Hybrid Taguchi-Differential Evolution Algorithm for Parameter Estimation of Differential Equation Models with Application to HIV Dynamics

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.


Introduction
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., 1-15 and the references therein .Many of the proposed approaches only work under the assumption that the parameters of the HIV dynamical models are known in advance.However, in the real world, it may be difficult to determine the parameters because of the complexity of HIV dynamical models.Therefore, in recent years, it has become a hot issue for estimating the parameters of the HIV dynamical models see, e.g., 16-25 and the references therein .On the other hand, for solving the problem of parameter estimation in the HIV dynamical models, the methods of estimation such as the Bayesian approach 17, 21 , the online estimation algorithm 18 , the Monte Carlo technique 20 , the numerical method with combined Adomian/Alienor approach 22 , the multiple time point method 23 , the nonlinear least squares method 24 , and the multistage smoothing-based method and spline-enhanced nonlinear least squares approach 25 have been presented in the literature.In particular, Miao et al. 24 introduced the differential evolution DE , a global optimization algorithm, to solve the nonlinear least squares problem to avoid the local minima and convergence difficulty in nonlinear optimization.
The DE 26 has received considerable attention concerning its potential as a new optimization technique for complex nonlinear problems and has been successfully used in various areas see, e.g., [27][28][29][30][31][32][33][34] and the references therein .The main specific feature of the DE as an optimization method is its implicit parallelism, which is a result of the evolution and the hereditary-like process.The advantages of using a DE for solving the optimization problems are its global solution finding property, simple but powerful search capability, easyto-understand concept, compact structure, having only a few control parameters, ease of use, and high convergence characteristics 35, 36 .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 37 can be further promoted, the purpose of this paper is to propose a new DE approach by introducing a systematic reasoning.This proposed new DE approach is named the hybrid Taguchi-differential evolution HTDE .The HTDE combines the DE 26 with the Taguchi method 38, 39 .The Taguchi method, a robust design approach, uses many ideas from statistical experimental design, where some of the factors or individuals are related, for evaluating and implementing improvements in products, processes, and equipment.Factors or individuals are called related when the desirable experimental region of some factors or individuals depends on the level settings of other factors or individuals .Two major tools used in the Taguchi method are i signal-to-noise ratio SNR which measures quality, and ii orthogonal arrays which are used to study many design parameters simultaneously 40, 41 .In the HTDE, the Taguchi method is to provide a new systematic crossover operation to replace the original crossover operation of DE.Then, the systematic reasoning ability of the Taguchi-method-based crossover operation is used to breed better individuals in order to generate the representative individuals to be the new potential offspring or trial vectors .So, the Taguchi-method-based crossover operation can enhance DEs, such that the HTDE can be robust, statistically sound, and quickly convergent.Thus, the HTDE can be an efficient optimization algorithm to deal with the problem of parameter estimation for an HIV dynamical model, which is also regarded an optimization problem.
This paper is organized as follows.Section 2 describes the HTDE for solving the parameter estimation problem of an HIV dynamical model.In Section 3, we evaluate the efficiency of the proposed HTDE by comparing our results with those obtained from the traditional DE approach and the numerical methods of Manseur et al. 22 respectively, by using the same example given by Manseur et al. 22 .Finally, Section 4 offers some conclusions.

