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Register a new userA qualitative mathematical model of the immune response under the effect of stress
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Maria Elena Gonzalez Herrero
Publication date
2021
fields
Biology
and research's language is
English
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In the last decades, the interest to understand the connection between brain and body has grown notably. For example, in psychoneuroimmunology many studies associate stress, arising from many different sources and situations, to changes in the immune system from the medical or immunological point of view as well as from the biochemical one. In this paper we identify important behaviours of this interplay between the immune system and stress from medical studies and seek to represent them qualitatively in a paradigmatic, yet simple, mathematical model. To that end we develop a differential equation model with two equations for infection level and immune system, which integrates the effects of stress as an additional parameter. We are able to reproduce a stable healthy state for little stress, an oscillatory state between healthy and infected states for high stress, and a burn-out or stable sick state for extremely high stress. The mechanism between the different dynamics is controlled by two saddle-node in cycle (SNIC) bifurcations. Furthermore, our model is able to capture an induced infection upon dropping from moderate to low stress, and it predicts increasing infection periods upon increasing before eventually reaching a burn-out state.
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We study a five-compartment mathematical model originally proposed by Kuznetsov et al. (1994) to investigate the effect of nonlinear interactions between tumour and immune cells in the tumour microenvironment, whereby immune cells may induce tumour cell death, and tumour cells may inactivate immune cells. Exploiting a separation of timescales in the model, we use the method of matched asymptotics to derive a new two-dimensional, long-timescale, approximation of the full model, which differs from the quasi-steady-state approximation introduced by Kuznetsov et al. (1994), but is validated against numerical solutions of the full model. Through a phase-plane analysis, we show that our reduced model is excitable, a feature not traditionally associated with tumour-immune dynamics. Through a systematic parameter sensitivity analysis, we demonstrate that excitability generates complex bifurcating dynamics in the model. These are consistent with a variety of clinically observed phenomena, and suggest that excitability may underpin tumour-immune interactions. The model exhibits the three stages of immunoediting - elimination, equilibrium, and escape, via stable steady states with different tumour cell concentrations. Such heterogeneity in tumour cell numbers can stem from variability in initial conditions and/or model parameters that control the properties of the immune system and its response to the tumour. We identify different biophysical parameter targets that could be manipulated with immunotherapy in order to control tumour size, and we find that preferred strategies may differ between patients depending on the strength of their immune systems, as determined by patient-specific values of associated model parameters.
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The last decade has seen an explosion in models that describe phenomena in systems medicine. Such models are especially useful for studying signaling pathways, such as the Wnt pathway. In this chapter we use the Wnt pathway to showcase current mathematical and statistical techniques that enable modelers to gain insight into (models of) gene regulation, and generate testable predictions. We introduce a range of modeling frameworks, but focus on ordinary differential equation (ODE) models since they remain the most widely used approach in systems biology and medicine and continue to offer great potential. We present methods for the analysis of a single model, comprising applications of standard dynamical systems approaches such as nondimensionalization, steady state, asymptotic and sensitivity analysis, and more recent statistical and algebraic approaches to compare models with data. We present parameter estimation and model comparison techniques, focusing on Bayesian analysis and coplanarity via algebraic geometry. Our intention is that this (non exhaustive) review may serve as a useful starting point for the analysis of models in systems medicine.
Simple ideas, endowed from the mathematical theory of control, are used in order to analyze in general grounds the human immune system. The general principles are minimization of the pathogen load and economy of resources. They should constrain the parameters describing the immune system. In the simplest linear model, for example, where the response is proportional to the load, the annihilation rate of pathogens in any tissue should be greater than the pathogens average rate of growth. When nonlinearities are added, a reference value for the number of pathogens is set, and a stability condition emerges, which relates strength of regular threats, barrier height and annihilation rate. The stability condition allows a qualitative comparison between tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response.
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