We aimed in this paper to use fuzzy logic approach to solve a hepatitis B virus optimal control problem. The approach efficiency is tested through a numerical comparison with the direct method by taking the values of determinant parameters of this disease for people administrating the drugs. Final results of both numerical methods are in good agreement with experimental data.
Hepatitis B is one of various diseases that are potentially life-threatening liver infection. Abbreviated in terms of HBV [
It has been scientifically found that hepatitis B is transmitted through contact with blood or bodily fluids from an individual infected with the hepatitis B virus (HBV). The virus mainly affects liver function; this is, it invades the liver cells (hepatocytes) and uses the cells’ machinery to replicate within it. The hepatitis B virion binds to the hepatocyte via the domain of the viral surface antigen. The cell then engulfs the virus in a process called endocytosis. As the infection occurs, the host immune response is triggered. The body’s immune system attacks the infected hepatocytes, which lead to liver injury at the same time as clearing the virus from the body. The liver damage associated with HBV infection is mainly caused by the adaptive immune response, particularly the virus-specific cytotoxic T lymphocytes (CTLs). These CTLs kill cells that contain the virus. Liver damage is also aggravated by the antigen-nonspecific inflammatory cells and activated platelets at the site of infection.
There are two possible phases of this infection [
1) Acute hepatitis B infection that lasts less than six months. In this case the immune system is usually able to clear the virus from the body, and the patient should recover completely within a few months. This kind of phase is the most case for adult people who acquire hepatitis B.
2) Chronic hepatitis B infection which lasts six months or longer. This infection manifests in the most infants infected with HBV at birth and many children infected between 1 and 6 years of age become chronically infected.
The chronic carrier people do not develop symptoms; these are taken as two-thirds of people with chronic HBV infection. In the all world more than 240 million people have chronic liver infections and about 600,000 people die every year due to the acute or chronic consequences of hepatitis B [
Mathematical models can be a useful tool in controlling hepatitis B virus in order to put down the infection from the population. It is in the manner that the simple mathematical model has been used by Anderson and May to illustrate the effects of carriers on the transmission of HBV [
Several drug therapies have been proposed for treating persons with chronic HBV including adefovir dipivoxil, alpha-interferon, lamivudine, pegylated interferon, entecavir, telbivudine, and tenofovir [
Optimal control theory has found wide-ranging applications in biological and ecological problems [
In this paper, the optimal control problem is presented and fuzzy logic strategy is used to solve it. It is organized as follows. Section 2 presents the model equations and optimal control problem. A short description of by fuzzy logic for solving optimal control problems is discussed in this section. Section 3 is interested in the application of the direct approach and the approach that integrates the fuzzy logic for solving an optimal control problem of an infected hepatitis B virus dynamics. The numerical simulation is presented in Section 4. Finally, we present concluding remarks in the last section.
In terms of constraints of our problem, the option was put on the model proposed in [
where T, I and V stand for the concentration of uninfected hepatocytes, infected hepatocytes and free virions respectively. In this context uninfected hepatocytes are produced at the constant rate s and die at the rate qT. For the purpose of describing the proliferation of existing T cells, we used the logistic function where a stands for the maximum proliferation rate of target cells and
If
Find
subject to the system (1)-(3).
The positive scalar coefficients
Let’s
The functions
where
Let us consider the following problem.
Find
subject to
where
The problem (7)-(8) can be solved by the dynamic programming method. This method has a fast convergence, its convergence rate is quadratic and the optimal solution is often represented as a state of control feedback [
Let’s consider the set of operating points
1) The approximation of order zero gives:
2) Using the first order of Taylor expansion series we obtain
To improve this approximation, we introduced the factor of the consequence for fuzzy Takagi-Sugeno system. This factor allows to minimize the error between the non linear function and the fuzzy approximation. If
If one replaces the term nl by its value approached in (8), the linearization around
where
Therefore, the optimal control problem (7)-(8) becomes a linear quadratic problem which the feedback control is given by the following expression [
where
is the feedback gain matrix and
Obviously the linearization around every operating point gives the system for which the equations have the form (12). Due to the presence of S operating points, S systems of that form should be formed and therefore according to the relation (13) S controls are determined. The defuzzyfication method [
Then, this transformation gives the following equation:
where
and where
To approximate the optimal control problem (4), (1)-(3), we propose to use the explicit Euler scheme because the stability of this scheme allows deal with some ordinary differential equations.
