**4. Conclusions**

collected on each unit is constant *ni* <sup>¼</sup> *<sup>m</sup>* and <sup>Δ</sup>*<sup>i</sup>*

**True values**

**Table 2.**

**104**

then we apply the GA after choosing the good algorithm parameters (N, EN, SR, CP, MP), and we get 5000 sets of parameters estimates. We repeat this for large and small

**(M,m) Mean and (Std)**

(0.0014)

(0.0059)

(0.0402)

(0.0012)

(0.0023)

(10,20) 0.015 (0.0031) 0.0938

(3 � <sup>10</sup>�<sup>6</sup> )

(4.4 � <sup>10</sup>�<sup>6</sup>

(0.8 � <sup>10</sup>�<sup>5</sup>

(1.2 � <sup>10</sup>�<sup>6</sup>

(1.6 � <sup>10</sup>�<sup>6</sup>

(1.35 � <sup>10</sup>�<sup>5</sup>

*Approximated maximum likelihood estimates and standard deviation from simulations of model Eq. (20),*

)

)

)

)

)

*σSI σ<sup>I</sup>*<sup>0</sup> *σ<sup>G</sup>*<sup>0</sup> *μSG σ*^*SI σ*^*<sup>I</sup>*<sup>0</sup> *σ*^*<sup>G</sup>*<sup>0</sup> *μ*^*SG*

(40,10) 0.000029

(10,20) 0.000016

(40,10) 0.00025

(10,20) 0.00037

*μSI μ<sup>I</sup>*<sup>0</sup> *μ<sup>G</sup>*<sup>0</sup> *μ*^*SI μ*^*<sup>I</sup>*<sup>0</sup> *μ*^*<sup>G</sup>*<sup>0</sup>

*σ*<sup>1</sup> *σ*<sup>2</sup> *σ*<sup>3</sup> *σSG σ*^<sup>1</sup> *σ*^<sup>2</sup> *σ*^<sup>3</sup> *σ*^*SG*

0.100 (0.0013)

0.209 (0.0295)

0.156 (0.046)

0.0616 (0.0011)

0.0708 (0.0060)

(1.0196)

44.98 (1.25)

43.94 (1.75)

41.09 (2.82)

94.20 (1.12)

92.05 (2.25)

122.51 (2.51)

0.00073 (2.02 � <sup>10</sup>�<sup>4</sup>

0.00068 (2.54 � <sup>10</sup>�<sup>4</sup>

> 0.00054 (0.0016)

> 0.0343 (0.0027)

> 0.0340 (0.0035)

> 0.0480 (0.0108)

321.2 (1.07)

318.6 (3.11)

281.5 (4.88)

45.96 (2.11) 0.0315

45.16 (2.53) 0.0349

44.64 (3.07) 0.0178

)

)

*p***<sup>2</sup>** *n γ h p*^**<sup>2</sup>** *n*^ ^*γ* ^

(40,10) 0.0784

(10,20) 0.0794

(40,10) 0.014

0.074 0.10 0.0007 90 (40,60) 0.0737

*Numerical Modeling and Computer Simulation*

0.01 0.06 0.03 0.006 (40,60) 0.009

0.000025 46 50 0.03 (40,60) 0.000021

0.0002 95 320 (40,60) 0.00021

ð Þ 40, 60 ; 40, 10 ð Þ and 10, 20 ð Þ; for each parameter the sample mean and standard deviation are reported in **Table 2**. The simulation study on small data is treated in order to see if the size of the sample influences on the results and if the number of measures taken over time has a negligible effect or not, in other words, to see if it is possible to select only the essential measuring moments without repeating the measurements several times to well simulate a subject. We treat this issue in relation to our model and its study context, since in epidemiology the availability of data (measurements) at any point of time is an interesting constraint. We note that quantities of *Gb* and *Ib* are randomly simulated from the normal range of healthy subjects.

In **Table 2**, we report the results obtained on large and small data by maximizing Eq. (22) using the GA. We notice that, for the case of the large data, the true values

data with different possibilities of repetition of the experiment ð Þ¼ *m*, *M*

*using the approximated transition density Eq. (11) with large and small DATA.*

*<sup>j</sup>* ¼ Δ for all 1≤*i* ≤ *M* and 1≤*j*≤ *ni*;

*h*

89.08 (0.031)

91.34 (0.712)

62.11 (1.13)

0.0061 (0.00010)

0.0073 (0.00030)

0.0067 (0.0006)

(0.0007)

(0.0036)

(0.0051)

