human
What is a the Python code you requested given the following:
1 One compartment model with absoprtion from the gut
A patient orally takes in a drug, which lands in the gut and is then absorbed into a central compartment (e.g. the blood). This process is described by a differential equation:
\begin{eqnarray}
  y_0' & = & -k_0 y_0, \
  y_1' & = & k_0 y_0 - k_1 y_1,
\end{eqnarray}
with each state corresponding to the drug mass in the gut and the central compartment.
This system can be solved analytically for initial conditions $(y_0^I, y_1^I)$ at time $t = 0$:
\begin{eqnarray}
  y_0 & = & y_0^I e^{-k_0 t}, \
  y_1 & = & \frac{e^{-k_1 t}}{k_0 - k_1} \left [ y_0^I k_0(1 - e^{(k_1 - k_0)t}) + (k_0 - k_1) y^I_1 \right ], 
\end{eqnarray}
provided $k_0 \neq k_1$.
We can also use on Jax's odeint and solve the equation numerically. This will set us up for more complicated problems. The data is noisy observation of $y_1$ (in practice we should use $y_1 / V$ where $V$ is the volume of the central compartment, but I'll omit this for now).
assistant
#Here is the Python code you requested.
# NOTE: need to pass the initial time as the first element of t.
t = np.array([0., 0.5, 0.75, 1, 1.25, 1.5, 2, 3, 4, 5, 6])
y0 = np.array([100.0, 0.0])

theta = np.array([1.5, 0.25])
def system(state, time, theta):
  k1 = theta[0]
  k2 = theta[1]

  return jnp.array([
    - k1 * state[0]  ,
    k1 * state[0] - k2 * state[1]
  ])

use_analytical_sln = True

if (use_analytical_sln):
  def ode_map(k1, k2):
    sln = jnp.exp(- k2 * t) / (k1 - k2) * (y0[0] * k1 * (1 - jnp.exp((k2 - k1) * t)) + (k1 - k2) * y0[1])
    return sln[1:]
else:
  def ode_map(k1, k2):
    theta = jnp.array([k1, k2])
    return odeint(system, y0, t, theta, mxstep = 1e6)[1:, 1]

states = ode_map(k1 = theta[0], k2 = theta[1])
random.normal(random.PRNGKey(37272710), (states.shape[0],))
jnp.log(states)