## 2 Sensitivity Analysis  
A sensitivity analysis of a mathematical model aims to quantify the relative importance of model inputs for a model output. In this paper, the global variance-based sensitivity analysis method (i.e., [PERSON]' method) [16; 17] is used. Model inputs are defined as independent random variables, and Sobol' indices quantify how much the variance of each input or input group contributes to the variance of the model output. It is implied that inputs or input groups that are more important for the model output contribute more to the output's variance and, therefore, have a higher Sobol' index. The principle idea behind Sobol' indices is to expend the mathematical model into summands of increasing dimensions and then decompose the model output variance into contributions from each summand. If \(y=f(\mathbf{x})\) is an integrable function (i.e., a mathematical model), where \(y\) is a scalar output and \(\mathbf{x}=x_{1},...,x_{n}\) is an input vector of \(n\) dimensions, \(f(\mathbf{x})\) can be decomposed as follows:  
\[f(\mathbf{x})=f_{0}+\sum_{i=1}^{n}f_{i}(x_{i})+\sum_{i=1}^{n}\sum_{j>i}^{n}f_ {ij}(x_{i}x_{j})+...+f_{1,2,...,n}(x_{1},x_{2},...,x_{n}) \tag{1}\]  
where \(f_{0}\) is a constant and the integrals of the summands with respect to their own variables are zero:  
\[\int f_{i_{1},...,i_{s}}(x_{1},...,x_{s})dx_{k}=0\quad for\quad 1\leq k\leq s \tag{2}\]  
Therefore, summands can be calculated by integrating \(f(\mathbf{x})\):\[\int f(\mathbf{x})d\mathbf{x}=f_{0} \tag{3}\]  
\[\int f(\mathbf{x})d\mathbf{x}_{\sim\ i}=f_{0}+f_{i}(x_{i}) \tag{4}\]  
\[\int f(\mathbf{x})d\mathbf{x}_{\sim\ i,j}=f_{0}+f_{i}(x_{i})+f_{j}(x_{j})+f_{ij} (x_{i}x_{j}) \tag{5}\]  
where \(\mathbf{x}_{\sim\ i}\) stands for vector \(\mathbf{x}\) that excludes its \(i\)-th element. The higher order summands are calculated in the same manner. If \(f(x)\) is square-integrable, it follows that all the summands are also square-integrable. Squaring the expansion and integrating over the input domain gives:  
\[ \begin{split}\int f^{2}(\mathbf{x})d\mathbf{x}-f_{0}^{2}=& \sum_{i=1}^{n}\int f_{i}^{2}(x_{i})dx_{i}\\ &+\sum_{i=1}^{n}\sum_{j>i}^{n}\int f_{ij}^{2}(x_{i}x_{j})dx_{i}dx_ {j}\\ &+\int f_{1,2,...,n}^{2}(x_{1},x_{2},...,x_{n})dx_{1},...,dx_{n} \end{split}  \tag{6}\]  
As the model inputs are random variables, the model output \(y\) is also a random variable whose variance (also called total variance) is given by the equation:  
\[D=\int f^{2}(\mathbf{x})d\mathbf{x}-f_{0}^{2} \tag{7}\]  
The total variance can then be decomposed into contributions from each group of inputs (also called partial variances):  
\[D=\sum_{i=1}^{n}D_{i}+\sum_{i=1}^{n}\sum_{j>i}^{n}D_{ij}+...+D_{1,2,...,n} \tag{8}\]  
where following the previous expansion from Equation (6) each partial variance is defined as:  
\[D_{i_{1},...,i_{s}}=\int f_{i_{1},...,i_{s}}^{2}(x_{i_{1}},...,x_{i_{s}})dx_{ 1},...,dx_{s},1\leq i_{1}<...<i_{s}\leq n,\quad s=1,..,n \tag{9}\]  
Normalizing the partial variances by the total variance, Sobol' indices are obtained:  
\[S_{i_{1},...,i_{s}}=\frac{D_{i_{1},...,i_{s}}}{D} \tag{10}\]  
Index \(S_{i_{1},...,i_{s}}\) describes how much of the model output variance is due to the set of model inputs \([i_{1},...,i_{s}]\). The indices are non-negative and sum up to 1. The integer \(s\) defines the order of the Sobol' index. For example, \(S_{1}\) is a first-order Sobol' index showing how much the variance of input 1 alone contributes to the variance of the output. Since one model input can belong to several input groups, total Sobol' indices are used to account for the total effect of an input [18]. The total Sobol' index of an input \(i\) represents the sum of all Sobol' indices containing input \(i\):  
\[S_{i}^{T}=\sum_{i\in[i_{1},...,i_{s}]}S_{i_{1},...,i_{s}} \tag{11}\]  
For example, a trivariate function \(y=f(x_{1},x_{2},x_{3})\) can be expended as follows:  
\[