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Chinchuluun, A. Pareto Optimality, Game Theory and Equilibria. Feasibility and Infeasibility in Optimization. Frames and Bases. Matrix Convolution Operators on Groups. Morphometrics with R. Integral Methods in Science and Engineering.

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Interactive and Dynamic Graphics for Data Analysis. Cowling, M. Representation Theory and Complex Analysis. Hamiltonian Dynamical Systems and Applications. Time Series Analysis.

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Cuyt, A. Functional and Operatorial Statistics. An Introduction to the Theory of Point Processes. Introductory Statistics with R. Asymptotic Theory of Statistics and Probability. From Gestalt Theory to Image Analysis. Deutsch, A. Institution-independent Model Theory. Algorithms in Algebraic Geometry. Introduction to Applied Optimization.

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Topological Degree Approach to Bifurcation Problems. Feichtinger, H. Pseudo-Differential Operators. Shape Analysis and Structuring. Statistics of Financial Markets. Field Arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. Tutorials in Mathematical Biosciences IV. Teoria delle Equazioni e Teoria di Galois. Hemodynamical Flows. Eisenstein Series and Applications. Topological Methods in Group Theory. Graphs, Dioids and Semirings. Universal Algebra.

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Braid Groups. Bayesian Networks and Influence Diagrams. Design and Analysis of Simulation Experiments. Volume 1. Probability Theory. Representation Theory and Automorphic Forms. Random Curves. Mathematical Modeling, Simulation, Visualization and e-Learning. Information Criteria and Statistical Modeling. Combinatorial Optimization. Country Risk Evaluation. Combinatorial Algebraic Topology. Geometric Integration Theory. Mathematics — Key Technology for the Future.

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Point Estimation of Root Finding Methods. Various discrete dynamic frameworks including Boolean networks [ 32 ], finite dynamical systems [ 33 ], difference equations [ 34 ], and Petri nets [ 35 ] have been used in modeling biological systems. Particularly, Boolean network models assume that each component has two qualitative states e. The active qualitative state can be interpreted as the concentration of an immune component that can induce downstream signaling.

Such network models, tracking the dynamics of more than 30 immune components including various cytokines and cells, have been constructed for two Bordetella pathogens [ 6 , 7 ], for which few quantitative parameters have been determined. These models reproduce the qualitative features, such as the number of peaks, of the experimental time-courses of various immune components such as neutrophils and dominant cytokines. Continuous-discrete hybrid models [ 7 , 36 , 37 ] are also developed with the aim to improve the representation of time while retaining the simplicity of switching functions.

These hybrid models have a relatively small number of parameters, such as activation thresholds and decay rates, which are at a higher, more coarse-grained level than the kinetics of elementary reactions. A hybrid Bordetella model [ 7 ] reveals that many parameter combinations are compatible with the existing experimental knowledge on the pathogenesis. The distribution of the parameter values for each immune component in the model tells us about its role in the pathogenesis. Recent experimental measurements validate the IL4 time-course predicted by the model [Pathak, A.

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Since the immune responses involve interactions at the site of infection, the maturation of T and B cells in the lymph nodes and the transport of cells through blood, capturing spatial dynamics may be critical for the success of a model. Approximations at various levels of detail are available that allow for the inclusion of some spatial information in the form of spatial compartments, coarse grids or reaction-diffusion processes.

For example, the follow-up models of Mtb and Bordetellae [ 7 , 20 ] define two compartments, the site of infection the lung and the site of T cell differentiation lymph node. A more detailed approach used by Gammack et. The decision to use qualitative or quantitative models is based on the density of observations over time, the number of molecular or cellular players participating in a particular process and the connectivity of the regulatory network formed by these players.

We note that both approaches necessitate knowledge of the causal or interaction network among components. Missing data and within-lab variations caused by the use of different experimental systems can introduce uncertainty in the determination of causal relationships; this issue is dealt with by the techniques of reverse engineering [ 40 ]. Observations taken at many time-points minimize the uncertainty about the behavior between the observations.

