In fintzij/BDAepimodel: Fit Stochastic Epidemic Models in R via Bayesian Data Augmentation
library(BDAepimodel)
library(coda)
library(gtools)
library(Rcpp)
library(pomp)
library(ggplot2)
This vignette contains code to fit SIR and SEIR models to binomially distributed prevalence counts sampled from an epidemic with time-varying dynamics. We simulated an epidemic with SEIR dynamics in a population of 400 individuals, 397 of whom were initially susceptible, 2 of whom were initially exposed, and 1 of whom were initially infectious. The dynamics of the epidemic varied over four different epochs are presented in the "model_misspecification" vignette. And the data is shown below:
data<-c(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,1,0,1,0,0,0,0,0,2,2,0,3,0,1,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,3,1,0,0,1,2,2,0,4,0,3,2,0,2,5,2,1,2,2,2,1,1,3,1,2,5,1,2,1,0,1,1,4,1,0,1,0,1,2,0,2,0,1,0,1,0,3,1,1,2,1,1,1,2,3,0,1,2,1,3,3,2,2,0,0,3,2,3,3,0,2,0,2,1,2,3,2,1,0,1,1,0,3,1,2,2,1,1,1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,0,0,1,0,3,0,1,1,0,0,0,1,1,1,1,0,0,2,0,1,1,1,0,1,0,0,1,1,0,1,0,1,1)
Fitting SEIR model:
We use the SEIR model to model the dataset. The hosts are divided into four classes, according to their status. The susceptible class (S); the exposed class (E); the infected class (I); and the removed class (R). Individuals in R are assumed to be immune against reinfection. It is natural to formulate this model as a continuous-time Markov process. To start with, we need to specify the measurement model.
rmeas<-"
cases=rbinom(I,exp(rho)/(1+exp(rho)));//represent the data
"
dmeas<-"
lik=dbinom(cases,I,exp(rho)/(1+exp(rho)),give_log); //return the loglikelihood
"
Here, we are using the $\textbf{cases}$ to refer to the data (number of reported cases) and $\textbf{I}$ to refer to the true incidence over the reporting interval. The binomial simulator rbinom and density function dbinom are provided by R. Notice that, in these snippets, we never declare the variables; pomp will ensure tht the state variable ($\textbf{I}$), observables ($\textbf{cases}$), parameters ($\textbf{rho}$, $\textbf{phi}$) and likelihood ($\textbf{lik}$) are defined in the contexts within which these snippets are executed.
And for the transition of the SEIR model, we use a approximating tau-leaping algorithm, one version of which is implemented in the pomp package via the $\bf{euler.sim}$ plug-in. The algorithm holds the transition rates constant over a small interval of time and simulates the numbers of transitions that occur over the interval. The functions $\textbf{reulermultinom}$ draw random deviates from such distributions. Then we need to first specify a function that advances the states from $t$ to $t+\triangle t$, and we give the trantition of the SIR model code below:
seir.step<-"
double rate[3];
double dN[3];
rate[0]=exp(beta)*I; //Infection
rate[1]=exp(gamma); //Rate of S to E
rate[2]=exp(mu); //Rate of E to I
reulermultinom(1,S,&rate[0],dt,&dN[0]); //generate the number of newly from S to E
reulermultinom(1,E,&rate[1],dt,&dN[1]); //generate the number of newly from E to I
reulermultinom(1,I,&rate[2],dt,&dN[2]); //generate the number of newly from I to R
if(!R_FINITE(S)) Rprintf(\"%lg %lg %lg %lg %lg %lg %lg %lg %lg\\n\",dN[0],rate[0],dN[1],rate[1],beta,mu,S,I,R);
S+=-dN[0]; //update the number of Susceptible
E+=dN[0]-dN[1]; //update the number of E
I+=dN[1]-dN[2]; //update the number of I
R+=dN[2]; //update the number of R
"
The pomp will ensure that the undeclared state variables and parameters are defined in the context within which the snippet is executed. And the $\textbf{rate}$ and $\textbf{dN}$ arrays hold the rates and numbers of transition events, respectively. Then we could construct the pomp objects below:
seir <- pomp(
