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deSolve.R
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deSolve.R
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library(deSolve)
SI_dyn <- function(t, var, par) {
Sc <- var[1]
Sb <- var[2]
Vc <- var[3]
Vb <- var[4]
Ic <- var[5]
Ib <- var[6]
Wc <- var[7]
Wb <- var[8]
CIc <- var[9]
CWc <- var[10]
u = par[[1]]
v = par[[2]]
removal = par[[3]]
vbeta = par[[4]]
vbetaCc = par[[5]]
Nc <- Sc + Vc + Ic + Wc
##print(Nc)
Nb <- Sb + Vb + Ib + Wb
##print(Nb)
S <- matrix(c(Sc, Sb, Vc, Vb))
I <- matrix(c(Ic, Ib, Wc, Wb))
Sdiag <- diag(c(Sc, Sb, Vc, Vb))
IdivN <- matrix(c((Ic/Nc), (Ib/Nb), (Wc/Nc), (Wb/Nb)))
N <- matrix(c(Nc, Nb, Nc, Nb))
ScVcdiag <- diag(c(Sc,Vc))
#print(N)
#print(u)
dS <- (u %*% N) - (v %*% S) - (Sdiag %*% vbeta %*% IdivN) + removal %*% I
dI <- (Sdiag %*% vbeta %*% IdivN) - ((removal+v) %*% I)
dC <- (ScVcdiag %*% vbetaCc %*% IdivN) + matrix(c(Ic,Wc))
return(list(c(dS, dI, dC)))
}
beta_scenarios = function(scenario) {
if (scenario == 1) {
betaCC <- 0.94*(0.7+vc)
betaBC <- 0.05*(vb)
betaCB <- 0.2*(0.7+vc)
betaBB <- 0.99*(vb)
## default scenario as argued by EBP & Wood (2015)
}
if (scenario == 2) {
betaCC <- 0.99*(0.7+vc)
betaBC <- 0.05*(vb)
betaCB <- 0.05*(0.7+vc)
betaBB <- 1.04*(vb)
## badger reservoir, with low inter-host transmission
}
if (scenario == 3) {
betaCC <- 1.04*(0.7+vc)
betaBC <- 0.05*(vb)
betaCB <- 0.05*(0.7+vc)
betaBB <- 0.99*(vb)
## cattle reservoir, with low inter-host transmission
}
if (scenario == 4) {
betaCC <- 0.94*(0.7+vc)
betaBC <- 0.14*(vb)
betaCB <- 0.07*(0.7+vc)
betaBB <- 0.99*(vb)
## van Tonder scenario
}
beta <- matrix(c(betaCC, betaBC, betaCB, betaBB), 2, 2, byrow = TRUE)
return(beta)
}
init_prevalence = function(scenario) {
#initial prevalence according to steady-state analysis
if (scenario==1) {
# default scenario
I0c = 0.01717
I0b = 0.1064
}
if (scenario==2) {
#badger reservoir scenario
I0c = 0.029
I0b = 0.093
}
if (scenario==3) {
#cattle reservoir scenario
I0c = 0.0576
I0b = 0.0976
}
if (scenario==4) {
#van Tonder scenario
I0c = 0.0324
I0b = 0.0866
}
return(c(I0c, I0b))
}
scenario = 2
pcattle <- 0.8
## proportion of cattle vaccinated
pbadgers <- 0
## proportion of badgers vaccinated
N0c <- 1
N0b <- 1
I0c <- init_prevalence(scenario)[1]
I0b = init_prevalence(scenario)[2]
epSus <- 0.4
## relative risk of infection (reduction in susceptibility due to vacc = 0.25)
epInf <- 0.75
## relative risk of transmission (reduction in infectiousness due to vacc = 0.36)
uc <- 0.1
## birth rate of cattle = birth rate (constant population size)
ub <- 0.2
## birth rate of badgers = birth rate (constant population size)
vc <- 0.1
