formatter

docs/src/tutorials/discrete_time.md

@@ -1,6 +1,6 @@
 # Identifiability of Discrete-Time Models (Local)
 
-Now we consider a discrete-time model in the state-space form 
+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)),\\
@@ -17,18 +17,19 @@ Again, we will distinguish two types of identifiability
 
   - **global** identifiability: the value can be recovered uniquely.
 
-Currently, `StructuralIdentifiability.jl` allows to assess only local identifiability for discrete-time models, 
+Currently, `StructuralIdentifiability.jl` allows to assess only local identifiability for discrete-time models,
 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:
 $\begin{cases}
-    S(t + 1) = S(t) - \beta S(t) I(t),\\
-    I(t + 1) = I(t) + \beta S(t) I(t) - \alpha I(t),\\
-    R(t + 1) = R(t) + \alpha I(t),\\
-    y(t) = I(t), 
+S(t + 1) = S(t) - \beta S(t) I(t),\\
+I(t + 1) = I(t) + \beta S(t) I(t) - \alpha I(t),\\
+R(t + 1) = R(t) + \alpha I(t),\\
+y(t) = I(t),
 \end{cases}$
 
 where the observable is `I`, the number of infected people.
 We start with creating a system as a `DiscreteSystem` from `ModelingToolkit`:
+
 ```@example discrete
 using ModelingToolkit
 using StructuralIdentifiability
@@ -42,29 +43,32 @@ eqs = [D(S) ~ S - β * S * I, D(I) ~ I + β * S * I - α * I, D(R) ~ R + α * I]
 ```
 
 Once the model is defined, we can assess identifiability by providing the formula for the observable:
+
 ```@example discrete
 assess_local_identifiability(sir; measured_quantities = [y ~ I])
 ```
+
 For each parameter or state, the value in the returned dictionary is `true` (`1`) if the parameter is locally identifiable and `false` (`0`) otherwise.
 We see that `R(t)` is not identifiable, which makes sense: it does not affect the dynamics of the observable in any way.
 
 In principle, it is not required to give a name to the observable, so one can write this shorter
+
 ```@example discrete
 assess_local_identifiability(sir; measured_quantities = [I])
 ```
 
 The `assess_local_identifiability` function has two 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.
+  - `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.
 
 ```@example discrete
 assess_local_identifiability(sir; measured_quantities = [I], funcs_to_check = [β * S])
 ```
 
-* `p` is the probability of correctness (default value `0.99`, i.e., 99%). The underlying algorithm is a Monte-Carlo algorithm, so in 
-  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).
+  - `p` is the probability of correctness (default value `0.99`, i.e., 99%). The underlying algorithm is a Monte-Carlo algorithm, so in
+    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).
 
 The implementation is based on a version of the observability rank criterion and will be described in a forthcoming paper.