Key words and phrases: immune tolerance, virus infection, effector immune cells, kinetic model.

Mikhail Kolev (Olsztyn) Iveta Nikolova (Blagoevgrad)

A mathematical model of some viral-induced autoimmune diseases.

Abstract We propose a new mathematical model of viral-induced autoimmune dis- eases. The model is described by a bilinear system of four integro-differential equa- tions of Boltzmann type. We present numerical results illustrating several typical outcomes of autoimmune diseases. In particular, special attention is devoted to the role of the ability of effector immune cells to destroy target cells for the development of autoimmune diseases.

2010 Mathematics Subject Classification: Primary: 92B05; Secondary: 92C50..

Key words and phrases: immune tolerance, virus infection, effector immune cells, kinetic model.

1. Introduction Usually the human immune system produces cells and

antibodies which are able to attack and neutralize foreign antigens and cancer

cells in the host organism. Antigens are foreign substances that can be dan-

gerous for the organism, i.e. they can be pathogenic, leading to disease. They

can be found on the surface of various invaders, food particles, pollen, cancer

cells etc. Some of the pathogens such as viruses, intracellular bacteria etc. are

able to invade host cells and use their resources for nutrition, proliferation and

so on. The host immune system can apply the mechanisms of the so-called

cellular immunity in order to fight such intracellular pathogens. The cellular

immune response is performed by specific white blood cells called T cells,

which are able to destroy the infected cells of the organism. Another group

of invaders such as extracellular bacteria do not need to invade host cells and

are able to live and proliferate in the extracellular environment of the host

organism using its fluids. They can secrete dangerous toxins causing various

diseases. In order to fight extracellular pathogens the host immune system

can apply the mechanisms of the so-called humoral immunity. The humoral

immune response is performed by specific antibodies, which are produced

by white blood cells called B cells. Antibodies can bind to an extracellular

antigen thus forming a complex antigen-antibody, which can be further elimi-

nated by other immune cells. Another possible mechanism used by antibodies

is the activation of complement proteins, which are able to neutralize foreign

antigens, rupture pathogenic cells, etc. [7].

During the process of development of lymphocytes in the organism, they should learn which cells belong to the own organism and consider them as safe. The self-cells should not be attacked and destroyed by the immune cells.

Only foreign antigens should be considered as dangerous and attacked by the immune system. The discrimination between self and non-self substances by the immune system and its ability to be tolerant to self and non-tolerant to non-self antigens is very important for the protection and maintenance of the health of the organism. The "education" of immune cells with respect to self-tolerance is a complex task that is performed by the use of various processes and mechanisms. In general, they can be divided into two groups depending on the place and time of their occurrence, namely into central and peripheral tolerance. The central tolerance is related to processes during the maturation of lymphocytes. They occur in thymus where T cells mature and in bone marrow where B cells mature. Potentially harmful lymphocytes are being eliminated via the processes of positive and negative selection, and clonal deletion. The survived mature lymphocytes are released into the blood circulation. However, some of the lymphocytes in the periphery can escaped clonal deletion being potentially harmful for the self cells. They can undergo clonal inactivation (anergy), suppression or apoptosis in order immune self- tolerance to be guaranteed [7].

Other processes and mechanisms are also used for achieving self-tolerance.

The interested reader is referred to [2] for more details. One can see that the achieving balance between proper tolerance to self substances and active im- mune response to foreign antigens is a very complex task and depends on many factors. If some of them are functioning improperly, this can lead to activity of self-reacting immune cells and possible destruction of healthy self cells, tissues and organs. Such autoimmune reactions can result in autoim- mune diseases, which are widely distributed nowadays. More than 80 diseases arise from abnormal activity of the immune system against healthy body tis- sues. Autoimmune diseases include some forms of diabetes, multiple sclerosis, rheumatoid arthritis, myocarditis, autoimmune hemolytic anemia, psoriasis, pancreatitis, Hashimoto’s thyroiditis and many others. Autoimmune diseases are characterized by slow progression and relapsing course. They can cause prolonged inflammation, subsequent tissue destruction and long lasting dis- ability. Among the possible results of autoimmune diseases are cardiovascular diseases, cancer development and increased mortality [6].

The aetiology of the autoimmune diseases is believed to be multifactorial,

including environmental and genetic factors, sedentary habits, socioeconomic

stress etc. [4]. The precise mechanisms by which breaking of self-tolerance

can occur, are not yet completely understood. However, many hypotheses

have been proposed in the literature to explain the mechanisms of develop-

ment of autoimmunity. Among them are the molecular mimicry theory, the

concepts related to sequestered antigens, cross-reaction, superantigen, deple-

tion of regulatory T cells etc. [5]. The mechanism of molecular mimicry is the following. The protein structure or the shape of some pathogens such as viruses and microbes may be very similar to those of normal self-proteins of the host organism. Because of this similarity, the immune response to the pathogen can also cross-react with self-antigens and kill healthy own cells.

