Computational and Mathematical Methods in Medicine

Volume 2018, Article ID 7930603, 10 pages

https://doi.org/10.1155/2018/7930603

## Stability Analysis of Mathematical Model including Pathogen-Specific Immune System Response with Fractional-Order Differential Equations

Kayseri University, Faculty of Applied Sciences, TR-38039 Kayseri, Turkey

Correspondence should be addressed to Bahatdin Daşbaşı; moc.liamtoh@nidtahab_isabsad

Received 31 July 2018; Revised 29 October 2018; Accepted 14 November 2018; Published 4 December 2018

Academic Editor: Chung-Min Liao

Copyright © 2018 Bahatdin Daşbaşı. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this study, the mathematical model examined the dynamics between pathogen and specific immune system cells (memory T cells) for diseases such as chronic infection and cancer in which nonspecific immune system cells are inadequate to destroy the pathogen and has been suggested by using a system of the fractional-order differential equation with multi-orders. Qualitative analysis of the proposed model reveals the equilibrium points giving important ideas about the proliferation of the pathogen and memory T cells. According to the results of this analysis, the possible scenarios are as follows: the absence of both pathogen and memory T cells, only the existence of pathogen, and the existence of both pathogen and memory T cells. The qualitative analysis of the proposed model has expressed the persistent situations of the disease where the memory T cells either do not be able to respond to the pathogen or continue to exist with the disease-causing pathogen in the host. Results of this analysis are supported by numerical simulations. In the simulations, the time-dependent size of the tumor population under the pressure of the memory T cells was tried to be estimated.

#### 1. Introduction

For three centuries, the theory of fractional derivatives was developed as a pure theoretical field of mathematics, useful only for mathematicians. But, the use of fractional-orders differential and integral operators in mathematical models has become increasingly common of late years. Therefore, various forms of fractional-order differential equations are suggested for standard models. In this sense, the fractional-order calculus plays an important role in physics [1], thermodynamics [2], viscoelasticity [3], electrical circuits theory [4], fractances [5], mechatronics systems [6], signal processing [7], chemical mixing [8], chaos theory [9], engineering [10], biological system [11], and other applications [12]. Also, a large number of literatures on the application of fractional-order differential equations (FODEs) in nonlinear dynamics have been improved. Especially, when the biological applications of FODEs have considered, it is a rich source for mathematical ideas [13].

The mathematical modeling of diseases in biological applications is a subject discussed in the literature. Such models are considered under two main headings as by modeling the size of the spread of infected individuals in a population () and modeling the population size of the pathogens such as the tumor in an individual as it is here. The word tumor simply refers to a mass. This is a general term that can refer to benign (generally harmless) or malignant (cancerous) growths. Many types of tumors are considered to be a major factor in many fatal diseases in human history. Fundamentally, it is said that this disease is a complex process for both tumor and host. Although different treatment strategies are proposed for tumors, the first and foremost role in disease progression belongs to the immune system of the individual (or host) [14]. The immune system is stated as a system of biological structures and processes in an organism that protects the body from the possible hazardous organism by recognizing and responding to antigens. In more detail, the immune system cells such as T cells are generally described in terms of two different types. These are the effector and the memory of T cells. The ordinary behaviour of the immune system is generally an acute infection, controlled initially by effector T cells (aspecific response or the innate immune system response), later by memory T cells (specific response or the adaptive immune response), until complete clearance of the pathogen. The adaptive immune response is much slower to respond to threats and infections than the innate immune response, which is primed and ready to fight at all times [15]. Especially, T cells are a crucial component of the adaptive immune response against malignancies. Antigen-experienced T cells specific for tumor antigens can be recovered from the blood, lymphoid organs, and tumors of both cancer patients and tumor-bearing mice [16]. Concordantly, the reactions of different hosts in case of the same disease may be different because of the immune system response given by host, and so, the disease progression varies from person to person. Within this context, dynamics of relevances between immune systems cells (T cells) and tumor are significant to find out the nature of the disease. The problem is to try to obtain the known biological features without making the mathematics too complicated.