data = data.frame(cases = data, time = seq(1, 1457, by = 7)), #"cases" is the dataset, "time" is the observation time
times = "time",
t0 = 1, #initial time point
dmeasure = Csnippet(dmeas),
rmeasure = Csnippet(rmeas),
rprocess = euler.sim(step.fun = Csnippet(seir.step), delta.t = 1/4), #delta.t is here
statenames = c("S", "E", "I", "R"), #state space variable name
paramnames = c("beta", "gamma", "mu", "rho", "theta1", "theta2", "theta3"), #parameters name
initializer = function(params, t0, ...) {
ps <-exp(params["theta1"]) / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"]))
pe <- exp(params["theta2"]) / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"]))
pi <- 1 / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"]))
pr <- exp(params["theta3"]) / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"])) #Initial proportion of S, E, I, R
return(setNames(as.numeric(rmultinom(1, 400, prob = c(ps, pe, pi, pr))), c("S", "E", "I", "R")))
},
params = c(
beta = log(0.00072),
gamma = log(0.5),
mu = log(0.2),
rho = log(1 / 4),
theta1 = log(90),
theta2 = log(2),
theta3 = log(7)
)
)
To carry out Bayesian inference we need to specify a prior distribution on unknown parame- ters. The pomp constructor function provides the rprior and dprior arguments, which can be filled with functions that simulate from and evaluate the prior density, respectively. Methods based on random walk Metropolis-Hastings require evaluation of the prior density (dprior), so we specify dprior for the SEIR model as follows.
trans<-function(a,b,c){
a_t<-exp(a)
b_t<-exp(b)
c_t<-exp(c)
return(a_t*b_t*c_t/((1+a_t+b_t+c_t)^4))
} #Jacobian for (theta1, theta2, theta3)
seir.dprior <- function(params, ..., log) {
f <- (
dgamma(exp(params[1]), 0.6, 10000, log = TRUE) + params[1] + #log prior for beta
dgamma(exp(params[2]), 1.5, 25, log = TRUE) + params[2] + #log prior for gamma
dgamma(exp(params[3]), 1.25, 70, log = TRUE) + params[3] + #log prior for mu
dbeta(exp(params[4]) / (1 + exp(params[4])), 5, 50, log = TRUE) +
params[4] - log((1 + exp(params[4])) ^ 2) + #log prior for rho
log(ddirichlet(c(exp(params[5]) / (1 + exp(params[5]) + exp(params[6]) + exp(params[7])),
exp(params[6]) / (1 + exp(params[5]) + exp(params[6]) + exp(params[7])),
1 / (1 + exp(params[5]) + exp(params[6]) + exp(params[7])),
exp(params[7]) / (1 + exp(params[5]) + exp(params[6]) + exp(params[7]))),
c(90, 0.5, 0.5, 0.01)) * trans(params[5], params[6], params[7])) #log prior for initial SEIR value
)
if (log) {
f
} else {
exp(f)
}
}
And starting value of the PMMH is specified below:
param.initial<- c(
beta = log(abs(rnorm(1, 0.000055, 1e-6))),
gamma = log(abs(rnorm(1, 0.03, 1e-3))),
mu = log(abs(rnorm(1, 0.01, 0.01))),
rho = -2,
theta1 = 6,
theta2 = -1,
theta3 = -2
)
And the following runs 1 PMMH chain for 2000 tunning steps (in the paper we used 2500 particles) to get a suitable multivariate normal random walk proposal diagonal variance-covariance matrix value.
pmcmc1 <- pmcmc(
pomp(seir, dprior = seir.dprior),
#given the prior function
start = param.initial,
#given the initial value of the parameters
Nmcmc = 10,
#number of mcmc steps
Np = 200,
max.fail = Inf,
proposal = mvn.rw.adaptive(
0.1 * c(
beta = 0.1,
#sampling variance for beta
gamma = 0.1,
#sampling variance for gamma
mu = 0.1,
#sampling variance for mu
rho = 0.1,
#sampling variance for rho
theta1 = 0.1,
#sampling variance for theta1
theta2 = 0.1,
#sampling variance for theta2
theta3 = 0.1 #sampling variance for theta3
),
scale.start = 1000,
shape.start = 1000,
max.scaling = 1.2
)
)
And the following runs 1 PMMH chain for 100000 steps (using 2500 particles) with suitable multivariate normal random walk proposal diagonal variance-covariance matrix value we get above to get the posterior distribution of the interested parameters.
