[Mathematical Modeling of HIV Replicaton and the Response of the Interferon System].

Developing physiologically meaningful mathematical models that describe multilevel regulation in a complex network of immune processes, in particular, of the system of interferon-regulated virus production processes, is a fundamental scientific problem, within the framework of an interdisciplinary systems approach to research in immunology. Here, we have presented a detailed high-dimensional model describing HIV (human immunodeficiency virus) replication, the response of type I interferon (IFN) to the virus infection of the cell, and suppression of the action of IFN-induced proteins by HIV accessory proteins. As a result, this model includes interactions of all three processes for the first time. The mathematical model is a system of 37 nonlinear ordinary differential equations including 78 parameters. Importantly, the model describes not only the processes of the IFN response of the cell to virus infection, but also the mechanisms used by the virus to prevent effects of the IFN system.

[Screening of patients with cerebral aneurysms: mathematical analysis and economic justification]. / Skrining patsientov s anevrizmami golovnogo mozga: matematicheskii analiz i ekonomicheskoe obosnovanie.

Subarachnoid hemorrhages due to rupture of cerebral aneurysms have a high risk of disability and mortality. Screening of the population to detect aneurysms in patients with risk factors is currently not carried out in Russia. However, the detection of clinically silent aneurysms and their subsequent prophylactic surgical treatment are justified, according to numerous studies. BACKGROUND: Demonstrate the clinical and economic feasibility of screening the population (including first-line relatives) for cerebral aneurysms using an economic and mathematical model of the RF virtual population. MATERIAL AND METHODS: Mathematical modeling was carried out using an algorithm that implements a discrete Markov chain. The virtual population consisted of 145 million people (the population of the Russian Federation). Magnetic resonance angiography 3DTOF was chosen as a screening method. Virtual patients underwent preventive surgical treatment in case of detection of aneurysm during screening. The number of aneurysms in the population, the number of aneurysmal subarachnoid hemorrhage (aSAH), the cost and outcomes of treatment, and the risk of disability were calculated. RESULTS: In the case of screening and preventive surgical treatment of aneurysms, there is a decrease in the number of aSAH by 14.3% (37.5% in first-line relatives (RPLR), which affects the reduction in mortality due to aSAH by 14.4% (24.1% in The total number of disabled people is reduced by 1.5% (5.1% for the RPHR). A shift in the structure of disability towards greater labor and social adaptation of patients was noted. An economic analysis for the entire population showed that screening saves 7.7 billion annually rubles, including in the population consisting of RPLR - 4.9 billion rubles. CONCLUSION: The created mathematical model of the virtual population demonstrated that screening and prophylactic treatment of cerebral aneurysms makes it possible to reduce the number of aSAH and associated mortality among the entire population and in the RPLR group. The number of individuals with severe disabilities is decreasing. Thus, population screening for the detection of cerebral aneurysms may be clinically effective and cost-effective in the general population, especially in RPCR.

Bifurcation analysis of multistability and hysteresis in a model of HIV infection.

