Multistability in a model for CTL response to HTLV-I infection and its implications to HAM/TSP development and prevention

Lyapunov functionals for a general time-delayed virus dynamic model with different CTL responses

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 Collapse

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MESH Headings

• T-Lymphocytes, Cytotoxic/immunology
• Humans
• Time Factors
• Viruses/immunology
• Virus Diseases/immunology
• Models, Immunological
• Models, Biological

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Grants Collapse

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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Stability analysis of general delayed HTLV-I dynamics model with mitosis and CTL immunity

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.

Dynamics of HIV-1/HTLV-I Co-Infection Model with Humoral Immunity and Cellular Infection

Human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses which infect the same target, CD4+ T cells. This type of cell is considered the main component of the immune system. Since both viruses have the same means of transmission between individuals, HIV-1-infected patients are more exposed to the chance of co-infection with HTLV-I, and vice versa, compared to the general population. The mathematical modeling and analysis of within-host HIV-1/HTLV-I co-infection dynamics can be considered a robust tool to support biological and medical research. In this study, we have formulated and analyzed an HIV-1/HTLV-I co-infection model with humoral immunity, taking into account both latent HIV-1-infected cells and HTLV-I-infected cells. The model considers two modes of HIV-1 dissemination, virus-to-cell (V-T-C) and cell-to-cell (C-T-C). We prove the nonnegativity and boundedness of the solutions of the model. We find all steady states of the model and establish their existence conditions. We utilize Lyapunov functions and LaSalle’s invariance principle to investigate the global stability of all the steady states of the model. Numerical simulations were performed to illustrate the corresponding theoretical results. The effects of humoral immunity and C-T-C transmission on the HIV-1/HTLV-I co-infection dynamics are discussed. We have shown that humoral immunity does not play the role of clearing an HIV-1 infection but it can control HIV-1 infection. Furthermore, we note that the omission of C-T-C transmission from the HIV-1/HTLV-I co-infection model leads to an under-evaluation of the basic HIV-1 mono-infection reproductive ratio.

Analysis of an HTLV/HIV dual infection model with diffusion

In the literature, several HTLV-I and HIV single infections models with spatial dependence have been developed and analyzed. However, modeling HTLV/HIV dual infection with diffusion has not been studied. In this work we derive and investigate a PDE model that describes the dynamics of HTLV/HIV dual infection taking into account the mobility of viruses and cells. The model includes the effect of Cytotoxic T lymphocytes (CTLs) immunity. Although HTLV-I and HIV primarily target the same host, CD4⁺T cells, via infected-to-cell (ITC) contact, however the HIV can also be transmitted through free-to-cell (FTC) contact. Moreover, HTLV-I has a vertical transmission through mitosis of active HTLV-infected cells. The well-posedness of solutions, including the existence of global solutions and the boundedness, is justified. We derive eight threshold parameters which govern the existence and stability of the eight steady states of the model. We study the global stability of all steady states based on the construction of suitable Lyapunov functions and usage of Lyapunov-LaSalle asymptotic stability theorem. Lastly, numerical simulations are carried out in order to verify the validity of our theoretical results.

Stability of an HTLV-HIV coinfection model with multiple delays and CTL-mediated immunity

In the literature, several mathematical models have been formulated and developed to describe the within-host dynamics of either human immunodeficiency virus (HIV) or human T-lymphotropic virus type I (HTLV-I) monoinfections. In this paper, we formulate and analyze a novel within-host dynamics model of HTLV-HIV coinfection taking into consideration the response of cytotoxic T lymphocytes (CTLs). The uninfected CD 4 + T cells can be infected via HIV by two mechanisms, free-to-cell and infected-to-cell. On the other hand, the HTLV-I has two modes for transmission, (i) horizontal, via direct infected-to-cell touch, and (ii) vertical, by mitotic division of active HTLV-infected cells. It is well known that the intracellular time delays play an important role in within-host virus dynamics. In this work, we consider six types of distributed-time delays. We investigate the fundamental properties of solutions. Then, we calculate the steady states of the model in terms of threshold parameters. Moreover, we study the global stability of the steady states by using the Lyapunov method. We conduct numerical simulations to illustrate and support our theoretical results. In addition, we discuss the effect of multiple time delays on stability of the steady states of the system.