start_time <- Sys.time(); #calculation of time
pmcmc1 <- pmcmc(
pmcmc1,
#given the prior function
start = param.initial,
#given the initial value of the parameters
Nmcmc = 10,
max.fail = Inf,
proposal = mvn.rw(covmat(pmcmc1))
)
end_time <- Sys.time(); #calculation of time
run_time <- difftime(end_time, start_time, units = "hours") #calculation of time
pomp_results <- list(time = run_time, results = pmcmc1)
Fitting SIR model:
We use the SIR model to model the dataset. The hosts are divided into three classes, according to their status. The susceptible class (S) contains those that have not yet been infected and are thereby still susceptible to it; the infected class (I) comprises those who are currently infected and, by assumption, infectious; the removed class (R) includes those who are recovered or quarantined as a result of the infection. Individuals in R are assumed to be immune against reinfection. It is natural to formulate this model as a continuous-time Markov process. To start with, we need to specify the measurement model.
# measurement process
rmeas<-"
cases=rbinom(I,exp(rho)/(1+exp(rho)));//represent the data
"
dmeas<-"
if(cases > 0 && I == 0) {
if(give_log) {
lik = R_NegInf;
} else {
lik = 0;
}
} else {
lik=dbinom(cases,I,exp(rho)/(1+exp(rho)),give_log); //return the loglikelihood
}
"
Here, we are using the $\textbf{cases}$ to refer to the data (number of reported cases) and $\textbf{I}$ to refer to the true incidence over the reporting interval. The binomial simulator rbinom and density function dbinom are provided by R. Notice that, in these snippets, we never declare the variables; pomp will ensure tht the state variable ($\textbf{I}$), observables ($\textbf{cases}$), parameters ($\textbf{rho}$, $\textbf{phi}$) and likelihood ($\textbf{lik}$) are defined in the contexts within which these snippets are executed.
And for the transition of the SIR model, we use a approximating tau-leaping algorithm, one version of which is implemented in the pomp package via the $\bf{euler.sim}$ plug-in. The algorithm holds the transition rates constant over a small interval of time and simulates the numbers of transitions that occur over the interval. The functions $\textbf{reulermultinom}$ draw random deviates from such distributions. Then we need to first specify a function that advances the states from $t$ to $t+\triangle t$, and we give the trantition of the SIR model code below:
sir.step<-"
double rate[2];
double dN[2];
rate[0]=exp(bet)*I; //Infection rate
rate[1]=exp(mu); //recovery rate
reulermultinom(1,S,&rate[0],dt,&dN[0]); //generate the number of newly infected people
reulermultinom(1,I,&rate[1],dt,&dN[1]); //generate the number of newly recovered people
if(!R_FINITE(S)) Rprintf(\"%lg %lg %lg %lg %lg %lg %lg %lg %lg\\n\",dN[0],rate[0],dN[1],rate[1],bet,mu,S,I,R);
S+=-dN[0]; //update the number of Susceptible
I+=dN[0]-dN[1]; //update the number of Infection
R+=dN[1]; //update the number of Recovery
if(I<0) Rprintf(\"%lg %lg %lg %lg %lg %lg %lg %lg %lg\\n\",dN[0],rate[0],dN[1],rate[1],bet,mu,S,I,R);
"
The pomp will ensure that the undeclared state variables and parameters are defined in the context within which the snippet is executed. And the $\textbf{rate}$ and $\textbf{dN}$ arrays hold the rates and numbers of transition events, respectively. Then we could construct the pomp objects below:
sir <- pomp(
data = data.frame(cases = data, time = seq(1, 1457, by = 7)), #"cases" is the dataset, "time" is the observation time
times = "time",
t0 = 1, #initial time point
dmeasure = Csnippet(dmeas),
rmeasure = Csnippet(rmeas), #return the likelihood
rprocess = euler.sim(step.fun = Csnippet(sir.step), delta.t = 1/3), # delta.t is here
statenames = c("S", "I", "R"), #state space variable name
paramnames = c("bet", "mu", "rho", "theta1", "theta2"), #parameters name
initializer = function(params, t0, ...) {
ps <-exp(params["theta1"]) / (1 + exp(params["theta1"]) + exp(params["theta2"])) # initial prob of susceptible
pi <- 1 / (1 + exp(params["theta1"]) + exp(params["theta2"])) # initial prob of infection
pr <-exp(params["theta2"]) / (1 + exp(params["theta1"]) + exp(params["theta2"])) # initial prob of recovery
return(setNames(as.numeric(rmultinom(
1, 400, prob = c(ps, pi, pr)
)), c("S", "I", "R")))
},
params = c( #for data generation(We didn't generate data here, but pomp need this part.)
