Here is a skeleton for the list of transitions and the function that returns the rates. You should write one line of code every where it is indicated.
SEITL_transitions <- list(
# fill transition for infection
# fill transition for onset of infectiousness
# fill transition for recovery + temporary immunity
# fill transition for long-term immunity
# fill transition for NO long-term immunity
)
SEITL_rateFunc <- function(state, theta, t) {
# define model parameters in term of the natural parameters
beta <- # fill HERE
epsilon <- # fill HERE
nu <- # fill HERE
alpha <- # fill HERE
tau <- # fill HERE
# create temporary variables for states to simplify the writing of the rates below
S <- state[["S"]]
E <- state[["E"]]
I <- state[["I"]]
T <- state[["T"]]
L <- state[["L"]]
N <- # fill HERE
# return rates
return(c(
# fill rate for infection
# fill rate for onset of infectiousness
# rfill rate for ecovery + temporary immunity
# fill rate for long-term immunity
# fill rate for NO long-term immunity
))
}
Once you have filled the gap you can come back to the practical. If you need more help in filling the gap, you can have a look at the solution.