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().

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))

Sensitivity Analysis

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)
#> List of 2
#>  $ Prey    : num [1:16, 1:51] 1.00e-02 -1.90e-02 2.46e-02 4.62e-05 -3.05e-05 ...
#>  $ Predator: num [1:16, 1:51] 1.00e-02 8.97e-03 5.96e-05 -1.80e-02 9.24e-03 ...

The first row of each state variable contains a copy of all timepoints. The other rows contain the Morris sensitivity indices μ, μ⋆, and σ for all 5 parameters and all 51 timepoints.

Plotting

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")