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.     

Multistability in a Model for CTL Response to HTLV-I Infection and Its Implications to HAM/TSP Development and Prevention

Human T-cell leukaemia/lymphoma virus type I (HTLV-I) is a retrovirus that has been identified as the causative agent of HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP) and other illnesses. HTLV-I infects primarily CD4⁺ T cells and the transmission occurs through direct cell-to-cell contact. HAM/TSP patients harbor higher proviral loads in peripheral blood lymphocytes than asymptomatic carriers. Also, HAM/TSP patients exhibit a remarkably high number of circulating HTLV-I-specific CD8⁺ cytotoxic T lymphocytes (CTLs) in the peripheral blood. While CTLs have a protective role by killing the infected cells and lowering the proviral load, a high level of CTLs and their cytotoxicity are believed to be a main cause of the development of HAM/TSP. A mathematical model for HTLV-I infection of CD4⁺ T cells that incorporates the CD8⁺ cytotoxic T-cell (CTL) response is investigated. Our mathematical analysis reveals that the system can stabilize at a carrier steady-state with persistent viral infection but no CTL response, or at a HAM/TSP steady-state at which both the viral infection and CTL response are persistent. We also establish two threshold parameters R ₀ and R ₁, the basic reproduction numbers for viral persistence and for CTL response, respectively. We show that the parameter R ₁ can be used to distinguish asymptomatic carriers from HAM/TSP patients, and as an important control parameter for preventing the development of HAM/TSP.     

Genetic control and dynamics of the cellular immune response to the human T-cell leukaemia virus, HTLV-I.

About 1% of people infected with the human T-cell leukaemia virus, type 1 (HTLV-I) develop a disabling chronic inflammatory disease of the central nervous system known as HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Patients with HAM/TSP have a vigorous immune response to HTLV-I, and it has been widely suggested that this immune response, particularly the HTLV-I-specific cytotoxic T-lymphocyte (CTL) response, causes the tissue damage that is seen in HAM/TSP. In this paper we summarize recent evidence that a strong CTL response to HTLV-I does in fact protect against HAM/TSP by reducing the proviral load of HTLV-I. We conclude that HTLV-I is persistently replicating at a high level, despite the relative constancy of its genome sequence. These results imply that antiretroviral drugs could reduce the risk of HAM/TSP by reducing the viral load, and that an effective anti-HTLV-I vaccine should elicit a strong CTL response to the virus. The dynamic nature of the infection also has implications for the epidemiology and the evolution of HTLV-I.     

Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models

The dynamics of HIV-1 infection consist of three distinct phases starting with primary infection, then latency and finally AIDS or drug therapy. In this paper we model the dynamics of primary infection and the beginning of latency. We show that allowing for time delays in the model better predicts viral load data when compared to models with no time delays. We also find that our model of primary infection predicts the turnover rates for productively infected T cells and viral totals to be much longer than compared to data from patients receiving anti-viral drug therapy. Hence the dynamics of the infection can change dramatically from one stage to the next. However, we also show that with the data available the results are highly sensitive to the chosen model. We compare the results using analysis and Monte Carlo techniques for three different models and show how each predicts rather dramatic differences between the fitted parameters. We show, using a chi(2) test, that these differences between models are statistically significant and using a jackknifing method, we find the confidence intervals for the parameters. These differences in parameter estimations lead to widely varying conclusions about HIV pathogenesis. For instance, we find in our model with time delays the existence of a Hopf bifurcation that leads to sustained oscillations and that these oscillations could simulate the rapid turnover between viral strains and the appropriate CTL response necessary to control the virus, similar to that of a predator-prey type system.     

Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model

Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.     

The cytotoxic T-lymphocyte response to HTLV-I: the main determinant of disease?

There is a powerful, chronically activated cytotoxic T-lymphocyte (CTL) response to the Tax protein of human T-cell leukaemia virus type I (HTLV-I) in most people infected with the virus. The CTL select variant sequences of Tax which escape immune recognition and interfere with recognition of the wild-type protein. This positive selection process is more efficient in healthy HTLV-I carriers than in patients with tropical spastic paraparesis, an inflammatory neurological disease associated with HTLV-I. The mean virus load is more than 10-fold greater in patients with this neurological disease than in healthy carriers of HTLV-I. We conclude that anti-Tax CTL play an important part in limiting the rate of replication of HTLV-I. We suggest that the outcome of infection with HTLV-I is primarily determined by the CTL response of the individual: low CTL responders to HTLV-I develop a high virus load, resulting in widespread chronic activation of T cells. The activated T cells then invade the tissues and cause bystander tissue damage, probably by releasing cytokines and other soluble substances. An efficient CTL response to HTLV-I limits the equilibrium virus load, and so reduces the chance of developing inflammatory disease.     

