Mathematical Analysis of an Autoimmune Diseases Model: Kinetic Approach

Kolev, Mikhail

doi:10.3390/math7111024

Cited by 9 publications

(8 citation statements)

References 82 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…At the same time, the macroscopic analogue derived from the kinetic system reflects the cellular behaviour on the global dynamical pattern of the populations and therefore allows us to relate both cellular and global performances. This duality is an advantage over the macroscopic models studied in previous papers, [6][7][8][9][10][11][12] where the individual behaviour of cells is not taken into account, and also over the kinetic models developed by Kolev and Nikolova, 11,12 where the relation to the macroscopic behaviour of the populations is not considered. Another very important aspect of our work is that, in contrast to the works by Kolev and Nikolova 11 and Zhang et al, 6,8 where the recurrent behaviour of the solution of the models is mainly due to the imbalance of the cells of the immune system, in our model, we show that factors external to the immune system could produce, as well, recurrent dynamics in the model solution.…”

Section: Discussionmentioning

confidence: 99%

“…A mathematical model of kinetic type was proposed by Kolev and Nikolova 11 and then reformulated by Kolev, 12 in view of developing some numerical simulations able to describe typical dynamics of autoimmune diseases. Such papers do not exploit the interplay between the kinetic description and its corresponding macroscopic analogue, but they are able to reproduce interesting numerical results describing typical behaviour of various autoimmune conditions, in particular absence of disease, mild symptoms, and chronic disease.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Mathematical modelling of oscillating patterns for chronic autoimmune diseases

Marca

Ramos

Ribeiro

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

Many autoimmune diseases are chronic in nature, so that in general, patients experience periods of recurrence and remission of the symptoms characterizing their specific autoimmune ailment. In order to describe this very important feature of autoimmunity, we construct a mathematical model of kinetic type describing the immune system cellular interactions in the context of autoimmunity exhibiting recurrent dynamics. The model equations constitute a nonlinear system of integro‐differential equations with quadratic terms that describe the interactions between self‐antigen presenting cells, self‐reactive T cells, and immunosuppressive cells. We consider a constant input of self‐antigen presenting cells, due to external environmental factors that are believed to trigger autoimmunity in people with predisposition for this condition. We also consider the natural death of all cell populations involved in our model, caused by their interaction with cells of the host environment. We derive the macroscopic analogue and show positivity and well‐posedness of the solution and then we study the equilibria of the corresponding dynamical system and their stability properties. By applying dynamical system theory, we prove that steady oscillations may arise due to the occurrence of a Hopf bifurcation. We perform some numerical simulations for our model, and we observe a recurrent pattern in the solutions of both the kinetic description and its macroscopic analogue, which leads us to conclude that this model is able to capture the chronic behaviour of many autoimmune diseases.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathematical modelling of oscillating patterns for chronic autoimmune diseases

Marca

Ramos

Ribeiro

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The present model is a generalization of the kinetic models of autoimmune diseases proposed recently in [22,23]. The model proposed in [23] describes only the populations of target cells, damaged cells and immune cells.…”

Section: Description Of the Mathematical Model Of A General Autoimmun...mentioning

confidence: 99%

“…The model proposed in [23] describes only the populations of target cells, damaged cells and immune cells. The model proposed in [22] consider the populations of target cells and damaged cells as homogeneous with respect to their activation states.…”

Section: Description Of the Mathematical Model Of A General Autoimmun...mentioning

confidence: 99%

Mathematical Modeling of Autoimmune Diseases

Kolev

2020

Symmetry

Self Cite

View full text Add to dashboard Cite

The human organism is a very complex system. To be in good health, its components must function properly. One of the most important systems of an organism is the immune system. It protects the body from the harmful effects of various external and internal agents. Sometimes, however, the immune system starts attacking its own healthy cells, tissues and organs. Then autoimmune diseases arise. They are widespread in recent decades. There is evidence that often autoimmune responses occur due to viral infections. In this paper, a new mathematical model of a general autoimmune disease is proposed. It describes the interactions between viral particles and host cells. The model is formulated by using integro-differential equations of Boltzmann type. This approach is typical for the nonequilibrium statistical mechanics. A preliminary qualitative and quantitative analysis of the model is presented.

show abstract

“…In 2003, Kuznetsov-Taylor's simplified model was put forward by Magda Galach [13], regarding one of the precursors in the emergence of cancer that enables the success of abnormal cells to coexist with normal cells for a prolonged period of time. Agent-based models (ABM) and kinetic theory have helped to explain the dynamics of a complex biological system [14,15] as well as pedestrian dynamics [16,17]. ABM are typically utilized in the cancer micro-environment for modeling tumor growth (drug response) and angiogenesis [18,19].…”

Section: Introductionmentioning

confidence: 99%

A New ODE-Based Model for Tumor Cells and Immune System Competition

Alharbi

Rambely

2020

Mathematics

View full text Add to dashboard Cite

Changes in diet are heavily associated with high mortality rates in several types of cancer. In this paper, a new mathematical model of tumor cells growth is established to dynamically demonstrate the effects of abnormal cell progression on the cells affected by the tumor in terms of the immune system’s functionality and normal cells’ dynamic growth. This model is called the normal-tumor-immune-unhealthy diet model (NTIUNHDM) and governed by a system of ordinary differential equations. In the NTIUNHDM, there are three main populations normal cells, tumor cell and immune cells. The model is discussed analytically and numerically by utilizing a fourth-order Runge–Kutta method. The dynamic behavior of the NTIUNHDM is discussed by analyzing the stability of the system at various equilibrium points and the Mathematica software is used to simulate the model. From analysis and simulation of the NTIUNHDM, it can be deduced that instability of the response stage, due to a weak immune system, is classified as one of the main reasons for the coexistence of abnormal cells and normal cells. Additionally, it is obvious that the NTIUNHDM has only one stable case when abnormal cells begin progressing into early stages of tumor cells such that the immune cells are generated once. Thus, early boosting of the immune system might contribute to reducing the risk of cancer.

show abstract

Mathematical Analysis of an Autoimmune Diseases Model: Kinetic Approach

Cited by 9 publications

References 82 publications

Mathematical modelling of oscillating patterns for chronic autoimmune diseases

Mathematical modelling of oscillating patterns for chronic autoimmune diseases

Mathematical Modeling of Autoimmune Diseases

A New ODE-Based Model for Tumor Cells and Immune System Competition

Contact Info

Product

Resources

About