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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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).