bet = -5,
mu = -2,
rho = 0,
theta1 = 6,
theta2 = 1
)
)
To carry out Bayesian inference we need to specify a prior distribution on unknown parame- ters. The pomp constructor function provides the rprior and dprior arguments, which can be filled with functions that simulate from and evaluate the prior density, respectively. Methods based on random walk Metropolis-Hastings require evaluation of the prior density (dprior), so we specify dprior for the SIR model as follows.
sir.dprior <- function(params, ..., log) {
f <- (dgamma(exp(params[1]), shape = 0.6, rate = 10000, log = TRUE) + params[1] +#log prior for log(infection rate) "beta"
dgamma(exp(params[2]), shape = 1.25, rate = 70, log = TRUE) +
params[2] + #log prior for log(recovery rate) "mu"
dbeta(exp(params[3]) / (1 + exp(params[3])), 5, 50, log = TRUE) +
params[3] - log((1 + exp(params[3])) ^ 2) + #log prior for logit(sampling probablity) "rho"
log(ddirichlet(c(exp(params[4]) / (1 + exp(params[4]) + exp(params[5])),
1 / (1 + exp(params[4]) + exp(params[5])),
exp(params[5]) / (1 + exp(params[4]) + exp(params[5]))
), c(90, 0.5, 0.01))) +
params[4] + params[5] - 3 * log(1 + exp(params[4]) + exp(params[5])) #log prior of logit(initial value)
)
if (log) {
f
} else {
exp(f)
}
}
And starting value of the PMMH is specified below:
param.initial <- c(
bet = log(abs(rnorm(1, 0.00007, 1e-5))),
mu = log(abs(rnorm(1, 0.005, 0.0001))),
rho = -2,
theta1 = 6,
theta2 = -1
)
And the following runs 1 PMMH chain for 2000 tunning steps (in the paper we used 2500 particles) to get a suitable multivariate normal random walk proposal diagonal variance-covariance matrix value.
pmcmc1 <- pmcmc(
pomp(sir, dprior = sir.dprior),
#given the prior function
start = param.initial,
#given the initial value of the parameters
Nmcmc = 10,
#number of mcmc steps
Np = 200,
max.fail = Inf,
proposal = mvn.rw.adaptive(0.1 * c(
bet = 0.1,
#sampling variance for beta
mu = 0.1,
#sampling variance for mu
rho = 0.1,
#sampling variance for rho
theta1 = 0.1,
#sampling variance for theta1
theta2 = 0.1 #sampling variance for theta2
),
scale.start = 100,
shape.start = 100,
max.scaling = 1.2)
)
And the following runs 1 PMMH chain for 100000 steps (using 5000 particles) with suitable multivariate normal random walk proposal diagonal variance-covariance matrix value we get above to get the posterior distribution of the interested parameters.
start_time <- Sys.time(); #calculation of time
pmcmc1 <- pmcmc(
pmcmc1,
#given the prior function
start = param.initial,
#given the initial value of the parameters
Nmcmc = 10,
max.fail = Inf,
proposal = mvn.rw(covmat(pmcmc1))
)
end_time <- Sys.time(); #calculation of time
run_time <- difftime(end_time, start_time, units = "hours") #calculation of time
pomp_results <- list(time = run_time, results = pmcmc1)
fintzij/BDAepimodel documentation built on Sept. 20, 2020, 1:44 p.m.
