Journal of Applied Mathematics

Volume 2015, Article ID 378753, 9 pages

http://dx.doi.org/10.1155/2015/378753

## A Fuzzy Delay Approach for HIV Dynamics Using a Cellular Automaton

^{1}Faculty of Mathematics, Federal University of Uberlândia, 38408-100 Uberlândia, MG, Brazil^{2}Faculty of Mechanical Engineering, Federal University of Uberlândia, 38408-100 Uberlândia, MG, Brazil^{3}Department of Applied Mathematics, IMECC, State University of Campinas, 13083-859 Campinas, SP, Brazil

Received 3 March 2015; Revised 12 May 2015; Accepted 14 May 2015

Academic Editor: Jianming Zhan

Copyright © 2015 R. Motta Jafelice et al. 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

The objective of this research is to study the evolution of CD4+ T lymphocytes infected with HIV in HIV-seropositive individuals under antiretroviral treatment utilizing a mathematical model consisting of a system of delay-differential equations. The infection rate of CD4+ T lymphocytes is a time-dependent parameter with delay. Such delay is given by a fuzzy number due to the uncertainty of the effects of both pharmacological and intracellular delays. A cellular automaton is utilized to estimate the parameters of the system. The effects of antiretroviral therapy in the cellular automaton are modeled using a fuzzy rule-based system with two inputs: the medication potency and the treatment adhesion for three hypothetical individuals. For each of them, we determine the infection rate of CD4+ T lymphocytes, which is different from zero, as opposed to other studies reported in the literature. As the infection rate is considered a fuzzy parameter, we determine the fuzzy and the defuzzified solutions for the infected CD4+ T lymphocytes. We obtain the maximum values of infected cells for individuals that receive low, medium, and high potency medication and treatment adhesion. The results obtained are in accordance qualitatively with what would be expected in a real situation.

#### 1. Introduction

AIDS (acquired immunodeficiency syndrome) is an infectious and contagious disease caused by HIV (human immunodeficiency virus) which leads to a progressive loss of infected individuals’ immunity. This syndrome is characterized by signals and symptoms which are due to the reduced rate of T lymphocytes within the bloodstream which are cells of paramount importance for the perfect function of the human immune system. Since HIV was identified, AIDS became a world epidemic of great proportions. Experts from the United Nations estimate that in 2013 about 35 million people were contaminated worldwide with HIV. In Latin American countries 1.6 million people are infected, which indicates that they face a biologic problem of great social and economic repercussion [1]. The virus produces three proteins that are of paramount importance for the replication process: reverse transcriptase, integrase, and protease. When the virus reaches the bloodstream, it attacks mainly the T lymphocytes. The quantity of T lymphocytes in peripheral blood presents important prognostic implication for the evolution of the viral infection. The antiretroviral treatment works through the inhibition of those three proteins. The inhibitors prevent free virus particles from infecting T lymphocytes delaying viral replication and allowing the organism to react naturally. The inhibition of these three enzymes has led to a substantial improvement in HIV therapy [2].

In the last decades, fuzzy set theory has contributed significantly to the mathematical modeling of uncertain phenomena such as in the case of the infection by HIV [3–5]. Zarei et al. [6] proposed a fuzzy mathematical model of HIV infection which consisted of a linear fuzzy differential equation system that describes the level of immune cells and the viral load due to the inherent fuzziness of the immune system’s strength. The authors also proposed a fuzzy optimal control problem that minimizes both the viral load and the systemic cost for the body.

The deterministic mathematical models described by delay-differential equations are sometimes more realistic than the ones with no delay. Herz et al. [7] developed a deterministic model to describe the behavior of uninfected and infected T lymphocytes and the plasma HIV load incorporating pharmacological and intracellular delays in individuals under antiretroviral therapy. Such delays are defined as, respectively, the time span during which the drugs reach an effective concentration and the time span between cell infection and the production of new virus particles. They also demonstrate that intracellular delay may affect the magnitude of the range of HIV decline causing variability or uncertainty in the response to the therapy with protease inhibitors. In many cases the equation parameters are uncertain and such equations should be reinterpreted. Traditionally such uncertainty has been treated by means of statistical methodologies. Mittler et al. [8] assume that the delay is characterized by a probability distribution. The adoption of distributed delay produced good results in the study of HIV dynamics under treatment. The authors above provide simulations close to the biological information given by Herz et al. (1996) in which the different phases of decay of the virus after treatment with antiretrovirals are represented graphically.

Jafelice et al. [9], using the model proposed by Herz et al. [7], modeled the virus rate decline in function of the delay at the onset of the treatment. The delay was considered a fuzzy parameter; therefore, a fuzzy solution was determined for the decline of the virus.

Cellular automata allow simulation of complex nonlinear dynamic systems simultaneously in time and space. The state of each cell of the automaton is modified according to its own state and the states of the cells in its neighborhood. These states correlate amongst themselves through simple rules that imitate the biological and physical laws which rule the behavior of the system [10]. Jafelice et al. (2009) simulated a cellular automaton, shaped like a torus, in which artificially uninfected and infected T lymphocytes, free virus particles, and virus-specific cytotoxic T lymphocytes (CTLs) that attack infected cells within an individual undergoing antiretroviral therapy coexist.

