Modifying SIR model to include stochasticity - r

Question

I am trying to establish a method of estimating infectious disease parameters by comparing real epidemic curves with simulations of a stochastic SIR model. To construct the stochastic SIR model, I am using the deSolve package and instead of using fixed parameter values I would like to draw the parameter value used in the equations at each time point from a Poisson distribution centered on the original parameter values.
Using the parameter beta as an example, beta represents the average number of transmission events per capita and is the product of the average number of contacts and the probability that transmission occurs upon contact. Realistically, there is variation in the number of contacts a person will have and since transmission is also a probabilistic event there is variation surrounding this too.
So even if the average transmission rate were to be 2.4 (for example), an individual can go on to infect 0, 1, 2 or 3 ... etc. people with varying probabilities.
I have tried to incorporate this into my code below using the rpois function and reassigning the parameters used in the equations to the outputs of the rpois.
I have run my code with the same initial values and parameters multiple times and all the curves are different indicating that SOMETHING "stochastic" is going on, but I am unsure whether the code is sampling using the rpois at each time point or just once at the beginning. I have only started coding very recently so do not have much experience.
I would be grateful if anyone more experienced than myself could verify what my code is ACTUALLY doing and whether it is sampling using rpois at each time point or not. If not I would be grateful for any suggestions for achieving this. Perhaps a loop is needed?
library('deSolve')
library('reshape2')
library('ggplot2')
#MODEL INPUTS
initial_state_values <- c(S = 10000,
I = 1,
R = 0)
#PARAMETERS
parameters <- c(beta = 2.4,
gamma = 0.1)
#POISSON MODELLING OF PARAMETERS
#BETA
beta_p <- rpois(1, parameters[1])
#GAMMA
infectious_period_p <- rpois(1, 1/(parameters[2]))
gamma_p <- 1/infectious_period_p
#TIMESTEPS
times <- seq(from = 0, to = 50,by = 1)
#SIR MODEL FUNCTION
sir_model <- function(time, state, parameters) {
with(as.list(c(state, parameters)), {
N <- S + I + R
lambda <- beta_p * I/N
dS <- -lambda * S
dI <- lambda*S - gamma_p*I
dR <- gamma_p*I
return(list(c(dS, dI, dR)))
})
}
output<- as.data.frame(ode(y= initial_state_values,
times = times,
func = sir_model,
parms = parameters))

Answer 1

The code given in the question runs the model with constant parameters over time. Here an example with parameters varying over time. However, this setting assumes that for a given time step, the parameters are equal for all indidividuals of the population. If you want to have individual variability, one can either use a matrix formulation for different sub-populations or use an individual model instead.
Model with fluctuating population parameters:
library('deSolve')
initial_state_values <- c(S = 10000,
I = 1,
R = 0)
parameters <- c(beta = 2.4, gamma = 0.1)
times <- seq(from = 0, to = 50, by = 1) # note time step = 1!
# +1 to add one for time = zero
beta_p <- rpois(max(times) + 1, parameters[1])
infectious_period_p <- rpois(max(times) + 1, 1/(parameters[2]))
gamma_p <- 1/infectious_period_p
sir_model <- function(time, state, parameters) {
# cat(time, "\n") # show time steps for debugging
with(as.list(c(state, parameters)), {
# this overwrites the parms passed via parameters
beta <- beta_p[floor(time) + 1]
gamma <- gamma_p[floor(time) + 1]
N <- S + I + R
lambda <- beta * I/N
dS <- -lambda * S
dI <- lambda * S - gamma * I
dR <- gamma * I
list(c(dS, dI, dR))
})
}
output <- ode(y = initial_state_values,
times = times,
func = sir_model,
parms = parameters)
plot(output)

Answer 2

Here another, slightly more generalized version. It is added as a second answer, to keep the original version compact and simple. The new version differs with respect to the following:
generalized, so that it can work with fixed parameters and stochastic forcing
pass parameters as list
run a basic Monte-Carlo simulation
library('deSolve')
sir_model <- function(time, state, parameters) {
with(as.list(c(state, parameters)), {
# this overwrites the parms passed via parameters
if (time_dependent) {
beta <- beta_p[floor(time) + 1]
gamma <- gamma_p[floor(time) + 1]
}
N <- S + I + R
lambda <- beta * I/N
dS <- -lambda * S
dI <- lambda * S - gamma * I
dR <- gamma * I
list(c(dS, dI, dR))
})
}
initial_state_values <- c(S = 10000, I = 1, R = 0)
times <- seq(from = 0, to = 50, by = 1) # note time step = 1!
## (1) standard simulation with constant parameters
parameters <- c(beta = 2.4, gamma = 0.1)
out0 <- ode(y= initial_state_values,
times = times,
func = sir_model,
parms = c(parameters, time_dependent = FALSE))
plot(out0)
## (2) single simulation with time varying parameters
beta_p <- rpois(max(times) + 1, parameters[1])
infectious_period_p <- rpois(times + 1, 1/(parameters[2]))
gamma_p <- 1/infectious_period_p
## here we need pass the vectorized parameters globally
## for simplicity, it can also be done as list
out1 <- ode(y = initial_state_values, times = times,
func = sir_model, parms = c(time_dependent = TRUE))
plot(out0, out1)
## (3) a sample of simulations
monte_carlo <- function(i) {
#parameters <- c(beta = 2.4, gamma = 0.1)
beta_p <- rpois(max(times) + 1, parameters[1])
infectious_period_p <- rpois(max(times) + 1, 1 / (parameters[2]))
gamma_p <- 1/infectious_period_p
ode(y = initial_state_values, times = times,
func = sir_model, parms = list(beta_p = beta_p,
gamma_p = gamma_p,
time_dependent = TRUE))
}
## run 10 simulations
out_mc <- lapply(1:10, monte_carlo)
plot(out0, out_mc, mfrow=c(1, 3))