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library(BDAepimodel) library(coda) library(gtools) library(Rcpp) library(pomp) library(ggplot2)
This vignette contains code to fit SIR and SEIR models to binomially distributed prevalence counts sampled from an epidemic with time-varying dynamics. We simulated an epidemic with SEIR dynamics in a population of 400 individuals, 397 of whom were initially susceptible, 2 of whom were initially exposed, and 1 of whom were initially infectious. The dynamics of the epidemic varied over four different epochs are presented in the "model_misspecification" vignette. And the data is shown below:
data<-c(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,1,0,1,0,0,0,0,0,2,2,0,3,0,1,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,3,1,0,0,1,2,2,0,4,0,3,2,0,2,5,2,1,2,2,2,1,1,3,1,2,5,1,2,1,0,1,1,4,1,0,1,0,1,2,0,2,0,1,0,1,0,3,1,1,2,1,1,1,2,3,0,1,2,1,3,3,2,2,0,0,3,2,3,3,0,2,0,2,1,2,3,2,1,0,1,1,0,3,1,2,2,1,1,1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,0,0,1,0,3,0,1,1,0,0,0,1,1,1,1,0,0,2,0,1,1,1,0,1,0,0,1,1,0,1,0,1,1)
Fitting SEIR model:
We use the SEIR model to model the dataset. The hosts are divided into four classes, according to their status. The susceptible class (S); the exposed class (E); the infected class (I); and the removed class (R). Individuals in R are assumed to be immune against reinfection. It is natural to formulate this model as a continuous-time Markov process. To start with, we need to specify the measurement model.
rmeas<-" cases=rbinom(I,exp(rho)/(1+exp(rho)));//represent the data " dmeas<-" lik=dbinom(cases,I,exp(rho)/(1+exp(rho)),give_log); //return the loglikelihood "
Here, we are using the $\textbf{cases}$ to refer to the data (number of reported cases) and $\textbf{I}$ to refer to the true incidence over the reporting interval. The binomial simulator rbinom and density function dbinom are provided by R. Notice that, in these snippets, we never declare the variables; pomp will ensure tht the state variable ($\textbf{I}$), observables ($\textbf{cases}$), parameters ($\textbf{rho}$, $\textbf{phi}$) and likelihood ($\textbf{lik}$) are defined in the contexts within which these snippets are executed.
And for the transition of the SEIR model, we use a approximating tau-leaping algorithm, one version of which is implemented in the pomp package via the $\bf{euler.sim}$ plug-in. The algorithm holds the transition rates constant over a small interval of time and simulates the numbers of transitions that occur over the interval. The functions $\textbf{reulermultinom}$ draw random deviates from such distributions. Then we need to first specify a function that advances the states from $t$ to $t+\triangle t$, and we give the trantition of the SIR model code below:
seir.step<-" double rate[3]; double dN[3]; rate[0]=exp(beta)*I; //Infection rate[1]=exp(gamma); //Rate of S to E rate[2]=exp(mu); //Rate of E to I reulermultinom(1,S,&rate[0],dt,&dN[0]); //generate the number of newly from S to E reulermultinom(1,E,&rate[1],dt,&dN[1]); //generate the number of newly from E to I reulermultinom(1,I,&rate[2],dt,&dN[2]); //generate the number of newly from I to R if(!R_FINITE(S)) Rprintf(\"%lg %lg %lg %lg %lg %lg %lg %lg %lg\\n\",dN[0],rate[0],dN[1],rate[1],beta,mu,S,I,R); S+=-dN[0]; //update the number of Susceptible E+=dN[0]-dN[1]; //update the number of E I+=dN[1]-dN[2]; //update the number of I R+=dN[2]; //update the number of R "
