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#' Simulation of a compartmental infectious disease transmission model including seasonality
#'
#' @description Simulation of a compartmental model with several different compartments:
#' Susceptibles (S), Infected and Pre-symptomatic (P),
#' Infected and Asymptomatic (A), Infected and Symptomatic (I),
#' Recovered and Immune (R) and Dead (D).
#'
#' This model includes natural births and deaths and waning immunity.
#' It also allows for seasonal variation in transmission.
#' The model is assumed to run in units of months.
#' This assumption is hard-coded into the sinusoidally varying
#' transmission coefficient, which is assumed to have a period of a year.
#'
#' @param S : initial number of susceptible hosts : numeric
#' @param P : initial number of infected, pre-symptomatic hosts : numeric
#' @param bP : level/rate of infectiousness for hosts in the P compartment : numeric
#' @param bA : level/rate of infectiousness for hosts in the A compartment : numeric
#' @param bI : level/rate of infectiousness for hosts in the I compartment : numeric
#' @param s : strength of seasonal/annual sigmoidal variation of transmission rate : numeric
#' @param gP : rate at which a person leaves the P compartment : numeric
#' @param gA : rate at which a person leaves the A compartment : numeric
#' @param gI : rate at which a person leaves the I compartment : numeric
#' @param f : fraction of pre-symptomatic individuals that have an asymptomatic infection : numeric
#' @param d : fraction of symptomatic infected hosts that die due to disease : numeric
#' @param w : rate at which recovered persons lose immunity and return to susceptible state : 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 timeunit : units of time in which the model should run (1=day, 2=week, 3=month, 4=year) : numeric
#' @param tmax : maximum simulation time, in units of months : 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. have I0 > PopSize or any 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_idpatterns_ode()
#' # To choose parameter values other than the standard one, specify them like such:
#' result <- simulate_idpatterns_ode(S = 2000, P = 10, tmax = 100, f = 0.1, d = 0.2, s = 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, 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_idpatterns_ode <- function(S = 1000, P = 1, bP = 0, bA = 0, bI = 0.001, s = 0, gP = 0.5, gA = 0.5, gI = 0.5, f = 0, d = 0, w = 0, m = 0, n = 0, timeunit = 1, tmax = 300)
{
############################################################
# start function that specifies differential equations used by deSolve
idpatternsode <- function(t, y, parms)
{
with(
as.list(c(y,parms)), #lets us access variables and parameters stored in y and pars by name
{
#seasonally varying transmission parameters
bPs <- bP * (1 + s * sin(2*pi*t/tunit) )
bAs <- bA * (1 + s * sin(2*pi*t/tunit) )
bIs <- bI * (1 + s * sin(2*pi*t/tunit) )
#the ordinary differential equations
dS = m - S * (bPs * P + bAs * A + bIs * I) + w * R - n *S; #susceptibles
dP = S * (bPs * P + bAs * A + bIs * I) - gP * P - n * P; #infected, pre-symptomatic
dA = f*gP*P - gA * A - n * A #infected, asymptomatic
dI = (1-f)*gP*P - gI*I - n * I #infected, symptomatic
dR = (1-d)*gI*I + gA * A - w * R - n * R #recovered, immune
dD = d*gI*I #dead
list(c(dS, dP, dA, dI, dR, dD))
}
) #close with statement
} #end function specifying the ODEs
############################################################
Y0 = c(S = S, P = P, A = 0, I = 0, R = 0, D = 0); #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)
tunitvec=c(365,52,12,1) #depending on choice of units (days/weeks/months/years), pick divisor for annual variation in transmission in the ODEs
tunit=tunitvec[timeunit]
#combining parameters into a parameter vector
pars = c(bP = bP, bA = bA, bI = bI, gP = gP , gA = gA, gI = gI, f = f, d = d, w = w, m = m, n = n, s = s , tunit = tunit);
#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
odeoutput = deSolve::ode(y = Y0, times = timevec, func = idpatternsode, parms=pars, method = "vode", atol=1e-12, rtol=1e-12);
result <- list()
result$ts <- as.data.frame(odeoutput)
return(result)
}
```

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