Modeling and analysis of a within-host HIV/HTLV-I co-infection

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the CD4 + T cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome (AIDS), while HTLV-I is the causative agent for adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. The present paper is concerned with the formulation and investigation of a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4 + T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell transmission. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Modeling and stability analysis of HIV/HTLV-I co-infection

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that infect the susceptible CD[Formula: see text]T cells. It is known that HIV and HTLV-I have in common a way of transmission through direct contact with certain body fluids related to infected individuals. Therefore, it is not surprising that a mono-infected person with one of these viruses can be co-infected with the other virus. In the literature, a great number of mathematical models has been presented to describe the within-host dynamics of HIV or HTLV-I mono-infection. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In this paper, we develop a new within-host HIV/HTLV-I co-infection model. The model includes the impact of Cytotoxic T lymphocytes (CTLs) immune response, which is important to control the progression of viral co-infection. The model describes the interaction between susceptible CD[Formula: see text]T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. We first show the nonnegativity and boundedness of the model’s solutions and then we calculate all possible equilibria. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle’s invariance principle. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we discuss the effect of HTLV-I infection on the HIV-infected patients and vice versa.

HTLV/HIV Dual Infection: Modeling and Analysis

Human T-lymphotropic virus type I (HTLV-I) and human immunodeficiency virus (HIV) are two famous retroviruses that share similarities in their genomic organization, and differ in their life cycle as well. It is known that HTLV-I and HIV have in common a way of transmission via direct contact with certain body fluids related to infected patients. Thus, it is not surprising that a single-infected person with one of these viruses can be dually infected with the other virus. In the literature, many researchers have devoted significant efforts for modeling and analysis of HTLV or HIV single infection. However, the dynamics of HTLV/HIV dual infection has not been formulated. In the present paper, we formulate an HTLV/HIV dual infection model. The model includes the impact of the Cytotoxic T lymphocyte (CTLs) immune response, which is important to control the dual infection. The model describes the interaction between uninfected CD4+T cells, HIV-infected cells, HTLV-infected cells, free HIV particles, HIV-specific CTLs, and HTLV-specific CTLs. We establish that the solutions of the model are non-negative and bounded. We calculate all steady states of the model and deduce the threshold parameters which determine the existence and stability of the steady states. We prove the global asymptotic stability of all steady states by utilizing the Lyapunov function and Lyapunov–LaSalle asymptotic stability theorem. We solve the system numerically to illustrate the our main results. In addition, we compared between the dynamics of single and dual infections.

Analysis of a within-host HIV/HTLV-I co-infection model with immunity

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the immune cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome, while HTLV-I is the causative agent for adult T-cell leukemia and HTLV-I-associated myelopathy/tropical spastic paraparesis. Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. In the present paper, we are concerned to formulate and analyze a new HIV/HTLV co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4⁺T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by two routes of transmission, virus-to-cell and cell-to-cell. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact, and (ii) vertical transmission through mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle's invariance principle. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

The stationary distribution and extinction of a double thresholds HTLV-I infection model with nonlinear CTL immune response disturbed by white noise

This paper investigates the stochastic HTLV-I infection model with CTL immune response, and the corresponding deterministic model has two basic reproduction numbers. We consider the nonlinear CTL immune response for the interaction between the virus and the CTL immune cells. Firstly, for the theoretical needs of system dynamical behavior, we prove that the stochastic model solution is positive and global. In addition, we obtain the existence of ergodic stationary distribution by stochastic Lyapunov functions. Meanwhile, sufficient condition for the extinction of the stochastic system is acquired. Reasonably, the dynamical behavior of deterministic model is included in our result of stochastic model when the white noise disappears.

