A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data

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.     

Sendai virus pneumonia: evidence for the early recruitment of gamma delta T cells during the disease course.

We previously reported that gamma delta T cells appeared and could play a protective role early in infections with intracellular bacteria such as Listeria monocytogenes, Mycobacterium bovis BCG, and Salmonella choleraesuis. To extend these findings to virus infection, we examined the developmental sequence of gamma delta T cells in bronchoalveolar lavage during the course of Sendai virus infection in C57BL/6 mice. To produce a natural but nonlethal infection course as far as possible, we used a sublethal dose of a wild-type virus which had not been subjected to serial passages in a chicken embryo, hence retaining full virulence for mice. Virus titers in lungs reached a peak on day 6 and then decreased to an undetectable level by day 10. This time course of virus reproduction was immediately and coincidentally followed by the developmental course of gamma delta T cells, in which the cell number peaked on day 7 and then decreased to a marginal level by day 10. On the other hand, the alpha beta T-cell number continued to increase until day 10 and remained at a high level thereafter. The early-appearing gamma delta T cells were CD4-, CD8-, IL-2R alpha- beta+, CD44+, Mel-14-, and LFA-1 alpha/beta+ in phenotype and used V gamma 1/2 and V gamma 4 and V delta 3, V delta 4, V delta 5, and V delta 6. The gamma delta T cells were responding to macrophages from infected mice when the cells were cultured in vitro. Furthermore, the expression of endogenous heat shock protein (hsp) was infection specific, and its level appeared to correlate with the gamma delta T-cell development. These results suggest that the early recruitment of gamma delta T cells, which proliferate in response to endogenous hsp+ cells, is also characteristic of this virus infection, although this view appears to be contradictory to earlier reports.     

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.     

Effect of cellular rearrangement time delays on the rheology of vertex models for confluent tissues

Large-scale tissue deformation during biological processes such as morphogenesis requires cellular rearrangements. The simplest rearrangement in confluent cellular monolayers involves neighbor exchanges among four cells, called a T1 transition, in analogy to foams. But unlike foams, cells must execute a sequence of molecular processes, such as endocytosis of adhesion molecules, to complete a T1 transition. Such processes could take a long time compared to other timescales in the tissue. In this work, we incorporate this idea by augmenting vertex models to require a fixed, finite time for T1 transitions, which we call the “T1 delay time”. We study how variations in T1 delay time affect tissue mechanics, by quantifying the relaxation time of tissues in the presence of T1 delays and comparing that to the cell-shape based timescale that characterizes fluidity in the absence of any T1 delays. We show that the molecular-scale T1 delay timescale dominates over the cell shape-scale collective response timescale when the T1 delay time is the larger of the two. We extend this analysis to tissues that become anisotropic under convergent extension, finding similar results. Moreover, we find that increasing the T1 delay time increases the percentage of higher-fold coordinated vertices and rosettes, and decreases the overall number of successful T1s, contributing to a more elastic-like—and less fluid-like—tissue response. Our work suggests that molecular mechanisms that act as a brake on T1 transitions could stiffen global tissue mechanics and enhance rosette formation during morphogenesis.     

Gamma delta T cell-deficient mice have a down-regulated CD8+ T cell immune response against Encephalitozoon cuniculi infection

Gamma(delta) T cells have been reported to play an essential effector role during the early immune response against a wide variety of infectious agents. Recent studies have suggested that the gamma(delta) T cell subtype may also be important for the induction of adaptive immune response against certain microbial pathogens. In the present study, an early increase of gamma(delta) T cells during murine infection with Encephalitozoon cuniculi, an intracellular parasite, was observed. The role of gamma(delta) T cells against E. cuniculi infection was further evaluated by using gene-knockout mice. Mice lacking gamma(delta) T cells were susceptible to E. cuniculi infection at high challenge doses. The reduced resistance of delta(-/-) mice was attributed to a down-regulated CD8+ immune response. Compared with parental wild-type animals, suboptimal Ag-specific CD8+ T cell immunity against E. cuniculi infection was noted in delta(-/-) mice. The splenocytes from infected knockout mice exhibited a lower frequency of Ag-specific CD8+ T cells. Moreover, adoptive transfer of immune TCR(alpha)beta+ CD8+ cells from the delta(-/-) mice failed to protect naive CD8(-/-) mice against a lethal E. cuniculi challenge. Our studies suggest that gamma(delta) T cells, due to their ability to produce cytokines, are important for the optimal priming of CD8+ T cell immunity against E. cuniculi infection. This is the first evidence of a parasitic infection in which down-regulation of CD8+ T cell immune response in the absence of gamma(delta) T cells has been demonstrated.     

