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Run_typhoid_calibration_revised.R
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Run_typhoid_calibration_revised.R
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# Manuscript: Comparison of strategies and incidence thresholds for Vi conjugate vaccines against typhoid fever: a cost-effectiveness modeling study
# Authored by: Nathan C Lo and colleagues
# Journal of Infectious Diseases (JID), 2018
# Coded by Nathan C. Lo (Stanford University)
# Email: [email protected]
# Last updated: 11/10/17
# Model summary
# This is the R code for the base case analysis in the manuscript. The analysis requires three sequential steps as follows.
# Note:
# 1) Model calibration, use "Run_typhoid_calibration_revised.R" to create posterior distribution of model parameters.
# 2) Model simulation of vaccine strategies, use "Run_typhoid_simulation_revised.R" to create transmission projections
# 3) Cost-effectiveness, use "Run_CEA_typhoid_vaccination.R" to compute cost-effective incidence thresholds
# The CSV age distribution file "Age_distriction_typhoid_111316" is required for models.
# **Note: You will need to change your directory files to load these excels.**
Run_typhoid_calibration <- function(total_incidence) {
file_mcmc_output <- paste("MCMC_chain_091617_", total_incidence, ".csv", sep = '')
Typhoid_model_calibration <- function(total_incidence, file_mcmc_output) {
####################################################################################################################################
# MODEL: Typhoid age-structured compartmental model with vaccination, CALIBRATION template
# Coded by: Nathan C. Lo (Stanford University)
# Email: [email protected]
# Methods:
# SIRCWRV compartmental model
# Time step of 1 month
# 50 year burn-in period to endemic levels
# What are we fitting?
# 1) beta- short cycle transmission coefficient
# 2) lambda- long cycle transmission coefficient
# 3) p- reporting/symptomatic fraction
# 4) w- rate of immune waning from natural immunity to susceptiblity
# 5) r- relative infectiousness of carrier
# Fitting procedure
# Bayesian MCMC-MH
####################################################################################################################################
# Load R packages
require(deSolve) # Differential equation solver
#library(xlsx)
#library(ggplot2)
library(matrixStats)
# Three setting age-distribution from meta-analysis
age_distribution <- read.csv(file="Age_distriction_typhoid_111316.csv", header=FALSE)
# Pick epi #################################################################################
total_incidence <- total_incidence # Annual incidence: per 100,000
############################################################################################
total_incidence_vec <- c(10, 50, 100, 200, 300, 400, 500, 1000)
if (total_incidence > 100) {
age_dist_iter <- age_distribution$V3
} else if (total_incidence < 10) {
age_dist_iter <- age_distribution$V1
} else if (total_incidence>=50 & total_incidence<=100) {
age_dist_iter <- ((total_incidence-50)/50)*age_distribution$V3 + (1-((total_incidence-50)/50))*age_distribution$V2
} else if (total_incidence>=10 & total_incidence<50) {
age_dist_iter <- ((total_incidence-10)/40)*age_distribution$V2 + (1-((total_incidence-10)/40))*age_distribution$V1
}
incidence_weight <- age_dist_iter/sum(age_dist_iter)
incidence_weight <- c(incidence_weight[1]*(1/5), incidence_weight[1]*(4/5), incidence_weight[-1])
# Age structure is 16 age groups (<1, 1-4, 5-9, 10-14, etc until 74)
age_weight <- c(1.542680, 6.103679, 7.588358, 7.556721, 7.513031, 7.451264, 7.384977, 7.308144, 7.210220, 7.080659, 6.885564, 6.588779, 6.139082, 5.476965, 4.606948, 3.562927)/100
# Morality rate (age-dep), monthly
u_age <- c(0.00288262, 0.000188349, 6.68004E-05, 5.00752E-05, 8.35424E-05, 0.000125471, 0.000142272, 0.000167506, 0.000226533, 0.000302733, 0.000439066, 0.000654345, 0.001013536, 0.001663672, 0.002630401, 0.004194527)
# Carriage risk (age-dep)
carrier_age <-c(0.3, 0.3, 0.3, 0.3, 0.3, 2.1, 2.1, 4.4, 4.4, 8.8, 8.8, 10.1, 10.1, 7.8, 7.8, 7.8)/100
total_pop <- 100000*(1/1.25) # Set population at 100,000, with population adjustment to account for demography
age_group <- length(age_weight)
# Time step is one month
# Compute initial conditions (month, time 0)
# Keep population fractions (not absolute numbers)
S.0 <- vector(length=age_group)
I.0 <- vector(length=age_group)
for (i in 1:age_group) {
I.0[i] <- ((total_incidence*incidence_weight[i])/12)/total_pop # Divide by 12 since converting from annual cases to monthly cases
S.0[i] <- (total_pop*age_weight[i] - (I.0[i]*total_pop))/total_pop # If not infected, then they are suceptibles
}
R.0 <- rep(0, age_group)
C.0 <- rep(0, age_group)
V.0 <- rep(0, age_group)
I_in.0 <- rep(0, age_group)