The basic of a most useful explanation of fractional calculus is memory concept. If the output of a system at each time depends only on the input at time , then such systems are said to be memoryless systems. On the contrary, if the system has to remember previous values of the input in order to determine the current value of the output, then such systems are said memory systems [17, 18]. Accordingly, the behaviour of most biological systems has memory or after-effects. The modeling of these systems by FODEs has more advantages than classical integer-order modeling, in which such effects are neglected. Also, FODEs are, at least, as stable as their integer order counterpart [11]. In the process of modeling real-life situations, the created mathematical models by using the fractional-order differential operations allow to display the some extra cases regarding the stability region of the equilibrium point of the mathematical model caused by parameters such as derivative orders. For this reason, the mathematical models formed by FODEs are more realistic and feasible [19]. Additionally, stability analysis of equilibrium points for mathematical models consisting of FODEs with multi‐orders and its systems is more general than those of the same-orders too.

Although there are many studies that examined the dynamics between tumor and immune system response, the proposed model in this study differs from them in terms of both mathematical structure such as the use of Holling function type-2 (functional and numerical responses) in the model consisting of the FODE system with multi-orders and examination of qualitative analysis of the proposed model. In this sense, it was tried to bring a different perspective from the previous studies.

In this study, a FODE model with multi-orders considering the basic mechanisms of tumor and the memory T cells having functional and numerical responses, respectively, has been constructed, and so, the qualitative analysis of the proposed model was performed. The reason for using the Holling function type-2 is to show the limit cycle behaviour of system [20]. The certain conditions dependent on the development of the tumor population under the pressure of memory T cells was obtained. In this respect, all of the possible scenarios related to the tumor size were tried to be explained as parameter-dependent. Additionally, numerical analysis of the model was given as to be compatible with the qualitative analysis.

#### 2. Preliminaries and Definitions

In here, the main definitions and properties of fractional derivative operators have been expressed. Also, the FODE systems with multi-orders have been introduced, and the properties such as stability and existence of the equilibrium points of such systems are given.

##### 2.1. Fractional Differential Operators

There are various definitions of a fractional derivative with the order . The definitions of Riemann–Liouville and Caputo are used most widely. The Caputo sense was used in this study. Taking into account the definition of Caputo sense, the fractional derivative of the function is identified asfor , , [21].

##### 2.2. The FODE System with Multi-orders

Let us consider that is the time parameter. We have assumed that the system of FODE with multi-orders is given as the following equation:where the variable , the initial conditions by , the functions by and for , and the derivative orders by .

Also, when it is considered as , indicates th-order fractional derivative in the Caputo sense. In this sense, it is . The multi-orders can be mathematically any real or complex vector. In this study, the real case was only taken into account. Throughout the paper, we restrict to a rational number in the interval [22].

*Remark 1. *From (2), we have assumed thatwhere the independent variable is not clearly seen in the function . The equilibrium point of (3) is the point obtained from the equations .

*Remark 2. *For each equilibrium point of the autonomous system in (3), the eigenvalues obtain from the following equation:where is the Jacobian matrix evaluated at the equilibrium point and is the smallest of the common multiples of the denominators of the rational numbers [23].

Theorem 1. *For each equilibrium point of system (3), is locally asymptotically stable (LAS), if the eigenvalues obtained from (4) satisfy Routh–Hurwitz Stability Criteria or the inequalities for . Here, has been defined in Remark 2 [22]. Because the 2-dimensional of system (3) is used in the proposed model in this study, the stability analysis of such systems are described in detail below.*

*Remark 3. *Let us assume that the autonomous system of FODE with multi-orders is as following:with the nonnegative initial conditionswhere the derivative orders and are rational numbers in the interval . The equilibrium point of system (5) is the point obtained from the equations for . To evaluate locally asymptotically stability (LAS) of equilibrium point, the Jacobian matrix, , is used. Considering Remark 2, the eigenvalues for are obtained from the following equation:Therefore, the characteristic equation for eigenvalues isIf the eigenvalues for satisfy Routh–Hurwitz stability criteria or the conditions,then the equilibrium point is the LAS point for system (5).

For the system of FODE with multi-orders and , the stability region is as shown in Figure 1 (where and are the real and imaginary parts of the eigenvalues, respectively, and ). By Figure 1, we openly see that the stability region of the equilibrium point of the FODE with multi-orders is greater than the stability regions of the integer-order case and the same fractional-order case [24].