# Transition probabilities of a death/birth-death process Computes the transition pobabilities of a death/birth-death process using the continued fraction representation of its Laplace transform ```r dbd_prob(t, a0, b0, mu1, lambda2, mu2, gamma, a = 0, 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` - `mu1`: death 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`: lower bound for the total number of type 1 particles (default `a = 0`) - `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 ```r ## Not run: data(Eyam) loglik_sir <- function(param, data) { alpha <- exp(param[1]) # Rates must be non-negative beta <- exp(param[2]) # Set-up SIR model drates1 <- function(a, b) { 0 } brates2 <- function(a, b) { 0 } drates2 <- function(a, b) { alpha * b } trans12 <- function(a, b) { beta * a * b } sum(sapply(1:(nrow(data) - 1), # Sum across all time steps k function(k) { log( dbd_prob( # Compute the transition probability matrix t = data$time[k + 1] - data$time[k], # Time increment a0 = data$S[k], b0 = data$I[k], # From: S(t_k), I(t_k) drates1, brates2, drates2, trans12, a = data$S[k + 1], B = data$S[k] + data$I[k] - data$S[k + 1], computeMode = 4, nblocks = 80 # Compute using 4 threads )[1, data$I[k + 1] + 1] # To: S(t_(k+1)), I(t_(k+1)) ) })) } loglik_sir(log(c(3.204, 0.019)), Eyam) # Evaluate at mode ## End(Not run) # Birth-death-shift model for transposable elements lam = 0.0188; mu = 0.0147; v = 0.00268; # birth, death, shift rates drates1 <- function(a, b) { mu * a } brates2 <- function(a, b) { lam * (a + b) } drates2 <- function(a, b) { mu * b } trans12 <- function(a, b) { v * a } # Get transition probabilities p <- dbd_prob(t = 1, a0 = 10, b0 = 0, drates1, brates2, drates2, trans12, a = 0, B = 50) ``` ## References Ho LST et al. 2016. "Birth(death)/birth-death processes and their computable transition probabilities with statistical applications". In review.

dbd_prob function