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This work emphasizes solving the problem of parameter estimation for a human immunodeficiency virus (HIV) dynamical model by using an improved differential evolution, which is called the hybrid Taguchi-differential evolution (HTDE). The HTDE, used to estimate parameters of an HIV dynamical model, can provide robust optimal solutions. In this work, the HTDE approach is effectively applied to solve the problem of parameter estimation for an HIV dynamical model and is also compared with the traditional differential evolution (DE) approach and the numerical methods presented in the literature. An illustrative example shows that the proposed HTDE gives an effective and robust way for obtaining optimal solution, and can get better results than the traditional DE approach and the numerical methods presented in the literature for an HIV dynamical model.

Recently, the mathematical modeling of the epidemiology and immunology dynamics of human immunodeficiency virus (HIV) has been proven to be valuable in understanding the HIV pathogenesis (see, e.g., [

The DE [

The DEs are designed by using the concept of evolutionary algorithms, such as multipoint searching, mutation operation crossover operation and selection operation, but crossover operation based on the random process is not a systematic reasoning way for breeding better offspring (or trial individual vectors). Therefore, in order to seek the optimal breeding in the DEs such that the efficiency of the DEs for estimating all the parameters in the HIV dynamical model [

This paper is organized as follows. Section

In this paper, the following three-dimensional model of HIV [

In clinical practice, it is difficult to measure the amount of healthy CD4+ T cells. The measurement of total CD4+ T cells in blood may not be representative of the total body pool of CD4+ T cells that are involved in HIV infection. In addition, the variation of total CD4+ T cells counts in blood is large [

When estimating the system parameters, suppose the structures of the system is known in advance, and thus the HIV dynamical model can be described as follows:

The parameter estimation problem of an HIV dynamical model considered here is to search for the optimal parameters

It is obvious that if one set of estimated parameters

The HTDE combines the traditional DE [

The population size

In the real coding representation, each individual is encoded as a vector of floating point numbers, with the same length as the vector of decision parameters. For convenience and simplicity, an individual

Generate a uniformly distributed random number

Let

Repeat the above two steps

Initialize the iteration index

Set the initial target index

In every generation, each individual vector

Set

Calculate the performance index

If

Calculate the effects of the various factors.

One optimal trial vector

Add one to the target index

Add one to the iteration index

Stop and display the optimal individual vector (i.e., the optimal parameters

In this section, we evaluated the performance of the proposed HTDE approach to an HIV dynamical model given by Manseur et al. [

The actual parameters of an HIV dynamical model with initial condition

For the HIV dynamical models (

Comparison of results for the estimated parameters and the performance index with 100 independent runs for the HIV dynamical model.

Approach | Best value | Mean value | Standard deviation | |
---|---|---|---|---|

HTDE | 1.0000 | 1.0000 | 0.0003 | |

0.8000 | 0.8000 | 0.0003 | ||

1.0000 | 1.0000 | 0.0002 | ||

0.8000 | 0.8000 | 0.0003 | ||

1.0000 | 1.0000 | 0.0005 | ||

0.01078 | 0.01079 | 0.0002 | ||

3.62 ^{-10} | 2.86 ^{-9} | 2.09 ^{-9} | ||

DE | 1.0209 | 1.1582 | 0.2750 | |

0.7636 | 0.9497 | 0.2692 | ||

1.0034 | 1.0544 | 0.1591 | ||

0.7919 | 0.8877 | 0.2848 | ||

1.0131 | 1.2349 | 0.2199 | ||

0.01073 | 0.10865 | 0.09900 | ||

0.0007924 | 0.0021460 | 0.0010318 | ||

Transformation 1 proposed by Manseur et al. [ | 0.964 | |||

0.856 | ||||

0.928 | ||||

0.715 | NA | NA | ||

0.98 | ||||

0.009 | ||||

0.0010 | ||||

Transformation 2 proposed by Manseur et al. [ | 0.724 | |||

0.582 | ||||

0.97 | ||||

0.3785 | NA | NA | ||

0.9409 | ||||

0.1058 | ||||

0.02036 | ||||

Levenberg-Marquardt method presented by Manseur et al. [ | 0.79 | |||

0.67 | ||||

0.85 | ||||

0.705 | NA | NA | ||

0.88 | ||||

0.021 | ||||

0.07 |

Comparison of results for the computational time (in minute) with 100 independent runs for the HIV dynamical model.

Approach | Best value | Mean value | Standard deviation |
---|---|---|---|

HTDE | 21.97 | 22.01 | 0.06 |

DE | 4.89 | 4.93 | 0.01 |

Average convergence results of performance index in 100 independent runs by using the HTDE and the DE, respectively, for the HIV dynamical model.

Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter

Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter

Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter

Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter

Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter

Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter

Responses of

Responses of

Responses of

From Table

An HTDE approach has been presented in this paper based on the Taguchi-method-based crossover operation for solving the problem of parameter estimation for an HIV dynamical model under the minimization of a performance index (

This work was in part supported by a Grant from the Chi-Mei Medical Center and Kaohsiung University Research Foundation (99CM-KMU-01) and the National Science Council, Taiwan, Republic of China, under Grant number NSC 99-2320-B-037-026-MY2. The authors also thank the editor and the reviewers for their suggestions and comments.