Simple MCMC under SIR

We describe how to set-up and run a simple Metropolis-Hastings-based Markov chain Monte Carlo (MCMC) sampler under the susceptible-infected-removed (SIR) model.

library(MultiBD)

This example uses the Eyam data that consist the population counts of susceptible, infected and removed individuals across several time points.

data(Eyam)
Eyam

##   time   S  I   R
## 1  0.0 254  7   0
## 2  0.5 235 14  12
## 3  1.0 201 22  38
## 4  1.5 153 29  79
## 5  2.0 121 20 120
## 6  2.5 110  8 143
## 7  3.0  97  8 156
## 8  4.0  83  0 178

The log likelihood function is the sum of the log of the transition probabilities between two consecutive observations. Note that, we will use (log α, log β) as parameters instead of (α, β). The rows and columns of the transition probability matrix returned by dbd_prob() correspond to possible values of S (from a to a0) and I (from 0 to B) respectively.

loglik_sir <- function(param, data) {
  alpha <- exp(param[1]) # Rates must be non-negative
  beta  <- exp(param[2])
  
  # Set-up SIR model
  drates1 <- function(a, b) { 0 }
  brates2 <- function(a, b) { 0 }
  drates2 <- function(a, b) { alpha * b     }
  trans12 <- function(a, b) { beta  * a * b }
  
  sum(sapply(1:(nrow(data) - 1), # Sum across all time steps k
             function(k) {
               log(
                 dbd_prob(  # Compute the transition probability matrix
                   t  = data$time[k + 1] - data$time[k], # Time increment
                   a0 = data$S[k], b0 = data$I[k],       # From: S(t_k), I(t_k)                                      
                   drates1, brates2, drates2, trans12,
                   a = data$S[k + 1], B = data$S[k] + data$I[k] - data$S[k + 1],
                   computeMode = 4, nblocks = 80         # Compute using 4 threads
                 )[1, data$I[k + 1] + 1]                 # To: S(t_(k+1)), I(t_(k+1))
               )
             }))
}

Here, we choose Normal(0, 100²) as the prior for both log α and log β.

logprior <- function(param) {
  log_alpha <- param[1]
  log_beta <- param[2]
  
  dnorm(log_alpha, mean = 0, sd = 100, log = TRUE) +
    dnorm(log_beta, mean = 0, sd = 100, log = TRUE)
}

We will use the random walk Metropolis algorithm implemented in the function MCMCmetrop1R() (MCMCpack package) to explore the posterior distribution. So, we first need to install the package and its dependencies.

source("http://bioconductor.org/biocLite.R")
biocLite("graph")
biocLite("Rgraphviz")
install.packages("MCMCpack", repos = 'http://cran.us.r-project.org')
library(MCMCpack)

The starting point of our Markov chain is the estimated value of (α, β) from Raggett (1982).

alpha0 <- 3.39
beta0  <- 0.0212

We discard the first 200 iterations and keep the next 1000 iterations of the chain.

post_sample <- MCMCmetrop1R(fun = function(param) { loglik_sir(param, Eyam) + logprior(param) },
                           theta.init = log(c(alpha0, beta0)),
                           mcmc = 1000, burnin = 200)

The trace plots of both log α and log β look good.

plot(as.vector(post_sample[,1]), type = "l", xlab = "Iteration", ylab = expression(log(alpha)))

plot(as.vector(post_sample[,2]), type = "l", xlab = "Iteration", ylab = expression(log(beta)))

We can visualize the joint posterior distribution of log α and log β using the ggplot2 package.

library(ggplot2)
x = as.vector(post_sample[,1])
y = as.vector(post_sample[,2])
df <- data.frame(x, y)
ggplot(df,aes(x = x,y = y)) +
  stat_density2d(aes(fill = ..level..), geom = "polygon", h = 0.26) + 
  scale_fill_gradient(low = "grey85", high = "grey35", guide = FALSE) +
  xlab(expression(log(alpha))) +
  ylab(expression(log(beta)))

We can also construct the 95% Bayesian credible intervals for α and β.

quantile(exp(post_sample[,1]), probs = c(0.025,0.975))

##     2.5%    97.5% 
## 2.721921 3.809780

quantile(exp(post_sample[,2]), probs = c(0.025,0.975))

##       2.5%      97.5% 
## 0.01649866 0.02327176