The availability of frequent measurements for all or almost all the immune components one wants to model facilitates the use of quantitative modeling. The unavailability of such data guides us to use qualitative models which will inform us about the sequence of events and ultimate outcomes rather than trying to interpolate between the existing sparse observations.

The assumption of switch-like regulatory relationships underlying qualitative models is a good approximation if the functional form of the regulatory relationship is sigmoidal. Qualitative and quantitative approaches detail the immune interactions at different levels. Generally speaking, quantitative models give a detailed description of a relatively small number of interactions whereas qualitative models incorporate more interactions but have fewer kinetic details. Quantitative models offer predictions of kinetic parameters and of how the system will behave at a given instance.

Qualitative models predict the response to knock-out or over-expression of components. An effective strategy to bridge these two approaches can be to iteratively refine qualitative models as more quantitative information becomes available through incorporation of more states, using a continuous-discrete hybrid formalism, or a fully quantitative description of an important sub-system.

Quantitative models require substantial prior knowledge and the interactions that require parameterization in these models have not yet been quantitatively characterized for most of the infections. The assumptions and estimations necessary to give values for the parameters may introduce unwanted artifacts in the model, reducing its usefulness. Since many molecular and cellular players of the immune cascades [ 41 , 42 ] are available for a range of infectious diseases, along with the outcomes of pathogen manipulation experiments, qualitative models can be constructed for less studied infectious diseases giving us insight about the dynamic interplay arising from the complex multi-scale interactions.

Qualitative models also lose their simplicity and usefulness if the number of components and interactions included in the network is too large since that dramatically increases the system's dynamic repertoire. Various network simplification methods are available which reduce the number of components, for instance based on shortening long linear paths or collapsing alternative paths between a pair of nodes [ 43 ].

The simple models developed to study parts of the immune system decipher parameters that reveal the regulation of immune responses and allow us to extrapolate the observations from experimental hosts to the natural hosts. The models developed to test the evolutionary fitness of pathogens reveal fundamental characteristics of the host-pathogen interactions and give useful insight into the pathogenesis of the infections.

Among the models which aim to describe most of the immune components important in the pathogenesis, we show that both qualitative and quantitative models can be used effectively to study the progression of the infections. Math Biosci. Essays Biochem. Effect of the parasite's red blood cell preference.

Proc Biol Sci. PLoS Comput Biol. J R Soc Interface. Louzoun Y: The evolution of mathematical immunology. Immunol Rev. Kirschner DE, Linderman JJ: Mathematical and computational approaches can complement experimental studies of host-pathogen interactions. Cell Microbiol.

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Bull Math Biol. J Virol. J Immunol. PLoS Med. J Theor Biol. Wigginton JE, Kirschner D: A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis. Csete M, Doyle J: Bow ties, metabolism and disease. Trends Biotechnol. Trends in Biotechnology.

BMC Syst Biol. PLoS One. Kauffman SA: Metabolic stability and epigenesis in randomly constructed genetic nets. Mathematical Modeling, Simulation, Visualization and e-Learning. May RM: Simple mathematical models with very complicated dynamics. Chaouiya C: Petri net modelling of biological networks. Brief Bioinform. Mendoza L, Xenarios I: A method for the generation of standardized qualitative dynamical systems of regulatory networks. Theor Biol Med Model. J Math Biol. SIAM journal of multiscale modeling and simulation.

Fomekong-Nanfack Y, Postma M, Kaandorp JA: Inferring Drosophila gap gene regulatory network: pattern analysis of simulated gene expression profiles and stability analysis. BMC Res Notes. Curr Opin Microbiol. Randall RE, Goodbourn S: Interferons and viruses: an interplay between induction, signalling, antiviral responses and virus countermeasures. J Gen Virol. Mol Biosyst. Syst Biol Stevenage. Immunity control in bacteriophage lambda.