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```
#' Simulation of a compartmental infectious disease transmission model to study the reproductive number
#'
#' @description Simulation of a basic SIR compartmental model with these compartments:
#' Susceptibles (S), Infected/Infectious (I),
#' Recovered and Immune (R).
#'
#' The model is assumed to be in units of months when run through the Shiny App.
#' However as long as all parameters are chosen in the same units,
#' one can directly call the simulator assuming any time unit.
#'
#' @param S : initial number of susceptible hosts : numeric
#' @param I : initial number of infected hosts : numeric
#' @param f : fraction of vaccinated individuals. Those individuals are moved from S to R at the beginning of the simulation : numeric
#' @param e : efficacy of vaccine, given as fraction between 0 and 1 : numeric
#' @param b : level/rate of infectiousness for hosts in the I compartment : numeric
#' @param g : rate at which a person leaves the I compartment : numeric
#' @param m : the rate at which new individuals enter the model (are born) : numeric
#' @param n : the rate of natural death (the inverse it the average lifespan) : numeric
#' @param w : rate at which recovered persons lose immunity and return to susceptible state : numeric
#' @param tmax : maximum simulation time : numeric
#' @return This function returns the simulation result as obtained from a call
#' to the deSolve ode solver.
#' @details A compartmental ID model with several states/compartments
#' is simulated as a set of ordinary differential
#' equations. The function returns the output from the odesolver as a matrix,
#' with one column per compartment/variable. The first column is time.
#' @section Warning:
#' This function does not perform any error checking. So if you try to do
#' something nonsensical (e.g. negative values or fractions > 1),
#' the code will likely abort with an error message.
#' @examples
#' # To run the simulation with default parameters just call the function:
#' result <- simulate_reproductivenumber2_ode()
#' # To choose parameter values other than the standard one,
#' # specify the parameters you want to change, e.g. like such:
#' result <- simulate_reproductivenumber2_ode(S = 2000, I = 10, tmax = 100, g = 0.5, n = 0.1)
#' # You should then use the simulation result returned from the function, like this:
#' plot(result$ts[ , "time"],result$ts[ , "S"],xlab='Time',ylab='Number Susceptible',type='l')
#' @seealso The UI of the app 'ReproductiveNumber 2', which is part of this package, contains more details on the model.
#' @references See e.g. Keeling and Rohani 2008 for SIR models and the
#' documentation for the deSolve package for details on ODE solvers
#' @author Andreas Handel
#' @export
simulate_reproductivenumber2_ode <- function(S = 1000, I = 1, f = 0.0, e = 0.0, b = 1e-2, g = 10, m = 0, n = 0, w = 0, tmax = 300){
############################################################
# start function that specifies differential equations used by deSolve
reproductivenumberode <- function(t, y, parms)
{
with(
as.list(c(y,parms)), #lets us access variables and parameters stored in y and parms by name
{
#the ordinary differential equations
dS = m - n*S - b*S*I + w*R; #susceptibles
dI = b*S*I - g*I - n*I; #infected/infectious
dR = g*I - n*R - w*R; #recovered
list(c(dS, dI, dR))
}
) #close with statement
} #end function specifying the ODEs
############################################################
S0eff = (1 - f*e) * S;
R = f*e * S; #initial number of recovered/removed (inlcudes vaccinated)
Y0 = c(S = S0eff, I = I, R = R); #combine initial conditions into a vector
dt = min(0.1, tmax / 1000); #time step for which to get results back
timevec = seq(0, tmax, dt); #vector of times for which solution is returned (not that internal timestep of the integrator is different)
#combining parameters into a parameter vector
pars = c(b = b, g = g, m = m, n = n, w = w);
#this line runs the simulation, i.e. integrates the differential equations describing the infection process
#the result is saved in the odeoutput matrix, with the 1st column the time, the 2nd, 3rd, 4th column the variables S, I, R
#This odeoutput matrix will be re-created every time you run the code, so any previous results will be overwritten
odeoutput = deSolve::ode(y = Y0, times = timevec, func = reproductivenumberode, parms=pars, method = "vode", atol=1e-8, rtol=1e-8);
result <- list()
result$ts <- as.data.frame(odeoutput)
return(result)
}
```

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