In this work, we have proposed a procedure to estimate the parameters of a mixed effects model containing stochastic differential equations, known by the SDME models, by proposing an approach to approximate its likelihood function to obtain the MLEs. This method has been evaluated by simulation studies on two SDME models in epidemiology: the two-dimensional Ornstein-Uhlenbeck process and stochastic minimal model. In fact, in models with SDEs instead of ODEs with random effects, the estimation of parameters is still not obvious even for one individual (one trajectory) because of the difficulties in deriving the transition densities, and difficulties become more interesting in using the population approach that treats the entire population simultaneously. The derivation of the exact density is not always possible for a stochastic and continuous process in an SDME model, so the search for an approximation of this density is an important step and requires an expensive calculation. This task is very interesting to give good results with good statistical properties of the estimators obtained by maximizing the likelihood function. In this work, the proposed estimation method has been applied to multidimensional and nonlinear SDME models with many random parameters normally distributed that can be extended to random parameters of any distribution. So, an approximation of the transition density *P*ð Þ *<sup>a</sup> <sup>Y</sup>* is obtained in a closed form using the Risken approximation for the formal solution of the Fokker-Planck equation proposed by Risken [34], and then the approximated likelihood is obtained using the Laplace approximation method and optimized using the genetic algorithm; these calculation procedures can be obtained using any numerical calculation software or with symbolic computing capabilities.

approximation method of integral computation for the case of an SDME model with several random parameters. We have treated two examples to illustrate the effectiveness of this approach using computer tools. Indeed, we believe that this type of models is very interesting and provides a powerful and flexible modeling approach for repeated measurement studies such as biological and pharmacokinetic/pharmacodynamic and financial studies, as they combine the good characteristics of mixed effects and stochastic increments in intra-subject dynamics for a good modeling of a

*Simulation and Parametric Inference of a Mixed Effects Model with Stochastic Differential…*

Z.T., H.E., and F.B conceived the presented idea and contributed to the design and implementation of the research. F.B., H.E., Z.T., and H.A contributed to the analysis of the results. Both F.B. and H.E authors developed the theory and performed the computations and the numerical simulations and verified the ana-

phenomenon.

**Acknowledgements**

**Conflict of interest**

**Abbreviations**

lytic methods and calculations.

*DOI: http://dx.doi.org/10.5772/intechopen.90751*

The author(s) declare no competing interests.

SDEs stochastic differential equations MCMC Markov Chain Monte Carlo FP Fokker-Planck equation

MLEs maximum likelihood estimators ODE ordinary differential equations NLME nonlinear mixed effects

OU Ornstein-Uhlenbeck GA genetic algorithm EN elite number SR selection rate

CP crossover probability MP mutation probability SMM stochastic minimal model

T2D type 2 diabetes

**107**

SDME stochastic differential mixed effects

The classical inference of SDME models implies the problem of the numerical evaluation of the integral for the given random effects in the likelihood function, which becomes complicated especially when the model contains more than two random parameters. In the literature, several methods have been proposed and tested for the approximation of the integral (see references in the introduction) and the following examples: [8] which proposes the Gaussian quadrature method to solve the integrals for the case of an SDME models with a single random effect and [20] where the study was revised for a general case with several random parameters using the Laplace approximation to compute the integral in Eq. (4) and Eq. (6) numerically. For the mixed effects framework, see [3, 16, 55, 56]. In the case of using the Laplace method, as in this chapter, the calculation of the Hessian matrix can be done analytically when it is possible, as the examples in Section 2, or with the help of a symbolic calculus software or the automatic differentiation (AD) tools [57].

The results of simulation studies are satisfactory and can be obtained either by using moderate values for the number of experimental units M and of observations n taken for each experimental unit or by using a small sample size but with a number of measurements taken for each subject of 10 at least; this is relevant for applications where large sets of data are not available, such as biomedical applications where the mixed effect theory is widely applied.

The advantages of this approach, compared to those proposed in the literature for multidimensional SDME models with more than one random parameter [26], are that the computation of the approximate density is very easy and does not require a lot of time to calculate it or to program it in a software; the only task that can be time-consuming is in the optimization step to search for the optimum solution of the likelihood function, and also the proposed method is effective even with large data with a MATLAB program on a common PC (Intel Pentium IV 3.0 GHz with 512 MB of RAM). Nevertheless, the method suffers some limitations, for example, when the conditions to use Eq. (11) are not verified when, e.g., the inverse of the diffusion term does not exist and when, in certain cases, it is not obvious to derive the gradients and Hessians terms. Another limitation is that measurement errors are not considered in this work, and for a good stochastic version, it will be better to include noise on process increments and noise on observations that may be significant compared to system noise. These limitations may provide a perspective towards a more elaborate extension of the statistical study for SDME models, particularly in the field of epidemiology.

In conclusion, in this work, we proposed a method of parameter estimation for a mixed effects models with SDEs proposing an approximation method for the transition density in the case when it cannot be obtained in a closed form, with an

*Simulation and Parametric Inference of a Mixed Effects Model with Stochastic Differential… DOI: http://dx.doi.org/10.5772/intechopen.90751*

approximation method of integral computation for the case of an SDME model with several random parameters. We have treated two examples to illustrate the effectiveness of this approach using computer tools. Indeed, we believe that this type of models is very interesting and provides a powerful and flexible modeling approach for repeated measurement studies such as biological and pharmacokinetic/pharmacodynamic and financial studies, as they combine the good characteristics of mixed effects and stochastic increments in intra-subject dynamics for a good modeling of a phenomenon.
