Updating the docs for the discrete case

docs/src/tutorials/discrete_time.md

@@ -3,11 +3,11 @@
 Now we consider a discrete-time model in the state-space form
 
 $\begin{cases}
-\mathbf{x}(t + 1) = \mathbf{f}(\mathbf{x}(t), \mathbf{p}, \mathbf{u}(t)),\\
+\Delta\mathbf{x}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{p}, \mathbf{u}(t)),\\
 \mathbf{y}(t) = \mathbf{g}(\mathbf{x}(t), \mathbf{p}, \mathbf{u(t)}),
-\end{cases}$
+\end{cases} \quad \text{where} \quad \Delta \mathbf{x}(t) := \mathbf{x}(t + 1) - \mathbf{x}(t)$
 
-where $\mathbf{x}(t), \mathbf{y}(t)$, and $\mathbf{u}(t)$ are time-dependent states, outputs, and inputs, respectively,
+and $\mathbf{x}(t), \mathbf{y}(t)$, and $\mathbf{u}(t)$ are time-dependent states, outputs, and inputs, respectively,
 and $\mathbf{p}$ are scalar parameters.
 As in the ODE case, we will call that a parameter or a states (or a function of them) is **identifiable** if its value can be recovered from
 time series for inputs and outputs (in the generic case, see Definition 3 in [^1] for details).
@@ -22,9 +22,9 @@ and below we will describe how this can be done.
 As a running example, we will use the following discrete version of the [SIR](https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIR_model) model:
 
 $\begin{cases}
-S(t + 1) = S(t) - \beta S(t) I(t),\\
-I(t + 1) = I(t) + \beta S(t) I(t) - \alpha I(t),\\
-R(t + 1) = R(t) + \alpha I(t),\\
+\Delta S(t) = S(t) - \beta S(t) I(t),\\
+\Delta I(t) = I(t) + \beta S(t) I(t) - \alpha I(t),\\
+\Delta R(t) = R(t) + \alpha I(t),\\
 y(t) = I(t),
 \end{cases}$
 
@@ -58,7 +58,7 @@ In principle, it is not required to give a name to the observable, so one can wr
 assess_local_identifiability(sir; measured_quantities = [I])
 ```
 
-The `assess_local_identifiability` function has two important keyword arguments:
+The `assess_local_identifiability` function has three important keyword arguments:
 
   - `funcs_to_check` is a list of functions for which one want to assess identifiability, for example, the following code
     will check if `β * S` is locally identifiable.
@@ -71,6 +71,9 @@ assess_local_identifiability(sir; measured_quantities = [I], funcs_to_check = [
     principle it may produce incorrect result but the probability of correctness of the returned result is guaranteed to be at least `p`
     (in fact, the employed bounds are quite conservative, so in practice incorrect result is almost never produced).
 
+  - `known_ic` is a list of the states for which initial conditions are known. In this case, the identifiability results will be valid not
+     at any time point `t` but only at `t = 0`.
+
 As other main functions in the package, `assess_local_identifiability` accepts an optional parameter `loglevel` (default: `Logging.Info`)
 to adjust the verbosity of logging.