Parameter Estimation of an HIV Dynamical Model with HTDE Approach
In this paper, the following three-dimensional model of HIV 22, 37 described by is considered, where x 1 t denotes amount quantity of healthy CD4 T cells, x 2 t denotes the infected CD4 T cells, x 3 t denotes the viral load, and the positive constants S, d, β, μ 1 , k, and μ 2 denote the system parameters for an HIV dynamical model described as the rate of production of the healthy cells, the death rate of the health calls, the infection rate of healthy cells CD4 by virus HIV, the death rate of infected cells, the rate of production of free virus, and the death rate of free virus, respectively.Equation 2.1a denotes the population dynamics of the healthy cells.In the presence of HIV, the healthy cells interact with the virus and its reproduction rate decreases according to the term −βx 1 t x 3 t .Equation 2.1b denotes the population dynamics of the infected cells.The growth of infected cells is proportional to the amount of healthy cells infected by virus and will be discounted by the amount of cells in destruction −μ 1 x 2 t .Equation 2.1c represents the dynamics of the concentration of free virus.The free virus increases in proportion to the infected cells kx 2 t , and depend on this natural decline rate −μ 2 x 3 t .
Remark 2.1.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 23 .Therefore, in this paper, for evaluating the performance of the proposed HTDE approach to an HIV dynamical model given by Manseur et al. 22 , and comparing the results obtained from our proposed HTDE approach with those from the traditional DE approach 26 and the numerical methods given by Manseur et al. 22 .By using the same example, we assume the measurements of viral load and T cells either the amount of healthy CD4 T cells or the total number of CD4 T cells are available.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: which is expected to match the actual HIV dynamical model 2.1a -2.1c , where the positive constants S, d, β, μ 1 , k, and μ 2 are the estimated parameters of an HIV dynamical model, and x i t i 1, 2, 3 denote the state variables of an HIV dynamical model obtained from the estimated parameters { S, d, β, μ 1 , k, μ 2 }.
The parameter estimation problem of an HIV dynamical model considered here is to search for the optimal parameters S, d, β, μ 1 , k, and μ 2 such that the performance index is minimized, where n is referred to as the sampling time point, T is the total number of sampling, and after a period of transient process, a set of state variables is selected as the initial state vector x 1 0 , x 2 0 , x 3 0 for the parameter estimation.This is equivalent to the optimization problem: , and a i are the positive real numbers, where x i and x i are the lower and upper bounds of a i given from the practical consideration, respectively.That is, the estimated parameters {a 1 , a 2 , . . ., a 6 } are generated at random from within userdefined bounds x j , x j .Due to there often being multiple local optima in the landscape of the performance index J, it is difficult to obtain the global optimal parameters.Therefore, the proposed HTDE approach will be applied to search for the global or close-to-global optimal parameters a i i 1, 2, . . ., 6 such that the performance index J is minimized.Therefore, the parameter estimation problem of an HIV dynamical model can become an optimization problem 2.4 , where 2.4 is a nonlinear function with the continuous variables.
Remark 2.2.It is obvious that if one set of estimated parameters {a 1 , a 2 , . . ., a 6 } is specified, then x i t i 1, 2, 3 can be determined from the HIV dynamical model 2.2a -2.2c , and thus the value of the performance index 2.3 corresponding to this set of {a 1 , a 2 , . . ., a 6 } can be calculated.Given another set of estimated parameters {a 1 , a 2 , . . ., a 6 }, there obtains another value of the performance index 2.3 .That is, the value of the performance index 2.3 is actually dependent on the set of estimated parameters {a 1 , a 2 , . . ., a 6 }.Therefore; it is assumed that there is no measurement error in this paper The HTDE combines the traditional DE 26 with the Taguchi method 38, 39 .In the HTDE, the Taguchi method is inserted between the crossover operations of a DE.Then, by using two major tools signal-to-noise ratio and orthogonal arrays of the Taguchi method, the systematic reasoning ability of the Taguchi method is incorporated in the crossover operations to systematically select the better genes to achieve the crossover operations, and consequently enhance the DEs 32 .The detailed steps of the HTDE are described as follows.The detailed description of the Taguchi method can be found in the books presented by Taguchi et al. 38 and Wu 39 .

Detailed Steps: HTDE
Step 1 parameter setting .The population size p s , the number of generations N g , the positive real mutation scaling factor F, and the user-defined bounds x i , x i i 1, 2, . . ., 6 .

Mathematical Problems in Engineering 5
Step 2 initialization .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 X G i is denoted as where G denotes the current generation, x G ij j 1, 2, . . ., 6 denote the estimated parameters i.e., {a 1 , a 2 , . . ., a 6 } , i 1, 2, . . ., p s .The initial values of x G ij are chosen at random from within user-defined bounds x j , x j .
Step 4. Let x G ij x j α x j − x j , where x G ij denotes the jth element of X G i in 2.5 .Repeat six times and produce an individual vector Step 5. Repeat the above two steps p s times and produce p s initial feasible individual vectors X G i i 1, 2, . . ., p s .
Step 6. Initialize the iteration index I 1.
Step 7. Set the initial target index i 1.
Step 8 mutation operation .In every generation, each individual vector X G i i 1, 2, . . ., p s in the population becomes a target vector.For each target vector X G i , DE applies a differential mutation operation to generate a mutated individual V G 1 i , according to where X G j , X G n , and X G q are randomly selected from the population such that j, n, and q belong to {1, 2, . . ., p s } and i / j / n / q; F is a mutation scaling factor which controls the evolution rate.In the event that mutation causes an element i.e., {a 1 , a 2 , . . ., a 6 } to shift outside the allowable interval The mutated vector V G 1 i will be used in the following Taguchi-method-based crossover operation as a donor vector for breeding a better offspring or trial vector U G 1 i .

Detailed Steps: Taguchi-Method-Based Crossover Operation
Substep 8.1.Set m 1.Take the three columns of the orthogonal array L 4 2 3 to allocate the three factors X G j , X G n , and X G q with two levels individuals , respectively.Step 9. Add one to the target index i.If i < p s , then go to Step 8.
Step 10.Add one to the iteration index I.If I < N g , then go to Step 7.
Step 11.Stop and display the optimal individual vector i.e., the optimal parameters {a 1 , a 2 , . . ., a 6 } and its performance index J.

