[R] Event-triggered change in value with a time-delay

[Hannah Meredith]

Hello,

I am solving a set of ODEs using deSolve and have run into a problem I
would appreciate some advice on. One of the parameters (m) in the ODEs
changes between two states when one of the variables (D) crosses a
threshold (D_T) for the first time in either direction. Additionally, when
the variable crosses the threshold (either by increasing or decreasing),
there is a time delay (delay) before the parameter shifts values.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Toy model:
# define parameters
P <- c(m = 10,
       R_1 = 0.5,
       R_2 = 0.1,
       D_0 = 50,
       D_T = 5,
       delay = 3
)
# set initial values
y_0 <- c(D = 0, Y = 1)

# set timesteps
times <- seq(0, 100, length = 101)

# define function
bit <- function(t,y,parms){
  with(as.list(c(parms,y)),{


    # How I would implement the change if there was no time delay
    # m <- ifelse(D>=D_T, m*.68, m)

    # Option 1: nested if statements with time delay
    # if (D >= D_T) {
    #   t_start1 <- t + delay
    #   if (t >= t_start1){
    #     m <- 6.8
    #   }
    # }
    # if (D < D_T) {
    #   t_start2<- t + delay
    #   if (t >= t_start2){
    #     m <- 10
    #   }
    # }


    dD <- D_0 * (R_1 / (R_1 - R_2)) * (-R_2 * exp(-R_2 * t) + R_1 *
exp(-R_1 * t))
    dY <- Y + m;
    res <- c(dD, dY)
    list(res, m = m)
  })
}


# Solve model
library(deSolve)
out <- ode(y = y_0, times = times, func = bit, parms = P)
out.df <- as.data.frame(out)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I have come up with a number of ideas, but I either haven't been able to
get them to work or don't know if R has a command for it:

1. Nested "if" statements with something to track time (as shown above).
One issue here is* t-start *is redefined with each time step in the ode
solver so *t* never reaches *t_start +delay*.

2. Event triggered changes. From the examples I have found (link
<https://cran.r-project.org/web/packages/deSolve/vignettes/deSolve.pdf> p.
24-25), the time at which the event happens is either pre-set or occurs at
a root of the ODE.  Also- would this strategy require multiple events? One
event in which the variable crossing the threshold triggers a timer,
another for the timer triggering the parameter change?

3. Lagvalue. This might take care of the time lag component, but I am still
struggling to understand (1) how lagvalue works and (2) how to integrate
both the threshold and timer components.

   if ( D < D_T )
     m<- ifelse(lagvalue(t-delay)<=0, 6.8, 10)
   end

4. Sigmoidal function. Someone suggested this to me earlier, but I am not
sure how this would work. Is it possible to have a sigmoidal function that
switches repeatedly between two states (as opposed to the characteristic S
curve that transitions once from one state to another)?

5. Mathmematica has a WhenEvent function- is there something similar in R?


Thank you for your time and any tips you might have!

	[[alternative HTML version deleted]]