Modeling the hiding–learning dynamics in large living systems

This paper deals with the modeling of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. Various mathematical structures of the hiding–learning dynamics are derived at the mesoscopic scale. Subsequently, these structures are further detailed referring to modeling issues and, in particular, to the learning-hiding competition among tumors and the immune system cells.     

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

A class of coupled cell–bulk ODE–PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius \(\epsilon \ll 1\) are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell–bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE–PDE cell–bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell–bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator–inhibitor reaction–diffusion systems in the limit of long-range inhibition and short-range activation.     

A dynamic network loading model for mesosimulation in transportation systems

Flow propagation models can be divided into static and dynamic network loading models. Different approaches to dynamic network loading problem formulated in the literature point out models that can be classified as disaggregate or aggregate.Applying aggregate models, it is possible to trace implicitly or explicitly vehicles movements. The second case concerns mesoscopic models. These models consider the traffic as a sequence of “packets” of vehicles. Two approaches can be followed:

• (a)continuous packets, where vehicles are distributed inside each packet, defined by the head and the tail points;
• (b)discrete packets, where all users belonging to a packet are grouped and represented by a single point, for instance the head.

In this paper, a mesoscopic model based on discrete packets has been developed, taking into account the vehicles acceleration. The proposed model, assuming discrete packets and uniformly accelerated movement, appears lifelike in the representation of outflow dynamics and quite easy to calculate.     

Tumor evasion from immune control: Strategies of a MISS to become a MASS

We biologically describe the phenomenon of the evasion of tumors from immune surveillance where tumor cells, initially constrained to exist in a microscopic steady state (MISS) elaborate strategies to evade from the immune control and to reach a macroscopic steady state (MASS). We, then, describe “evasion” as a long term loss of equilibrium in a framework of prey–predator-like models with adiabatic varying parameters, whose changes reflect the evolutionary adaptation of the tumor in a “hostile” environment by means of the elaboration of new strategies of survival. Similarities and differences between the present work and the interesting seminal paper [Kuznetsov VA, Knott GD. Modeling tumor regrowth and immunotherapy. Math Comput Model 2001;33:1275–87] are discussed. We also propose and study a model of clonal resistance to the immune control with slowly varying adaptive mutation parameter.     

Analysis of a kinetic cellular model for tumor-immune system interaction

Starting at a kinetic level from the equations for the evolution of dominance in populations of interacting organisms, and taking proliferative and destructive encounters into account, a simple model describing the competition between tumor cells and immune system is studied in some detail. Under reasonable assumptions, a closed set of macroscopic balance equations for macroscopic observables is derived by a moment procedure, and analyzed in the frame of the theory of dynamical systems. It is shown that a transcritical bifurcation of equilibria generates a region in the phase space in which, according to the model, the immune system defeats the tumor and leads to its depletion. Numerical results are presented and briefly commented on.     

Modelling of tumor cells regression in response to chemotherapeutic treatment

The proposed model describes the interaction among normal, immune and tumor cells in a tumor with a chemotherapeutic drug, using a system of four coupled partial differential equations. The dimensions of the tumor and initial conditions of tumor cells are chosen under the assumption that the tumor is already large enough in size to be detectable with the available clinical devices. The pattern of distribution of tumor cells is drafted on the basis of clinical observations. The stability of the system is established with tumor and tumor-free equilibria. The process of tumor regression with the introduction of different diffusion coefficients of tumor and immune cells is considered along with normal cells of tissue without any diffusive movement. It is shown that the results of chemotherapy treatment are in agreement with Jeff’s phenomenon. The response of three different levels of immune system strength to the pulsed chemotherapy are investigated. It is observed that the tumor performs better if a chemotherapeutic drug is injected near the invasive fronts of the tumor.     

一类带有有限维控制器的弹性系统的反馈控制

本文讨论一类弹性系统的反馈控制问题,它在工程上,特别是在大宇航飞行器设计中的应用是非常活跃的.我们的主要结果是:利用谱分析方法和无条件基理论证明了这类系统是严格耗散的分布参数系统,给出了该系统的降维模型可控可观的充分条件,从而设计一个有限维控制器对系统进行反馈控制,使得闭环系统具有更好的稳定性能,将文献[2,3]中的结果推广到非常一般的情形.     