Results and Comparisons
In this section, we evaluated the performance of the proposed HTDE approach to an HIV dynamical model given by Manseur et al. 22 and compared the results obtained from our proposed HTDE approach with those from the traditional DE approach 26 , and the numerical methods with combined Adomian/Alienor approach and the classical optimization method with Levenberg-Marquardt approach given by Manseur et al. 22 by using the same example.Besides, for the sake of fair comparisons between our proposed HTDE approach and the traditional DE approach, the same evolutionary environments used in this paper are the population size p s 100, the same maximum number of generations N g 100, the mutation scaling factor F 0.8, and the same search interval x j , x j 0, 2 j 1, 2, . . ., 6 for the estimated parameters {a 1 , a 2 , . . ., a 6 }, respectively.
The actual parameters of an HIV dynamical model with initial condition x 1 0 x 2 0 x 3 0 1 0.2 0.8 that will be estimated are assumed to be S 1, d 0.8, β 1, μ 1 0.8, k 1, and μ 2 0.01078 22 .In the following, we will apply the  proposed HTDE to find the optimal parameters { S, d, β, μ 1 , k, μ 2 } i.e., {a 1 , a 2 , . . ., a 6 } such that the performance index J 2.3 is minimized, where the sampling time is 0.1 second, and the total sampling number T is 10.
For the HIV dynamical models 2.1a -2.1c and 2.2a -2.2c , after using the proposed HTDE to execute 100 independent runs, we can get the best values, the mean  values, and the standard deviation of the obtained data for the estimated parameters, the performance index and the computational time with Intel R Core TM 2 Duo CPU 2.80 GHz and 2.00 GB RAM as shown in Tables 1 and 2, respectively.On the other hand, if we employ the traditional DE to execute 100 independent runs, we can also get the best values, the mean values, the standard deviation of the obtained data for the estimated parameters, the   1 and 2, respectively.Besides, the estimated parameters and the performance indices are obtained from two transformations by using the combined Adomian/Alienor method and a class optimization method with the Levenberg-Marquardt method 22 , respectively, are also shown in Table   average convergence results of performance index J and estimated parameters {a 1 , a 2 , . . ., a 6 } with respect to the number of generations in 100 independent runs by using the proposed HTDE and the traditional DE, respectively, for the HIV dynamical model.The responses of the x 1 t , x 2 t , and x 3 t for the HIV dynamical model with the mean values of the estimated parameters in 100 independent runs obtained from the proposed HTDE and the traditional DE are, respectively, shown in Figures 8, 9, and 10.From Table 1 and Figures 1-10, the following results can be observed: i the proposed HTDE can obtain better results including the best values of estimated parameters and performance index, and the mean values of estimated parameters and performance index than the traditional DE and the numerical methods presented by Manseur et al. 22 ; ii the proposed HTDE gives a smaller standard deviation of the obtained data for the estimated parameters and the performance index than the traditional DE, so the proposed HTDE has a stable solution quality; iii the average convergence results for the proposed HTDE are better than those of the traditional DE in 100 independent runs regarding the estimated parameters and the performance index as well as the proposed HTDE has a better solution convergence quality; iv although the computational time of the proposed HTDE is about four times the traditional DE, the better estimated parameters and performance index are concerned rather than the computational time in the problem of parameter estimation for an HIV dynamical model.Thus, it can be concluded that the proposed HTDE approach can give a more effective and robust way for finding the actual parameters than the traditional DE approach 26 and the numerical methods presented by Manseur et al. 22 for the HIV dynamical model.

Conclusions
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 2.3 .The HTDE can be easily employed to find the system parameters of an HIV dynamical model.An illustrative example regarding an HIV dynamical model has shown that the proposed HTDE approach is effective and robust to estimate the system parameters of an HIV dynamical model, and outperforms

Figure 1 :Substep 8 . 4 .Substep 8 . 5 . 1 i
Figure 1: Average convergence results of performance index in 100 independent runs by using the HTDE and the DE, respectively, for the HIV dynamical model.

1 Figure 2 :
Figure 2: Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter a 1 of the HIV dynamical model.

2 Figure 3 :
Figure 3: Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter a 2 of the HIV dynamical model.

3 Figure 4 :
Figure 4: Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter a 3 of the HIV dynamical model.

4 Figure 5 :
Figure 5: Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter a 4 of the HIV dynamical model.

5 Figure 6 :
Figure 6: Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter a 5 of the HIV dynamical model.

6 Figure 7 :
Figure 7: Average convergence results of 100 independent runs by using the HTDE and the DE, respectively, for the estimated parameter a 6 of the HIV dynamical model.

Figure 8 :
Figure 8: Responses of x 1 t by using the HTDE and the DE, respectively, for the HIV dynamical model.

Figure 9 :
Figure 9: Responses of x 2 t by using the HTDE and the DE, respectively, for the HIV dynamical model.

Figure 10 :
Figure 10: Responses of x 3 t by using the HTDE and the DE, respectively, for the HIV dynamical model.

Table 1 :
Comparison of results for the estimated parameters and the performance index with 100 independent runs for the HIV dynamical model.traditionalDEapproach 26 and the numerical methods presented byManseur et al. 22 .Although this HTDE approach is proposed under the framework of HIV dynamical model, it should be generally applicable to any optimization problem. the

Table 2 :
Comparison of results for the computational time in minute with 100 independent runs for the HIV dynamical model.