Global Dynamics of an Immunosuppressive Infection Model Based on a Geometric Approach

Global Dynamics of an Immunosuppressive Infection Model Based on a Geometric Approach

Zohreh Dadi (University of Bojnord, Iran)
DOI: 10.4018/978-1-5225-2515-8.ch007
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By clinical data, drug treatment sometimes is ineffective to eradicate the infection completely from the host in some human pathogens such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), and human T cell lymphotropic virus type I. Therefore, mathematical modeling can play a significant role to understand the interactions between viral replication and immune response. In this chapter, the author investigates the global dynamics of antiviral immune response in an immunosuppressive infection model which was studied by Dadi and Alizade (2016). In this model, the global asymptotic stability of an immune control equilibrium point is proved by using the Poincare–Bendixson property, Volterra–Lyapunov stable matrices, properties of monotone dynamical systems and geometric approach. The analysis and results which are presented in this chapter make building blocks towards a comprehensive study and deeper understanding of the dynamics of immunosuppressive infection model.
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In mathematical modeling of biological phenomena, one of complex systems is the immune system. The immune system is human body primary defense against pathogenic organisms and cells that have become malignantly transformed. Also, it involves multiple cell types and hundreds of soluble mediators and different receptor ligand interactions. The immune system can produce different types and intensity of responses, learn from experience and exhibit memory. Therefore, the modeling of immune system needs the knowledge about cells, molecules, and genes that make up that. This knowledge is based on the Human Genome Project progresses which uncover the genes and molecules that influence the behavior of single lymphocytes.

It is important to understand the behavior of the cells of immune system and how every cell of immune system interacts with other cells to generate an immune response. Furthermore, it should be noted that modeling in immunology is in contrast to the field of neurophysiology. In fact, the behavior of a single cell of immune system is not described in immunology. This means that the equivalents of the Hodgkin-Huxley (1952) equations do not exist in immunology. Hence, understanding interactions among the elementary components of a system in immunology is the major part in modeling. Quantitative results which are obtained by mathematical modeling can help researchers to modify and complete their understanding of immunological phenomena, (Barnes et al., 2002; Bekkering, Stalgis, McHutchison, Brouwer, & Perelson, 2001; Borghans, De Boer, Sercarz, & Kumar, 1998; Borghans, Noest, & De Boer, 1999; Borghans, Taams, Wauben, & De Boer, 1999; Butler & Waltman, 1986; Canabarro, Gléeria, & Lyra, 2004; Celada & Seiden, 1992; Celada & Seiden, 1996; Chun et al. 1997; Coppel,1965; Dadi & Alizade, 2016; De Boer & Perelson,1993; Detours & Perelson, 1999; Detours & Perelson, 2000; Diepolder et al. 1998; Fenton, Lello, & Bonsall, 2006; Hlavacek, Redondo, Metzger, Wofsy, & Goldstein, 2001; Kalams & Walker, 1998; Kepler & Perelson, 1993; Kesmir & De Boer, 1999; Komarova, Barnes, Klenerman, & Wodarz, 2003; Lang & Li, 2012; Lechner et al., 2000; Lechner et al., 2000b; Lewin et al. 2001; Li & Shu, 2010a, 2010b, 2011, 2012; Lifson et al., 2000, 2001; Liu & Wang, 2010; Lohr et al., 1998; Maini, & Bertoletti, 2000; McKeithan, 1995; McLean, Rosado, Agenes, Vasconcellos, & Freitas, 1997; Mukandavire, Garira, & Chiyaka, 2007; Nelson, Murray, & Perelson, 2000; Nelson & Perelson, 2002; Neumann et al., 1998; Ortiz et al. 2002; Percus, J.K., Percus, O.E. & Perelson, 1993; Perelson, 2002; Perelson, & Oster, 1979; Perelson & Weisbuch, 1997; Pugliese & Gandolfi, 2008; Rosenberg et al., 2000; Segel & BarOr, 1999; Shamsara, Mostolizadeh, & Afsharnezhad, 2016; Shu, Wang, &Watmough, 2014; Smith, Forrest, Ackley, & Perelson, 1999; Tam, 1999; Wang, Wang, Pang, & Liu, 2007; Whalley et al., 2001; Wodarz et al., 2000; Wodarz & Nowak, 1999; Zhu & Zou, 2008).

Key Terms in this Chapter

Monotone Dynamical Systems: Let X be a metric space with an ordering A dynamical system with flow on X is called monotone if this order is preserved by the flow: for each .

Immune Control Equilibrium Point: An equilibrium point of immunosuppressive infection model is called immune control equilibrium point if the number of antiviral immune responses is not zero.

Poincare-Bendixson Property: Let be an autonomous differential equation, f be a C 1 function on D , and D be an open set. If any nonempty compact omega limit set of this system which does not contain any equilibrium points is a closed orbit, then this system satisfies the Poincare-Bendixson property. This property is used to determine the global behavior of bounded trajectories of the system.

Volterra-Lyapunov Stable: A square matrix A is Volterra-Lyapunov stable if there exists a positive diagonal matrix M such that MA + A T M T is symmetric negative definite. Volterra-Lyapunov matrix properties can be used for the studying dynamical behavior of nonlinear models.

Equilibrium Point: Let be an autonomous differential equation. If , then is called an equilibrium point. This means that there is a constant function which .

Geometric Approach: Geometric approach is the approach which is based on Poincare-Bendixson property for studying the global stability of equilibrium points in a nonlinear autonomous differential equation.

Global Stability: Let where D is the feasible region of this system. Also, consider as an equilibrium point. This point is globally stable if the omega limit set of every point of D is .

Immunosuppressive Infection Model: An ordinary differential equations system which shows the dynamics between immunosuppressive infection and antiviral immune responses is called immunosuppressive infection model.

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