SIRTermTimeForcing: SIR model with corrected term-time forcing (P 5.2).
In EpiDynamics: Dynamic Models in Epidemiology

Description Usage Arguments Details Value References See Also Examples

Solves a SIR model with corrected term-time forcing of the transmission rate.

SIRTermTimeForcing(
  pars = NULL,
  init = NULL,
  term.times = terms,
  cicles = 10,
  low.term.first = TRUE,
  ...
)

`pars`	`list` with 4 values: the mean transmission rate, a scalar (or a `vector` to create bifurcations) with the amplitude of sinusoidal forcing, the removal recovery rate, and the per capita death rate. The names for these values must be: "beta0", "beta1", "gamma", and "mu", respectively. All parameters must be positive and beta1 <= 1.
`init`	`vector` with 3 values: the initial proportion of proportion of susceptibles, infectious and recovered. The names of these values must be "S", "I" and "R", respectively. S + I + R <= 1.
`term.times`	`vector` indicating the term times (see details and example).
`cicles`	value indicating how many times `term.times` must be simulated (see details and example).
`low.term.first`	logical. If `TRUE` (default), the first term-time is considered -1, the second 1, the tirth -1 and so on. When `FALSE`, the first term-time is 1, the second -1, and so on (see example).
`...`	further arguments passed to ode function.

This is the R version of program 5.2 from page 171 of "Modeling Infectious Disease in humans and animals" by Keeling & Rohani.

This model is based on the behaviour os measles and other child-hood diseases. Transmission rate is low during term == -1 (e.g. holydas term) and high during term == 1 (e.g. school term). We can define the year as the temporal unit of cicles and each cicle is composed by a term-time sequence (see example).

list. The first element, *$model, is the model function. The second, third and fourth elements are vectors (*$pars, *$init, *$time, respectively) containing the pars, init and time arguments of the function. The fifth element *$results is a data.frame with up to as many rows as elements in time. First column contains the time. Second, third and fourth columns contain the proportion of susceptibles, infectious and recovered.

Keeling, Matt J., and Pejman Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2008.

ode.

## Parameters and initial conditions.
initials <- c(S = 1/17, I = 1e-4, R = 1 - 1/17 - 1e-4)
parameters <- list(beta0 = 17 / 13, beta1 = 0.25,
                gamma = 1 / 13, mu = 1 / (50 * 365))

## Term-times and cycles
# In a year-unit cicle, holidays happen for example
# between days 1 and 6, 101 and 115, 201 and 251,
# 301 and 307 and 308 and 365. 
# Setting low.term.first == TRUE (default) we define the
# previous term-times as low terms.
# Simulate 10 years.
terms <- c(1, 6, 100, 115, 200, 251, 300, 307, 356, 365)
cicles <- 10

# Solve and plot.
sir.term.time.forcing <- SIRTermTimeForcing(pars = parameters,
                                            init = initials,
                                            term.times = terms,
                                            cicles = 10)
PlotMods(sir.term.time.forcing)

# Solve bifurcation dynamics for 20 years.
# If the number of time-units per cicle (e.g. days) times
# the number of cicles (e.g. number of days) is less
# than 3650, bifurcation dynamics are solved for 3650
# time-steps
parameters2 <- list(beta0 = 17 / 13,
                beta1 = seq(0, 0.3, by = 0.001),
                gamma = 1 / 13, mu = 1 / (50 * 365))
# Uncomment the following lines (running it takes more than a few seconds):
# bifur <- SIRTermTimeForcing(pars = parameters2, init = initials,
#                             term.times = terms, cicles = 10)
# PlotMods(bifur, bifur = TRUE)