A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay

 We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases. We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1). Received: 18 November 2000 / Published online: 28 February 2003 RID="*" ID="*" Research was partially supported by the NSERC and MITACS of Canada and a start-up fund from the College of Arts and Sciences at the University of Miami. On leave from Dalhousie University, Halifax, Nova Scotia, Canada. Current address: Department of Mathematics, Clarke College, Dubuque, Iowa 52001, USA Key words or phrases: HIV-1 – Cell-to-cell spread – Time delay – Stability – Hopf bifurcation – Periodicity     

Mean-square stability of a stochastic model for bacteriophage infection with time delays

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.     

Estimating time since infection in early homogeneous HIV-1 samples using a poisson model

Background  

The occurrence of a genetic bottleneck in HIV sexual or mother-to-infant transmission has been well documented. This results in a majority of new infections being homogeneous, i.e., initiated by a single genetic strain. Early after infection, prior to the onset of the host immune response, the viral population grows exponentially. In this simple setting, an approach for estimating evolutionary and demographic parameters based on comparison of diversity measures is a feasible alternative to the existing Bayesian methods (e.g., BEAST), which are instead based on the simulation of genealogies.     

Nonlinear control of HIV-1 infection with a singular perturbation model

Using a singular perturbation approximation, a nonlinear state-space model of HIV-1 infection, having as state variables the number of healthy and infected CD4+T cells and the number of virion particles, is simplified and used to design a control law. The control law comprises an inner block that performs feedback linearizing of the virus dynamics and an outer block implementing an LQ regulator that drives the number of virion particles to a number below the specification. A sensitivity analysis of the resulting law is performed with respect to the model parameter to the infection rate, showing that the controlled system remains stable in the presence of significant changes of this parameter with respect to the nominal value.     

For protection from HIV-1 infection, more might not be better: a systematic analysis of HIV Gag epitopes of two alleles associated with different outcomes of HIV-1 infection

A subset of women in the Pumwani Sex Worker Cohort, established in 1985 in Nairobi, Kenya, remains uninfected despite repeated high-risk exposure (HIV-exposed, seronegative [HESN]) through active sex work. This HESN phenotype is associated with several alleles of human leukocyte antigens (HLAs) and specific CD8(+) and CD4(+) T cell responses to HIV-1. The associations of HLA alleles with differential HIV-1 infection are most likely due to their different abilities to present antigen and the different immune responses they induce. The characteristics of epitopes of HLA alleles associated with different outcomes of HIV-1 infection might therefore point to a vital clue for developing an effective vaccine. In this study, we systematically analyzed HIV-1 clade A and D Gag CD8(+) T cell epitopes of two HLA class I alleles associated with different outcomes of HIV-1 infection. Binding affinity and off-rates of the identified epitopes were determined. Gamma interferon (IFN-γ) enzyme-linked immunospot (ELISpot) assays with patient peripheral blood mononuclear cells (PBMCs) validated the epitopes. Epitope-specific CD8(+) T cells were further phenotyped for memory markers with tetramer staining. Our study showed that the protective allele A*01:01 recognizes only three Gag epitopes. By contrast, B*07:02, the allele associated with susceptibility, binds 30 epitope variants. These two alleles differ most importantly in the spectrum of Gag epitopes they can present and not in affinity, off-rates, the location of the epitopes, or epitope-specific Tem/Tcm frequencies. The binding of more epitopes and strong IFN-gamma ELISpot responses are associated with susceptibility to HIV-1 infection, while more focused antigen recognition of multiple subtypes is protective. Rational vaccine design should take these observations into account.     