W.0 <- 0
# Enter initial parameter list
# Time step of one month!
params <- c(S.0=c(S.0), I.0= c(I.0), R.0= c(R.0), C.0= c(C.0), V.0= c(V.0), W.0=W.0, I_in.0=c(I_in.0), lambda=NA, beta=NA, p=NA, w=NA, r=NA, ksi=NA, w_vacc=1/(19.2*12), a1=0, a2=0, v=(1/(12*10)), N=total_pop, Bi=(21.2/1000/12), u_i=0.005, gamma=(1/(21/30)), psi=(1/(21/30))) # ksi=4.5e9 Vol=(15*30*total_pop)
# Note, states with multiple transitions (infected -> recovred, carrier) are corrected to ensure average duration is constant
# Create test typhoid endemic data
num_months <- 12
annual_incidence <- total_incidence
calibration_years <- 50+10
incidence1 <- rep(annual_incidence/num_months*incidence_weight[1],num_months*calibration_years)
incidence2 <- rep(annual_incidence/num_months*incidence_weight[2],num_months*calibration_years)
incidence3 <- rep(annual_incidence/num_months*incidence_weight[3],num_months*calibration_years)
incidence4 <- rep(annual_incidence/num_months*incidence_weight[4],num_months*calibration_years)
incidence5 <- rep(annual_incidence/num_months*incidence_weight[5],num_months*calibration_years)
incidence6 <- rep(annual_incidence/num_months*incidence_weight[6],num_months*calibration_years)
incidence7 <- rep(annual_incidence/num_months*incidence_weight[7],num_months*calibration_years)
incidence8 <- rep(annual_incidence/num_months*incidence_weight[8],num_months*calibration_years)
incidence9 <- rep(annual_incidence/num_months*incidence_weight[9],num_months*calibration_years)
incidence10 <- rep(annual_incidence/num_months*incidence_weight[10],num_months*calibration_years)
incidence11 <- rep(annual_incidence/num_months*incidence_weight[11],num_months*calibration_years)
incidence12 <- rep(annual_incidence/num_months*incidence_weight[12],num_months*calibration_years)
incidence13 <- rep(annual_incidence/num_months*incidence_weight[13],num_months*calibration_years)
incidence14 <- rep(annual_incidence/num_months*incidence_weight[14],num_months*calibration_years)
incidence15 <- rep(annual_incidence/num_months*incidence_weight[15],num_months*calibration_years)
incidence16 <- rep(annual_incidence/num_months*incidence_weight[16],num_months*calibration_years)
month <- seq(1,num_months*calibration_years)
data<- data.frame(month, incidence1, incidence2, incidence3, incidence4, incidence5, incidence6, incidence7, incidence8, incidence9, incidence10, incidence11, incidence12, incidence13, incidence14, incidence15, incidence16)
# Compartmental model
typhoid.model <- function (t, y, params) { # Note: this input format is required for ode function (used in "prediction" function)
# x- initial guesses
age_groups<- age_group # <1, 1-4, 5-9, etc (5 year groups) til 74
S <- vector(length=age_groups)
I <- vector(length=age_groups)
R <- vector(length=age_groups)
C <- vector(length=age_groups)
V <- vector(length=age_groups)
W <- vector(length=1)
dS <- vector(length=age_groups)
dI <- vector(length=age_groups)
dR <- vector(length=age_groups)
dC <- vector(length=age_groups)