The infectious disease caused by human immunodeficiency virus type 1 (HIV-1) remains a serious threat to hu- man health. The current approach to HIV-1 treatment is based on the use of highly active antiretroviral therapy, which has side effects and is costly. For clinical practice, it is highly important to create functional cures that can enhance immune control of viral growth and infection of target cells with a subsequent reduction in viral load and restoration of the immune status. HIV-1 control efforts with reliance on immunotherapy remain at a conceptual stage due to the complexity of a set of processes that regulate the dynamics of infection and immune response. For this reason, it is extremely important to use methods of mathematical modeling of HIV-1 infection dynamics for theoretical analysis of possibilities of reducing the viral load by affecting the immune system without the usage of antiviral therapy. The aim of our study is to examine the existence of bi-, multistability and hysteresis properties with a meaningful mathematical model of HIV-1 infection. The model describes the most important blocks of the processes of interaction between viruses and the human body, namely, the spread of infection in productively and latently infected cells, the appearance of viral mutants and the develop- ment of the T cell immune response. Furthermore, our analysis aims to study the possibilities of transferring the clinical pattern of the disease from a more severe state to a milder one. We analyze numerically the conditions for the existence of steady states of the mathematical model of HIV-1 infection for the numerical values of model parameters correspond- ing to phenotypically different variants of the infectious disease course. To this end, original computational methods of bifurcation analysis of mathematical models formulated with systems of ordinary differential equations and delay differ- ential equations are used. The macrophage activation rate constant is considered as a bifurcation parameter. The regions in the model parameter space, in particular, for the rate of activation of innate immune cells (macrophages), in which the properties of bi-, multistability and hysteresis are expressed, have been identified, and the features characterizing transi- tion kinetics between stable equilibrium states have been explored. Overall, the results of bifurcation analysis of the HIV-1 infection model form a theoretical basis for the development of combination immune-based therapeutic approaches to HIV-1 treatment. In particular, the results of the study of the HIV-1 infection model for parameter sets corresponding to different phenotypes of disease dynamics (typical, long-term non-progressing and rapidly progressing courses) indicate that an effective functional treatment (cure) of HIV-1-infected patients requires the development of a personalized ap- proach that takes into account both the properties of the HIV-1 quasispecies population and the patient's immune status.

[Mathematical Modeling of the Intracellular Regulation of Immune Processes].

The modern era of research in immunology is characterized by an unprecedented level of detail about structural characteristics of the immune system and the regulation of activities of its numerous components, which function together as a whole distributed-parameter system. Mathematical modeling provides an analytical tool to describe, analyze, and predict the dynamics of immune responses by applying a reductionist approach. In modern systems immunology and mathematical immunology as a new interdisciplinary field, a great challenge is to formulate the mathematical models of the human immune system that reflect the level achieved in understanding its structure and describe the processes that sustain its function. To this end, a systematic development of multiscale mathematical models has to be advanced. An appropriate methodology should consider (1) the intracellular processes of immune cell fate regulation, (2) the population dynamics of immune cells in various organs, and (3) systemic immunophysiological processes in the whole host organism. Main studies aimed at modeling the intracellular regulatory networks are reviewed in the context of multiscale mathematical modelling. The processes considered determine the regulation of the immune cell fate, including activation, division, differentiation, apoptosis, and migration. Because of the complexity and high dimensionality of the regulatory networks, identifying the parsimonious descriptions of signaling pathways and regulatory loops is a pressing problem of modern mathematical immunology.

Interplay between reaction and diffusion processes in governing the dynamics of virus infections.

Spreading of viral infection in the tissues such as lymph nodes or spleen depends on virus multiplication in the host cells, their transport and on the immune response. Reaction-diffusion systems of equations with delays in cell proliferation and death by apoptosis represent an appropriate model to study this process. The properties of the cells of the immune system and the initial viral load determine the spatiotemporal regimes of infection spreading. Infection can be completely eliminated or it can persist at some level together with a certain chronic immune response in a spatially uniform or oscillatory mode. Finally, the immune cells can be completely exhausted leading to a high viral load persistence in the tissue. It has been found experimentally, that virus proteins can affect the immune cell migration. Our study shows that both the motility of immune cells and the virus infection spreading represented by the diffusion rate coefficients are relevant control parameters determining the fate of virus-host interaction.

Pathogenesis and treatment of HIV infection: the cellular, the immune system and the neuroendocrine systems perspective.

Infections with HIV represent a great challenge for the development of strategies for an effective cure. The spectrum of diseases associated with HIV ranges from opportunistic infections and cancers to systemic physiological disorders like encephalopathy and neurocognitive impairment. A major progress in controlling HIV infection has been achieved by highly active antiretroviral therapy (HAART). However, HAART does neither eliminate the virus reservoirs in form of latently infected cells nor does it completely reconstitute immune reactivity and physiological status. Furthermore, the failure of the STEP vaccine trial and the only marginal efficacies of the RV144 trial together suggest that the causal relationships between the complex sets of viral and immunological processes that contribute to protection or disease pathogenesis are still poorly understood. Here, we provide an up-to-date overview of HIV-host interactions at the cellular, the immune system and the neuroendocrine systems level. Only by integrating this multi-level knowledge one will be able to handle the systems complexity and develop new methodologies of analysis and prediction for a functional restoration of the immune system and the health of the infected host.