## death rate of cattle = birth rate (constant population size)
vb <- 0.2
## death rate of badgers = birth rate (constant population size)
tau <- 0.7
## removal rate of infected cattle
## assumes DIVA test has equal efficacy in vaccinated and unvaccinated cattle
beta <- beta_scenarios(scenario)
vbeta <- matrix(0, 4, 4)
vbeta[1,1] <- beta[1,1]
vbeta[1,2] <- beta[1,2]
vbeta[2,1] <- beta[2,1]
vbeta[2,2] <- beta[2,2]
vbeta[3,1] <- beta[1,1]*epSus
vbeta[3,2] <- beta[1,2]*epSus
vbeta[4,1] <- beta[2,1]
vbeta[4,2] <- beta[2,2]
vbeta[1,3] <- beta[1,1]*epInf
vbeta[1,4] <- beta[1,2]
vbeta[2,3] <- beta[2,1]*epInf
vbeta[2,4] <- beta[2,2]
vbeta[3,3] <- beta[1,1]*epSus*epInf
vbeta[3,4] <- beta[1,2]*epSus
vbeta[4,3] <- beta[2,1]*epInf
vbeta[4,4] <- beta[2,2]
## 4x4 matrix for values of beta (including effects of vaccination)
vbetaCc <- matrix(nrow=2,ncol=4)
vbetaCc[1,1] <- beta[1,1]
vbetaCc[1,2] <- beta[1,2]
vbetaCc[1,3] <- epInf*beta[1,1]
vbetaCc[1,4] <- beta[1,2]
vbetaCc[2,1] <- epSus*beta[1,1]
vbetaCc[2,2] <- epSus*beta[1,2]
vbetaCc[2,3] <- epSus*epInf*beta[1,1]
vbetaCc[2,4] <- epSus*beta[1,2]
u <- diag(c(((1-pcattle)*(uc)), (1-pbadgers)*(ub), pcattle*(uc), pbadgers*(ub)))
v <- diag(c(vc, vb, vc, vb))
removal <- diag(c((tau), 0, (tau), 0))
SI.par <- list(u, v, removal, vbeta, vbetaCc)
SI.init <- c(N0c - I0c, N0b - I0b,
0, 0,
I0c, I0b,
0, 0,
I0c, 0)
SI.t <- seq(0, 400, by = 1)
## since beta and other parameters are measured p/a, timescale should be in years
## e.g. 50 years
SI.sol <- lsoda(SI.init, SI.t, SI_dyn, SI.par)
TIMES <- SI.sol[,1]
Sc <- SI.sol[,2]
Sb <- SI.sol[,3]
Vc <- SI.sol[,4]
Vb <- SI.sol[,5]
Ic <- SI.sol[,6]
Ib <- SI.sol[,7]
Wc <- SI.sol[,8]
Wb <- SI.sol[,9]
Cc <- SI.sol[,10]+SI.sol[,11]
plot(TIMES, Sc, col = "blue",
pch = ".",
ylab = "Proportion of Individuals",
xlab = "Time (years)", ylim=c(0,1))
lines(TIMES, Sc, col = "blue")
lines(TIMES, Sb, col = "purple")
lines(TIMES, Vc, col = "green")
lines(TIMES, Vb, col = "brown")
lines(TIMES, Ic, col = "red")
lines(TIMES, Ib, col = "pink")
lines(TIMES, Wc, col = "yellow")
lines(TIMES, Wb, col = "grey")
legend("center",
legend=c("Sc", "Sb", "Vc","Vb","Ic","Ib","Wc","Wb"),
col=c("blue","purple","green","brown","red","pink","yellow","grey"),
pch=c("-","-","-","-","-","-","-","-"),
box.lty = 1,
cex=0.85,
ncol=4)
#CUMULATIVE CASES PLOT
plot(TIMES, Cc, col = "red",
pch = ".",
ylab = "Cumulative cases in cattle (Cc)",
xlab = "Time (years)",
ylim=c(0,Cc[length(Cc)]))
lines(TIMES, Cc, col = "red")
for (i in 1:length(Cc)) {
if (diff(c(Cc))[i] < 0.0001) {
#print(SI.t[i]/(1/364))
#prints number of days to incidence > 0.01%
print(SI.t[i])
break
}
}