dV <- vector(length=age_groups)
dW <- vector(length=1)
dI_in <- vector(length=age_groups)
for (i in 1:length(y)) {
if (i<=age_groups) {
S[i] <- y[i]
} else if (i>age_groups & i<=(2*age_groups)) {
I[i-age_groups] <- y[i]
} else if (i>(2*age_groups) & i<=(3*age_groups)) {
R[i-(2*age_groups)] <- y[i]
} else if (i>(3*age_groups) & i<=(4*age_groups)) {
C[i-(3*age_groups)] <- y[i]
} else if (i>(4*age_groups) & i<=(5*age_groups)) {
V[i-(4*age_groups)] <- y[i]
} else if (i==(5*age_groups + 1)) {
W[1] <- y[i]
}
}
N_curr <- (sum(S) + sum(I) + sum(R) + sum(C) + sum(V))*params["N"]
with(
as.list(params),
{
for (i in 1:age_groups) {
human_contr_temp <- 0
water_contr_temp <- 0
for (j in 1:age_groups) {
dS[i] <- -beta*S[i]*I[j] - beta*S[i]*C[j]*r + dS[i]
dI[i] <- beta*S[i]*I[j] + beta*S[i]*C[j]*r + dI[i]
dR[i] <- dR[i]
dC[i] <- dC[i]
dV[i] <- dV[i]
dI_in[i] <- ( beta*S[i]*I[j] + beta*S[i]*C[j]*r) + dI_in[i]
}
dS[i] <- -lambda*S[i]*W + w*R[i] + w_vacc*V[i] - u_age[i]*S[i] + dS[i]
dI[i] <- lambda*S[i]*W - (1-carrier_age[i])*gamma*I[i] - carrier_age[i]*gamma*I[i] - (u_age[i]+u_i)*I[i] + dI[i]
dC[i] <- carrier_age[i]*gamma*I[i] - v*C[i] - u_age[i]*C[i] + dC[i]
dR[i] <- (1-carrier_age[i])*gamma*I[i] - w*R[i] + v*C[i] - u_age[i]*R[i] + dR[i]
dV[i] <- -w_vacc*V[i] - u_age[i]*V[i] + dV[i]
# dW <- ksi*I[i] + ksi*r*C[i] - psi*W + dW
dW <- ksi*I[i] + ksi*r*C[i] + dW
if (i==1) {
dW <- -psi*W + dW
}
dI_in[i] <- lambda*S[i]*W + dI_in[i]
#dI_water[i] <- lambda*S[i]*(W/(k+W)) + dI_water[i]
# Enter aging process
# Age structure (16 groups)
# <1 yr
# 1-4 year
# 5-9 year
# 10-14 yr
# etc
# max age, 74 yrs
if (i==1) {
# Age group <1
# 1) Model births. Add to suceptibles compartment at birth rate.
# 2) Age up the infant cohort; 1/12 per month since 1 year group
dS[i] <- dS[i] + Bi - (1/12)*S[i]
dI[i] <- dI[i] - (1/12)*I[i]
dR[i] <- dR[i] - (1/12)*R[i]
dC[i] <- dC[i] - (1/12)*C[i]
dV[i] <- dV[i] - (1/12)*V[i]
} else if (i==2) {
# Age group 1-4
# 1) Age upthe cohort (per year) in all compartments; 1/48 per month since 4 year group
dS[i] <- dS[i] + (1/12)*S[i-1] - (1/48)*S[i]
dI[i] <- dI[i] + (1/12)*I[i-1] - (1/48)*I[i]
dR[i] <- dR[i] + (1/12)*R[i-1] - (1/48)*R[i]
dC[i] <- dC[i] + (1/12)*C[i-1] - (1/48)*C[i]
dV[i] <- dV[i] + (1/12)*V[i-1] - (1/48)*V[i]
} else if (i==3) {
# Age group 5-9
# 1) Age upthe cohort (per year) in all compartments; 1/48 per month since 4 year group
dS[i] <- dS[i] + (1/48)*S[i-1] - (1/60)*S[i]
dI[i] <- dI[i] + (1/48)*I[i-1] - (1/60)*I[i]
dR[i] <- dR[i] + (1/48)*R[i-1] - (1/60)*R[i]
dC[i] <- dC[i] + (1/48)*C[i-1] - (1/60)*C[i]
dV[i] <- dV[i] + (1/48)*V[i-1] - (1/60)*V[i]
} else {
# Age groups (5-year categories) from 10 until 74. Then all die (yikes)
# 1) Age upthe cohort (per year) in all compartments; 1/60 per month since 5 year group
dS[i] <- dS[i] + (1/60)*S[i-1] - (1/60)*S[i]
dI[i] <- dI[i] + (1/60)*I[i-1] - (1/60)*I[i]
dR[i] <- dR[i] + (1/60)*R[i-1] - (1/60)*R[i]
dC[i] <- dC[i] + (1/60)*C[i-1] - (1/60)*C[i]
dV[i] <- dV[i] + (1/60)*V[i-1] - (1/60)*V[i]