Thus viruses resembling self-antigens can activate self-reacting immune cells and cause autoimmune reaction. During this process other self-antigens could be released leading to continuation and strengthening of the immune attack.

Due to the fact that self-antigens are continuously produced by the organism, the auto-aggressive immune response continues leading to the development of an autoimmune disease. The mechanism of molecular mimicry due to virus infection is shown to be one of the main causes in many cases of such au- toimmune diseases as multiple sclerosis, insulin-dependent diabetes mellitus type-1, primary biliary cirrhosis, systemic lupus erythematosus etc. [5, 6, 9].

In order to study the possible role of viral infection for the development of autoimmune diseases, in our paper we propose a new mathematical model, described in the following Section 2. In Section 3 we present results of nu- merical simulations and discuss them. Some concluding remarks are given in Section 4.

2. Model description In our model we consider a particular case of an autoimmune disease with only four interacting populations: the population of target cells, denoted by the subscript T, the population of damaged cells, denoted by the subscript D, the population of effector immune cells, denoted by the subscript I and the population of viral agents with molecular mimicry, denoted by the subscript V. We assume that some target cells (that can be healthy cells) are damaged due to virus infection. The proteins of the damaged cells can be captured by antigen presenting cells and presented by them to the immune system as self-antigens. As a result cross-reactive immune response can be elicited: specific lymphocytes or other effector immune cells can be produced and activated, which can attack and destroy healthy cells possessing the same antigen. There is evidence implicating on the one side the possible increase in the destructive ability of effector immune cells against target cells and on the other side the possible decrease in such a destructive ability of effector immune cells, which in some cases can be controlled by regulatory T cells [1, 5].

Over the past years, many mathematical models have been proposed for the description and analysis of autoimmune diseases. Some of the models describe specific autoimmune phenomena [5, 10]. In our paper we propose a new mathematical model for general autoimmune disease.

Many of the proposed mathematical models are formulated as systems

of ordinary differential equations considering the interacting populations as

homogeneous. In our paper, we propose a model, which is formulated in

the framework of the so-called kinetic theory of active particles, which uses

ideas typical for non-equilibrium statistical mechanics and generalized ki- netic theory for the description of biological entities, see [3] for reference. In kinetic models, some of the interacting populations are considered as non- homogeneous with respect to their biological activity or the ability to express their main functions. In our model we denote such an internal state of the population of effector immune cells by variable u chosen to belong to the in- terval [0, 1]. In our model, the state of activity of the effector immune cells denotes their ability to destroy target cells: more active effector immune cells are assumed to be able to destroy higher amounts of target cells.

The kinetic models describe the time evolution of the statistical distribu- tions over the microscopic activation state of the interacting populations (or of the respective concentrations if some of the populations are assumed to be homogeneous with respect to their activation states). In our model, the distribution density of the population of immune cells labeled by the index I at time t is denoted by:

f _I (t, u), f _I : [0, ∞) × [0, 1] → R ⁺ .

The concentrations of the populations are denoted by n _s (t). For the pop- ulation of immune cells the following relation holds:

n _I (t) = Z 1

0

f _I (t, u)du, n _I : [0, ∞) → R ⁺ . (1) For the sake of simplicity, we assume that the remaining populations of target cells, of damaged cells and of viral agents are homogeneous with respect to their biological activity.

Our model of an autoimmune disease is given by the following system of partial and ordinary integro-differential equations:

d

dt n T (t) = S T (t) + n T (t)



p T − _T ^p

M ax

n T (t)



−d _{T T} n _T (t) − d _{T I} n _T (t)

1

Z

0

uf _I (t, u)du,

(2)

d

dt n D (t) = d T I n T (t)

1

Z

0

uf I (t, u)du − d DD n D (t), (3)

∂f

∂t (t, u) = (1 − u) p _ID n _D (t) + p _IV n _V (t)
− d _II f _I (t, u) +c IR 2

u

Z

0

(u − v)f I (t, v)dv − (1 − u) ² f I (t, u)

!

+c IL 2

1

Z

u

(v − u)f I (t, v)dv − u ² f I (t, u)

! ,

(4)

d

dt n V (t) = p V n V (t) − d V V n V (t) − d IV n V (t)

1

Z

0

f I (t, u)du (5)

with nonnegative initial conditions

n _i (0) = n ⁽⁰⁾ _i , i = 1, 2, 4, f ₃ (0, u) = f ₃ ⁽⁰⁾ (u).

All parameters of the system (2)-(5) are assumed to be nonnegative.