The pomp will ensure that the undeclared state variables and parameters are defined in the context within which the snippet is executed. And the $\textbf{rate}$ and $\textbf{dN}$ arrays hold the rates and numbers of transition events, respectively. Then we could construct the pomp objects below:
seir <- pomp( data = data.frame(cases = data, time = seq(1, 1457, by = 7)), #"cases" is the dataset, "time" is the observation time times = "time", t0 = 1, #initial time point dmeasure = Csnippet(dmeas), rmeasure = Csnippet(rmeas), rprocess = euler.sim(step.fun = Csnippet(seir.step), delta.t = 1/4), #delta.t is here statenames = c("S", "E", "I", "R"), #state space variable name paramnames = c("beta", "gamma", "mu", "rho", "theta1", "theta2", "theta3"), #parameters name initializer = function(params, t0, ...) { ps <-exp(params["theta1"]) / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"])) pe <- exp(params["theta2"]) / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"])) pi <- 1 / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"])) pr <- exp(params["theta3"]) / (1 + exp(params["theta1"]) + exp(params["theta2"]) + exp(params["theta3"])) #Initial proportion of S, E, I, R return(setNames(as.numeric(rmultinom(1, 400, prob = c(ps, pe, pi, pr))), c("S", "E", "I", "R"))) }, params = c( beta = log(0.00072), gamma = log(0.5), mu = log(0.2), rho = log(1 / 4), theta1 = log(90), theta2 = log(2), theta3 = log(7) ) )
To carry out Bayesian inference we need to specify a prior distribution on unknown parame- ters. The pomp constructor function provides the rprior and dprior arguments, which can be filled with functions that simulate from and evaluate the prior density, respectively. Methods based on random walk Metropolis-Hastings require evaluation of the prior density (dprior), so we specify dprior for the SEIR model as follows.
trans<-function(a,b,c){ a_t<-exp(a) b_t<-exp(b) c_t<-exp(c) return(a_t*b_t*c_t/((1+a_t+b_t+c_t)^4)) } #Jacobian for (theta1, theta2, theta3) seir.dprior <- function(params, ..., log) { f <- ( dgamma(exp(params[1]), 0.6, 10000, log = TRUE) + params[1] + #log prior for beta dgamma(exp(params[2]), 1.5, 25, log = TRUE) + params[2] + #log prior for gamma dgamma(exp(params[3]), 1.25, 70, log = TRUE) + params[3] + #log prior for mu dbeta(exp(params[4]) / (1 + exp(params[4])), 5, 50, log = TRUE) + params[4] - log((1 + exp(params[4])) ^ 2) + #log prior for rho log(ddirichlet(c(exp(params[5]) / (1 + exp(params[5]) + exp(params[6]) + exp(params[7])), exp(params[6]) / (1 + exp(params[5]) + exp(params[6]) + exp(params[7])), 1 / (1 + exp(params[5]) + exp(params[6]) + exp(params[7])), exp(params[7]) / (1 + exp(params[5]) + exp(params[6]) + exp(params[7]))), c(90, 0.5, 0.5, 0.01)) * trans(params[5], params[6], params[7])) #log prior for initial SEIR value ) if (log) { f } else { exp(f) } }
And starting value of the PMMH is specified below:
param.initial<- c( beta = log(abs(rnorm(1, 0.000055, 1e-6))), gamma = log(abs(rnorm(1, 0.03, 1e-3))), mu = log(abs(rnorm(1, 0.01, 0.01))), rho = -2, theta1 = 6, theta2 = -1, theta3 = -2 )
And the following runs 1 PMMH chain for 2000 tunning steps (in the paper we used 2500 particles) to get a suitable multivariate normal random walk proposal diagonal variance-covariance matrix value.
