# ODEsensitivity

## Introduction

The goal of sensitivity analysis is to examine how sensitive a mathematical model responds to variations in its input variables. Here we focus on the sensitivity analysis of ordinary differential equation (ODE) models via Morris screening.

If the assumption of a uniform distribution on the domain intervals doesn’t hold, the Morris screening method cannot be used and the variance-based Sobol’ method should be considered instead. In this case, simply switch from using the function ODEmorris to the function ODEsobol.

## Analyse the Lotka-Volterra Equations

The Lotka-Volterra equations describe a predator and its prey’s population development and go back to Lotka (1925) and Volterra (1926). The prey’s population at time $$t$$ (in days) will be denoted with $$P(t)$$ and the predator’s (or rather consumer’s) population with $$C(t)$$. $$P(t)$$ and $$C(t)$$ are called state variables. This ODE model is two-dimensional, but it should be noted that ODE models of arbitrary dimensions (including one-dimensional ODE models) can be analyzed with ODEsensitivity.

### Model Definition

Now we define the model according to the definition in deSolve::ode().

Each of the five parameter names, their lower and upper boundaries, the initial values for the state variables and the timepoints of interest are saved in separate vectors:

### Sensitivity Analysis

The sensitivity analysis of a general ODE model can be performed by using the generic function ODEsensitivity::ODEmorris().

Let’s take a look at the output LVres_morris.

The first row of each state variable contains a copy of all timepoints. The other rows contain the Morris sensitivity indices $$\mu$$, $$\mu^\star$$, and $$\sigma$$ for all 5 parameters and all 51 timepoints.

### Plotting

ODEsensitivity provides a plot() method for objects of class ODEmorris:

plot.ODEmorris() has two important arguments: pars_plot and state_plot. Using pars_plot, a subset of the parameters included in the sensitivity analysis can be selected for plotting (the default is to use all parameters). state_plot gives the name of the state variable for which the sensitivity indices shall be plotted (the default being the first state variable):