Your equation in B(t) is just-about separable since you can divide out B(t), from which you can get that
B(t) = C * exp{-p5 * t} * (p2 + B(t)) ^ {of_interest * p1 * p3}
This is an implicit solution for B(t) which we'll solve point-wise.
You can solve for C given your initial value of B. I suppose t = 0 initially? In which case
C = B_0 / (p2 + B_0) ^ {of_interest * p1 * p3}
This also gives a somewhat nicer-looking expression for A(t):
dA(t) / dt = B_0 / (p2 + B_0) * p1 * p3 * (1 - of_interest) *
exp{-p5 * t} * ((p2 + B(t) / (p2 + B_0)) ^
{of_interest * p1 * p3 - 1} - p4 * A(t)
This can be solved by integrating factor (= exp{p4 * t}), via numerical integration of the term involving B(t). We specify the lower limit of the integral as 0 so that we never have to evaluate B outside the range [0, t], which means the integrating constant is simply A_0 and thus:
A(t) = (A_0 + integral_0^t { f(tau; parameters) d tau}) * exp{-p4 * t}
The basic gist is B(t) is driving everything in this system -- the approach will be: solve for the behavior of B(t), then use this to figure out what's going on with A(t), then maximize.
First, the "outer" parameters; we also need nleqslv to get B:
library(nleqslv)
t_min <- 0
t_max <- 10000
t_N <- 10
#we'll only solve the behavior of A & B over t_rng
t_rng <- seq(t_min, t_max, length.out = t_N)
#I'm calling of_interest ttheta
ttheta_min <- 0
ttheta_max <- 1
ttheta_N <- 5
tthetas <- seq(ttheta_min, ttheta_max, length.out = ttheta_N)
B_0 <- 1.4
A_0 <- 28
#No sense storing this as a vector when we'll only ever use it as a list
parameters <- list(p1 = 0.028, p2 = 0.3, p3 = 0.5,
p4 = 0.0002, p5 = 0.001)
From here, the basic outline is:
- Given the parameter values (in particular
ttheta), solve for BB over t_rng via non-linear equation solving
- Given
BB and the parameter values, solve for AA over t_rng by numerical integration
- Given
AA and your expression for dAdt, plug & maximize.
derivs <-
sapply(tthetas, function(th){
#append current ttheta
params <- c(parameters, ttheta = th)
#declare a function we'll use to solve for B (see above)
b_slv <- function(b, t)
with(params, b - B_0 * ((p2 + b)/(p2 + B_0)) ^
(ttheta * p1 * p3) * exp(-p5 * t))
#solving point-wise (this is pretty fast)
# **See below for a note**
BB <- sapply(t_rng, function(t) nleqslv(B_0, function(b) b_slv(b, t))$x)
#this is f(tau; params) that I mentioned above;
# we have to do linear interpolation since the
# numerical integrator isn't constrained to the grid.
# **See below for note**
a_int <- function(t){
#approximate t to the grid (t_rng)
# (assumes B is monotonic, which seems to be true)
# (also, if t ends up negative, just assign t_rng[1])
t_n <- max(1L, which.max(t_rng - t >= 0) - 1L)
idx <- t_n:(t_n+1)
ts <- t_rng[idx]
#distance-weighted average of the local B values
B_app <- sum((-1) ^ (0:1) * (t - ts) / diff(ts) * BB[idx])
#finally, f(tau; params)
with(params, (1 - ttheta) * p1 * p3 * B_0 / (p2 + B_0) *
((p2 + B_app)/(p2 + B_0)) ^ (ttheta * p1 * p3 - 1) *
exp((p4 - p5) * t))
}
#a_int only works on scalars; the numeric integrator
# requires a version that works on vectors
a_int_v <- function(t) sapply(t, a_int)
AA <- exp(-params$p4 * t_rng) *
sapply(t_rng, function(tt)
#I found the subdivisions constraint binding in some cases
# at the default value; no trouble at 1000.
A_0 + integrate(a_int_v, 0, tt, subdivisions = 1000L)$value)
#using the explicit version of dAdt given as flux1 - flux2
max(with(params, (1 - ttheta) * p1 * p3 * BB / (p2 + BB) - p4 * AA))})
Finally, simply run `tthetas[which.max(derivs)]` to get the maximizer.
Note:
This code is not optimized for efficiency. There are a few places where there are some potential speed-ups:
- probably faster to run the equation solver recursively, as it'll converge faster with better initial guesses -- using the previous value instead of the initial value is surely better
- Will be faster to simply use Riemann sums to integrate; the tradeoff is in accuracy, but should be fine if you have a dense enough grid. One beauty of Riemann is you won't have to interpolate at all, and numerically they're simple linear algebra. I ran this with
t_N == ttheta_N == 1000L and it ran within a few minutes.
- Probably possible to vectorize
a_int directly instead of just sapplying on it, which concomitant speed-up by more direct appeal to BLAS.
- Loads of other small stuff. Pre-compute
ttheta * p1 * p3 since it's re-used so much, etc.
I didn't bother including any of that stuff, though, because you're honestly probably better off porting this to a faster language -- Julia is my own pet favorite, but of course R speaks well with C++, C, Fortran, etc.