pmcmc1 <- pmcmc( pomp(seir, dprior = seir.dprior), #given the prior function start = param.initial, #given the initial value of the parameters Nmcmc = 10, #number of mcmc steps Np = 200, max.fail = Inf, proposal = mvn.rw.adaptive( 0.1 * c( beta = 0.1, #sampling variance for beta gamma = 0.1, #sampling variance for gamma mu = 0.1, #sampling variance for mu rho = 0.1, #sampling variance for rho theta1 = 0.1, #sampling variance for theta1 theta2 = 0.1, #sampling variance for theta2 theta3 = 0.1 #sampling variance for theta3 ), scale.start = 1000, shape.start = 1000, max.scaling = 1.2 ) )
And the following runs 1 PMMH chain for 100000 steps (using 2500 particles) with suitable multivariate normal random walk proposal diagonal variance-covariance matrix value we get above to get the posterior distribution of the interested parameters.
start_time <- Sys.time(); #calculation of time pmcmc1 <- pmcmc( pmcmc1, #given the prior function start = param.initial, #given the initial value of the parameters Nmcmc = 10, max.fail = Inf, proposal = mvn.rw(covmat(pmcmc1)) ) end_time <- Sys.time(); #calculation of time run_time <- difftime(end_time, start_time, units = "hours") #calculation of time pomp_results <- list(time = run_time, results = pmcmc1)
Fitting SIR model:
We use the SIR model to model the dataset. The hosts are divided into three classes, according to their status. The susceptible class (S) contains those that have not yet been infected and are thereby still susceptible to it; the infected class (I) comprises those who are currently infected and, by assumption, infectious; the removed class (R) includes those who are recovered or quarantined as a result of the infection. Individuals in R are assumed to be immune against reinfection. It is natural to formulate this model as a continuous-time Markov process. To start with, we need to specify the measurement model.
# measurement process rmeas<-" cases=rbinom(I,exp(rho)/(1+exp(rho)));//represent the data " dmeas<-" if(cases > 0 && I == 0) { if(give_log) { lik = R_NegInf; } else { lik = 0; } } else { lik=dbinom(cases,I,exp(rho)/(1+exp(rho)),give_log); //return the loglikelihood } "
Here, we are using the $\textbf{cases}$ to refer to the data (number of reported cases) and $\textbf{I}$ to refer to the true incidence over the reporting interval. The binomial simulator rbinom and density function dbinom are provided by R. Notice that, in these snippets, we never declare the variables; pomp will ensure tht the state variable ($\textbf{I}$), observables ($\textbf{cases}$), parameters ($\textbf{rho}$, $\textbf{phi}$) and likelihood ($\textbf{lik}$) are defined in the contexts within which these snippets are executed.
And for the transition of the SIR model, we use a approximating tau-leaping algorithm, one version of which is implemented in the pomp package via the $\bf{euler.sim}$ plug-in. The algorithm holds the transition rates constant over a small interval of time and simulates the numbers of transitions that occur over the interval. The functions $\textbf{reulermultinom}$ draw random deviates from such distributions. Then we need to first specify a function that advances the states from $t$ to $t+\triangle t$, and we give the trantition of the SIR model code below:
sir.step<-" double rate[2]; double dN[2]; rate[0]=exp(bet)*I; //Infection rate rate[1]=exp(mu); //recovery rate reulermultinom(1,S,&rate[0],dt,&dN[0]); //generate the number of newly infected people reulermultinom(1,I,&rate[1],dt,&dN[1]); //generate the number of newly recovered people if(!R_FINITE(S)) Rprintf(\"%lg %lg %lg %lg %lg %lg %lg %lg %lg\\n\",dN[0],rate[0],dN[1],rate[1],bet,mu,S,I,R); S+=-dN[0]; //update the number of Susceptible I+=dN[0]-dN[1]; //update the number of Infection R+=dN[1]; //update the number of Recovery if(I<0) Rprintf(\"%lg %lg %lg %lg %lg %lg %lg %lg %lg\\n\",dN[0],rate[0],dN[1],rate[1],bet,mu,S,I,R); "
