The Logistic ODE hazard function
The logistic ODE is defined by (Christen and Rubio 2023)
\[ h'(t) = \lambda h(t) \left(1 - \dfrac{h(t)}{\kappa}\right); \quad h(0) = h_0, \]
where \(\lambda > 0\) represents the intrinsic growth rate of the hazard function, \(\kappa > 0\) represents the upper bound of the hazard function, and \(h_0 > 0\) is the value of the hazard function at \(t=0\). This ODE has the following analytic solution \[ h(t \mid \lambda, \kappa, h_0) = \frac{\kappa h_0 e^{\lambda t}}{\kappa + h_0 (e^{\lambda t} - 1)} . \] The solution \(h(t)\), \(t \in [0,\infty)\) is positive, and \(\lim_{t \rightarrow \infty} h(t) = \kappa\). To obtain the cumulative hazard function, \(H(t)\), one may define the following system of ODEs, \[ \begin{cases} h'(t) = \lambda h(t) \left(1 - \dfrac{h(t)}{\kappa}\right), & h(0) = h_0\\ H'(t) = h(t), & H(0) = 0. \end{cases} \] The corresponding cumulative hazard function is \[ H(t \mid \lambda, \kappa, h_0) = \dfrac{\kappa}{\lambda } \log \left(\dfrac{\kappa + h_0 \left(e^{\lambda t}-1\right)}{\kappa} \right). \] And it is straightforward to see that its corresponding pdf is \[ f( t \mid \lambda, \kappa, h_0) = \frac{\kappa^2 h_0 e^{\lambda t - \kappa/\lambda}} {\left( \kappa + h_0 (e^{\lambda t} - 1) \right)^2} . \]
Routines
source("routines.R")Examples
Hazard function
# Parameter values
l <- 1
k <- 10
h0s <- c(1/10,2,1)*k
haz1 <- Vectorize(function(t) hlogisode(t, l, k, h0s[1]))
curve(haz1, 0, 10, ylim = c(0,20), xlab = "time", ylab = "Hazard",
cex.axis = 1.5, cex.lab = 1.5, lwd = 2, lty = 1, n = 250)
haz2 <- Vectorize(function(t) hlogisode(t, l, k, h0s[2]))
curve(haz2, 0, 10, ylim = c(0,20), xlab = "time", ylab = "Hazard",
cex.axis = 1.5, cex.lab = 1.5, lwd = 2, lty = 2, add = TRUE, n = 250)
haz3 <- Vectorize(function(t) hlogisode(t, l, k, h0s[3]))
curve(haz3, 0, 10, ylim = c(0,20), xlab = "time", ylab = "Hazard",
cex.axis = 1.5, cex.lab = 1.5, lwd = 2, lty = 3, add = TRUE, n = 250)
legend("topright", legend = c(expression(paste(h[0], " = ", kappa/10)),
expression(paste(h[0], " = ", 2*kappa)),
expression(paste(h[0], " = ", kappa))),
lty = c(1,2,3), lwd = c(2,2,2))
Survival function
# Parameter values
l <- 1
k <- 10
h0s <- c(1/10,2,1)*k
surv1 <- Vectorize(function(t) exp(-chlogisode(t, l, k, h0s[1])))
curve(surv1, 0, 2, ylim = c(0,1), xlab = "time", ylab = "Survival",
cex.axis = 1.5, cex.lab = 1.5, lwd = 2, lty = 1, n = 250)
surv2 <- Vectorize(function(t) exp(-chlogisode(t, l, k, h0s[2])))
curve(surv2, 0, 10, ylim = c(0,1), xlab = "time", ylab = "Survival",
cex.axis = 1.5, cex.lab = 1.5, lwd = 2, lty = 2, add = TRUE, n = 250)
surv3 <- Vectorize(function(t) exp(-chlogisode(t, l, k, h0s[3])))
curve(surv3, 0, 10, ylim = c(0,1), xlab = "time", ylab = "Survival",
cex.axis = 1.5, cex.lab = 1.5, lwd = 2, lty = 3, add = TRUE, n = 250)
legend("topright", legend = c(expression(paste(h[0], " = ", kappa/10)),
expression(paste(h[0], " = ", 2*kappa)),
expression(paste(h[0], " = ", kappa))),
lty = c(1,2,3), lwd = c(2,2,2))
Density function and simulated data
# Parameter values
l <- 1
k <- 10
h0s <- c(1/10,2,1)*k
n <- 10000
set.seed(1234)
dat1 <- rlogisode(n, l, k, h0s[1])
den1 <- Vectorize(function(t) dlogisode(t, l, k, h0s[1]))
hist(dat1, breaks = 50, probability = TRUE, xlab = "time", ylab = "Density",
cex.axis = 1.5, cex.lab = 1.5, main = "", ylim = c(0,h0s[1]) )
curve(den1, 0, 3, lwd = 2, lty = 1, n = 250, add = TRUE)
box()
set.seed(1234)
dat2 <- rlogisode(n, l, k, h0s[2])
den2 <- Vectorize(function(t) dlogisode(t, l, k, h0s[2]))
hist(dat2, breaks = 50, probability = TRUE, xlab = "time", ylab = "Density",
cex.axis = 1.5, cex.lab = 1.5, main = "", ylim = c(0,h0s[2]) )
curve(den2, 0, 0.5, lwd = 2, lty = 1, n = 250, add = TRUE)
box()
set.seed(1234)
dat3 <- rlogisode(n, l, k, h0s[3])
den3 <- Vectorize(function(t) dlogisode(t, l, k, h0s[3]))
hist(dat3, breaks = 50, probability = TRUE, xlab = "time", ylab = "Density",
cex.axis = 1.5, cex.lab = 1.5, main = "", ylim = c(0,h0s[3]) )
curve(den3, 0, 1, lwd = 2, lty = 1, n = 250, add = TRUE)
box()