## ----echo=T,eval=T-------------------------------------------------------
library(BDAepimodel)
library(coda)
library(Rcpp)
library(pomp)
library(gtools)
library(ggplot2)
## ----echo=T,eval=T-------------------------------------------------------
set.seed(1834)
data<-c(0,0,0,1,0,1,1,1,2,1,2,1,0,3,1,0,2,0,2,1,2,2,2,1,0,2,1,0,0,2,0,1,1,0,1,1,1,0,1,2,2,1,1,2,4,2,4,3,5,3,1,1,5,5,4,3,1,1,3,2,0,0,2,2,2,3,2,5,4,1,3,1,4,3,1,2,2,5,2,4,2,1,2,1,3,1,1,3,1,1,0,2,2,3,5,1,0,1,0,1,0,0,0,0,1)
## ----echo=T,eval=T-------------------------------------------------------
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
"
## ----echo=T,eval=T-------------------------------------------------------
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
"
## ----echo=T,eval=T,warning=F---------------------------------------------
seir <- pomp(
data = data.frame(cases = data, time = seq(1, 729, 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 = 1/12), # tau-leaping over 2 hour intervals
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, ...) {
#Initial proportion of S, E, I, R
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"]))
return(setNames(as.numeric(rmultinom(1, 500, 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)
)
)
## ----echo=T,eval=T-------------------------------------------------------
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))
}
seir.dprior <- function(params, ..., log) {
f <- (dgamma(exp(params[1]), 1, 10000, log = TRUE) + params[1] + #log prior for beta
dgamma(exp(params[2]), 1, 11, log = TRUE) + params[2] + #log prior for gamma
dgamma(exp(params[3]), 3.2, 100, log = TRUE) + params[3] + #log prior for mu
dbeta(exp(params[4]) / (1 + exp(params[4])), 3.5, 6.5, 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(100, 0.01, 0.4, 0.01)) * trans(params[5], params[6], params[7])) #log prior for initial SEIR value
)
if (log) {
f
} else {
exp(f)
}
}
## ----echo=T,eval=T-------------------------------------------------------
param.initial<- c(
beta = log(abs(rnorm(1, 0.00005, 1e-6))),
gamma = log(abs(rnorm(1, 0.15, 0.01))),
mu = log(abs(rnorm(1, 0.04, 0.001))),
rho = -1,
theta1 = 5,
theta2 = 1,
theta3 = 1
)
## ----echo=T,eval=T,warning=F---------------------------------------------
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(
0.5 * 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 = 100,
shape.start = 100
)
)
## ----echo=T,eval=T,warning=F---------------------------------------------
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)
## ----echo=T,eval=T-------------------------------------------------------
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
"
## ----echo=T,eval=T-------------------------------------------------------
# define the rate function, stoichiometry matrix, and rate-event dependency matrix
SEIR_stoich <- cbind(exposure = c(-1, 1, 0, 0), # exposures yield S-1, E+1
infection = c(0, -1, 1, 0), # infections yield E-1, I+1
recovery = c(0, 0, -1, 1)) # recoveries yield I-1, R+1
SEIR_depmat <- cbind(exposure = c(1, 0, 1, 0), # exposure rate updated by changes to S and I
infection = c(0, 1, 0, 0), # infection rate updated by changes to E
recovery = c(0, 0, 1, 0)) # recovery rate updated by changes to I
# SIR rate function
# j the number of the elementary event (1 = infection, 2 = recovery)
# x named numeric vector with the value of the state of the process at time t
# t time
# params named numeric vector containing the parameters
# returns single numerical value with the rate of the elementary event
SEIR_rates <- function(j, x, t, params, ...) {
switch(j,
exp(params["beta"]) * x["S"] * x["I"], # exposure
exp(params["gamma"]) * x["E"], # infection
exp(params["mu"]) * x["I"] # recovery
)
}
# instatiate the gillespie stepper function
SEIR_sim <- gillespie.sim(rate.fun = SEIR_rates,
v = SEIR_stoich,
d = SEIR_depmat)
## ----echo=T,eval=T,warning=F---------------------------------------------
seir <- pomp(
data = data.frame(time = seq(1, 729, by = 7), cases = data), #"cases" is the dataset, "time" is the observation time
times = "time",
t0 = 1, # initial time point
dmeasure = Csnippet(dmeas), # evaluates the density of the measurement process
rmeasure = Csnippet(rmeas), # simulates from the measurement process
rprocess = SEIR_sim, # simulates from the latent process
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, ...) {
#Initial proportion of S, E, I, R
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"]))
return(setNames(as.numeric(rmultinom(1, 500, 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)
)
)
## ----echo=T,eval=T-------------------------------------------------------
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]), 1, 10000, log = TRUE) + params[1] + #log prior for beta
dgamma(exp(params[2]), 1, 11, log = TRUE) + params[2] + #log prior for gamma
dgamma(exp(params[3]), 3.2, 100, log = TRUE) + params[3] + #log prior for mu
dbeta(exp(params[4]) / (1 + exp(params[4])), 3.5, 6.5, 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(100, 0.01, 0.4, 0.01)) * trans(params[5], params[6], params[7])) #log prior for initial SEIR value
)
if (log) {
f
} else {
exp(f)
}
}
## ----echo=T,eval=T-------------------------------------------------------
param.initial<- c(
beta = log(abs(rnorm(1, 0.00005, 1e-6))),
gamma = log(abs(rnorm(1, 0.15, 0.01))),
mu = log(abs(rnorm(1, 0.04, 0.001))),
rho = -1,
theta1 = 5,
theta2 = 1,
theta3 = 1
)
## ----echo=T,eval=T,warning=F---------------------------------------------
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(0.5 * 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 = 100,
shape.start = 100)
)
## ----echo=T,eval=T,warning=F---------------------------------------------
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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