Analyzing Interval Systems of Human T-Cell Lymphotropic Virus Type I Infection of CD4+ T-Cells

Analyzing Interval Systems of Human T-Cell Lymphotropic Virus Type I Infection of CD4+ T-Cells

Zohreh Dadi (University of Bojnord, Iran)
DOI: 10.4018/978-1-5225-2515-8.ch006
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Abstract

Human T-cell lymphotropic virus type I (HTLV-I) infects a type of white blood cell called a T lymphocyte. HTLV-I infection is seen in diverse region of the world such as the Caribbean Islands, southwestern Japan, southeastern United States, and Mashhad (Iran). This virus is the etiological agent of two main types of disease: HTLV-I-associated myelopathy/tropical spastic paraparesis and adult T cell leukemia. Also, the role of HTLV-I in the pathogenesis of autoimmune diseases such as HTLV-I associated arthropathy and systemic lupus erythematosus is under investigation. In this chapter, the author considers an ODE model of T-cell dynamics in HTLV-I infection which was proposed by Stilianakis and Seydel in 1999. Mathematical analysis of the model with fixed parameters has been done by many researchers. The author studies dynamical behavior (local stability) of this model with interval uncertainties, called interval system. Also, effective parameters in the local dynamics of model are found. For this study, interval analysis and particularly of Kharitonov's stability theorem are used.
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Introduction

Recently, attention of many researchers has been attracted to the study of population dynamics of infectious diseases, such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), and human T cell lymphotropic virus type I (HTLV-I) (Arafa, Rida, & Khalil, 2011; Asquith & Bangham, 2008; Asquith et al., 2005; Atay, Başbük, & Eryılmaz, 2016; Bangham, 2000; Bangham & Osame, 2005; Bangham et al., 2009; Blattner et al., 1982; Blattner et al., 1983; Bofill et al., 1992; Cai, Li, & Ghosh, 2011; Cann & Chen, 1996; Chiavetta et al., 2003; Dadi & Alizade, 2016; DeBoer & Perelson, 1998; Elaiw, 2010; Eshima et al., 2009; Eshima, Tabata, Okada, & Karukaya, 2003; Gokdogan & Merdan, 2011; Gómez-Acevedo & Li, 2005; Katri & Ruan, 2004; Lang, 2009; Lang & Li, 2012; Lim & Maini, 2014; Mortreux, Gabet, & Wattel, 2003; Mortreux, Kazanji, Gabet, de Thoisy, & Wattel, 2001; Murphy et al., 1991; Nelson, Murray, & Perelson, 2000; Nowak & Bangham, 1996; Oguma, 1990; Olavarria, Gomes, Kruschewsky, Galvão-Castro & Grassi, 2012; Perelson & Nelson, 1999; Poiesz, 1980; Ramirez, Cartier, Torres, & Barria, 2007; Ribeiro, Mohri, Ho, & Perelson, 2002; Richardson et al., 1997; Richardson, Edwards, Cruickshank, Rudge, & Dalgleish, 1990; Robbins, 2010; Seigel, Nash, Poiesz, Moore, & O'Brien, 1986; Seydel, & Kramer, 1996; Seydel & Stilianakis, 2000; Shirdel et al., 2013; Song & Li, 2006; Stilianakis & Seydel, 1999; Sun & Wei, 2013; Tortevoye, Tuppin, Carles, Peneau, & Gessain, 2005; Vieira, Cheng, Harper, & Senna, 2010; Wang, Fan, & Torres, 2010; Wang, Li, & Kirschner, 2002; Wattel, Vartanian, Pannetier, &Wain-Hobson, 1995; Williams, Fang, & Slamon, 1988; Yamamoto, Okada, Koyanagi, Kannagi, & Hinuma, 1982; Yu, Nieto, Torres, & Wang, 2009).

Key Terms in this Chapter

Kharitonov’s Stability Theorem: Kharitonov’s stability theorem discusses on Hurwitz stability of an interval polynomial by using interval analysis. This theorem shows that Hurwitz stability of an interval polynomial is equivalent to Hurwitz stability of four polynomials with fixed coefficients.

Infection Equilibrium Point: An equilibrium point is called infection equilibrium point if the population of four subsets of CD4 + T cells (susceptible, latently infected, actively infected and leukemia cells) is not zero.

Equilibrium Point: Let be an autonomous differential equation. If f ( x 0 ) = 0, then x 0 is called an equilibrium point. This means that there is a constant function which .

Infection-Free Equilibrium Point: An equilibrium point is called infection-free equilibrium point if the number of latently infected cells, actively infected cells and leukemia cells are zero.

Modeling: The modeling of HTLV-I infection is based on the assumption cell-to-cell infection of CD4 + T cells. The model of HTLV-I infection introduces the dynamics between four subsets of CD4 + T cells; susceptible, latently infected, actively infected and leukemia cells.

Interval System: A dynamical system with interval parameters is called interval system.

Human T-Cell Lymphotropic Virus Type I: Human T-Cell Lymphotropic Virus Type I is the first human retrovirus and the etiological agent of HTLV-I associated myelopathy/tropical spastic paraparesis and adult cell leukemia.

Leukemic Equilibrium Point: An equilibrium point is called leukemic equilibrium point if the number of latently infected cells and actively infected cells are zero, but the population of leukemia cells is not zero.

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