[Parvovirus B-19 infections in pregnant women in day care facilities: health economic analysis of prohibition to employ seronegative women]. / Parvovirus-B19-Infektionen bei Schwangeren in der Kinderbetreuung. Gesundheitsökonomische Analyse eines Beschäftigungsverbots.

German protective legislation during pregnancy and maternity prohibit employing pregnant women if occupational activities endanger the health of either the mother-to-be or the fetus. This applies for parvovirus B19 seronegative women caring for children <6 years. Here we present a cost-effectiveness analysis from the view of the society for the prohibition to employ B19-seronegative women in day care. Prohibition of employment starting at the first day of pregnancy may prevent 1.4 cases of fetal death (mortality) and 1.7 cases of hydrops fetalis (morbidity) per year resulting in costs of 30 million (22 million /live birth). The incidence of B19 infection, the elevated occupational risk and the fetal death rate were varied in sensitivity analyses. This resulted in 0.2-3.1 fetal deaths prevented per year and costs between 10 million and 150 million per live birth. Indeed, the protective effect is assumed to be even lower since 30% of fetal deaths occur after infection during the first 8 weeks of pregnancy. During that time prohibition of employment is often unrealistic since the majority of women are not aware of pregnancy. In conclusion a small number of fetal lives can be saved by prohibiting employment in contrast to the extremely high costs. The regulations for maternal protection should be revised.

Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis.

Many important viruses persist at very low levels in the body in the face of host immunity, and may influence the maintenance of this state of 'infection immunity'. To analyse low level viral persistence in quantitative terms, we use a mathematical model of antiviral cytotoxic T lymphocyte (CTL) response to lymphocytic choriomeningitis virus (LCMV). This model, described by a non-linear system of delay differential equations (DDEs), is studied using numerical bifurcation analysis techniques for DDEs. Domains where low level LCMV coexistence with CTL memory is possible, either as an equilibrium state or an oscillatory pattern, are identified in spaces of the model parameters characterising the interaction between virus and CTL populations. Our analysis suggests that the coexistence of replication competent virus below the conventional detection limit (of about 100 pfu per spleen) in the immune host as an equilibrium state requires the per day relative growth rate of the virus population to decrease at least 5-fold compared to the acute phase of infection. Oscillatory patterns in the dynamics of persisting LCMV and CTL memory, with virus population varying between 1 and 100 pfu per spleen, are possible within quite narrow intervals of the rates of virus growth and precursor CTL population death. Whereas the virus replication rate appears to determine the stability of the low level virus persistence, it does not affect the steady-state level of the viral population, except for very low values.

Predicting the dynamics of antiviral cytotoxic T-cell memory in response to different stimuli: cell population structure and protective function.

This paper examines the numerical and functional consequences of various stimuli on antiviral CD8+ T-cell memory using a mathematical model. The model is based upon biological evidence from the murine model of infection with lymphocytic choriomeningitis virus (LCMV) that the phenotype of immunological memory represents low-level responses driven by various stimuli, and the memory CTL population is partitioned between resting, cycling and effector cells. These subpopulations differ in their lifespan, their potential to mediate antiviral protection and in the stimuli needed for their maintenance. Three types of maintenance stimuli are examined: non-antigen-specific (bystander) stimulation, persisting antigen stimulation and reinfection-mediated stimulation. The modelling predicts that: (i) stable persistence of CTL memory requires the presence of either bystander or antigen-specific stimulation above a certain threshold depending on the sensitivity of memory CTL to stimulation and their life-span; (ii) a relatively low level of stimuli (approximately 10(4) fold less on a per CTL basis compared to acute infection) is needed to stabilize the expanded memory CTL population; (iii) the presence of CTL subsets in the memory pool of different activation states and lifespans ensures the robustness of memory persistence in the face of temporal variation in the low-level stimuli and; (iv) an 'optimal' population structure of the memory CTL pool, in terms of immediate protection, requires the presence of both activated cycling and effector CTL. For this, persisting antigen alone or synergistically with bystander signals provide the appropriate stimulation, so that the stimuli equivalent to approximately 30 p.f.u. of LCMV in the spleen are sufficient to maintain approximately 10(5)-10(6) specific CTL in the memory pool. These observations are relevant both to our understanding of natural protective immunity and to vaccine design.