Maintenance of internal pH and an electrochemical gradient in Trypanosoma brucei

The internal pH value (pHi) of the long-slender bloodstream form of Trypanosoma brucei was estimated from the distribution of 14C-labeled 5,5-dimethyl-2,4-oxazolidinedione or 14C-labeled methyl amine between the intracellular space of the cells and the medium. The pHi of T. brucei remained relatively constant at 7.0-7.2 throughout an extracellular pH (pHo) range of 6.0-8.0. The maintenance of an internal pH more acidic than the environment appears to be a unique feature. Preincubation of T. brucei with carbonyl cyanide m-chlorophenyl hydrazone (CCCP) or CCCP + valinomycin had no appreciable effect on the delta pH across the T. brucei membrane when the external pH was 8.0. However, when the external pH was 6.0, CCCP abolished the observed delta pH. Nigericin significantly dissipated the delta pH across the T. brucei membrane at all pHo values. These data suggest that under physiological conditions, the maintenance of a delta pH across the bloodstream-form T. brucei membrane may be by a mechanism other than an energy-dependent gradient, whereas an energy-dependent pump may be needed for maintaining the pHi in an acidic environment. The electrical potential (delta psi) across the trypanosomal plasma membrane was also estimated using the lipophilic cation, [3H]tetraphenyl-phosphonium bromide. It appears dependent on both the external pH and the external salt conditions. Under ionic conditions similar to the host bloodstream, it ranges from -76 to -160 mV over an external pH range of 6.0 to 8.0, with an estimated value of -155.5 +/- 0.7 at the physiological pH.(ABSTRACT TRUNCATED AT 250 WORDS)     

Reduction in adhesiveness to extracellular matrix components, modulation of adhesion molecules and in vivo migration of murine macrophages infected with Toxoplasma gondii

Toxoplasma gondii is an obligate intracellular parasite, able to disseminate into deep tissues and cross biological barriers, reaching immunoprivileged sites such as the brain and retina. In order to investigate whether the parasite uses leukocyte trafficking to disseminate throughout the host, the adhesive potential to extracellular matrix components, the expression of adhesion molecules and the in vivo migration of murine macrophages infected with RH strain of T. gondii were investigated. Cellular adhesion to fibronectin, laminin and collagen IV decreased after 24 h of T. gondii infection. However, the decrease in adhesion of infected macrophages observed at early infection was reversed after 48 h. Moreover, decreased adhesion was dependent on active penetration, since heat-killed parasites were unable to reproduce it. Expression of integrins alphaL, alpha4 and alpha5 chains was downmodulated early postinfection, but a progressive regain of expression was observed after 12 h of infection. Expression of beta2, alphav and alpha4 integrins by peritoneal macrophages at late infection was also gradually reestablished. The assessment of in vivo migration of infected macrophages labeled with the fluorescent dye 5-chloromethylfluorescein diacetate showed a 48-h delay in migration to cervical lymph nodes when compared to LPS pre-stimulated macrophages. Furthermore, cells that migrate to distal lymph nodes were loaded with live parasites. Taken together, these results provide insights about T. gondii escape from the host immune response, placing the macrophage as a "Trojan horse", contributing to parasite dissemination and access to immunoprivileged sites.     

Effects of infection-induced migration delays on the epidemiology of avian influenza in wild mallard populations

Wild waterfowl populations form a natural reservoir of Avian Influenza (AI) virus, and fears exist that these birds may contribute to an AI pandemic by spreading the virus along their migratory flyways. Observational studies suggest that individuals infected with AI virus may delay departure from migratory staging sites. Here, we explore the epidemiological dynamics of avian influenza virus in a migrating mallard (Anas platyrhynchos) population with a specific view to understanding the role of infection-induced migration delays on the spread of virus strains of differing transmissibility. We develop a host-pathogen model that combines the transmission dynamics of influenza with the migration, reproduction and mortality of the host bird species. Our modeling predicts that delayed migration of individuals influences both the timing and size of outbreaks of AI virus. We find that (1) delayed migration leads to a lower total number of cases of infection each year than in the absence of migration delay, (2) when the transmission rate of a strain is high, the outbreak starts at the staging sites at which birds arrive in the early part of the fall migration, (3) when the transmission rate is low, infection predominantly occurs later in the season, which is further delayed when there is a migration delay. As such, the rise of more virulent AI strains in waterfowl could lead to a higher prevalence of infection later in the year, which could change the exposure risk for farmed poultry. A sensitivity analysis shows the importance of generation time and loss of immunity for the effect of migration delays. Thus, we demonstrate, in contrast to many current transmission risk models solely using empirical information on bird movements to assess the potential for transmission, that a consideration of infection-induced delays is critical to understanding the dynamics of AI infection along the entire flyway.     

Stability analysis of within-host parasite models with delays

We provide a global analysis of systems of within-host parasitic infections. The systems studied have parallel classes of different length of latently infected target cells. These systems can also be thought as systems arising from within-host parasitic systems with distributed continuous delays. We compute the basic reproduction ratio R0 for the systems under consideration. If R0< or =1 the parasite is cleared, if R0>1 and if a sufficient condition is satisfied we conclude to the global asymptotic stability (GAS) of the endemic equilibrium. For some generic class of models this condition reduces to R0>1. These results make possible to revisit some parasitic models including intracellular delays and to study their global stability.