}
# EPI vaccination: ages <1 group only
# Note: Infected, carriers, and water (Directly) are not relevant here.
if (i==1) {
dS[i] <- -a1*S[i] + dS[i] # Vaccinate suceptibles and recovered at equal rate
#dR[i] <- -a1*R[i] + dR[i]
dV[i] <- a1*S[i] + dV[i] # + a1*R[i]
}
# Vaccination of school-aged children: ages 5-9 and 10-14
# Note: Infected, carriers, and water (Directly) are not relevant here.
if (i>=3 & i<=4) {
dS[i] <- -a2*S[i] + dS[i] # Vaccinate suceptibles and recovered at equal rate
#dR[i] <- -a2*R[i] + dR[i]
dV[i] <- a2*S[i] + dV[i] # + a2*R[i]
}
}
dx<- c(dS, dI, dR, dC, dV, dW, dI_in)
list(dx)
#print(t)
}
)
}
prediction <- function (params, times) {
xstart <- params[c("S.01","S.02","S.03","S.04","S.05","S.06","S.07","S.08","S.09","S.010","S.011","S.012","S.013","S.014","S.015","S.016","I.01","I.02","I.03","I.04","I.05","I.06","I.07","I.08","I.09","I.010","I.011","I.012","I.013","I.014","I.015", "I.016","R.01","R.02","R.03","R.04","R.05","R.06","R.07","R.08","R.09","R.010","R.011","R.012","R.013","R.014","R.015","R.016","C.01","C.02","C.03","C.04","C.05","C.06","C.07","C.08","C.09","C.010","C.011","C.012","C.013","C.014","C.015","C.016","V.01","V.02","V.03","V.04","V.05","V.06","V.07","V.08","V.09","V.010","V.011","V.012","V.013","V.014","V.015","V.016","W.0", "I_in.01", "I_in.02", "I_in.03", "I_in.04", "I_in.05", "I_in.06", "I_in.07", "I_in.08", "I_in.09", "I_in.010", "I_in.011", "I_in.012", "I_in.013", "I_in.014", "I_in.015", "I_in.016")] # ODE function
# y- initial conditions
# func- set of ODEs
# params- passed to function
out <- ode(
func=typhoid.model,
y=xstart,
times= c(0, times),
parms=params,
method="rk4"
)
tail(out[, 83:98], num_months*10) # return the I variable only
#out <- data.frame(out)
}
prediction2 <- function (params, times) {
xstart <- params[c("S.01","S.02","S.03","S.04","S.05","S.06","S.07","S.08","S.09","S.010","S.011","S.012","S.013","S.014","S.015","S.016","I.01","I.02","I.03","I.04","I.05","I.06","I.07","I.08","I.09","I.010","I.011","I.012","I.013","I.014","I.015", "I.016","R.01","R.02","R.03","R.04","R.05","R.06","R.07","R.08","R.09","R.010","R.011","R.012","R.013","R.014","R.015","R.016","C.01","C.02","C.03","C.04","C.05","C.06","C.07","C.08","C.09","C.010","C.011","C.012","C.013","C.014","C.015","C.016","V.01","V.02","V.03","V.04","V.05","V.06","V.07","V.08","V.09","V.010","V.011","V.012","V.013","V.014","V.015","V.016","W.0", "I_in.01", "I_in.02", "I_in.03", "I_in.04", "I_in.05", "I_in.06", "I_in.07", "I_in.08", "I_in.09", "I_in.010", "I_in.011", "I_in.012", "I_in.013", "I_in.014", "I_in.015", "I_in.016")] # ODE function