In vitro analysis of complement-dependent HIV-1 cell infection using a model system

Previous studies based on the use of human serum as a source of C have provided evidence for the C-dependent enhancement of cell infection by HIV-1. The present study was undertaken to distinguish C from other serum factors and to identify the proteins and the mechanisms involved in C-dependent cell infection by HIV-1. The classical C activation pathway was reconstituted from the proteins C1q, C1r, C1s, C4, C2, C3, factor H, and factor I; each were purified to homogeneity. A mixture of these proteins at physiological concentrations was shown to reproduce the ability of normal human serum to enhance the infection of MT2 cells by HIV-1 at low doses of virus. This enhancing effect was abolished when heat-inactivated serum and C2- or C3-depleted serum were used, and was restored upon addition of the corresponding purified proteins. A mixture of two synthetic peptides corresponding to positions 10-15 and 90-97 of human C receptor type 2 (CD21) as well as soluble CD4 both inhibited the C-dependent infection process. These data provide unambiguous evidence that HIV-1 triggers a direct activation of the classical C pathway in vitro and thereby facilitates the infection of MT2 cells at low doses of virus. These findings are consistent with a mechanism involving increased interaction between the virus opsonized by C3b-derived fragment(s) and the CD21 cell receptors and subsequent virus entry through CD4 receptors.     

An age-structured within-host HIV-1 infection model with virus-to-cell and cell-to-cell transmissions

In this paper, a within-host HIV-1 infection model with virus-to-cell and direct cell-to-cell transmission and explicit age-since-infection structure for infected cells is investigated. It is shown that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number. By analysing the corresponding characteristic equations, the local stability of an infection-free steady state and a chronic-infection steady state of the model is established. By using the persistence theory in infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if the basic reproduction number is less than unity, the infection-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the chronic-infection steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.     

Gender differences in HIV-1 diversity at time of infection

To develop an HIV-1 vaccine with global efficacy, it is important to identify and characterize the viruses that are transmitted, particularly to individuals living in areas of high incidence. Several studies have shown that virus from the blood of acutely infected adults was homogeneous, even when the virus population in the index case was genetically diverse. In contrast to those results with mainly male cohorts in America and Europe, in several cases a heterogeneous virus population has been found early in infection in women in Africa. Thus, we more closely compared the diversity of transmitted HIV-1 in men and women who became infected through heterosexual contact. We found that women from Kenya were often infected by multiple virus variants, whereas men from Kenya were not. Moreover, a heterogeneous virus was present in the women before their seroconversion, and in each woman it was derived from a single index case, indicating that diversity was most likely to be the result of transmission of multiple variants. Our data indicate that there are important differences in the transmitted virus populations in women and men, even when cohorts from the same geographic region who are infected with the same subtypes of HIV-1 are compared.     

Modeling control of foot and mouth disease with two time delays

We develop a delay ordinary differential equation model that captures the effects of prophylactic vaccination, reactive vaccination, prophylactic treatment and reactive culling on the spread of foot and mouth disease (FMD) with time delays. Simulation results from the study suggest that increasing time delay while increasing the control strategies decreases the burden of FMD. Further, the results reveal, that decreasing time delay while decreasing the control strategies increases the burden of FMD. The intermediate scenarios of either (i) increasing time delay while decreasing control or (ii) decreasing time delay while increasing control have intermediate effects on burden reduction. Thus, the implementation of effective control strategies combination can play an important role in mitigating against the FMD burden.     

A mathematical model of two-trophic-level aquatic systems with two complementary nutrients

A model of phytoplankton, zooplankton, and two complementary nutrients, nitrogen and phosphorus, that simulates the dynamics of their interactions in lakes or oceans is presented. The paper studies a mathematical model, based on a modified Michaelis-Menten kinetics, for the phytoplankton uptake of the two nutrients, and a herbivore-density dependent grazing rate. Given the parameters of the system, we answer the basic questions about persistence, extinction, and limiting behavior of the microorganisms. The analytical studies demonstrate the role of the total system nutrients in governing system behavior and in turn in governing water quality control. Features of the model system include behavior determined solely by the total system nutrients, nitrogen and phosphorus. A conclusion robust for the model is the coupling of nitrogen and phosphorus cycles in more realistic dynamics that increase the effectiveness of water quality-control programs.     

