Author: Chris Rackauckas
Date: 2024-12-04

# Solving Delay Differential Equations (DDE) in R with diffeqr

#### Chris Rackauckas

#### 2024-12-04

A delay differential equation is an ODE which allows the use of previous values. In this case, the function needs to be a JIT compiled Julia function. It looks just like the ODE, except in this case there is a function `h(p,t)` which allows you to interpolate and grab previous values.

We must provide a history function `h(p,t)` that gives values for `u` before `t0`. Here we assume that the solution was constant before the initial time point. Additionally, we pass `constant_lags = c(20.0)` to tell the solver that only constant-time lags were used and what the lag length was. This helps improve the solver accuracy by accurately stepping at the points of discontinuity. Together this is:

    f <- JuliaCall::julia_eval("function f(du, u, h, p, t)
      du[1] = 1.1/(1 + sqrt(10)*(h(p, t-20)[1])^(5/4)) - 10*u[1]/(1 + 40*u[2])
      du[2] = 100*u[1]/(1 + 40*u[2]) - 2.43*u[2]
    end")
    h <- JuliaCall::julia_eval("function h(p, t)
      [1.05767027/3, 1.030713491/3]
    end")
    u0 <- c(1.05767027/3, 1.030713491/3)
    tspan <- c(0.0, 100.0)
    constant_lags <- c(20.0)
    JuliaCall::julia_assign("u0", u0)
    JuliaCall::julia_assign("tspan", tspan)
    JuliaCall::julia_assign("constant_lags", tspan)
    prob <- JuliaCall::julia_eval("DDEProblem(f, u0, h, tspan, constant_lags = constant_lags)")
    sol <- de$solve(prob,de$MethodOfSteps(de$Tsit5()))
    udf <- as.data.frame(t(sapply(sol$u,identity)))
    plotly::plot_ly(udf, x = sol$t, y = ~V1, type = 'scatter', mode = 'lines') %>% plotly::add_trace(y = ~V2)

delay

Notice that the solver accurately is able to simulate the kink (discontinuity) at `t=20` due to the discontinuity of the derivative at the initial time point! This is why declaring discontinuities can enhance the solver accuracy.