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Computational and Mathematical Methods in Medicine
Volume 2018, Article ID 7930603, 10 pages
https://doi.org/10.1155/2018/7930603
Research Article

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].

Figure 1: Stability region of the equilibrium point of system (5).

Remark 4. Let in system (5). In this case, we have the systemwith the nonnegative initial conditions . From the equations for , we have presumed that the equilibrium point of system (10) is . If the eigenvalues and obtained from the equationprovide the conditionsthen the equilibrium point is the LAS point for system (10).
Conditions expressed in (12) can be detailed as the followings. Characteristic equation of (11) is the following generalized polynomial:When both the conditions (12) and the polynomial (13) are considered together, the conditions for LAS of the equilibrium point are either Routh–Hurwitz conditions [25, 26]:or

3. Model Formulation

The proposed model is particularly well suited for describing diseases such as chronic infection and cancer in which the nonspecific immune system cells are inadequate to destroy the pathogen. Consequently, it has been proposed, and another extension of the models in [20, 2733] has been analyzed.

It has been identified mathematically as a pathogen load, specifically tumor population, and level of memory T cells, namely, the adaptive immune response or specific response, in an individual. In this sense, we have assumed that the population densities of pathogen and memory T cells at time are denoted by and , respectively. Additionally, the memory T cells predate the tumor cells by a Holling function type-2.

Under the assumptions aforementioned, we have proposed the following system of FODE with multi-orders and :where for indicates th-order fractional derivatives in the Caputo sense, it is and , and the parameters have the following properties:

In addition that, system (16) has to be finished with positive initial conditions and . The parameters used in the model are defined as follows.

It is presumed that the pathogen follows a logistic growth rule with the carrying capacity and the growth rate . The memory T cells proliferate proportionally to the pathogen load by the Holling function type-2. Since the pathogen capture rate of memory T cells is assumed to be proportional to the per capita growth rate of memory T cells, the constant represents the maximum growth rate for memory T cells and the constant is the pathogen population size at which the growth rate of memory T cells in half of its maximum. These situations are very suitable for the growth of memory T cells especially in case of chronic infection or tumor. Memory T cells have per capita natural death rate . Moreover, the pathogen die due to the action of the memory T cells, and we have presumed that the effect of these cells on pathogen is modeled using a saturating response, , subject to a maximum killing rate and the level of memory T cells required for the half maximum effect, .

Remark 5. Rate of replication of the immune system cells is higher than its death rate, at least every time the pathogen load is very high [20]. In this case, we have by (16). Therefore, the following inequality is obtained:by this limit.

Proposition 1. System (16) provides the followings. The free-disease equilibrium point and the equilibrium point , where only the pathogen exists and always exist. In addition to and , there exists a third equilibrium point as forwhen .

Proof. The steady states of the model (16) are again the intersection of null clines and in (16). We have accepted that the solutions of theses equations consist of the pairs . Then, we have the following system:From the first equation of (20), it is or . Let , and then . Therefore, the system (16) has the free-disease equilibrium point . On the contrary, let , that is, . If the value is rewritten in the second equation of system (20), then we have found the equilibrium points and . Considering (19), if is rewritten, the point is obtained. always exists due to (17). Let us consider . is positive due to (17) and (18). On the contrary, is positive due to (17), whenTherefore, we have a positive equilibrium point where and are in (19).
In Table 1, biological existence conditions of equilibrium points of system (16) are showed.

Table 1: Biological existence conditions for the equilibria of system (16).

Proposition 2. In system (16), let us consider derivative orders aswhere the smallest common multiple of and is . System (16) satisfies the following:(a) is a unstable point(b) is LAS, when . Also, if , then this point is a unstable point(c) where and defined in (19) is LAS, when all roots for found from the equationsatisfy Routh–Hurwitz stability criteria or the condition . In here, it is