Methane flux in Peltandra virginica (Araceae) wetlands: comparison of field data with a mathematical model

We present a mathematical model of the diffusive flux of methane through Peltandra virginica. Data on the diurnal changes in both the petiolar [CH₄] gradient and the values of the radial bulk exchange coefficient, E_r, are entirely consistent with this model and the assertion that changes in stomatal conductance regulate the rate of methane efflux in P. virginica. The differences between the values of E_r calculated for daytime and nighttime conditions are -40% for the submerged condition and -54% for the emergent condition. The axial diffusivity of CH₄ through the petiole of P. virginica is estimated in vitro to be 0.771 cm² min^-1. Using our model, we estimate the equilibrium rate of methane efflux under daytime (97 ng CH₄ min ¹ petiole^-1) and nighttime (65 ng CH₄ min^-1 petiole^-1) emergent conditions. Numerical solutions of the model equations in the time domain offer a way of providing a dynamic model of the gas exchange responses of P. virginica to changing environmental conditions.     

Analysis of an SEIRS epidemic model with two delays

 A disease transmission model of SEIRS type with exponential demographic structure is formulated. All newborns are assumed susceptible, there is a natural death rate constant, and an excess death rate constant for infective individuals. Latent and immune periods are assumed to be constants, and the force of infection is assumed to be of the standard form, namely proportional to I(t)/N(t) where N(t) is the total (variable) population size and I(t) is the size of the infective population. The model consists of a set of integro-differential equations. Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportion equilibrium, are investigated. The stability results are stated in terms of a key threshold parameter. More detailed analyses are given for two cases, the SEIS model (with no immune period), and the SIRS model (with no latent period). Several threshold parameters quantify the two ways that the disease can be controlled, by forcing the number or the proportion of infectives to zero. Received 8 May 1995; received in revised form 7 November 1995     

A mathematical model of tumor growth. III. comparison with experiment

In this paper a model of nonuniform inhibitor production is used to discuss some experimental results obtained by Folkman and Hochberg for multicellular spheroids (of V-79 Chinese hamster lung tissue). Built into the mathematical model is a parameter b which is a measure of the degree of nonuniformity of the inhibitor production rate. The “inverse problem” is solved by finding the value of b necessary to account for results with which the uniform model is incompatible [6]. The resulting value of b enables a good estimate to be made for the observed width of peripheral mitotic zones in the V-79 spheroids, and the resulting stability parameters lie in appropriate ranges for the model to be a significant improvement over uniform models. Further improvements are discussed, including a heuristic model for estimating the destabilizing effect of vascularization on tissue growth.     

A semiparametric model for regression analysis of interval-censored failure time data

Left-, right-, and interval-censored response time data arise in a variety of settings, including the analyses of data from laboratory animal carcinogenicity experiments, clinical trials, and longitudinal studies. For such incomplete data, the usual regression techniques such as the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model are inapplicable. In this paper, we present a method for regression analysis which accommodates interval-censored data. We present applications of this methodology to data sets from a study of breast cancer patients who were followed for cosmetic response to therapy, a small animal tumorigenicity study, and a clinical trial.     

Therapeutic perspectives in HIV-1 infection from recent advances in HIV-1 pathogenesis: it is time to move on

The recent availability of highly active antiretroviral combination therapy (HAART) has significantly influenced the natural history of the infection, delaying the progression to overt AIDS and prolonging survival. At the same time, the increasing knowledge on the pathogenic mechanisms an the use of HAART have challenged the most widely accepted theories about HIV-1 disease, in particular about the feasibility to eradicate the virus. This review will outline what is and what is not achievable by HAART, and will discuss new concepts in the immunopathogenesis of HIV-1 infection that provide the rationale for the design of new therapeutic approaches in the management of HIV-1 disease and AIDS.     

Global stability of an SIR epidemic model with time delays

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.Research partly supported by the Ministry of Education, Science and Culture, Japan, Grant 05640256     

Virological responses in a patient with recent HIV-1 infection experiencing an EBV reactivation

The dynamics of interactions between HIV and other viral agents and their reciprocal influence on the cellular immune response is not fully understood. A clinical report is here described regarding an EBV reactivation occurring during a recent HIV infection. The two viruses appear to act in a sequential manner, mutually influencing each other in their replication and leading to determine a clinical outcome in the patient under study.