My ordinary differential equation (ODE) system contains 386 compartments with all but one rates fixed. I assigned a normal prior for the non-fixed rate “theta” and runned my ODE system for 120 timesteps. All the rates are known: the model is not fitted. I only used the bayesian (and Stan) framework in order to vary theta according to its prior. To have an idea of the computational time needed by Stan (and more particularly the `integrate_ode_rk45`

), runned the model for 1 iteration, no warm-up and 1 chain. Surpringly, I found a big difference in the time declared by R and by Stan! The processing time calculated by R is 32 min, but when looking at Stan message, the elapsed time is only 275 secondes (approx. 4 min) (see below). According to me, these two times should be approximatively the same, as the .stan file has been already compilled before this simulation. Moreover, as I specified only 1 iteration, the ODE system needs to be solved only a few times until one value of the theta is accepted. I already solved this ODE system in R and this should not take 32 minutes.

As it is a quite complex model, I cannot unfortunately provide the code.

```
time.start_nuts1 <- Sys.time()
fit = sampling(mod3, data = list_data, init = 0, chains = 1, warmup = 0, iter = 1)
SAMPLING FOR MODEL 'diff1c' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 171.198 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.71198e+006 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: WARNING: No variance estimation is
Chain 1: performed for num_warmup < 20
Chain 1:
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0 seconds (Warm-up)
Chain 1: 275.864 seconds (Sampling)
Chain 1: 275.864 seconds (Total)
Chain 1:
Warning messages:
1: There were 1 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
2: Examine the pairs() plot to diagnose sampling problems
3: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#bulk-ess
4: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
http://mc-stan.org/misc/warnings.html#tail-ess
time.end_nuts1 <- Sys.time()
time.end_nuts1 - time.start_nuts1
```

**EDIT** : When I try to run the model with all parameters fixed (no “theta”) using the “Fixed_param” algorithm in the `sampling()`

function, I also end up with big difference in computational times calculated in R and Stan:

time.start_nuts1 <- Sys.time()

fit = sampling(mod2, data = list_data, init = 0, chains = 1, warmup = 0, iter = 1, seed=13219, algorithm = “Fixed_param”)

```
SAMPLING FOR MODEL 'diff1b' NOW (CHAIN 1).
Chain 1: Iteration: 1 / 1 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0 seconds (Warm-up)
Chain 1: 1.551 seconds (Sampling)
Chain 1: 1.551 seconds (Total)
Chain 1:
time.end_nuts1 <- Sys.time()
time.end_nuts1 - time.start_nuts1
Time difference of 7.666541 mins
```