Induction of CD4+, human T lymphotropic virus type-1-specific cytotoxic T lymphocytes from patients with HAM/TSP. Recognition of an immunogenic region of the gp46 envelope glycoprotein of human T lymphotropic virus type-1.

Although the humoral response to human T lymphotropic virus type-1 (HTLV-I) has been well characterized in patients with HTLV-I-associated neurologic disease (HAM/TSP), little is known about a functional HTLV-I-specific human T cell response, such as CTL, in these patients. To define both the phenotype of the responding CTL and the fine specificity of this response, long term T cell lines were generated from two HAM/TSP patients who were from two different countries. Patient's peripheral blood lymphocytes were repeatedly stimulated in vitro with an HTLV-I expressing autologous T cell line. The resultant long term T cell culture was shown to be CD4+ and cytotoxic for targets expressing HTLV-I Ag. Using a panel of synthetic peptides that span hydrophilic regions of the HTLV-I gp46 envelope glycoprotein, the CTL lines generated from both patients were shown to recognize the same region of the HTLV-I envelope between amino acids 196-209 as defined by the synthetic peptide sp4a1. Interestingly, this sequence overlaps a region of HTLV-I envelope that had also been shown to elicit a strong B cell response in HAM/TSP patients (amino acids 190-203). One CTL line recognized this HTLV-I epitope in the context of HLA DQ5 whereas the other CTL line was restricted by HLA DRw16. The generation of two independent CTL lines from two HAM/TSP patients from different geographic areas that recognize the same region of the HTLV-I envelope glycoprotein highlights the immunogenic nature of this envelope region.     

一类具有CTL免疫和时滞的HTLV-I传染病模型的动力学性质(英文)

主要研究了一类具有CTL免疫和时滞的HTLV-I传染的数学模型.通过构造Lyapunov泛函,分别证明了当R₀≤1,R₁≤10,R1>1时,系统（1.1）的无病平衡点E₀,无免疫平衡点E₁及地方病平衡点E₂是全局吸引的.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Backward bifurcation in a model for HTLV-I infection of CD4+ T cells

Human T-cell Lymphotropic Virus Type I (HTLV-I) primarily infects CD4⁺ helper T cells. HTLV-I infection is clinically linked to the development of Adult T-cell Leukemia/Lymphoma and of HTLV-I Associated Myelopathy/Tropical Spastic Paraparesis, among other illnesses. HTLV-I transmission can be either horizontal through cell-to-cell contact, or vertical through mitotic division of infected CD4⁺ T cells. It has been observed that HTLV-I infection has a high proviral load but a low rate of proviral genetic variation. This suggests that vertical transmission through mitotic division of infected cells may play an important role.We consider and analyze a mathematical model for HTLV-I infection of CD4⁺ T cells that incorporates both horizontal and vertical transmission. Among interesting dynamical behaviors of the model is a backward bifurcation which raises many new challenges to effective infection control.     

Time-delayed model of RNA interference

RNA interference (RNAi) is a fundamental cellular process that inhibits gene expression through cleavage and destruction of target mRNA. It is responsible for a number of important intracellular functions, from being the first line of immune defence against pathogens to regulating development and morphogenesis. In this paper we consider a mathematical model of RNAi with particular emphasis on time delays associated with two aspects of primed amplification: binding of siRNA to aberrant RNA, and binding of siRNA to mRNA, both of which result in the expanded production of dsRNA responsible for RNA silencing. Analytical and numerical stability analyses are performed to identify regions of stability of different steady states and to determine conditions on parameters that lead to instability. Our results suggest that while the original model without time delays exhibits a bi-stability due to the presence of a hysteresis loop, under the influence of time delays, one of the two steady states with the high (default) or small (silenced) concentration of mRNA can actually lose its stability via a Hopf bifurcation. This leads to the co-existence of a stable steady state and a stable periodic orbit, which has a profound effect on the dynamics of the system.     

