-digamma(1) is the Euler's Constant in R.

e, (exp(1) in R), which is the natural base of the natural logarithm

Euler's Constant. Euler's Number

Making the code more generic and modular (separating out the numerical methods), we can plot the solution to the epidemic model initial value problem (differential equation):

Euler_update <- function(xn, delta_t, f) {
  return (xn + delta_t * f(xn))
}

Improved_Euler_update <- function(xn, delta_t, f) {
  k1 <- f(xn) * delta_t
  k2 <- f(xn + k1) * delta_t
  return (xn + 1/2 *  (k1 + k2))
}

Runge_Kutta_4_update <- function(xn, delta_t, f) {
  k1 <- f(xn) * delta_t
  k2 <- f(xn + k1/2) * delta_t
  k3 <- f(xn + k2/2) * delta_t
  k4 <- f(xn + k3) * delta_t
  return (xn + 1/6 * (k1 + 2*k2 + 2*k3 + k4))
}

logistic_function <- function(x, r=1) {
  return (r * x * (1 - x))
}

analytic_solution <- function(x0, ts, r=1) {
  return (1 / (1 + (1-x0)/x0*exp(-r*ts)))  
}

solve_logistic <- function(N0, r = 1, delta_t = 0.01, times = 1000, update_fn=Euler_update) {
  t <- rep(0, times)
  N <- rep(N0, times)
  
  for (i in seq(2, times)) {
    # Euler
    t[i] <- t[i-1] + delta_t
    N[i] <- update_fn(N[i-1], delta_t, logistic_function)
  }
  
  return (list(t, N))
}

solution_df <- NULL
N0 <- 0.1
for (delta_t in c(0.01,0.1,0.5,1,1.25)) {
  sol <- solve_logistic(N0, delta_t=delta_t, times=10/delta_t) #, update_fn = Runge_Kutta_4_update)
  solution_df <- rbind(solution_df, data.frame(delta_t=delta_t, t=sol[[1]], N_t=sol[[2]]))
}
solution_df$delta_t <- as.factor(solution_df$delta_t)
ts <- seq(0,10,0.01)
solution_df <- rbind(solution_df, data.frame(delta_t='analytic', t=ts, N_t=analytic_solution(N0, ts)))

library(dplyr)
library(ggplot2)
solution_df %>% ggplot(aes(t, N_t, col=delta_t)) + geom_line(lwd=1) + theme_bw() + ggtitle('Euler method')

Just by changing the function name for the corresponding algorithm, we can obtain the solution with improved Euler and Runge-Kutta order-4 methods, improving the accuracy (in terms of deviance from the analytic solution) much more and faster convergence,, as shown in the figures below.