the way, we point out that the partial derivative fv(x,0) is increasing (but not necessarily strictly) in x>0 for the general monotonic incidence f(x,v). However, some papers defaulted that the partial derivative was strictly increasing. Our main results show that if the basic reproduction number R0≤1, the infection-free equilibrium E0 is globally asymptotically stable (GAS); if CTL stimulation function cyg2(z)=0 for z=0 and the CTL threshold parameter R1≤1
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Affiliation(s)
- Ke Guo
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
| | - Songbai Guo
- School of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China
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2
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Bai N, Xu R. Mathematical analysis of an HIV model with latent reservoir, delayed CTL immune response and immune impairment. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:1689-1707. [PMID: 33757205 DOI: 10.3934/mbe.2021087] [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: 06/12/2023]
Abstract
In this paper, an in-host HIV infection model with latent reservoir, delayed CTL immune response and immune impairment is investigated. By using suitable Lyapunov functions and LaSalle's invariance principle, it is shown that when time delay is equal to zero, the immunity-inactivated reproduction ratio is a threshold determining the global dynamics of the model. By means of the persistence theory for infinite dimensional systems, it is proven that if the immunity-inactivated reproduction ratio is greater than unity, the model is permanent. Choosing time delay as the bifurcation parameter and analyzing the corresponding characteristic equation of the linearized system, the existence of a Hopf bifurcation at the immunity-activated equilibrium is established. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics.
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Affiliation(s)
- Ning Bai
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, China
| | - Rui Xu
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, China
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Liu C, Kong L. Dynamics of an HIV model with cytotoxic T-lymphocyte memory. ADVANCES IN DIFFERENCE EQUATIONS 2020; 2020:581. [PMID: 33101401 PMCID: PMC7568027 DOI: 10.1186/s13662-020-03035-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/19/2020] [Accepted: 10/06/2020] [Indexed: 06/11/2023]
Abstract
We consider a four-dimensional HIV model that includes healthy cells, infected cells, primary cytotoxic T-lymphocyte response (CTLp), and secondary cytotoxic T-lymphocyte response (CTLe). The CTL memory generation depends on CD4+ T-cell help, and infection of CD4+ T cells results in impaired T-cell help. We show that the system has up to five equilibria. By the Routh-Hurwitz theorem and central manifold theorem we obtain some sufficient conditions for the local stability, globally stability of the equilibria, and the bifurcations. We still discover the bistability case where in the system there may coexist two stable equilibria or a stable equilibrium together with a stable limit cycle. Several numerical analyses are carried out to illustrate the validity of our theoretical results.
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Affiliation(s)
- Chunhua Liu
- School of Mathematics and Statistics, Yangtze Normal University, Fuling district, 408100 Chongqing city, P.R. China
| | - Lei Kong
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guizhou, 550025 P.R. China
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Abstract
Some HIV-infected patients (the so-called post-treatment controllers) can control the virus after cessation of antiretroviral therapy. A small fraction of patients can even naturally maintain undetectable viral load without therapy (they are called elite controllers). The immune response may play an important role in viral control in these patients. In this paper, we analyze a within-host model including immune response to study the virus dynamics in HIV-infected patients. We derived two threshold values for the immune cell proliferation parameter. Below the lower immune proliferation rate, the model has a stable immune-free steady state, which predicts that patients have a high viral load. Above the higher immune proliferation rate, the model has a stable low infected steady state, which indicates that patients are under elite control. Between the two immune thresholds, the model exhibits the dynamic behavior of bistability, which suggests that patients either undergo viral rebound after treatment termination or achieve the post-treatment control. These results may explain the different virus dynamics in HIV-infected patients.
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Affiliation(s)
- SHAOLI WANG
- School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475001, P. R. China
| | - FEI XU
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada
| | - LIBIN RONG
- Department of Mathematics, University of Florida, Gainesville FL 32611, USA
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Abstract
In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte (CTL) cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.
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Affiliation(s)
- Jianhua Pang
- School of Science, Guangxi University of Science and Technology, Liuzhou 545006, P. R. China
| | - Jing-An Cui
- School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, P. R. China
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Sutimin, Chirove F, Soewono E, Nuraini N, Suromo LB. A model incorporating combined RTIs and PIs therapy during early HIV-1 infection. Math Biosci 2017; 285:102-111. [PMID: 28108293 DOI: 10.1016/j.mbs.2017.01.004] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2014] [Revised: 05/26/2015] [Accepted: 01/13/2017] [Indexed: 01/20/2023]
Abstract
We develop a within host mathematical model of HIV-1 infection describing the effects of combined RTIs and PIs treatments on early HIV-1 infection when treatment is captured using periodic functions of pharmacokinetics type. We use an alternative of the basic reproduction number to analyze endemicity level of HIV-1 infection. Various treatment scenarios incorporating perfect and imperfect drug adherence in drug administration are explored. Our results show that pharmacokinetics treatment is a more realistic way of administering the treatment. Apart from confirming that PIs drugs are more effective than RTIs drugs and that combined RTIs and PIs therapy is more effective than monotherapy of RTIs or PIs, our results show that imperfect drug adherence leads to the increase of viral in the absence of mutation even though the drug is good.
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Affiliation(s)
- Sutimin
- Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia; Department of Mathematics, Diponegoro University, Semarang 50275, Indonesia.
| | - F Chirove
- School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa
| | - E Soewono
- Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia
| | - N Nuraini
- Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia
| | - L B Suromo
- Faculty of Medicine, Diponegoro University, Semarang 50275, Indonesia
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Li F, Ma W, Jiang Z, Li D. Stability and Hopf Bifurcation in a Delayed HIV Infection Model with General Incidence Rate and Immune Impairment. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2015; 2015:206205. [PMID: 26413141 PMCID: PMC4539976 DOI: 10.1155/2015/206205] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/16/2015] [Accepted: 06/29/2015] [Indexed: 01/26/2023]
Abstract
We investigate the dynamical behavior of a delayed HIV infection model with general incidence rate and immune impairment. We derive two threshold parameters, the basic reproduction number R 0 and the immune response reproduction number R 1. By using Lyapunov functional and LaSalle invariance principle, we prove the global stability of the infection-free equilibrium and the infected equilibrium without immunity. Furthermore, the existence of Hopf bifurcations at the infected equilibrium with CTL response is also studied. By theoretical analysis and numerical simulations, the effect of the immune impairment rate on the stability of the infected equilibrium with CTL response has been studied.
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Affiliation(s)
- Fuxiang Li
- Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
| | - Wanbiao Ma
- Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
| | - Zhichao Jiang
- Fundamental Science Department, North China Institute of Aerospace Engineering, Langfang, Hebei 065000, China
| | - Dan Li
- Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
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