Proof. For the stability analysis of the equilibrium points, the functions in system (16) are assigned asIn this respect, the Jacobian matrix evaluated at each equilibrium point showed in Table 1 isfor .(a)For , the Jacobian matrix in (26) isNow, we will investigate whether the inequality (9) has been achieved. By (27), the eigenvalues have been found from the following determinant:In this respect, the characteristic equation obtained from (28) isThus, we have and . is real positive due to (17). Moreover, it is obtained asFor the eigenvalues in (30), it is for . These eigenvalues are positive real number on the right side of the complex plane, and so, it is . Since the stability condition is not supplied, the equilibrium point is a unstable point for system (16).(b)From (24), the Jacobian matrix related to isFrom the equation , the characteristic equation of eigenvalues isTherefore, it is and . These equations are examined as the following:
(1) is a negative real number due to (17). By De Moivre’s formula, we have , such that . Considering (9), the stability condition for is due to . This condition has been always provided since in (16).(2)On the other hand, we have considered the equation . If , then for is obtained from by De Moivre formulas. In this respect, the stability condition is by . This condition has been always provided from (16). Additionally, If , then the eigenvalues are positive real number due to for . In this sense, we have . The stability condition is not provided due to . Therefore, the equilibrium point is an unstable point for system (16).Consequently, if the inequalityis provided, then the equilibrium point is LAS, and if , then this point is an unstable point for system (16). When (33) is rearranged, the stability condition of is(c)Jacobian matrix evaluated at isthat is,where and are defined in (24).Let us give more details for and . Accordingly, it issince the value , the pathogen size may take, is less than or equal to its carrying capacity . In addition, it isdue to (17), and the components of the equilibrium point in are positive.
From (36), we have the characteristical equation as follows:To be LAS of , it should be that all roots for found from the (39) satisfy the inequalities in (9).
Proposition is proved.

Corollary 1. Equation (39) can be examined in more detail as shown below. This equation can be rewritten by De Moivre formulas such thatwhere , angle , and . By (40), (39) transforms toand so,By arranging (42), there is the following system:and so,From the first equation in system (44), we have foundBy substituting (45) in the second equation in (44), it is foundConsequently, if the angles for obtained from (46) satisfy Routh–Hurwitz stability criteria or the condition (9) , then is LAS.
For equilibria of system (16), the conditions found for LAS and biological existence are summarized in Table 2.

Table 2: LAS and biological existence conditions for the equilibria of system (16).

Corollary 2. is an unstable point, when exists biologically. Therefore, these equilibrium points cannot be stable when together. Similarly, is biologically meaningless, when is LAS. These circumstances appeared are also seen in Table 2.

Corollary 3. Let us consider the special case of for . In this case, we have Remark 4. The characteristical equation obtained from isThe conditions for LAS of the equilibrium point are either Routh–Hurwitz stability conditions.due to (37) and (38), or the conditionsIn addition that, by considering equation (46), there may be found a similar condition for stability of this point. In this sense, it isby formulas of half angle. Thereby, the LAS condition of iswhere . Corollary 3 is summarized in Table 3.

Table 3: The LAS conditions for , in case of .

4. Numerical Analysis for Model

The proposed model in this study summarizes the general dynamics of the pathogen-immune system. The immune system cells described herein are memory T cells specially produced by the host against the pathogen. Hence, the proposed model is suitable for modeling of diseases such as chronic infections or tumors in which the nonspecific immune system cells of the host at the beginning of the disease have failed to destroy the pathogen.

The conditions found in Table 2 have been supported by numerical studies shown below. In this section, the time-dependent sizes of the tumor and memory T cells for cancer tried to be estimated by giving the different values to the parameters in the proposed model. The reason for this is to be able to obtain different scenarios and to better demonstrate the results of qualitative analysis. The values of parameters used in system (16) are shown in Table 4.

Table 4: The interpretation and considered values of the parameters in the proposed model.

The values calculated by Table 4 are given in Table 5.

Table 5: The values calculated from Table 4 according to Table 2.

Through the values in the first columns of Table 4, the stability of the equilibrium point , where the tumor exists and it approaches its carrying capacity, is obtained as shown in Figure 2. This happens within at least 200 days.

Figure 2: In case of the first column data in Table 4, temporary course of population size of the tumor and memory T cells.

When the values in the second columns of Table 4 are used, a situation where the system behaviour is limit cycle and is an unstable point is obtained as shown in Figure 3.

Figure 3: In case of the second column data in Table 4, temporary course of population size of the tumor and memory T cells.

Finally, let us consider the third column in Table 4. In here, for the tumor has a value of and for memory T cells has a value of . This occurs in at least 100 days as seen in Figure 4. Therefore, these two types of cells stay permanently in the host.

Figure 4: In case of the third column data in Table 4, temporary course of population sizes of the tumor and memory T cells.

5. Conclusions

In this study, the mathematical model examining the changes in the pathogen population size under pressure of specific immune system response in case of cancer or chronic infection has been constructed by the FODE system with multi-orders. According to the results of analysis of model, the pathogen causing disease never disappears in host unless an additional treatment is provided, since the disease is continued by a pathogen, and the free-disease equilibrium point is an unstable point. This case is very suitable for the presumed diseases in the proposed model.

