bbd_prob function

Transition probabilities of a birth/birth-death process

Computes the transition pobabilities of a birth/birth-death process using the continued fraction representation of its Laplace transform

bbd_prob(t, a0, b0, lambda1, lambda2, mu2, gamma, A, B, nblocks = 256,
  tol = 1e-12, computeMode = 0, nThreads = 4, maxdepth = 400)

Arguments

• t: time
• a0: total number of type 1 particles at t = 0
• b0: total number of type 2 particles at t = 0
• lambda1: birth rate of type 1 particles (a two variables function)
• lambda2: birth rate of type 2 particles (a two variables function)
• mu2: death rate function of type 2 particles (a two variables function)
• gamma: transition rate from type 2 particles to type 1 particles (a two variables function)
• A: upper bound for the total number of type 1 particles
• B: upper bound for the total number of type 2 particles
• nblocks: number of blocks
• tol: tolerance
• computeMode: computation mode
• nThreads: number of threads
• maxdepth: maximum number of iterations for Lentz algorithm

Returns

a matrix of the transition probabilities

Examples

## Not run:

data(Eyam)

# (R, I) in the SIR model forms a birth/birth-death process

loglik_sir <- function(param, data) {
  alpha <- exp(param[1]) # Rates must be non-negative
  beta  <- exp(param[2])
  N <- data$S[1] + data$I[1] + data$R[1]
  
  # Set-up SIR model with (R, I)
  
  brates1 <- function(a, b) { 0 }
  brates2 <- function(a, b) { beta  * max(N - a - b, 0)  * b }
  drates2 <- function(a, b) { 0 }
  trans21 <- function(a, b) { alpha * b }
  
  sum(sapply(1:(nrow(data) - 1), # Sum across all time steps k
             function(k) {
               log(
                 bbd_prob( # Compute the transition probability matrix
                   t  = data$time[k + 1] - data$time[k], # Time increment
                   a0 = data$R[k], b0 = data$I[k],       # From: R(t_k), I(t_k)
                   brates1, brates2, drates2, trans21,
                   A = data$R[k + 1], B = data$R[k + 1] + data$I[k] - data$R[k],
                   computeMode = 4, nblocks = 80         # Compute using 4 threads
                 )[data$R[k + 1] - data$R[k] + 1, 
                   data$I[k + 1] + 1]                    # To: R(t_(k+1)), I(t_(k+1))
               )
             }))
}

loglik_sir(log(c(3.204, 0.019)), Eyam) # Evaluate at mode
## End(Not run)

References

Ho LST et al. 2015. "Birth(death)/birth-death processes and their computable transition probabilities with statistical applications". In review.