Direct quantitation of rapid elimination of viral antigen-positive lymphocytes by antiviral CD8(+) T cells in vivo.

Lysis of infected cells by CD8(+) T cells is an important mechanism for the control of virus infections, but remains difficult to quantify in vivo. Here, we study the elimination kinetics of viral antigen-positive lymphocytes by antiviral CD8(+) T cells using flow cytometry and mathematical analysis. In mice acutely infected with lymphocytic choriomeningitis virus, more than 99.99 % of target cells were eliminated each day, corresponding to a half-life of 1.4 h. Even in mice exposed to virus 300 days previously, and with no ex vivo killing activity, 84 % of the target cells were eliminated per day. Unexpectedly, the elimination kinetics of antigen-positive lymphocytes was not significantly impaired in mice deficient in either perforin-, CD95 ligand- or TNF-mediated cytotoxicity. For viruses with a particular tropism for lymphocytes, such as Epstein-Barr virus or HIV, our results illustrate how effectively CD8(+) T cell-mediated elimination of target cells can potentially contribute to virus control and immunosuppression.

Modelling and analysis of time-lags in some basic patterns of cell proliferation.

In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population.

The impact of variation in the number of CD8(+) T-cell precursors on the outcome of virus infection.

We investigated the role of varying the initial number of naive antiviral CTL precursors on the dynamics of LCMV-DOCILE infection. C57BL/6 mice, exhibiting LCMV-specific CTLp frequencies of about 50, are protected against virus persistence over a range of infectious doses up to 10(4) pfu. With 10-fold higher doses, a 100-fold increase in CTLp is required to restore virus control. With doses above 10(6) pfu, elevation of the initial CTLp number leads only to lethal immunopathology. Similarly, a 1000-fold increase in the number of initial naïve CTLp enhances the overall kinetics of virus elimination, but cannot limit early virus spread within the first 48 h after low-dose infection (500 pfu). Increases in initial naïve virus-specific CTLp numbers are of limited benefit in antiviral control. In addition to the number of virus-specific T cells, the time period needed to reach cytolytic effector function is a limiting parameter.

Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses.

Lymphocytic choriomeningitis virus (LCMV) infection in mice provides an example of an extraordinarily dynamic process with an extreme sensitivity of phenotype of infection to parameters of virus/host interaction. A mathematical model is developed to examine the dynamics of virus-specific cytotoxic T lymphocyte (CTL) response for LCMV infection in mice. The model, formulated by a system of nonlinear delay-differential equations, considers the interacting populations of viruses, precursor CTLs, terminally differentiated effector CTLs and total virus antigen load. Clonal elimination of virus-specific cytotoxic T cells in high-dose LCMV-Docile infection represents an example of the classical phenomenon--high zone tolerance. To describe both conventional and exhaustive CTL responses in the acute phase of LCMV-D infection two mechanisms are invoked: the high virus antigen load inhibition of T-cells proliferation via energy induction and the activation-induced cell death by apoptosis. Parameters of the model, characterizing the rates of virus and CTL production and elimination in spleen, are estimated by assimilating with the model data on the LCMV-D infection in C57BL/6 mice for low-, moderate- and high-dose infections. It is suggested that not only the clonal expansions have to be described in mathematical models as being virus regulated but also the later phases of primary immune response. Down-regulation of the primary CTL response is controlled by a network of mechanisms inducing anergy and apoptosis in activated T cells. The model is used to investigate the effect of variations in virus and CTL response parameters on LCMV infection outcome and suggest predictions for experimental studies, in particular the phenotype of LCMV-WE infection in C57BL/6 as a function of initial virus doses.