When the existence condition of and the stability condition of in Table 2 is rearranged, thenis obtained, respectively. The term is the rate of growth of the memory T cells when the tumor reaches its carrying capacity . The parameter is the natural death rate of memory T cells. Accordingly, the value can be interpreted as the proliferative power of the memory T cells against the tumor. When this value is negative as seen in Figure 2, the memory T cells cannot reproduced as enough, and consequently, it can be mentioned from the stability of the equilibrium point , that only the tumor exists, and the tumor approaches its carrying capacity. Let us consider that this value is positive, and some additional conditions in Table 2 are met. As can be seen in Figure 4, it can be said that the positive equilibrium point , in which both tumor and memory T cells exist, is stable.

In numerical studies, we tried to estimate the timing and magnitude of the development of the tumor. If treatment procedure for the individual has not been applied, then the results obtained from the proposed model emphasize the fact that either the tumor reaches its maximum size and the memory T cells collapse or the tumor and memory T cells continue to stay together in the host. In the last case mentioned above that the memory T cells of the individual does not collapse, the tumor maintains its presence in the host in a limited manner. The results obtained from analysis are quite consistent with the scenarios of real situations related to the tumor.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The author read and approved the final manuscript.

References

  1. J. F. Gomez-Aguilar, R. Razo-Hernandez, and D. Granados-Lieberman, “A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response,” Revista Mexicana de Física, vol. 60, pp. 32–38, 2014. View at Google Scholar
  2. R. P. Meilanov and R. A. Magomedov, “Thermodynamics in fractional calculus,” Journal of Engineering Physics and Thermophysics, vol. 87, no. 6, pp. 1521–1531, 2014. View at Publisher · View at Google Scholar · View at Scopus
  3. C.-Q. Fang, H.-Y. Sun, and J.-P. Gu, “Application of fractional calculus methods to viscoelastic response of amorphous shape memory polymers,” Journal of Mechanics, vol. 31, no. 4, pp. 427–432, 2015. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Sikora, “Fractional derivatives in electrical circuit theory–critical remarks,” Archives of Electrical Engineering, vol. 66, no. 1, pp. 155–163, 2017. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. Liu, Y.-F. Pu, X.-D. Shen, and J.-L. Zhou, “Design of 1/2 (n) order analog fractance approximation circuit based on continued fractions decomposition,” Journal of Circuits, Systems and Computers, vol. 44, no. 3, pp. 153–158, 2012. View at Google Scholar
  6. C. M. Ionescu, R. Caponetto, Y.-Q. Chen, J. Bourgeois, and M. Labachelerie, “Fractional order modeling and control in mechatronics 2. Design, control, and software implementation for distributed MEMS,” Mechatronics, vol. 23, no. 7, pp. 739–916, 2013. View at Publisher · View at Google Scholar · View at Scopus
  7. N. Miljković, N. Popović, O. Djordjević, L. Konstantinović, and T. B. Šekara, “ECG artifact cancellation in surface EMG signals by fractional order calculus application,” Computer Methods and Programs in Biomedicine, vol. 140, pp. 259–264, 2017. View at Publisher · View at Google Scholar · View at Scopus
  8. N. A. Sheikh, F. Ali, M. Saqib et al., “Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction,” Results in Physics, vol. 7, pp. 789–800, 2017. View at Publisher · View at Google Scholar · View at Scopus
  9. Z. Odibat, N. Corson, M. A. Aziz-Alaoui, and A. Alsaedi, “Chaos in fractional order cubic chua system and synchronization,” International Journal of Bifurcation and Chaos, vol. 27, no. 10, pp. 1–13, 2017. View at Publisher · View at Google Scholar · View at Scopus
  10. T. J Faber, A. Jaishankar, and G. H. McKinley, “Describing the firmness, springiness and rubberiness of food gels using fractional calculus. Part II: measurements on semi-hard cheese,” Food Hydrocolloids, vol. 62, pp. 325–339, 2017. View at Publisher · View at Google Scholar · View at Scopus
  11. F. A. Rihan, “Numerical modeling of fractional-order biological systems,” Abstract and Applied Analysis, vol. 2013, Article ID 816803, 11 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  12. V. V. Tarasova and V. E. Tarasov, “Elasticity for economic processes with memory: fractional differential calculus approach,” Fractional Differential Calculus, vol. 6, no. 2, pp. 219–232, 2016. View at Publisher · View at Google Scholar
