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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 R : initial number of recovered hosts : numeric
#' @param b : rate of new infections : numeric
#' @param g : rate of recovery : numeric
#' @param f : strength of intervention effort : numeric
#' @param tstart : time at which intervention effort starts : numeric
#' @param tend : time at which intervention effort ends : numeric
#' @param tnew : time at which new infected enter : 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.
#' The model implement basic processes of infection at rate b and recovery at rate g.
#' Treatment is applied, which reduces b by the indicated proportion, during times tstart and tend.
#' At time intervals given by tnew, a new infected individual enters the population.
#' The simulation also monitors the number of infected and when they drop below 1, they are set to 0.
#' @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_idcontrolmultioutbreak_ode()
#' # To choose parameter values other than the standard one,
#' # specify the parameters you want to change, e.g. like such:
#' result <- simulate_idcontrolmultioutbreak_ode(S = 2000, I = 10, tmax = 100, g = 0.5)
#' # 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 'Multi Outbreak ID Control', which is part of the DSAIDE package, contains more details.
#' @author Andreas Handel
#' @export
simulate_idcontrolmultioutbreak_ode <- function(S = 1000, I = 1, R = 0, b = 1e-3, g = 1, f = 0.3, tstart = 10, tend = 50, tnew = 50, tmax = 100){
############################################################
# start function that specifies differential equations used by deSolve
idcontrolmultioutbreak_ode <- function(t, y, parms)
{
with(
as.list(c(y,parms)), #lets us access variables and parameters stored in y and parms by name
{
f = ifelse( (t>=tstart && t<=tend) ,f,0) #set f to zero if outside treatment interval
#the ordinary differential equations
dS = - b*(1-f)*S*I #susceptibles
dI = b*(1-f)*S*I - g*I #infected/infectious
dR = g*I #recovered
list(c(dS, dI, dR))
}
) #close with statement
} #end function specifying the ODEs
############################################################
############################################################
# functions that monitor I and sets it to 0 if it crosses the 1 threshold
checkinfected <- function(t,y,parms)
{
y["I"]-0.99
}
zeroinfected <- function(t,y,parms)
{
y["I"] = 0
return(y)
}
Y0 = c(S = S, I = I, R = R); #combine initial conditions into a vector
dt = min(0.1, tmax / 1000); #time step for which to get results back
#combining parameters into a parameter vector
pars = c(b = b, g = g, f = f, tstart = tstart, tend = tend, tnew = tnew);
#if times of immigration is set larger than max simulation time, set them to simulation time max
if (tnew>tmax) {tnew=tmax}
newinftimes = seq(tnew,tmax,by=tnew) #times at which a new infected enters the population
#since the desolve/ode do not handle multiple events/roots (as far as I know)
#we need to do the entry of new infected at times tnew 'by hand' and integrate piecewise
#easiest to do with a loop
odeoutput = NULL
ts = 0; #start pieces of integration at 0
for (n in 1:length(newinftimes))
{
tf = newinftimes[n] #simulate to time of first new infected entering
timevec = seq(ts,tf,dt)
odetmp = deSolve::ode(y = Y0, times = timevec, func = idcontrolmultioutbreak_ode, parms=pars, method = "lsoda", events = list(func = zeroinfected, root = TRUE), rootfun = checkinfected, atol=1e-8, rtol=1e-8);
odeoutput = rbind(odeoutput,odetmp) #not very efficient but ok for here way to save all results
ts = tf #new starting time is last ending time
Y0 = odetmp[nrow(odetmp),-1] #values of variables at last step of simulation
Y0["I"] = Y0["I"] + 1 #add one infected
}
#one more simulation bit to the end
if (ts<tmax)
{
timevec = seq(ts,tmax,dt)
odetmp = deSolve::ode(y = Y0, times = timevec, func = idcontrolmultioutbreak_ode, parms=pars, method = "lsoda", events = list(func = zeroinfected, root = TRUE), rootfun = checkinfected, atol=1e-8, rtol=1e-8);
odeoutput = rbind(odeoutput,odetmp) #not very efficient but ok for here way to save all results
}
#browser()
#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
result <- list()
result$ts <- as.data.frame(odeoutput)
return(result)
}
```

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