MULTIPLE HIV-1 INFECTION OF CELLS AND THE EVOLUTIONARY DYNAMICS OF CYTOTOXIC T LYMPHOCYTE ESCAPE MUTANTS

Cytotoxic T lymphocytes (CTL) are an important branch of the immune system, killing virus-infected cells. Many viruses can mutate so that infected cells are not killed by CTL anymore. This escape can contribute to virus persistence and disease. A prominent example is HIV-1. The evolutionary dynamics of CTL escape mutants in vivo have been studied experimentally and mathematically, assuming that a cell can only be infected with one HIV particle at a time. However, according to data, multiple virus particles frequently infect the same cell, a process called coinfection. Here, we study the evolutionary dynamics of CTL escape mutants in the context of coinfection. A mathematical model suggests that an intermediate strength of the CTL response against the wild-type is most detrimental for an escape mutant, minimizing overall virus load and even leading to its extinction. A weaker or, paradoxically, stronger CTL response against the wild-type both lead to the persistence of the escape mutant and higher virus load. It is hypothesized that an intermediate strength of the CTL response, and thus the suboptimal virus suppression observed in HIV-1 infection, might be adaptive to minimize the impact of existing CTL escape mutants on overall virus load.     

Modeling the T-cell dynamics and pathogenesis of HTLV-I infection

Human T-cell lymphotropic virus type I (HTLV-I) infection in humans causes a chronic infection of CD4⁺ T cells, and is associated with various disease outcomes, among them with the development of adult T-cell leukemia (ATL). The T-cell dynamics after HTLV-I infection can be described in a mathematical model with coupled differential equations. The infection process is modeled assuming cell-to-cell infection of CD4⁺ T cells. The model allows for CD4⁺ T cell subsets of susceptible, latently infected and actively infected cells as well as for leukemia cells. Latently infected T cells may harbor the virus for several years until they become activated and able to infect susceptible T cells. Uncontrolled proliferation of CD4⁺ T cells with monoclonal DNA-integration of HTLV-I results in the development of ATL. The model describes basic features that characterize HTLV-I infection; the chronic infection of CD4⁺ T cells, the increasing number of abnormal cells and the possible progression to ATL.     

Human T-cell leukemia virus type I infection of CD4+ or CD8+ cytotoxic T-cell clones results in immortalization with retention of antigen specificity.

The human T-cell leukemia virus type I (HTLV-I) is capable of chronically infecting various types of T cells and nonlymphoid cells. The effects of chronic infection on the specific functional activities and growth requirements of mature cytotoxic T lymphocytes (CTL) have remained poorly defined. We have, therefore, investigated the results of HTLV-I infection of both CD4+ and CD8+ human CTL clones. HTLV-I infection resulted in the establishment of functional CTL lines which propagated indefinitely in culture many months longer than the uninfected parental clone. The infected cells became independent of the need for antigen (target cell) stimulation as a requirement for proliferation and growth. Like their uninfected counterparts, however, these HTLV-I-infected clones remained strictly dependent on conditioned medium from mitogen-stimulated T lymphocytes for their growth. This growth factor requirement was not fulfilled by recombinant interleukin-2 alone. Furthermore, the infected lines remained functionally identical to their uninfected parental CTL clones in their ability to specifically recognize and lyse the appropriate target cells. Our findings indicate that the major effects of HTLV-I infection on mature CTL consist of (i) the capacity for proliferation in the absence of antigen stimulation and (ii) a prolonged or immortal survival in vitro, but they also indicate that the fine specificity and cytolytic capacity of these cells remain unaffected.     

Genetic stability of human T lymphotropic virus type I despite antiviral pressures by CTLs

Human T lymphotropic virus type I (HTLV-I)-associated myelopathy/tropical spastic paraparesis (HAM/TSP) is an inflammatory neurological disease. Patients with HAM/TSP show high proviral load despite increased HTLV-I Tax-specific CTL. It is still unknown whether the CTL efficiently eliminate the virus in vivo and/or whether a naturally occurring variant virus becomes predominant by escaping from the CTL. To address these issues, we sequenced a large number of HTLV-I tax genes from HLA-A*02 HAM/TSP patients and estimated synonymous and nonsynonymous changes of the genes to detect positive selection pressure on the virus. We found the pressures in three of six CTL epitopes in HTLV-I Tax, where amino acid substitutions preferentially occurred. Although some of variant viruses were not recognized by the CTL, no variant viruses accumulated within 3-8 years, indicating genetic stability of HTLV-I tax gene. These results suggest that CTL eliminate the infected cells in vivo and naturally occurring variant viruses do not predominate. As Tax is a regulatory protein which controls viral replication, the amino acid substitutions in Tax may reduce viral fitness for replication. Viral fitness and host immune response may contribute to the viral evolution within the infected individuals. Furthermore, the genetic stability in the epitopes despite the antiviral pressures suggests that the three epitopes can be the candidate targets for HTLV-I vaccine development.     

