Applying the Wiener-Hermite Random Technique to Study the Evolution of Excess Weight Population in the Region of Valencia (Spain) ()
1. Introduction
In the physical, engineering, economical or epidemiological sciences, random differential equations arise in a quite natural manner in the description of models. In fact, numerous phenomena of interest in these areas, which are very important for scientific and technological progress, have been traditionally formulated through mathematical models based on ordinary or partial differential equations, where the data (initial conditions, source term and/or coefficients) are expressed by means of numerical values or deterministic functions. Nevertheless, scientists really set these data from measurements, which always are subject to error. Depending on the quality of these measurements (which frequently can take a lot of time and high cost), the results obtained from the model may be satisfactory. In addition to measurement errors, we should consider the random character of complex external factors that can affect the system, such as pressure, temperature and humidity in Meteorology; the composition of the land in Seismology; investor tendency and economical policy of countries and companies in Finance; the environmental and genetical factors in Epidemiology;
etc. These circumstances make more advisable to consider the data as random magnitudes. The consideration of these facts leads to the reformulation of the traditional deterministic models, which, in order to improve them, should be replaced by random models.
However, even recognizing the necessity of considering a random approach in the formulation of such models, often it is very difficult, if not impossible, to establish a suitable way to model uncertainty due to the complexity involved in the specific problem under study. Based on the central limit theorem, a popular and successful way to model randomness is through a Gaussian process having additional mathematical properties such as white noise does. White noise is a stationary Gaussian stochastic process with mean value zero and a constant spectral density on the entire real axis. Interesting contributions in the modeling of applied problems in different fields using white noise can be found in [1-6], for instance.
In this paper we are interested in forecasting the evolution of excess and normal weight population in the region of Valencia (Spain) by means of a random model that involves white noise. Due to the lack of statistical data about the problem under study, this stochastic model takes advantage of certain information obtained from a deterministic SIS-type epidemic model. In fact, as we will see, the random model takes as starting point useful conclusions provided by the classical deterministic approach. In this way, the random model improves the deterministic one because it considers uncertainty in its formulation, what it is considered more realistic. In addition, this approach provides valuable information such as average and variance functions of the approximate solution stochastic process.
As it is shown in Section 2, this epidemiological model can be described as a particular case of the following random differential equation
(1)
where coefficients
, 
and initial condition 
are deterministic, 
is a small parameter and 
is a white noise process, which intensity is modulated by parameter
. By
, we denote a random outcome of a probability space
, where 
is a sample space, 
is a 
-algebra associate to 
and, 
is a probability measure.
As we will see throughout the subsequent development, the method used to obtain the approximate solution to Equation (1) is based on the so-called Wiener-Hermite expansion (WHE). WHE constitutes a powerful technique to represent any stochastic process in terms of the so-called Wiener-Hermite polynomials as well as certain deterministic kernels to be calculated. Interesting contributions where this technique have been used successfully to solve other class of random differential equations can be found in references [7-10] and other contained therein.
The article is organized as follows. In Section 2 we establish a random model of type (1) in order to study excess weight population aged between 24 - 65 years old in the region of Valencia (Spain). The stochastic model arises in a natural way by introducing uncertainty in the corresponding deterministic SIS-type epidemiological model. In Section 3 we first summarize the main results about the WHE method and then, we apply it to derive a coupled integro-differential system that is satisfied by the involved kernels. This section concludes with the application of the perturbation technique to conduct the resolution of such a system. Section 4 is devoted to solve the random SIS-type epidemiological model presented in Section 2 by taking advantage of development given in Section 3. Conclusions are discussed in Section 5.
2. Motivating the Mathematical Model
Some mathematical models to deal with the evolution over time of excess weight populations have been recently developed [11,12]. In [12] it is presented a deterministic differential mathematical model to predict the future evolution of the 3 - 5 years old infant excess weight population in the region of Valencia (Spain) over a finite time. In [11] the study is developed for the whole population and an asymptotic behavior analysis is presented. Both papers consider obesity as a health concern that spreads by social peer pressure and social contact through unhealthy lifestyle habits [11-13]. These contributions are based on epidemiological models [14]. Although more complex deterministic models to study excess weight population have been proposed [12,15,16], in this paper we want, in a first step, to explore by means of a simple but representative type-model, the ability of Wiener-Hermite expansion to provide a suitable approach to deal with such a class of models that include uncertainty in their formulation. It could permit the extension to this approach more sophisticated models in future works.
In order to motivate the statement of the random model, we first take the corresponding SIS-epidemiological deterministic one as starting point. Hereinafter, we concentrate on population aged between 24 - 65 in the region of Valencia (Spain). In this study, we consider that population is partitioned into two subpopulations, 
and
, that denote the proportion of normal and excess weight individuals at time
, respectively. Without loss of generality, we assume that the whole population is normalized to unit, i.e., 
for all time
. Following an analogous reasoning as it is given in [11,12], the model can be represented by the following two-state dynamical coupled nonlinear system:
(2.1)
with initial conditions 
and
. Time invariant parameters for system (2.1) are:
• 
, average stay time in the system of 24 - 65 year old adults.
• 
, rate at which an excess weight individual moves to normal weight subpopulation.
• 
, transmission rate due to social pressure to adopt an unhealthy lifestyle (TV, friends, family, job, ).
• 
, proportion of normal weight population coming from the 23 years old age group.
• 
, proportion of excess weight population coming from the 23 years old age group.
System (2.1) can be interpreted as a SIS-type compartmental model which dynamic of transits between subpopulations is depicted in Figure 1.
Since
, system (2.1) can be simplified to only one nonlinear differential equation involving as unique unknown the percentage of normal weight people
Figure 1. Flow diagram of the deterministic model for the dynamic of obesity prevalence in the population.
(2.2)
for a given initial condition
, being
, 
,
.
Following an analogous methodology as in [12,15,16], parameter 
is estimated by fitting the model with data from the Health Survey of the Region of Valencia 2000 and 2005 [17,18]. The other parameters are estimated using the same Health Survey and [19]. Table 1 collects these values where time variable t is measured in weeks.
However, note that the aforementioned deterministic model does not take into account nor the inherent errors in the measured data provided by the Health Survey neither inherent complexity of obesity such as individual behavior, geographical conditions, genetic aspects, health advertising campaigns, etc. When data are available to inform us about the best choice for data distribution, the parameter assignment is easily made. However, in the lack of data of this sort of information on the distribution for a specific parameter or, even more, for the randomness affecting a complex problem as obesity is, the specification of such information is very difficult, if not impossible, to get. White noise stochastic process has demonstrated to be a powerful tool to model properly general uncertainty [20,21]. In this paper, based on this consideration, we propose to modify the obesity model (2.1), considering that the dynamic of normal weight subpopulation is described by the random differential equation:
(2.3)
where 
is a white noise process, which intensity is given by parameter
. This model is just a particular case of (1), where coefficients are now assumed to be time-independent. Notice that in the real problem we are interested to apply the random model, from Table 1, it is plausible to assume that 
is a small parameter.
The previous exposition leads us to face several new problems that need to be answered. Firstly, we now have to solve the random differential Equation (2.3) or the more general, (1). Secondly, taking the deterministic model as a starting point, we have to fit parameter 
in order to provide a complete description of obesity model through random approach. This motivates the next section which is devoted to obtain an approximate solution
Table 1. Initial conditions and parameter values for the SIS model (2.1).