Inhibitory killer cell immunoglobulin-like receptors strengthen CD8+ T cell-mediated control of HIV-1, HCV, and HTLV-1

Killer cell immunoglobulin-like receptors (KIRs) are expressed predominantly on natural killer cells, where they play a key role in the regulation of innate immune responses. Recent studies show that inhibitory KIRs can also affect adaptive T cell-mediated immunity. In mice and in human T cells in vitro, inhibitory KIR ligation enhanced CD8+ T cell survival. To investigate the clinical relevance of these observations, we conducted an extensive immunogenetic analysis of multiple independent cohorts of HIV-1-, hepatitis C virus (HCV)-, and human T cell leukemia virus type 1 (HTLV-1)-infected individuals in conjunction with in vitro assays of T cell survival, analysis of ex vivo KIR expression, and mathematical modeling of host-virus dynamics. Our data suggest that functional engagement of inhibitory KIRs enhances the CD8+ T cell response against HIV-1, HCV, and HTLV-1 and is a significant determinant of clinical outcome in all three viral infections.

Modeling brain lentiviral infections during antiretroviral therapy in AIDS

Understanding HIV-1 replication and latency in different reservoirs is an ongoing challenge in the care of patients with HIV/AIDS. A mathematical model was created to describe and predict the viral dynamics of HIV-1 and SIV infection within the brain during effective combination antiretroviral therapy (cART). The mathematical model was formulated based on the biology of lentiviral infection of brain macrophages and used to describe the dynamics of transmission and progression of lentiviral infection in brain. Based on previous reports quantifying total viral DNA levels in brain from HIV-1 and SIV infections, estimates of integrated proviral DNA burden were made, which were used to calibrate the mathematical model predicting viral accrual in brain macrophages from primary infection. The annual rate at which susceptible brain macrophages become HIV-1 infected was estimated to be 2.90×10^-7-4.87×10^-6 per year for cART-treated HIV/AIDS patients without comorbid neurological disorders. The transmission rate for SIV infection among untreated macaques was estimated to be 5.30×10^-6-1.37×10^-5 per year. An improvement in cART effectiveness (1.6-48%) would suppress HIV-1 infection in patients without neurological disorders. Among patients with advanced disease, a substantial improvement in cART effectiveness (70%) would eradicate HIV-1 provirus from the brain within 3-32 (interquartile range 3-9) years in patients without neurological disorders, whereas 4-51 (interquartile range 4-16) years of efficacious cART would be required for HIV/AIDS patients with comorbid neurological disorders. HIV-1 and SIV provirus burdens in the brain increase over time. A moderately efficacious antiretroviral therapy regimen could eradicate HIV-1 infection in the brain that was dependent on brain macrophage lifespan and the presence of neurological comorbidity.

Analysis of a hepatitis B viral infection model with immune response delay

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.

Dynamics of an infection model with two delays

In this paper, a HTLV-I infection model with two delays is considered. It is found that the dynamics of this model are determined by two threshold parameters R0 and R1, basic reproduction numbers for viral infection and for CTL response, respectively. If R0 < 1, the infection-free equilibrium P0 is globally asymptotically stable. If R1 < 1 < R0, the asymptomatic-carrier equilibrium P1 is globally asymptotically stable. If R1 > 1, there exists a unique HAM/TSP equilibrium P2. The stability of P2 is changed when the second delay τ2 varies, that is there exist stability switches for P2.

Construction of Lyapunov functions for some models of infectious diseases in vivo: from simple models to complex models

We present a constructive method for Lyapunov functions for ordinary differential equation models of infectious diseases in vivo. We consider models derived from the Nowak-Bangham models. We construct Lyapunov functions for complex models using those of simpler models. Especially, we construct Lyapunov functions for models with an immune variable from those for models without an immune variable, a Lyapunov functions of a model with absorption effect from that for a model without absorption effect. We make the construction clear for Lyapunov functions proposed previously, and present new results with our method.