The objective of this research is to propose a model of the evolution of infected T lymphocytes in HIV-seropositive individuals under antiretroviral treatment. From a cellular automaton [11] it is possible to obtain an approximate function of the infection rate coherent with the HIV dynamic behavior. In this paper we study HIV dynamics by means of a system of differential equations with uncertain delay in three hypothetical individuals submitted to three levels of treatment adhesion and three medication potencies. The parameters and the infection rate values for this system are obtained from the cellular automaton. The infection rate values are adjusted by the least squares method. Afterward we determine the numeric solution of the fuzzy model considering the delay as a triangle fuzzy number. Finally, we attain the defuzzified numeric solution for this model for each one of the three individuals. A different aspect of our research is that we consider the infection rate as different from zero, whereas in the model by Herz et al. (1996), set for simplification, such rate was equal to zero. Considering the infection rate equal to zero implies that the antiretroviral treatments are 100% effective, a situation that does not occur in reality since there is no definitive control of AIDS.

#### 2. Methodology

##### 2.1. Basic Concepts of Fuzzy Set Theory

First we define a* fuzzy subset * of given from a function , called* the membership function*, which is associated with the fuzzy set , where is a nonempty set. The value represents the degree of membership of . A* classic subset * of is a particular fuzzy set for which the membership function is the characteristic function [12].

The -cuts of the fuzzy set are denoted by and defined as , ; , where is the support of . A fuzzy set is called a* fuzzy number* when and all -cuts of are nonempty, all -cuts of are closed intervals of , and the support of is bounded.

Secondly we remind the reader that fuzzy rule-based systems (FRBS) is a concept that plays a key role in fuzzy set theory. They are broadly used in model problems, control, and classification [13] and may be considered as “expert systems” due to their basic composition that includes an inference machine and a rule base, both under the direct influence of human expert knowledge [14]. Four components comprise a FBRS: an input processor that performs the fuzzification of the input data, a collection of nebulous rules called rule bases, an inference machine, and an output processor that provides a real number. Once rule bases of the type “If…then…” are established, a FRBS can be understood as a mapping between an input and an output of the formula , and .

Thirdly a concept used also in this paper is Zadeh’s Extension Principle. Its role is to obtain images of fuzzy sets via classic functions.

Let and be two nonempty universal sets and a function . Given a fuzzy set , the fuzzy set , with membership function given byis called Zadeh’s Extension Principle of by . Note that if is a classical set of .

##### 2.2. Classic Models for HIV Dynamics

Herz et al. [7] present a model which incorporates an antiretroviral treatment that contains three time-dependent variables: the population of uninfected cells , the population of infected cells that produce virus , and the plasma viral load , the three of them dying at the respective rates of , , and . In this model it is assumed that uninfected cells are continually produced by the human body at a constant influx . The infected cells produce virus particles at rate . In order to incorporate the virus life cycle in the intracellular phase of the model, Herz et al. [7] assume a time delay for virus production after cell infection. This implies that the recruitment of virus-producing cells at time is given by the density of cells that were newly infected at time and are still alive at time . Moreover, we assumed a constant death rate for infected but not yet virus-producing cells. The probability of survival from to time is only . More generally, the probability of survival is given by a nonincreasing function with . Thus, the model can be written as

Based on the work by [15], we include in system (2) an equation for CTL as variable and a mortality rate caused by CTLs of infected cells in the second equation, as follows:The rate of CTL proliferation in response to antigen is . In the absence of simulation CTLs decay at rate . The qualitative behavior of the HIV infection rate in relation to the lymphocyte T is also studied. The variables and parameters are dimensionless since the system of delay-differential equations (3) is solved numerically according to the parameter values and initial conditions of the variables which are further described in Section 3.1.

In the next subsection we depict the cellular automaton and the FRBS used for attaining the parameters of the system.

##### 2.3. Cellular Automaton and Fuzzy Rule-Based System

In the cellular automaton (CA) described by Jafelice et al. [11], uninfected and infected T lymphocytes, free virus particles, and virus-specific CTL (cytotoxic T lymphocyte) coexist in the bloodstream of an individual under antiretroviral treatment. The CA is two-dimensional with Moore neighborhood [16] and periodic (toroidal) boundary [17]. The cells move in eight directions: north, east, south, west, northeast, northwest, southeast, and southwest. The changes in the states of the cells are implemented immediately; therefore the updating is asynchronous and at random. The CA model utilizes the output of the FRBS for the simulation.

The input variables of the FRBS are the adhesion to treatment and the medication potency (efficacy). The following linguistic values attributed to the variables are* very low*,* low*,* medium*,* high*, and* very high*. For adhesion to treatment, the interval is defined where 0 means none and 1 total adhesion. Based on [18] we define the medication potency interval as . The output variables are the percentage of infected T lymphocytes and the period of virus replication. For the former the linguistic terms are* very low*,* low*,* medium*,* high*, and* very high* and for the latter are* very rapid*,* rapid*,* medium*,* slow*, and* very slow*. For percentage of infected T lymphocytes the interval is and for the period of virus replication is . The lower limits of these two intervals correspond to the numbers of the iterations under the best treatment and the upper limits to the numbers of the iterations under no treatment [11]. The fuzzy rule base is constructed based on expert medical knowledge (Tables 1 and 2) utilizing the Mamdani inference method with centroid defuzzification [13].