Modeling treatment of cancer using virotherapy with generalized logistic growth of tumor cells

Abstract

Virotherapy is an effective strategy in cancer treatment. It eliminates tumor cells without harming the healthy cells. In this article, a deterministic mathematical model to understand the dynamics of tumor cells in response to virotherapy is formulated and analyzed by incorporating cytotoxic T lymphocytes (CTLs). The basic reproduction number and the immune response reproduction number are computed and different equilibria of the proposed model are found. The local stability of different equilibria is discussed in detail. Further, the proposed model is extended to stochastic model. Numerical simulation is performed for both deterministic and stochastic models. It is observed that when both the reproduction numbers are greater than one, which corresponds to existence of unique nontrivial equilibrium point, dynamics of deterministic and stochastic models are almost same. The deterministic model shows a very complex dynamics when one or both the reproduction numbers are below one. The system exhibits both backward bifurcation and Hopf-bifurcation for suitable sets of parameters and in this situation it is not easy to predict the dynamics of cancer cells and virus particles. The existence of backward bifurcation demonstrates the fact that partial success of virotherapy can be achieved even if the immune response reproduction number is less than one.     

Solitary Waves, periodic peakons, pseudo-peakons and compactons given by three ion-acoustic wave models in electron plasmas

The nonlinear ion-acoustic oscillations models are governed by three partial differential equation systems. Their travelling wave equations are three first class singular traveling wave systems depending on different parameter groups, respectively. By using the method of dynamical system and the theory of singular traveling wave systems, in this paper, it is shown that there exist parameter groups such that these singular systems have solitary wave solutions, pseudo-peakons, periodic peakons and compactons as well as kink and anti-kink wave solutions. The results of this paper complete the studies of three papers [5,13] and [14].     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

Fixed Energy Universality for Generalized Wigner Matrices

We prove the Wigner‐Dyson‐Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.© 2016 Wiley Periodicals, Inc.     

Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features

By adapting a pre-existing model to include the effects of Vascularization within a tumor or multicell spheroid, a predator-prey system describing the cell populations of a solid tumor and reactive lymphocytes is formulated. The paper serves as a review of the minimal deterministic approach to tumor-host immune system interactions while examining, in a qualitative manner, the modifications to the dynamics induced by a simple representation of the vascularized tumor. In addition, the possibility of limit-cycle behavior is studied by regarding each of six parameters present in the model as a bifurcation parameter. Thus, in principle, well-defined and periodic oscillations in both lymphocyte and tumor cell populations may occur under appropriate circumstances; whether or not such oscillations are sustainable by the host, and their stability, amplitude and period depend on aquisition of more quantitative information concerning the relevant parameter ranges.     

Studies on the Interaction Mechanism between the mRNA Vaccine against SARS-CoV-2 and the Immune System

Vaccines are an effective tool in the fight against infectious diseases. However, mathematical models of SARS-CoV-2 focus on the macroscopic situation, while articles on vaccines focus on effectiveness and safety. We develop four mathematical models to investigate the immune system and the microdynamics of antigens and viruses in individuals injected with mRNA vaccines. We first theoretically analyze the optimal model, calculate all equilibria, and prove that the disease-free equilibrium is globally asymptotically stable while the others are unstable. This suggests that after a certain period after vaccination, the infected cells and antigens will no longer exist in vivo and will be eliminated by the immune system over time or will die naturally. This theoretically proves the safety of the mRNA vaccines. Then, we use the differential algebra to analyze the structural identifiability of the models. We find that two of them are globally identifiable while the other two are unidentifiable, but once a certain parameter is fixed, then they are identifiable as well. To select the optimal model among four models, we use the Affine Invariant Ensemble Markov Chain Monte Carlo algorithm for data fitting and parameter estimation. We find that the roles of memory cells in killing infected cells and promoting immune cells and neutralizing antibodies in the process of mRNA vaccination are not significant and can be ignored in the modeling. On the other hand, the innate immunity of the human body plays an important role in this process. In addition, we also analyze the practical identifiability of the parameters of the optimal model. The results show that even if the structure of the system is globally identifiable, it does not ensure that all the parameters are practically identifiable. After random sampling and simulating the four unidentifiable parameters, we find that only two variables, infected cells II and antibodies, are sensitive to these unidentifiable parameters, but the results are still within acceptable ranges. This suggests that our fitting results are generally reliable. Finally, we simulate multiple booster injections and find that booster injections are indeed effective in maintaining antibody levels in vivo, which could otherwise gradually die off over time. Therefore, booster injections are beneficial to help the human body increase and maintain immunity.     