Equation (2) describes the dynamics of the concentration n _T (t) of tar- get cells. The function S _T (t) denotes the production rate of target cells from sources within the organism, for example the thymus. In our numerical ex- periments we suppose that S _T (t) is constant. In addition, following [5] we suppose that target cells can also be produced by proliferation of existing target cells. In our model we assume that the proliferation of existing target cells is described by the logistic-like gain term n _T (t) 

p _T − _T ^p

M ax

n _T (t)  . The parameter p _T characterizes the maximal rate of proliferation of the target cells. The parameter T _{M ax} characterizes the concentration of the the target cells at which proliferation shuts off. The parameter d _{T T} characterizes the natural death of the target cells. The parameter d _{T I} characterizes the rate of destruction of the target cells by the effector immune cells, which is assumed to be proportional to the concentration of the target cells as well as to the activation state of the effector immune cells (see equation (2)). In healthy organisms cell populations constituting normal tissues and organs are be- ing kept on some physiological level maintained by proper balance between destructive and proliferative processes. When this balance is disturbed with predominance of destruction processes, an autoimmune or other disease can occur.

Equation (3) describes the evolution of the concentration n _D (t) of dam- aged cells. The parameter d _DD characterizes the natural death of the damaged cells.

Equation (4) describes the evolution of the distribution density f _I (t, u) of effector immune cells. Parameter p _ID characterizes the rate of production of effector immune cells due to self-antigens presented by damaged cells. It is assumed to be proportional to the concentration of the damaged cells.

Parameter p _IV characterizes the rate of production of immune cells due to

presence of viral agents. It is assumed to be proportional to the concentration

of the viral agents. We assume that the state of activity of the newly produced

effector immune cells is low. This is the meaning of the factor (1 − u) in the

both gain terms. The parameter d _II characterizes the natural death of the

immune cells. Moreover, the possible increase in the activity of the effector

immune cells due to their development as well as the possible decrease in

their activation due to the activity of some regulatory cells are described in

0 200 400 600 800 1000 0

0.5 1 1.5 2 2.5 3 3.5 x 10

t Target cells

Damaged cells

Figure 1: Concentrations of target cells and damaged cells for c _IR = 0, c IL = 0

equation (4) by conservative terms. Parameter c _IR characterizes the steady progress of effector immune cells towards increasing their activation state while parameter c _IL characterizes the possible lowering of the activity of effector immune cells by regulatory cells.

Equation (5) describes the evolution of the concentration n _V (t) of viral agents. Parameter p _V characterizes the rate of production of viral particles.

It is assumed to be proportional to the concentration of the viral agents. The parameter d _{V V} characterizes the natural death of the viral particles. The parameter d _IV characterizes the rate of destruction of the viral particles due to the immune response. It is assumed to be proportional to the concentrations of the viral particles and of the effector immune cells.

Our model is a generalization of our previous model [8], where only the populations of target cells, damaged cells and effector immune cells are con- sidered. In this previous model we did not take into account the role of the viruses and the possibility of change of the activation state of effector immune cells.

3. Numerical results The equation (4) of the system (2)-(5) was dis- cretized with respect to the activation state u ∈ [0, 1] by uniform mesh. The appearing integrals were approximated by using the composite Simpson’s rule.

Thus we obtained a system of ordinary differential equations, which was solved

for various parameter sets by using the solver ode15s from the Matlab ODE

suite [11]. ode15s is a multistep solver for initial value problems based on nu-

merical differentiation formulas [11]. The error control properties were defined

by setting the relative accuracy RelT ol = 10 ⁻³ and absolute error tolerance

0 200 400 600 800 1000 0

0.5 1 1.5 2 2.5 3 3.5

4 x 10

t Target cells

Damaged cells

Figure 2: Concentrations of target cells and damaged cells for c _IR = 0, c _IL = 1

0 200 400 600 800 1000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

5 x 10

t Target cells

Damaged cells

Figure 3: Concentrations of target cells and damaged cells for c _IR = 0, c IL = 5

AbsT ol = 10 ⁻⁴ .

As initial conditions we assumed the presence of a certain amount of target cells, a small amount of damaged cells and a very small amount of viruses and effector immune cells with very low activation state:

n _T (0) = 100, n _D (0) = 0.5, n _V (0) = 0.01,

f _I (0, 0) = 0.01, f _I (0, u) = 0.0, ∀u ∈ (0, 1].

We have performed numerical experiments with the following values of parameters:

S _T (t) = 10, ∀t ≥ 0, T _{M ax} = 100000,

p T = 0.5, d T T = 0.2, d T I = 0.1, d DD = 1.1,

p ID = 0.001, p IV = 0.001, d II = 0.1,

p V = 10.0, d V V = 4.0, d IV = 4.0 and various values of parameters c _IR and c _IL .