By letting the following variable change
the system (1)-(3) becomes
The discretization of the constraints (1)-(3) is done using the first order explicit Euler method. The first order of explicit Euler’s method gives following system
where
Using the approximation
that is
Since the system (21) has nonlinear factors let us designate these points as
To simplify, we consider only the Taylor expansion of first order around the operating points
Assuming that
where
and
To approximate the objective function of the problem (4), we use the rectangular method. Hence, we obtain
where
Finally, the optimal control problem (4), (1)-(3) can be formulated as follows.
Find
subject to
It easy to note that the problem (29)-(30) is a linear quadratic (LQ). Since there are three linear state systems, the solution leads to three feedback controls of the form
where
To approximate the system (1)-(3), we consider
a linear B-splines basis functions on the uniform grid
such that
Let us introduce the vector space
1)
2)
Let us consider
satisfying
We verify easily that
Therefore, the system (1)-(3) can be approached by the following form.
Find
such that
The discretization of the optimal problem (4) is done as follows.
where
with
We are looking for
Therefore the cost function (45) becomes
where (47) is determined using rectangular method such that the discretization is done on a regular grid
The discrete formulation of optimal problem (4) subject to (1)-(3) is written as follows.
where
Finally, the optimal control problem (4), (1)-(3) is a minimisation problem with constraint. The discreet formulation of such problem can be written as follows.
Find
subject to
where
Taking in account of the mechanism of linealisation of the nonlinear terms of the system (20), we applied the fuzzy approach where the concentration of uninfected hepatocytes for health person has been considered to be
If T, I and are respectively included in the middle of 200 and 1500 cells/dl, 0 and 300cells/dl and 0 and 500, we suppose that the concentration of uninfected hepatocytes (UH), infected hepatocytes (IH) and free virions (FV) are normal. While if
According the relation (19), we have
The operating points associated to those linguistic variables are given in the
Let us set
The relations (24), (25) and (26) give respectively the following matrices
Variable | Operating Points |
---|---|
T | [−500; 0; 500] |
I | [0; 150; 300] |
V | [0; 0; 250:500] |
Parameter | a | q | s | σ | β | c | p | b | Tmax |
---|---|---|---|---|---|---|---|---|---|
Value | 0.108 | 0.072 | 36 | 0.5 | 0.001 | 3 | 5 | 0.01 | 1500 |
It is easy to note that the problem (29)-(30) 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 obtaining one system. Consequently, for the system (30) this technique gives the following system
where A and B are 3 × 3 and 3 × 2 matrices and c a 3 × 1 matrix.
In the same way, from the matrixes K1, K2 and K3 the defuzzification process allows to have one matrix K. We propose the following procedure.
The rows of matrixes
1) We consider the degree of membership of the entry uninfected hepatocytes, infected hepatocytes and free virions respectively. These values are respectively 400 cells/dl (see the
2) Each equation of the system (22) has nonlinear factor.
Considering these hypothesis and from the relation (18) where we take degrees of membership as given in the
Variable | ω1 | ω2 | ω3 |
---|---|---|---|
T | 0 | 0.85 | 0.15 |
I | 0.80 | 0.20 | 0 |
V | 0.72 | 0.28 | 0 |
The numerical simulation gives the graphical results given in the
The
It is known that acute hepatitis B infection (short-term inflammation of the liver) goes away on its own since the immune system is able to clear the virus from the body. The patient of acute hepatitis B infection may not need treatment. The main aim of treatment for chronic hepatitis B is to suppress HBV replication before there is irreversible liver damage. Furthermore, the role of drugs on chronic hepatitis B virus is to reduce the risk of liver disease and prevent you from passing the infection to others. The controls variation of hepatitis B virus are represented in
In this work, we have been dealing with an optimal control problem related to an uninfected hepatitis B virus dynamics. To handle that problem, two numerical approaches have been compared to determine the optimal trajectories of uninfected hepatocytes, infected hepatocytes and free virions as response to hepatitis B virus controls that is two drugs, interferon and ribavirin. The findings show that those two used methods are satisfactory and provide the closer results. Consequently, the approach that involves fuzzy logic approach can be seen to play an important role for the resolution of the optimal control problem. In particular, it gives the optimal trajectories and in the same way it ensures healthy.
Jean MarieNtaganda,MarcelGahamanyi, (2015) Fuzzy Logic Approach for Solving an Optimal Control Problem of an Uninfected Hepatitis B Virus Dynamics. Applied Mathematics,06,1524-1537. doi: 10.4236/am.2015.69136