Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the firstand the secondorder moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model.
Epidemiological mathematical models based on ordinary differential equations are usually used to understand the processes involved in the transmission of diseases [
Differential equations where some or all of the coefficients are considered random variables or that incorporate stochastic effects (usually in the form of white noise) have been increasingly used in the last few decades to deal with errors and uncertainty see [
Monte Carlo methods [
In this paper, we will use the polynomial chaos approach to study this type of epidemiological models with randomness, due to its simplicity. The computational cost can be high if many random parameters are considered, and highorder expansions are used. But in our problem this was not the case. The polynomial chaos method applied to a system of ordinary differential equations with random equations is based on expanding the random coefficients and the unknown variables in terms of orthogonal polynomials of random variables. For example, if a random coefficient has a normal distribution, the Hermite polynomials should be used since they form an orthogonal basis with the normal distribution as the weight. These expansions are then substituted into the differential equations, and the orthogonality is used to obtain a system of the differential equations of the same form as the deterministic model for the unknown coefficients of the expansions. These equations can then be solved using the same numerical methods used for the deterministic case. More details are given in Section
This polynomial chaos technique allows us to consider that the transmission parameters in an epidemiological model are random variables and obtain the evolution of the epidemic and its predictions considering the effects of these randomness. Additionally, the quantification of the effects of the random transmission parameters on the variance of the response of the epidemiological model can also be analyzed calculating the polynomial chaosbased Sobol’s indices. These indices are based on the decomposition of the variance of the output as a sum of contributions of each input variable. Taking into account this decomposition, Sobol’s indices allow us to quantify the rate defined by the variance related to each parameter and the total variance of the output.
Therefore, this approach is useful to predict the evolution of an epidemic considering the effects of the randomness and to quantify the effects of the random transmission parameters on epidemic evolution (sensitivity analysis).
This paper is structured as follows. In Section
Classical models of disease dynamics rely on systems of differential equations that divide the number of individuals in various categories through continuous variables allowing for infinitesimal population densities. The origin of these models is commonly traced back to the wellknown pioneer work of Hethcote [
Some of the assumptions in this type of models are: (i) The number of individuals grows without bound in a Malthusian way; this is modeled by a linear term. (ii) The effect of the disease (the transit to
The population with excess weight is growing at a worrying rate in developed and developing countries [
The obesity model used to present the possibilities of the polynomial chaos approach was proposed in [
The values of these parameters for the region of Valencia (Spain) were determined by health survey for the region of Valencia, Spain, year 2000 and year 2005 [
Estimated parameters for the region of Valencia, Spain.
Parameter  Value 


0.00085 

0.000469 

0.0003 

0.000004 

0.000035 

0.704 

0.25 

0.046 
Parameters,
The initial conditions of the system are also defined by health survey for the region of Valencia, Spain, year 2000. In this case,
Note that taking into account the differential equations of the model (
It is necessary to introduce randomness in the model (
In many situations, the number of data points available is very small, so it is not possible to establish the type of distributions satisfied by the random parameters. This is also true in our case where the number of data points for the transmission parameters is so scarce that is not possible to have a welldefined type of distribution. In this work, we only have the information of one value to estimate the probabilistic distribution of the parameters, the values shown in Table
Probability distributions of the transmission parameters.
Parameter  Value  Distribution 


0.00085  Uniform (0, 0.0017) 

0.0003  Uniform (0, 0.0006) 

0.000004  Uniform (0, 0.000008) 

0.000035  Uniform (0, 0.00007) 
Therefore, we consider that the transmission parameters of the model
In order to perform numerical simulations of the dynamical model (
In this context, polynomial chaoses can be arranged in a sequence
In this paper, since the probability distributions of the transmission parameters are uniform see Table
For
A proper description of the random transmission parameters in terms of the independent chaos variables
We are now ready to develop the differential equations used in the numerical study. Considering the equations of the mathematical model (
Considering that we define the model in the restricted region
For notational convenience, we consider a onetoone correspondence between the Legendre polynomials
To obtain a system of ordinary differential equations for the unknown coefficients with only one derivative of an unknown per equation, we use the orthogonality of the basis functions. In particular, taking the inner product of (
For the second equation of system (
Equations (
A sensitivity analysis is also performed in order to quantify the output uncertainty due to the randomness in each of the transmission parameters. Polynomial chaosbased Sobol’s indices are used. This method is based on the decomposition of the variance of the output as a sum of contributions of each input variable, or combinations thereof see [
In order to compute the sensitivity indices based on the polynomial chaos expansions of the output stochastic processes it is necessary to consider the coefficients of these expansions, that is,
The idea behind the construction of polynomial chaosbased Sobol’s indices is simple: once the polynomial chaos representation of the output stochastic process is available (the expansion coefficients are known, i.e., the solution of system (
Note that
Note that numerator in (
Figure
Prevalence prediction for obese subpopulation in the region of Valencia. Note that
Figure
Evolution of excess weight population for the next few years. Predictions are shown by standard deviation intervals. Additionally, mean values are presented.
Year  Overweight population  Obese population 

2010  36.51%  13.16% 

[33.54%, 39.49%]  [8.13%, 18.18%] 
2011  36.52%  13.18% 

[33.33%, 39.70%]  [7.96%, 18.39%] 
2015  36.54%  13.21% 

[33.52%, 40.51%]  [7.36%, 19.05%] 
Prevalence prediction for overweight and normalweight subpopulation in the region of Valencia. Note that
We can observe that polynomial chaos approach quantifies the output uncertainty due to the randomness in the input parameters. The definition of the output confidence interval by secondorder moment evaluation allows us to predict the epidemic evolution with more accuracy than in deterministic approach. As it is described in [
Evolution of excess weight population for the next few years using deterministic model (
Year  Overweight population  Obese population 

2010  37.86%  15.20% 
2011  37.99%  15.52% 
2015  38.14%  15.92% 
Figure
Influence of transmission parameters uncertainty on obesity epidemic prediction. Polynomial chaosbased Sobol indices. (a)
In this paper, we have shown the possibilities of polynomial chaos related to epidemiological models. It is shown how polynomial chaos can be a useful tool to consider the effects of randomness on the evolution of the epidemics and to perform sensitivity analysis (by polynomial chaosbased Sobol’s indices) in order to propose optimal policies to control epidemics.
As an example, we have studied an obesity model. As it is usual in social epidemic models, the transmission parameters involved in these types of mathematical model cannot be determined exactly, and it is necessary to introduce randomness. In this work, randomness in the transmission parameters is considered, and the resulting system of random coefficient differential equations has been solved approximately using the method of polynomial chaos.
We have shown how the application of polynomial chaos approach to an epidemiological model allows us to determine the epidemic evolution with more realism than in deterministic approach. Since in this case, it is possible to define a confidence interval to the epidemic evolution. Additionally, taking into account this approach, sensitivity analysis (an useful tool for policy makers and healthy planners) is easy to perform. Sensitivity indices based on polynomial chaos expansion may be computed with no additional cost.
To the best of our knowledge, this work is one of the first applications of polynomial chaos approach to epidemiological models based on ordinary differential equations although evidences detected make the method a good candidate to be employed in the study of epidemics.