The pomp will ensure that the undeclared state variables and parameters are defined in the context within which the snippet is executed. And the $\textbf{rate}$ and $\textbf{dN}$ arrays hold the rates and numbers of transition events, respectively. Then we could construct the pomp objects below:
sir <- pomp( data = data.frame(cases = data, time = seq(1, 1457, by = 7)), #"cases" is the dataset, "time" is the observation time times = "time", t0 = 1, #initial time point dmeasure = Csnippet(dmeas), rmeasure = Csnippet(rmeas), #return the likelihood rprocess = euler.sim(step.fun = Csnippet(sir.step), delta.t = 1/3), # delta.t is here statenames = c("S", "I", "R"), #state space variable name paramnames = c("bet", "mu", "rho", "theta1", "theta2"), #parameters name initializer = function(params, t0, ...) { ps <-exp(params["theta1"]) / (1 + exp(params["theta1"]) + exp(params["theta2"])) # initial prob of susceptible pi <- 1 / (1 + exp(params["theta1"]) + exp(params["theta2"])) # initial prob of infection pr <-exp(params["theta2"]) / (1 + exp(params["theta1"]) + exp(params["theta2"])) # initial prob of recovery return(setNames(as.numeric(rmultinom( 1, 400, prob = c(ps, pi, pr) )), c("S", "I", "R"))) }, params = c( #for data generation(We didn't generate data here, but pomp need this part.) bet = -5, mu = -2, rho = 0, theta1 = 6, theta2 = 1 ) )
To carry out Bayesian inference we need to specify a prior distribution on unknown parame- ters. The pomp constructor function provides the rprior and dprior arguments, which can be filled with functions that simulate from and evaluate the prior density, respectively. Methods based on random walk Metropolis-Hastings require evaluation of the prior density (dprior), so we specify dprior for the SIR model as follows.
sir.dprior <- function(params, ..., log) { f <- (dgamma(exp(params[1]), shape = 0.6, rate = 10000, log = TRUE) + params[1] +#log prior for log(infection rate) "beta" dgamma(exp(params[2]), shape = 1.25, rate = 70, log = TRUE) + params[2] + #log prior for log(recovery rate) "mu" dbeta(exp(params[3]) / (1 + exp(params[3])), 5, 50, log = TRUE) + params[3] - log((1 + exp(params[3])) ^ 2) + #log prior for logit(sampling probablity) "rho" log(ddirichlet(c(exp(params[4]) / (1 + exp(params[4]) + exp(params[5])), 1 / (1 + exp(params[4]) + exp(params[5])), exp(params[5]) / (1 + exp(params[4]) + exp(params[5])) ), c(90, 0.5, 0.01))) + params[4] + params[5] - 3 * log(1 + exp(params[4]) + exp(params[5])) #log prior of logit(initial value) ) if (log) { f } else { exp(f) } }
And starting value of the PMMH is specified below:
param.initial <- c( bet = log(abs(rnorm(1, 0.00007, 1e-5))), mu = log(abs(rnorm(1, 0.005, 0.0001))), rho = -2, theta1 = 6, theta2 = -1 )
And the following runs 1 PMMH chain for 2000 tunning steps (in the paper we used 2500 particles) to get a suitable multivariate normal random walk proposal diagonal variance-covariance matrix value.
pmcmc1 <- pmcmc( pomp(sir, dprior = sir.dprior), #given the prior function start = param.initial, #given the initial value of the parameters Nmcmc = 10, #number of mcmc steps Np = 200, max.fail = Inf, proposal = mvn.rw.adaptive(0.1 * c( bet = 0.1, #sampling variance for beta mu = 0.1, #sampling variance for mu rho = 0.1, #sampling variance for rho theta1 = 0.1, #sampling variance for theta1 theta2 = 0.1 #sampling variance for theta2 ), scale.start = 100, shape.start = 100, max.scaling = 1.2) )
And the following runs 1 PMMH chain for 100000 steps (using 5000 particles) with suitable multivariate normal random walk proposal diagonal variance-covariance matrix value we get above to get the posterior distribution of the interested parameters.
start_time <- Sys.time(); #calculation of time pmcmc1 <- pmcmc( pmcmc1, #given the prior function start = param.initial, #given the initial value of the parameters Nmcmc = 10, max.fail = Inf, proposal = mvn.rw(covmat(pmcmc1)) ) end_time <- Sys.time(); #calculation of time run_time <- difftime(end_time, start_time, units = "hours") #calculation of time pomp_results <- list(time = run_time, results = pmcmc1)
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