Convergence properties of the least squares estimation algorithm for multivariable systems

This paper focuses on the convergence properties of the least squares parameter estimation algorithm for multivariable systems that can be parameterized into a class of multivariate linear regression models. The performance analysis of the algorithm by using the stochastic process theory and the martingale convergence theorem indicates that the parameter estimation errors converge to zero under weak conditions. The simulation results validate the proposed theorem.     

Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells

The studies of nonlinear models in epidemiology have generated a deep interest in gaining insight into the mechanisms that underlie AIDS-related cancers, providing us with a better understanding of cancer immunity and viral oncogenesis. In this article, we analyze an HIV-1 model incorporating the relations between three dynamical variables: cancer cells, healthy CD4 ⁺ T lymphocytes, and infected CD4 ⁺ T lymphocytes. Recent theoretical investigations indicate that these cells interactions lead to different dynamical outcomes, for instance to periodic or chaotic behavior. Firstly, we analytically prove the boundedness of the trajectories in the system’s attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. Our calculations reveal that the highest observable variable is the population of cancer cells, thus indicating that these cells could be monitored in future experiments in order to obtain time series for attractor’s reconstruction. We identify different dynamical behaviors of the system varying two biologically meaningful parameters: r ₁, representing the uncontrolled proliferation rate of cancer cells, and k ₁, denoting the immune system’s killing rate of cancer cells. The maximum Lyapunov exponent is computed to identify the chaotic regimes. Considering very recent developments in the literature related to the homotopy analysis method (HAM), we calculate the explicit series solutions of the cancer model and focus our analysis on the dynamical variable with the highest observability index. An optimal homotopy analysis approach is used to improve the computational efficiency of HAM by means of appropriate values for the convergence control parameter, which greatly accelerate the convergence of the series solution. The approximated analytical solutions are used to compute density plots, which allow us to discuss additional dynamical features of the model.     

Rotation of a Neutron in the Coat of Helium-5 as a Classical Particle for a Relatively Large Value of the Hidden Parameter <Emphasis Type="Italic">t</Emphasis><Subscript>meas</Subscript>

Rotation of a neutron in the coat of helium-5 as a classical particle for a relatively large value of the hidden parameter (measurement time) t_meas = h/E_ms is considered. In consideration of the asymptotics as N → 0, equations for the mesoscopic energy E_ms are given. A model for the helium nucleus is introduced and the values of the mesoscopic parameters M_ms, and E_ms for helium-4 are calculated.     

Mathematical modeling of the immune system response to pathogens

The detection and elimination of pathogens in an organism are the main tasks of its immune system. The most important cells involved in these processes are neutrophils and macrophages. These processes might have two resolutions: The first is the possibility of pathogen elimination, and the other the possibility of the inflammation resolution. In this work, we present several mathematical models involving immune cell densities and inflammation levels. Our general goal is to exhibit the possible pathogen eradication or the inflammation resolution. We use bifurcation techniques in order to analyze how parameter variations may change the system evolution. Our results indicate that the elimination of apoptotic neutrophils by macrophages has a dichotomy effect: It contributes to the decrease of the inflammation level, but it may hinder the pathogen elimination. Also, an increment of the average neutrophil life can improve healthy outcomes. Moreover, we find scenarios when pathogens cannot be eliminated, as well as conditions for their successful eradication.     

Macro‐hybrid mixed variational two‐phase flow through fractured porous media

Macro‐hybrid mixed variational models of two‐phase flow, through fractured porous media, are analyzed at the mesoscopic and macroscopic levels. The mesoscopic models are treated in terms of nonoverlapping domain decompositions, in such a manner that the porous rock matrix system and the fracture network interact across rock–rock, rock–fracture, and fracture–fracture interfaces, with flux transmission conditions dualized. Alternatively, the models are scaled to a macroscopic level via an asymptotic process, where the width of the fractures tends to zero, and the fracture network turns out to be an interface system of one less spatial dimension, with variable high permeability. The two‐phase flow is characterized by a fractional flow dual mixed variational model. Augmented two‐field and three‐field variational reformulations are presented for regularization, internal approximations, and macro‐hybrid mixed finite element implementation. Also abstract proximal‐point penalty‐duality algorithms are derived and analyzed for parallel computing. Copyright © 2016 John Wiley & Sons, Ltd.     

Hopf bifurcation of a tumor immune model with time delay

In this paper, a tumor immune model with time delay is studied. First, the stability of nonnegative equilibria is analyzed. Then the time delay τ is selected as a bifurcation parameter and the existence of Hopf bifurcation is proved. Finally, by using the canonical method and the central manifold theory, the criteria for judging the direction and stability of Hopf bifurcation are given.