Mathematical model of antiviral immune response. III. Influenza A virus infection.

We present an approach to studying theoretically the regularities and the kinetic characteristics of influenza A virus (IAV) infection in man. The estimates of the "numbers" (Zinkernagel et al., 1985) characterizing evolutionary established interferon and immune responses in uncomplicated IAV infection are explored by developing a multiparameter mathematical model which allows direct quantitative references to the biological reality. The system of equations of the mathematical model of antiviral immune response, applied earlier to acute hepatitis B virus infection (Marchuk et al., 1991a, b), is modified and extended to describe the joint reaction of the interferon and immune systems in IAV infection. Macrophages infiltrating the airway's epithelium are considered to be the principal source of interferon that induces antiviral resistance in lung epithelial cells. The model is formulated as a delay-differential system with about 60 parameters characterizing the rates of various processes contributing to the typical course of IAV infection. The key aspect of the adjustment between the model and various data on the immunity to influenza is the derivation of a consistent data set--the generalized picture of uncomplicated IAV infection. It serves as a consistent theoretical definition of the structure of the normal course of the infection and the antiviral immune response suitable for model fitting. The parameter estimates for the processes considered in the model are carefully discussed. The quantitative model is used to study the organization and dynamic properties of the processes contributing to IAV infection. The threshold condition for immune protection of virus-free host to infection with IAV is analyzed. The relative roles of humoral, cellular and interferon reactions for the kinetics of the uncomplicated IAV infection are studied. The contribution of parameters of virus-sensitive tissue, interferon and IAV-specific immune processes to the variations of duration and severity of the infection is quantitatively estimated by sensitivity studies. It is shown that the variations in the parameters of a virus-epithelial cell system are more influential on the severity of the infection rather than that of the antiviral immune response. The need for fine co-ordination of the kinetics of the non-specific interferon response and the adaptive antigen-specific immune reactions to provide recovery from the infection is illustrated.

Mathematical model of antiviral immune response. I. Data analysis, generalized picture construction and parameters evaluation for hepatitis B.

The present approach to the mathematical modelling of infectious diseases is based upon the idea that specific immune mechanisms play a leading role in development, course, and outcome of infectious disease. The model describing the reaction of the immune system to infectious agent invasion is constructed on the bases of Burnet's clonal selection theory and the co-recognition principle. The mathematical model of antiviral immune response is formulated by a system of ten non-linear delay-differential equations. The delayed argument terms in the right-hand part are used for the description of lymphocyte division, multiplication and differentiation processes into effector cells. The analysis of clinical and experimental data allows one to construct the generalized picture of the acute form of viral hepatitis B. The concept of the generalized picture includes a quantitative description of dynamics of the principal immunological, virological and clinical characteristics of the disease. Data of immunological experiments in vitro and experiments on animals are used to obtain estimates of permissible values of model parameters. This analysis forms the bases for the solution of the parameter identification problem for the mathematical model of antiviral immune response which will be the topic of the following paper (Marchuk et al., 1991, J. theor. Biol. 15).

Mathematical model of antiviral immune response. II. Parameters identification for acute viral hepatitis B.

Considering the mathematical model of antiviral immune response, we describe a method of fitting the model to the data characterizing acute viral hepatitis B. The corresponding procedure employs an idea of sequential parameter estimation to make the problem of fitting manageable. The underlying mechanisms responsible for the quantitative manifestations of the four basic phases of acute hepatitis B are used to select the model parameters. The identified model of acute hepatitis B is then tested with regard to the following situations: the effect of HBsAg-specific antibodies on HBV challenge; the vaccination and the resistance to challenge using live hepatitis B virus; the dose of viruses--the incubation time relationships. The sensitivity of the model with respect to parameters variations is then analysed. The developed model allows us to quantitatively simulate the basic features of the antiviral immune response during acute hepatitis B and some closely related phenomena.