# On a Mathematical Model of a General Autoimmune Disease

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## Abstract

**:**

## 1. Introduction

## 2. Basic Immunological Concepts

## 3. Mathematical Modeling in Medicine

## 4. The Mathematical Model

- (i)
- The first population of target cells;
- (ii)
- The second population of damaged cells;
- (iii)
- The third population of immune cells;
- (iv)
- The fourth population of viral particles causing mimicry.

- Immune cells: The activation state refers to their ability to destroy target cells.
- Viruses: The activation state refers to their ability to trigger the production of immunological cells. Increasing values of the activation state denote higher amounts of newly produced immunological cells.

- ${S}_{1}\left(t\right)$ models the production rate of target cells from sources within the organism;
- ${p}_{11}$ characterizes the rate of proliferation of the target cells;
- ${T}_{Max}$ refers to the concentration of the target cells at which their proliferation turns off;
- ${d}_{11}$ describes the rate of the natural death of the target cells;
- ${d}_{13}$ describes the damaging rate of the target cells attacked by the immune cells.

- ${p}_{32}$ is the rate of production of immune cells due to the presence of self-antigens presented by damaged cells;
- ${p}_{34}$ is the rate of production of immune cells due to the presence of viruses;
- ${d}_{33}$ is the rate of the natural death of the immune cells;
- The factor $1-u$ is related to the assumption that the state of activity of the newly produced immune cells is low and they need time for activation.

- ${p}_{4}$ is the rate of production of viruses;
- ${d}_{43}$ is the rate of destruction of the viruses due to the immune response;
- ${d}_{44}$ is the rate of natural death of the viruses.

**Theorem**

**1.**

**Proof.**

## 5. Results of Simulations

`ode15s`from the Matlab ODE suite [37].

## 6. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Kolev, M.; Netov, N.; Nikolova, I.; Naskinova, I.; Kuneva, V.; Milev, M.
On a Mathematical Model of a General Autoimmune Disease. *Axioms* **2023**, *12*, 1021.
https://doi.org/10.3390/axioms12111021

**AMA Style**

Kolev M, Netov N, Nikolova I, Naskinova I, Kuneva V, Milev M.
On a Mathematical Model of a General Autoimmune Disease. *Axioms*. 2023; 12(11):1021.
https://doi.org/10.3390/axioms12111021

**Chicago/Turabian Style**

Kolev, Mikhail, Nikolay Netov, Iveta Nikolova, Irina Naskinova, Velika Kuneva, and Marian Milev.
2023. "On a Mathematical Model of a General Autoimmune Disease" *Axioms* 12, no. 11: 1021.
https://doi.org/10.3390/axioms12111021