  13. H. A. El-Saka, E. Ahmed, M. I. Shehata, and A. M. A. El-Sayed, “On stability, persistence, and Hopf bifurcation in fractional order dynamical systems,” Nonlinear Dynamics, vol. 56, no. 1-2, pp. 121–126, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. M. R. Shurin, “Cancer as an immune-mediated disease,” ImmunoTargets and Therapy, vol. 1, pp. 1–6, 2012. View at Publisher · View at Google Scholar
  15. L. B. Nicholson, “The immune system,” Essays in Biochemistry, vol. 60, no. 3, pp. 275–301, 2016. View at Publisher · View at Google Scholar · View at Scopus
  16. E. D. Thompson, H. L. Enriquez, Y.-X. Fu, and V. H. Engelhard, “Tumor masses support naive T cell infiltration, activation, and differentiation into effectors,” Journal of Experimental Medicine, vol. 207, no. 8, pp. 1791–1804, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. M. A. Matlob and Y. Jamali, “The concepts and applications of fractional order differential calculus in modelling of viscoelastic systems: a primer,” 2018, https://arxiv.org/abs/1706.06446. View at Google Scholar
  18. M. Du, Z. Wang, and H. Hu, “Measuring memory with the order of fractional derivative,” Scientific Reports, vol. 3, no. 3431, pp. 1–3, 2013. View at Publisher · View at Google Scholar · View at Scopus
  19. S. S. Ray, A. Atangana, S. C. O. Noutchie, M. Kurulay, N. Bildik, and A. Kilicman, “Fractional calculus and its applications in applied mathematics and other sciences,” Mathematical Problems in Engineering, vol. 2014, Article ID 849395, 2 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  20. A. Pugliese and A. Gandolfi, “A simple model of pathogen–immune dynamics including specific and non-specific immunity,” Mathematical Biosciences, vol. 214, no. 1-2, pp. 73–80, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. C. Li, D. Qian, and Y.-Q. Chen, “On Riemann-Liouville and Caputo derivatives,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 562494, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  22. W. Deng, C. Li, and Q. Guo, “Analysis of fractional differential equations with multi-orders,” Fractals, vol. 15, no. 2, pp. 173–182, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. M. Zaid, “Odibat analytic study on linear systems of fractional differential equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1171–1183, 2010. View at Google Scholar
  24. N. D. Cong, T. S. Doan, S. Siegmund, and H. T. Tuan, “Linearized asymptotic stability for fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 39, pp. 1–13, 2016. View at Publisher · View at Google Scholar · View at Scopus
  25. L. J. S. Allen, An Introduction to Mathematical Biology, Pearson Education, London, UK, 2007, ISBN 0-13-035216-0.
  26. H. El-Saka and A. El-Sayed, Fractional Order Equations and Dynamical Systems, Lambert Academic Publishing, Saarbrücken, Germany, 2013, ISBN 978-3-659-40197-8.
  27. S. Arshad, D. Baleanu, J. Huang, Y. Tang, and M. M. Al Qurashi, “Dynamical analysis of fractional order model of immunogenic tumors,” Advances in Mechanical Engineering, vol. 8, no. 7, pp. 1–13, 2016. View at Publisher · View at Google Scholar · View at Scopus
  28. F. Bozkurt, “Mathematical modeling and stability analysis of the brain tumor glioblastoma multiforme (GBM),” International Journal of Modeling and Optimization, vol. 4, no. 4, pp. 257–262, 2014. View at Publisher · View at Google Scholar
  29. A. Eladdadi, P. Kim, and D. Mallet, “Mathematical models of tumor-immune system dynamics,” in Proceedings in Mathematics & Statistics, p. 107, Springer, Berlin, Germany, 2014. View at Google Scholar
  30. T. Kostova, “Persistence of viral infections on the population level explained by an immunoepidemiological model,” Mathematical Biosciences, vol. 206, no. 2, pp. 309–319, 2007. View at Publisher · View at Google Scholar · View at Scopus
  31. I. N’Doye, H. Voos, and M. Darouach, “Chaos in a fractional-order cancer system,” in Proceedings of European Control Conference (ECC), pp. 171–176, Strasbourg, France, 2014.
  32. A. Sohail, S. Arshad, and S. Javed, “Numerical analysis of fractional-order tumor model,” International Journal of Biomathematics, vol. 8, no. 5, Article ID 1550069, 2015. View at Publisher · View at Google Scholar · View at Scopus
  33. F. A. Rihan, A. Hashish, F. Al-Maskarai et al., “Dynamics of tumor-immune system with fractional-order,” Journal of Tumor Research, vol. 2, no. 1, pp. 109–115, 2016. View at Google Scholar