Estimation of the HIV-1 infection rate and the basic reproductive ratio
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Estimation of the HIV-1 infection rate and the basic reproductive ratio Nara Bobko1 · Jorge Passamani Zubelli2
Received: 29 December 2016 / Revised: 17 July 2017 / Accepted: 18 August 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract This paper proposes and analyzes an alternative method to estimate important parameters of the HIV-1 infection: the infection rate and the basic reproductive ratio (in the presence and absence of the immune response). These parameters are of great relevance, impacting on the infection chronicity. The method consists in comparing clinical data in the chronic stage with predicted equilibrium points of a mathematical model which describes the dynamics of HIV-1 in the acute infection phase. The fact that the method uses only clinical data relating to the chronic stage is an important aspect since most patients are diagnosed at this stage, and thus only have data available for this stage. A comparison between the proposed method and other estimation methods, as well as between numerical simulations, is presented and the results are quite satisfactory. In addition, by applying the proposed method for data of 31 patients, we estimate the basic reproductive ratio in the presence of the immune response and the HIV-1 infection rate for which we obtained a median of 1.95 (IQR 0.55) and 1.96 × 10−5 mm3 virions−1 day−1 (IQR 3.52 × 10−5 ), respectively. Keywords HIV-1 dynamics · Parameters’ estimation · Infection rate · Basic reproductive ratio · Global stability Mathematics Subject Classification Primary 92C50; Secondary 34D23
Communicated by Eduardo Souza de Cursi.
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Jorge Passamani Zubelli [email protected] Nara Bobko [email protected]
1
Universidade Tecnológica Federal do Paraná, Curitiba, Brazil
2
IMPA, Rio de Janeiro, Brazil
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N. Bobko, J. P. Zubelli
1 Introduction The human immunodeficiency virus type 11 (HIV-1), causes the depletion and dysregulation of the immune system and leads to life-threatening tumors and opportunistic infections (Douek et al. 2009; Cohen et al. 2008; Munier and Kelleher 2006). Due to the high complexity and severity, the HIV-1 infection has been the subject of intense research. But some aspects of its dynamics remain unknown and, therefore, requires global focus and investment (Cohen et al. 2008). Mathematical models of the HIV-1 infection constitute a valuable means to study the complex dynamics involving the viral particles and the cells of the infected organism (Ho et al. 1995; Perelson et al. 1996; Gilmore et al. 2013; Nowak and May 2000; Asquith and Bangham 2003), being an important ally to clinical research. Many important features of its dynamics have been understood through the analysis of these models (Perelson et al. 1993; Perelson and Nelson 1999; Frost and McLean 1994; Kirschner 1996; Nelson and Perelson 2002; Nowak and May 2000; Korobeinikov 2004; Souza and Zubelli 2011; Pastore 2005; Bobko 2010; Bonhoeffer et al. 1997; Smith and Leenheer 2003; Wang and Song 2007; Stafford et al. 2000; Bobko and Zub
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