Interval analysis of the HIV dynamics model solution using type-2 fuzzy sets

Abstract

The objective of this study is to analyze the numerical intervals of the Human Immunodeficiency Virus (HIV) using the solution of a dynamics model at each time iteration, considering the infection rate of $CD4+$ T lymphocytes and the production rate of the virus as outputs of an interval type-2 Fuzzy Rule-Based System (FRBS). The mathematical model consists of a system of ordinary differential equations in the case of the HIV-seropositive individuals who are under antiretroviral treatment. The input variables of FRBS are adherence to treatment and medication potency, which are important factors in the treatment of HIV-seropositive individuals. If the infection rate of $CD4+$ T lymphocytes and the production rate of the virus are output variables of a type-1 FRBS, then we obtain curves for the solution of the HIV dynamic model. When the rates are outputs obtained through the interval type-2 FRBS, we can obtain ranges. This means that a type-2 FRBS provides more information about the behavior of the uninfected and infected $CD4+$ T lymphocytes, free virus particles, and virus-specific cytotoxic T lymphocytes (CTL) that attack infected cells in the bloodstream of a HIV-seropositive individual. The theory of the single level constraint interval arithmetic allows us to obtain a single interval that containing the intervals of all iterations for uninfected and infected $CD4+$ T lymphocytes, free virus particles, and CTL. Thus, the novel contribution of this research is twofold. Firstly, the use of interval type-2 fuzzy rule-base analysis to obtain more informative and robust results not possible with standard approaches is new.  Secondly, the use of the fuzzy constraint interval representation for fuzzy type-2 intervals in the context of the type-2 fuzzy rule base is new.  Moreover, constraint intervals unify the theory and its use in single-level constraint representations simplifies the calculations, though this aspect is not pursued in detail here.

Introduction

Acquired Immunodeficiency Syndrome (AIDS) is a set of symptoms caused by the HIV. According United Nations [39], the latest statistics on the status of the AIDS epidemic, affirm that, in 2019, there were 38 million people living with HIV worldwide. In this same year, the statistics show that around 67% of all people living with HIV had access to antiretroviral treatment. This treatment inhibits the function of the virus proteins. The important proteins for the replication process of HIV are Reverse Transcriptase, Integrase and Protease. These inhibitors prevent free virus particles to infect $CD4+$ T lymphocytes and they delay the viral replication. The $CD4+$ T lymphocytes are the main cells attacked by HIV when these reach the bloodstream [12]. Mathematical modeling of diseases that cause obstructions in blood flow and other pathologies, which are extremely complex systems, typically use differential equations [1].

This research uses parameters of a system of ordinary differential equations obtained as outputs of type-1 FRBS and of interval type-2 FRBS via single level constraint interval arithmetic. Fuzzy set theory originated in 1965 with Zadeh’s publication [43]. This theory has been used in many mathematical analyses of imprecise phenomena, as for example in [2], [13], [14] and [16], among others. Increasingly, the theory of fuzzy sets has been applied in a variety of areas, including in the uncertainties of controlled robotic systems that have provided promising results [34] and [20]. To mathematically model problems that require input data and structures based on opinions of experts or when graduality is present, Zadeh, in 1975, introduced some fundamentals of fuzzy type-2 set theory [44]. In the late 1990s the research of Karnik and Mendel presented a complete theory of type-2 FRBS, including operations, type-reducer and defuzzification methods. Recall that defuzzification are mappings from fuzzy sets to real numbers. In [21] is found the theory of interval type-2 SBRF, with several applications. However, the execution of the type-reducer method has a very high computational cost. In [40] computational algorithms that improved this negative aspect were presented. Currently, many studies have been carried out with this theory, including medical diagnoses [31], [45] and [38], among others. In other areas of knowledge, this methodology has been applied, showing its potential and performance in results [5], [6], [24], [32], [37] and [23].

We note that the use of type-2 FRBS in HIV analysis places the mathematical modeling process closer to how this medical model occurs in reality. Clearly, many different mathematical HIV models provide insight into the complexity found in the medical process. However, it is our thesis that our type-2 FRBS model is not only closer to the way the problem is found, but also provides a richer set of resultant, output, information as will be demonstrated in the sequel. Thus, the novel contribution of this research is twofold. Firstly, the use of interval type-2 fuzzy rule-base analysis to obtain more informative and robust results not possible with standard approaches is new.  Secondly, the use of the fuzzy constraint interval representation for fuzzy type-2 intervals in the context of the type-2 fuzzy rule base is new.  Furthermore, constraint intervals unify the theory and its use in single-level constraint representations simplifies the calculations, though this aspect is not presented in detail in this work. Moreover, fuzzy type-2 interval, in our view, capture the uncertainty associated with the parameters of the model and the acquisition of the data. The relationships between cause (not taking mediation at the intervals specified time or skipping one or two times, for example) and effect (decreased immunity, increase vulnerability to various diseases, for example) are not crisp relationships nor reliably modeled by a single-valued membership function.  Thus, type-2 fuzzy intervals are used in this study since they encode this characteristic of the problem, data, and relationships.