Periodic orbits arising from Hopf bifurcations in a Volterra prey-predator model

 In this paper we derive a formula which enables the stability of periodic solutions to a Volterra integro-differential system to be determined. This system which has been studied by Cushing [1], models a predator-prey interaction with distributed delays. The results are obtained by using the algorithm developed by Kazarinoff, Wan and van den Driessche [2] based on the centre manifold formulas of Hassard and Wan [3]. We discuss an example of the formula for the case of weak kernels and show that under certain conditions stable periodic solutions arising from Hopf bifurcations at different critical values of the parameters can exist together. Received 30 December 1994; received in revised form 12 December 1995     

An altered peptide ligand antagonizes antigen-specific T cells of patients with human T lymphotropic virus type I-associated neurological disease

Human T lymphotropic virus type I (HTLV-I)-associated myelopathy/tropical spastic paraparesis (HAM/TSP) is an inflammatory neurologic disease associated with HTLV-I infection, in which chronically activated, HTLV-I-specific CD8+ CTL have been suggested to be immunopathogenic. In HLA-A2 HAM/TSP patients, CD8+ HTLV-I-specific CTLs recognize an immunodominant peptide of the HTLV-I Tax protein, Tax11-19. We examined the functional outcome on activation of both cloned peripheral blood and cerebrospinal spinal fluid-derived CTL and bulk PBMC from HAM/TSP patients by altered peptide ligands (APL) derived from HTLV-I Tax11-19. In CTL clones generated from PBMC and CSF of HLA-A2 HAM/TSP patients, an APL substituted at position 5 significantly decreased CTL responses when compared with the native peptide. Moreover, these ligands were also shown to inhibit CTL responses to the native peptide in bulk PBMC of HLA-A2 HAM/TSP patients. These data suggest that a modification of an antigenic peptide at the central position can manipulate the T cell responses in bulk PBMC from different individuals with an inflammatory disease. Additionally, these results have implications for the potential use of APL-based immunotherapy in this T cell-mediated CNS disease.     

Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R ₀ for the viral infection, and establish that the global dynamics are completely determined by the values of R ₀. If R ₀≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R ₀>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R ₀>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.     

Oscillatory phenomena in a model of infectious diseases

Oscillations of the number of cases around an average endemic level are common in several infectious diseases. In this paper we study simple deterministic models, where the oscillations arise either solely from periodically varying contact rates or from the combined effect of large initial perturbation, small periodic variation of the contact rate, and the destabilizing nature of infectious and latent periods when described as time delays. The main results are: (a) For a model with a periodically varying contact rate and a recovery rate, a threshold amplitude of variation is found by numerical and analytic methods at which 2-year subharmonic resonance appears. (b) Approximate analytic relationships are derived for the amplitude and phase of the forced 1-year oscillations below this threshold and for the 2-year oscillations above it—in terms of the reproduction rate of the infection. (c) Similar calculations are performed when the recovery rate is replaced by a fixed infectious period represented by a pure time delay. The threshold amplitude of variation in the contact rate is found here to be smaller than in the recovery rate model. (d) A model with a fixed infectious period and a constant contact rate is considered. The nontrivial steady state is shown to be locally stable for the parameter range of interest. However, the ratio of the imaginary to real parts of the eigenvalues in the characteristic equation is increased as compared to the corresponding model with a recovery rate. (e) For the model with a fixed infectious period and a constant contact rate an approximation method indicates consistency in a certain range of contact rates with the existence of an unstable periodic solution about the locally stable steady state. The actual existence of such a solution is not verified. The interpretation is that the destabilizing effect of the introduction of a pure delay into the model becomes more significant as the distance in the variables space from the endemic steady state is increased. (f) For a fixed infectious period and very small subthreshold variation in the contact rate, two different types of solutions are found numerically: yearly small-amplitude oscillations about an endemic average and large-amplitude oscillations of a subharmonic period. The pattern seen depends on the initial conditions. For a sufficiently large initial deviation from the endemic level even very small seasonal variations lead to regular recurrent outbreaks of the disease. The effect of latent periods and of changing the form of the interaction are also considered.