knitr::opts_chunk$set( collapse = TRUE , comment = "#>" , fig.width = 7 ) library(ODEsensitivity)

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`

.

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.

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

.

LVmod = function(Time, State, Pars) { with(as.list(c(State, Pars)), { Ingestion <- rIng * Prey * Predator GrowthPrey <- rGrow * Prey * (1 - Prey/K) MortPredator <- rMort * Predator dPrey <- GrowthPrey - Ingestion dPredator <- Ingestion * assEff - MortPredator return(list(c(dPrey, dPredator))) }) }

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:

LVpars = c("rIng", "rGrow", "rMort", "assEff", "K") LVbinf = c(0.05, 0.05, 0.05, 0.05, 1) LVbsup = c(1.00, 3.00, 0.95, 0.95, 20) LVinit = c(Prey = 1, Predator = 2) LVtimes = c(0.01, seq(1, 50, by = 1))

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

.

set.seed(1618) LVres_morris = ODEmorris(mod = LVmod, pars = LVpars, state_init = LVinit , times = LVtimes, binf = LVbinf, bsup = LVbsup )

Let's take a look at the output `LVres_morris`

.

str(LVres_morris, give.attr = FALSE)

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.

ODEsensitivity provides a `plot()`

method for objects of class `ODEmorris`

:

```
plot(LVres_morris)
```

`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):

plot(LVres_morris, pars_plot = c("rIng", "rMort", "assEff"), state_plot = "Predator")

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