# func- set of ODEs
# params- passed to function
out <- ode(
func=typhoid.model,
y=xstart,
times= c(0, times),
parms=params,
method="rk4"
)
#tail(out[, 17:31], num_months*10) # return the I variable only
out <- data.frame(out)
}
# Define likelihood function with poisson distribution (count data)
poisson.loglik <- function (params, data) {
times <- data$month # convert to time-scale year
pred <- prediction(params,times)*params["p"]*params["N"]
pred_I2 <- matrix(nrow=dim(pred)[1], ncol=dim(pred)[2])
pred_I2[1,] <- rep(0,dim(pred)[2])
for (j in 2:dim(pred)[1]) {
pred_I2[j,] <- as.numeric(pred[j,] - pred[j-1,])
}
pred_I <- matrix(nrow=(dim(pred)[1]/12), ncol=dim(pred)[2])
for (j in 1:(dim(pred)[1]/12)) {
pred_I[j,] <- colSums(pred_I2[((j-1)*12+1):(j*12),])
}
pred_I[1,] <- pred_I[1,]/(11/12)
# For MCMC
# Use subset of age groups (use >1 case per month)
x1 <- sum(dpois(x=(round(tail(data$incidence1*12, 10))+1),lambda=((pred_I[,1])+1), log=TRUE))
x2 <- sum(dpois(x=(round(tail(data$incidence2*12, 10))+1),lambda=((pred_I[,2])+1), log=TRUE))
x3 <- sum(dpois(x=(round(tail(data$incidence3*12, 10))+1),lambda=((pred_I[,3])+1), log=TRUE))
x4 <- sum(dpois(x=(round(tail(data$incidence4*12, 10))+1),lambda=((pred_I[,4])+1), log=TRUE))
x5 <- sum(dpois(x=(round(tail(data$incidence5*12, 10))+1),lambda=((pred_I[,5])+1), log=TRUE))
x6 <- sum(dpois(x=(round(tail(data$incidence6*12, 10))+1),lambda=((pred_I[,6])+1), log=TRUE))
x7 <- sum(dpois(x=(round(tail(data$incidence7*12, 10))+1),lambda=((pred_I[,7])+1), log=TRUE))
x8 <- sum(dpois(x=(round(tail(data$incidence8*12, 10))+1),lambda=((pred_I[,8])+1), log=TRUE))
x9 <- sum(dpois(x=(round(tail(data$incidence9*12, 10))+1),lambda=((pred_I[,9])+1), log=TRUE))
x10<- sum(dpois(x=(round(tail(data$incidence10*12, 10))+1),lambda=((pred_I[,10])+1), log=TRUE))
x11<- sum(dpois(x=(round(tail(data$incidence11*12, 10))+1),lambda=((pred_I[,11])+1), log=TRUE))
x12<- sum(dpois(x=(round(tail(data$incidence12*12, 10))+1),lambda=((pred_I[,12])+1), log=TRUE))
x13<- sum(dpois(x=(round(tail(data$incidence13*12, 10))+1),lambda=((pred_I[,13])+1), log=TRUE))
x14<- sum(dpois(x=(round(tail(data$incidence14*12, 10))+1),lambda=((pred_I[,14])+1), log=TRUE))
x15<- sum(dpois(x=(round(tail(data$incidence15*12, 10))+1),lambda=((pred_I[,15])+1), log=TRUE))
x16<- sum(dpois(x=(round(tail(data$incidence16*12, 10))+1),lambda=((pred_I[,16])+1), log=TRUE))
totes <- sum(x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 , x11 , x12 , x13 , x14 , x15, x16)
return(totes)
