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This paper aims at the development of an approach integrating the fuzzy logic strategy for a therapeutic hepatitis C virus dynamics optimal control problem. To test the efficiency of this strategy, the authors propose a numerical comparison with the direct method by taking the values of determinant parameters of this disease for people administrating the drugs. The results are in good agreement with experimental data.

In the 1970s and 1980s, serological tests developed hepatitis A virus (HAV) and B viruses (HBV) which indicated that most transfusion-associated hepatitis was not caused by either HAV or HBV, which was therefore named non-A, non-B hepatitis (NANBH). After detection of the first NANBH-specific clone, the entire viral genome of the now termed hepatitis C virus (HCV) was sequenced. Hepatitis C is a liver infection caused by the Hepatitis C virus (HCV). HCV is a small positive-strand ribonucleic acid (RNA) virus in the Flaviviridae family where it forms its own genus hepacivirus [

HCV can be transmitted by transfusion of blood and blood products, transplantation of solid organs from infected donors, injection drug abuse, unsafe therapeutic injections, and occupational exposure to blood (primarily contaminated needles) [

To date, an active or passive vaccination against HCV is not yet available. The main factor that hampers the development of an efficient vaccine is the considerable genetic heterogeneity of this positively-stranded RNA virus. However, better understanding of the natural immunity to HCV and the proof of vaccine efficacy in the chimpanzee challenge model allow some optimism about the development of an at least partly effective vaccine against this heterogeneous pathogen [

The ultimate goal of CHC treatment is to reduce the occurrence of end-stage liver disease and its complications including decompensated cirrhosis, liver transplantation, and hepatocellular carcinoma (HCC). However, because progression of liver disease occurs over a long period of time, clinicians use sustained virologic response (SVR), defined as lack of detection of HCV RNA in blood several months after completing a course of treatment, to determine treatment success. SVR is considered a virologic cure [

Mathematical modelling and quantitative analysis of hepatitis C infections has been explored extensively over the last decade. Most of the modelling has been restricted to the short term dynamics of the model. One of the earliest models was proposed by Neumann et al. [

Control theory has found wide ranging applications in biological and ecological problems [

In this paper we are interested in the role of drugs and how they play a crucial role in controlling HCV diseases through a bicompartmental model such that the controls are those drugs. Therefore, the formulation of optimal control problem is done. There are the numerous methods that allow solving this kind of problem. We prefer to make a comparative study of direct method with another approach based on the fuzzy logic strategy.

This paper is organized as follows. Section 2 presents the model equations and optimal control problem. A short description of strategy approach by fuzzy logic for solving optimal control problems is discussed in this section. Section 3 is interested in presentation of the direct approach and the approach integrating the fuzzy logic for solving an optimal control problem of hepatitis C. The numerical simulation is presented in Section 4. Finally, we present concluding remarks in Section 5.

In terms of constraints of our problem, we consider a two compartmental mathematical model proposed in [

This mathematical model is formulated from a diagram given in

The model equations are as follows:

where

If

Find

subject to the system (1) - (2).

The positive scalar coefficients

To describe fuzzy logic strategy approach, we consider a linear quadratic problem which can be formulated as follows. For two positive defined matrices

subject to

The methods with a fast convergence can be used to solve the problem (6) - (7). One of these methods is dynamic programming method which is quadratically convergent. Furthermore, a state of control feedback is an optimal solution for this mathematical method [

If

1) The approximation of order zero which satisfies:

2) Taylor expansion series of first order verifying

To minimize the error between the non linear function and the fuzzy approximation, we introduce the factor of the consequence for fuzzy Takagi-Sugeno system that allows the improvement of approximation (8) or (9). Taking

The following expression is obtained from the replacement of nonlinear term

where

The outcome of approximation of nonlinear term

where

is the feedback gain matrix and

From (12) S controls are determined and the defuzzyfication method [

where

and where

Since explicit Euler scheme is stable, it is used to approximate the optimal control problem (5) subject to (1) - (2). This method is an advantage to approach some ordinary differential equations. The following system is obtained from Taylor expansion around

where

Using the variable change

the system (18) becomes

On the uniform grid and from the system (20) yields the following approximated system

where

To determine the Takagi-Sugeno fuzzy system and using the form given by expression (8) or (9), now we focus on linearization of two nonlinear factors of the system (21) that is

This mechanism is done by taking the points

Let us note

where

Now we are interested in approximating the objective function of the problem (5). Hence, using rectangular method the following expression yields

where matrices

such that the optimal control problem () subject to () - () can be formulated as follows.