The other mathematical theory used in this study is the interval arithmetic. One of the founders of this arithmetic was R. E. Moore. Moore developed interval arithmetic, which is the usual approach to interval arithmetic we call standard interval arithmetic (SIA) and it is the approach to interval arithmetic in common use [29]. In 1999, Lodwick defined the notion of interval arithmetic called of Constraint Interval Arithmetic (CIA). This arithmetic redefines an interval as a real value function of one variable [22]. Subsequently, [8] introduced a subset of arithmetic proposed by Lodwick (1999) in which the same level for each interval involved in the operations is used. This new arithmetic is Single Level Constraint Interval Arithmetic (SLCIA) and simplifies calculations arising from CIA.

The aim of this research is to study the behavior of the uninfected and infected $CD4+$ T lymphocytes, free virus particles, and virus-specific cytotoxic T lymphocytes (CTL) that attack infected cells in the bloodstream of a HIV-seropositive individual under antiretroviral treatment. The outputs of a interval type-2 Fuzzy Rule-Based System (FRBS) are the infection rate of CD4+ T lymphocytes and the production rate of virus. The input variables of FRBS are levels of adherence to treatment and medication potency. In [42] a fuzzy discrete event system scheme modeling HIV is presented. The four clinical parameters relevant to the approach of this model are decisions of the antiretroviral therapy, as follows: anticipated potency of the regimen; anticipated adherence for the patient under the regimen; prognosis for adverse events under the regimen; and expected future drug options due to the potential for development of resistance to the regimen. Of these four factors, two directly influence the infection rate of $CD4$ T lymphocytes by virus and the production rate of virus; they are adherence to treatment and medication potency. Our model is distinct from other research. In particular this study, the rates are outputs obtained through interval type-2 FRBS. Moreover, we use constraint intervals to encode interval type-2 entities.

The constraint interval (CI) representation is a particularly useful approach to model interval-valued uncertainty. This is because the uncertainty associated with the data is symbolically propagated in the modeling processallowing for:

• Computation of the associated uncertainty in results in a direct way;

• Defuzzification to be the last step since the uncertainty is carried through the calculations, symbolically via the uncertainty parameter, as will be seen.

What is meant by the second point above is that since constraint intervals propagate the uncertainty via a parameter, the results will have the resultant uncertainty encoded, so that defuzzification does not have to occur until necessary, which is the last step of the analysis.

A subset of constraint interval representation is the single level constraint interval representation [8].  It uses just one parameter for all the distinct variables instead of a different parameter for each distinct variable.  This simplifies our analysis.  The properties and algebraic structure of SLCI can be found in [8].  The associated arithmetic and mathematical analysis using SLCI, CI, and standard interval representations is denoted by SLCIA, CIA, SIA respectively.  What is important is for this research is that

$$SLCIA\subset CIA\subset SIA.$$

The resulting output from a SLCI analysis is a “swath” which is quite informative in that it represents the range of possible solutions given uncertainty in the input data.  SLCI is particularly useful in this regard.

Our fuzzy type-2 constraint interval representation uses $SLCIA$ at each $\alpha -level$.  Fuzzy type-2 fuzzy numbers are able to capture and encode the uncertainty that is typical of the way experts’ verbal descriptions and definitions are made describing a HIV patient’s response to treatment or lack thereof. Experts in our case, are physicians and researchers who deal with HIV in this case. For example, “at least half” of the patients who do not take their prescription doses for “less than 50% of the time” will have “adverse effects” is not an uncommon description coming from an HIV physician. This new fuzzy type-2 approach provides more information about the behavior of the uninfected and infected $CD4+$ T lymphocytes, free virus particles, and virus-specific cytotoxic T lymphocytes (CTL) that attack infected cells in the bloodstream of a HIV-seropositive individual than the outputs obtained through the type-1 FRBS. Also, we compute a single interval that containing the intervals of all iterations for uninfected and infected $CD4+$ T lymphocytes, free virus particles, and CTL.

This paper is organized as follows: in Section 2 fundamental concepts are presented; in Section 3 we present the model of HIV dynamics; in Section 4 we explain the model fuzzy model to HIV dynamics; in Section 5 we present the results and finally in Section 6 we state the conclusion.