}
# f
# Objective: How does a set of model parameters perform for poisson negative log-likelihood?
likelihood <- function (par_initial) {
par <- params
par[c("beta")] <- par_initial[1]
par[c("p")] <- par_initial[2]
par[c("lambda")] <- par_initial[3]
par[c("w")] <- par_initial[4]
par[c("r")] <- par_initial[5]
par[c("ksi")] <- par_initial[6]
cat("beta",par[c("beta")], "lambda", par[c("lambda")], "p",par[c("p")], "w",par[c("w")], "r",par[c("r")], "ksi",par[c("ksi")])
poiss <- poisson.loglik(par,data) # smaller numer is better fit (20 better than 200); should find global min
print(poiss)
poiss # min with negative
}
####################################################################################################################################
# MCMC-MH
####################################################################################################################################
# Prior distribution
prior <- function(par_initial){
beta = par_initial[1]
p = par_initial[2]
lambda = par_initial[3]
w = par_initial[4]
r = par_initial[5]
ksi = par_initial[6]
if (total_incidence<100) { # 10 incidence
beta_prior = dnorm(beta, mean=3, sd=1.5, log=T)
p_prior = dnorm(p, mean=0.2, sd = 0.15, log=T)
lambda_prior = dnorm(lambda, mean=3, sd = 1.5, log=T)
w_prior = dunif(w, min=0, max = (1/12*2), log=T)
r_prior = dunif(r, min=0.005, max = 0.4, log=T)
ksi_prior = dunif(ksi, min=0, max = 5, log=T)
} else if (total_incidence>=100 & total_incidence<500) { # 200 incidence
beta_prior = dnorm(beta, mean=3, sd=1.5, log=T)
p_prior = dnorm(p, mean=0.3, sd = 0.15, log=T)
lambda_prior = dnorm(lambda, mean=3, sd = 1.5, log=T)
w_prior = dunif(w, min=0, max = (1/12*2), log=T)
r_prior = dunif(r, min=0.005, max = 0.4, log=T)
ksi_prior = dunif(ksi, min=0, max = 5, log=T)
} else if (total_incidence>=500) { # 500 incidence
beta_prior = dnorm(beta, mean=3, sd=1.5, log=T)
p_prior = dnorm(p, mean=0.3, sd = 0.15, log=T)
lambda_prior = dnorm(lambda, mean=3, sd = 1.5, log=T)
w_prior = dunif(w, min=0, max = (1/12*2), log=T)
r_prior = dunif(r, min=0.005, max = 0.4, log=T)
ksi_prior = dunif(ksi, min=0, max = 5, log=T)
}
return((beta_prior+p_prior+lambda_prior+w_prior+r_prior+ksi_prior))
}
# Posterior
posterior <- function(param){
return (likelihood(param) + prior(param))
}
# MCMC run
run_MCMC <- function(startvalue, iterations){
chain = array(dim = c(iterations+1,7)) # 5 unknown parameters + likelihood store
chain[1,1:6] <- startvalue
check_iter <- 0
chain[1,7] <- posterior(chain[1,1:6])
for (i in 1:iterations){
# proposal function
# Generate
while (check_iter==0) {
proposal <- vector(length=6)
proposal[1:3] <- rnorm(3,mean = chain[i,1:3], sd= c(1.5/3, 0.15/3, 1.5/3) )
proposal[4] <- runif(1, min=max(0,chain[i,4] - 0.00005), max= (chain[i,4] + 0.00005) )
proposal[5] <- runif(1, min=max(0,chain[i,5] - 0.005), max= (chain[i,5] + 0.005) )
proposal[6] <- 1 #runif(1, min=max(0,chain[i,6] - 0.005), max= (chain[i,6] + 0.005) )
if (sum(proposal<0)==0) {
check_iter <- 1
}
}
check_iter <- 0
posterior_proposal <- posterior(proposal)
# MCMC- MH
probab <- exp(posterior_proposal - chain[i,7]) # add syntax to prevent NaN trigger
# MCMC- MH
if (runif(1) < as.numeric(max(na.omit(c(probab, 0)))) ) {
chain[i+1,1:6] <- proposal
chain[i+1,7] <- posterior_proposal
cat("Accepted parameters:", chain[i+1,])
} else {
chain[i+1,1:6] <- chain[i,1:6]
chain[i+1,7] <- chain[i,7]
cat("Accepted parameters:", chain[i+1,])
write.csv(chain, file=file_mcmc_output)
} }
return(chain)
}
# Inputs to MCMC-MH
sim_size <- 5000
if (total_incidence<100) {
startvalue = c(1, 0.1, 2, 0.0001, 0.05, 1) # modify to initials from MCMC-MH
} else if (total_incidence>=100 & total_incidence<500) {
startvalue = c(3, 0.1, 4, 0.0001, 0.05, 1) # modify to initials from MCMC-MH
} else if (total_incidence>=500) { # 500 incidence
startvalue = c(3.5, 0.25, 5, 0.0001, 0.05, 1) # modify to initials from MCMC-MH
}
posterior_results <- run_MCMC(startvalue,sim_size)
write.csv(posterior_results, file=file_mcmc_output)
}
# Run the typhoid model
Typhoid_model_calibration(total_incidence, file_mcmc_output)
}