Find

subject to

The problem (28) - (29) is a linear quadratic (LQ) such that for each

where

Firstly, the direct method focuses on approximation of the system (1) - (2) on uniform grid

where we define a linear B-splines basis functions of the form

such that we have the relation

where

This approximation is done by introducing the vector space

1)

2)

Assuming

satisfying

we are able to verify easily that

Furthermore, the approximation of the system (1)-(2) can be formulated as follows.

Find

such that

Secondly, the direct method deals with the discretization of the optimal problem (5) which becomes

where

with

Taking

Using rectangular method such that the discretization is done on a regular grid

Therefore, the discrete formulation of optimal problem (5) subject to (1) - (2) is written as follows.

where

Taking the approximated objective function (45) and (37) - (38) satisfying (40) and (41) into account, the optimal control problem (5) subject to (1) - (2) is a minimization problem with constraint such that discreet formulation of such problem can be formulated as follows.

Find

subject to

where

Using the mechanism of linearization of the nonlinear terms of the system (21), we apply the fuzzy strategy and we consider the case of health person where we take

We consider a universe of discourse

According the relation (19) and equilibrium values, we have

Let us set

Variable | Operating Points |
---|---|

H | [−500; 0; 500] |

I | [0; 150; 300] |

It easy to note that the problem (28) - (29) is a linear quadratic (LQ). Since there are three linear state systems, the solution leads to three feedback controls of the form

where

The implementation can be made in several platforms. Here we use MATLAB package. Taking

The defuzzification transformation allows to obtain one system. Consequently, for the system (29) this technique gives the following system

where

In the same way, from the matrixes

The first (respectively second) line of matrixes

i) We consider the degree of membership of the entry uninfected hepatocytes (respectively infected hepatocytes). According to variable change (19), this value is 400 cells/dl (see

[resp.

ii) The nonlinear factor

is used only in the first (respectively second) equation of the system (22).

Considering these hypothesis, we have the following matrixes.

The solutions of the optimal control problem (28) - (29) and (47) - (48) can be determined in several platform. The implementation of these solutions is made using MATLAB packages.

To solve the problem (28) - (29) by fuzzy logic strategy only one program is enough. Using direct approach, the solutions of the problem (47) - (48) are given by a succession of programs based on MATLAB function used in optimization that is fmincon. This function is a MATLAB program which allows solving minimization problem with constraints.

In this section, we note by AHLF, ADIR to designate respectively the hybrid approach integrating fuzzy logic strategy and direct approach. Consequently,

AHLF | ADIR | |
---|---|---|

Jopt | 0.0524 | 14.2379 |

T (second) | 0.0149 | 51.08034 |

MATLAB program for AHLF, ADIR respectively. The results are obtained using a Processor Intel (R) core (TM) 2 Duo CPU, 2.20 GHZ.

Considering a patient who is administrating the drugs during 12 months, the variations of the optimal parameters is obtained using the hybrid approach integrating fuzzy logic strategy (AHLF) and the direct approach (ADIR). Numerical results of fuzzy logic strategy are obtained from the resolution of the problem (28) - (29) such that the controls are calculated using the relation (30). Solving the problem (47) - (48), we have the numerical results of direct approach.

It is known that the main aim of treatment for chronic hepatitis C is to suppress HCV replication before there is irreversible liver damage. Furthermore, the role of drugs on chronic hepatitis C virus is to reduce the risk of liver disease and prevent the patient from passing the infection to others. The variation of controls given in

This work deals with an optimal control problem related to uninfected hepatocytes and infected hepatocytes of hepatitis C virus. To handle that problem, two numerical approaches have been compared to determine the optimal trajectories of these determinant parameters which respond to two controls (interferon and ribavirin) of this disease hepatitis C virus for a patient who is administrating drugs during 12 months. The results show that two used methods are satisfactory and closed. The findings also show that, in terms of time, the hybrid approach integrating the fuzzy logic strategy has an advantage on the direct approach in terms of time. Consequently, it constitutes an important approach for the resolution of the optimal control problem. In particular, it gives the optimal trajectories of uninfected hepatocytes and infected hepatocytes in the same way that it ensures their

performance.

Jean MarieNtaganda,Mahamat Saleh DaoussaHaggar,BenjaminMampassi, (2015) Fuzzy Logic Strategy for Solving an Optimal Control Problem of Therapeutic Hepatitis C Virus Dynamics. Open Journal of Applied Sciences,05,527-541. doi: 10.4236/ojapps.2015.59051