The aim of our numerical experiments was to study the role of the possible change of the activation state of effector immune cells for the viral-induced autoimmune disease. In Fig. (1) we present a case when the activation state of immune cells does not change, setting c _IR = 0 and c IL = 0. In this case we observe initial growth of the concentration of target cells followed by an associated growth of damaged cells and effector immune cells. The target cells are attacked by effector immune cells, which leads to a decrease in the concentration of target cells and further of damaged cells and effector immune cells. However, an amount of target cells remains alive, which leads to a new increase in their concentration followed by an increase in the concentration of damaged cells. This cycle continues with damping oscillations tending to an equilibrium state.

Further, we considered situations when regulatory immune cells or some other factors are able to lower the activation state of the effector immune cells. In Fig. (2) and Fig. (3) we present results of simulations for c _IL = 1 and c _IL = 5 keeping c _IR = 0. We observe a similar course of competition with higher peaks of target cells and decreasing number of cycles. The system tends to higher equilibrium for target cells and lower equilibrium for damaged cells for higher values of parameter c _IL .

Further, we considered situations with increasing activation states of ef-

fector immune cells setting c _IL = 0. The results of simulations for c _IR = 0.1

0 200 400 600 800 1000 0

0.5 1 1.5 2 2.5 3 3.5 x 10

t Target cells

Damaged cells

Figure 4: Concentrations of target cells and damaged cells for c _IR = 0.1, c IL = 0

0 200 400 600 800 1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2 x 10

t Target cells

Damaged cells

Figure 5: Concentrations of target cells and damaged cells for c _IR = 0.2,

c _IL = 0

and c _IR = 0.2 are presented in Fig. (4) and Fig. (5). Here, we observe an in- crease in the number of cycles representing periodic symptoms and repeated flare-ups.

We can conclude that when the effector immune cells can increase their ability to destroy target cells, the organism can develop an autoimmune dis- ease with repeated flare-ups. Such cases are observed in patients with multiple sclerosis [5].

On the contrary, when the ability of the effector immune cells to destroy target cells decreases, a relatively high amount of target cells remains alive in the phase of stabilization, which can be considered as mild symptoms or absence of an autoimmune disease.

4. Conclusions In our paper we propose a new mathematical model of an autoimmune disease. The model takes into account the role of viral agents as well as of regulatory cells changing the ability of the effector immune cells to destroy target cells for the occurrence and development of autoimmune diseases. The modeled problem is solved numerically. The numerical solutions represent several typical dynamics of an autoimmune disease.

Our future plans include the analysis of the role of other model parameters as well as the development of the model through inclusion of activation state variables for the populations of target cells, damaged cells and viral agents.

In addition, we plan to study the problem whether viral infections can lead to a decrease of autoimmune reaction. We also plan to apply our model to biological data.

5. Acknowledgements

The authors wish to express their gratitude to Dr. Violeta Stefanova for her valuable medical comments as well as to anonymous referees for their useful remarks which led to the improvements of the paper.

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Matematyczny model niektórych wywołanych wirusami chorób autoimmunologicznych.

Mikhail Kolev i Iveta Nikolova

Streszczenie W pracy został zaproponowany nowy matematyczny model chorób autoimmunologicznych wywołanych przez wirusy. Model opisany jest dwuliniowym układem czterech równań całkowo-różniczkowych typu Boltzmanna. Zaprezentowane są wyniki obliczeń numerycznych ilustrujące kilka typowych przebiegów takich cho- rób. Szczególną uwagę poświęca się roli zdolności efektorowych komórek odpornościo- wych do niszczenia komórek docelowych w rozwoju chorób autoimmunologicznych.

Klasyfikacja tematyczna AMS (2010): 92B05; 92C50.

Słowa kluczowe: tolerancja immunologiczna, infekcja wirusowa, odpornościowe ko-

mórki efektorowe, model kinetyczny.

Mikhail Kolev is an associate professor of computer science at the University of Warmia and Mazury in Olsztyn. He works in the field of mathematical modelling of biomedical phenomena, mainly related to tumour growth, viral infections and treatment.

However, he is also interested in other applications, like modelling autoimmune diseases, and lastly – bacterial infections.

Iveta Nikolova holds Master degree in Mathematics, currently she is a PhD student in Mathematics and writes a PhD thesis about mathematical modelling of autoimmune diseases. She is also interested in the role of various diets in autoimmune disease development.

Mikhail Kolev

University of Warmia and Mazury

Faculty of Mathematics and Computer Science Słoneczna 54, Olsztyn 10-710, Poland E-mail: kolev@matman.uwm.edu.pl Iveta Nikolova

South-West University "Neofit Rilski"

Faculty of Mathematics, Informatics and Mechanics Bulgaria

E-mail: iveta.nikolova@abv.bg

Communicated by: Urszula Foryś

(Received: 21th of May 2018; revised: 23rd of July 2018)