Problem running RStan with simulated data in R

You mean log1p() vs log(), right? Did you try init=0?

Try this, it should give a different error:

real surv_loglogistic_lpdf(vector t, vector d, vector p, matrix kernel,
                           vector exp_x, array[] int county, real shape,
                           vector scale) {
  vector[num_elements(t)] log_lik;
  real prob;
  vector[num_elements(t)] ls_s;
  vector[num_elements(t)] ls_f;
  for (i in 1 : num_elements(t)) {
    if (t[i] <= 0.0) {
      reject("t[", i, "] is ", t[i]);
    }
  }
  ls_s = log_S(t, p, kernel, exp_x, county, shape, scale);
  ls_f = log_f(t, p, kernel, exp_x, county, shape, scale);
  log_lik = d .* log_f(t, p, kernel, exp_x, county, shape, scale)
            + (1 - d) .* log_S(t, p, kernel, exp_x, county, shape, scale);
  for (i in 1 : num_elements(log_lik)) {
    if (is_inf(log_lik[i]) || is_nan(log_lik[i])) {
      reject("log_lik[", i, "] is ", log_lik[i]);
    }
  }
  prob = sum(log_lik);
  return prob;
}