GLOBAL PROPERTIES FOR A VIRUS DYNAMICS MODEL WITH LYTIC AND NON-LYTIC IMMUNE RESPONSES, AND NONLINEAR IMMUNE ATTACK RATES

We consider a mathematical model that describes a viral infection with lytic and non-lytic immune responses. One of the main features of the model is that it includes a rate of linear activation of cytotoxic T lymphocytes (CTLs) immune response, a constant production rate of CTLs export from thymus, and a nonlinear attack rate for each immune effector mechanism. Stability of the infection-free equilibrium, and existence, uniqueness and stability of an immune-controlled equilibrium, are investigated. The stability results are given in terms of the basic reproductive number. We use the method of Lyapunov functions to study the global stability of the infection-free equilibrium and the immune-controlled equilibrium. We give a sufficient condition on the non-lytic-immune attack rate for the global asymptotic stability of the immune-controlled equilibrium. By theoretical analysis and numerical simulations, we show that the lytic and non-lytic activities are required to combat a viral infection.

HTLV-I infection: a dynamic struggle between viral persistence and host immunity

Human T-lymphotropic virus type I (HTLV-I) causes chronic infection for which there is no cure or neutralising vaccine. HTLV-I has been clinically linked to the development of adult T-cell leukaemia/lymphoma (ATL), an aggressive blood cancer, and HAM/TSP, a progressive neurological and inflammatory disease. Infected individuals typically mount a large, persistently activated CD8(+) cytotoxic T-lymphocyte (CTL) response against HTLV-I-infected cells, but ultimately fail to effectively eliminate the virus. Moreover, the identification of determinants to disease manifestation has thus far been elusive. A key issue in current HTLV-I research is to better understand the dynamic interaction between persistent infection by HTLV-I and virus-specific host immunity. Recent experimental hypotheses for the persistence of HTLV-I in vivo have led to the development of mathematical models illuminating the balance between proviral latency and activation in the target cell population. We investigate the role of a constantly changing anti-viral immune environment acting in response to the effects of infected T-cell activation and subsequent viral expression. The resulting model is a four-dimensional, non-linear system of ordinary differential equations that describes the dynamic interactions among viral expression, infected target cell activation, and the HTLV-I-specific CTL response. The global dynamics of the model is established through the construction of appropriate Lyapunov functions. Examining the particular roles of viral expression and host immunity during the chronic phase of HTLV-I infection offers important insights regarding the evolution of viral persistence and proposes a hypothesis for pathogenesis.

Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection

The cytotoxic T lymphocyte (CTL) response to the infection of CD4+ T cells by human T cell leukemia virus type I (HTLV-I) has previously been modelled using standard response functions, with relatively simple dynamical outcomes. In this paper, we investigate the consequences of a more general CTL response and show that a sigmoidal response function gives rise to complex behaviours previously unobserved. Multiple equilibria are shown to exist and none of the equilibria is a global attractor during the chronic infection phase. Coexistence of local attractors with their own basin of attractions is the norm. In addition, both stable and unstable periodic oscillations can be created through Hopf bifurcations. We show that transient periodic oscillations occur when a saddle-type periodic solution exists. As a consequence, transient periodic oscillations can be robust and observable. Implications of our findings to the dynamics of CTL response to HTLV-I infections in vivo and pathogenesis of HAM/TSP are discussed.

Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection

Stable periodic oscillations have been shown to exist in mathematical models for the CTL response to HTLV-I infection. These periodic oscillations can be the result of mitosis of infected target CD4(+) cells, of a general form of response function, or of time delays in the CTL response. In this study, we show through a simple mathematical model that time delays in the CTL response process to HTLV-I infection can lead to the coexistence of multiple stable periodic solutions, which differ in amplitude and period, with their own basins of attraction. Our results imply that the dynamic interactions between the CTL immune response and HTLV-I infection are very complex, and that multi-stability in CTL response dynamics can exist in the form of coexisting stable oscillations instead of stable equilibria. Biologically, our findings imply that different routes or initial dosages of the viral infection may lead to quantitatively and qualitatively different outcomes.