### A Pluto.jl notebook ###
# v0.20.4

#> [frontmatter]
#> order = "33"
#> title = "6. Uncertainty fermenter monod"
#> date = "2025-08-06"
#> tags = ["exercises"]
#> description = "Uncertainty fermenter monod"
#> layout = "layout.jlhtml"
#> 
#>     [[frontmatter.author]]
#>     name = "Gauthier Vanhaelewyn"

using Markdown
using InteractiveUtils

# ╔═╡ 9d8acf40-e635-4dc4-9938-ec63ab68e3bd
using Pkg; Pkg.activate("../../pluto-deployment-environment")

# ╔═╡ 1ec7def2-0c43-11ef-0850-ddefa680d2a4
using Markdown

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using InteractiveUtils

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using Catalyst

# ╔═╡ 0720732e-105a-40c2-bf45-f8fbbc10e5bc
using OrdinaryDiffEq, StatsPlots

# ╔═╡ 28bb60c9-ab56-497f-b039-0d6577501be5
using Measurements

# ╔═╡ 437364b6-a953-428b-a891-e17ffd531656
md"""
### Exercise: Fermenter - Monod kinetics - Uncertainty analysis
"""

# ╔═╡ c44e10f3-8452-4a60-b840-f4a7e6ec3238
md"""
In one of the previous practicals we were introduced to a fermenter in which biomass $X$ [$g/L$] grows by breaking down substrate $S$ [$g/L$]. The reactor is fed with a inlet flow rate $Q_{in}$ [$L/h$], which consist of a (manipulable) input concentration of substrate $S_{in}$ [$g/L$]. This process was modelled using Monod kinetics, resulting in the model below:

$$\begin{eqnarray*}
S + X \xrightarrow[\quad\quad]{k} (1 + Y) \, X \quad\quad\quad\quad \textrm{with} \quad k = \cfrac{\mu_{max}}{S + K_s}
\end{eqnarray*}$$
"""

# ╔═╡ 356822df-bad3-41f9-8556-de04234b0a09
md"""
The *reaction network object* for this model could be set-up as:
"""

# ╔═╡ c551c33e-95b0-4ea8-85d4-6c68db72b8bc
fermenter_monod = @reaction_network begin
    μmax/(S+Ks), S + X --> (1 + Y)*X
    Q/V, (S, X) --> 0
    Q/V*Sin, 0 --> S
end

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md"""
which resulted in the following differential equations:
"""

# ╔═╡ 2806654f-7a24-4f8e-8b5a-3129b2431c95
md"""
$$\begin{eqnarray*}
\cfrac{dS}{dt} &=& \cfrac{Q}{V} \left(S_{in} - S \right) - \mu_{max}\cfrac{S}{S + K_s} X\\
\cfrac{dX}{dt} &=& -\cfrac{Q}{V} X + Y \mu_{max}\cfrac{S}{S + K_s} X
\end{eqnarray*}$$
"""

# ╔═╡ 5d6a881d-421e-4dea-9668-38b3f3378bf1
# osys = missing

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md"""
Assume the uncertainties in the following parameter values:

-  $\mu_{max} = 0.40 \pm 0.06\;h^{-1}$
-  $K_s = 0.015 \pm 0.003 \;g/L$
-  $S_{in} = 0.022 \pm 0.004\;g/L$

and that the uncertainty in the other parameters values: $Y = 0.67$, $Q = 2.0\;L/h$, $V = 40.0\;L$ are negligible. Suppose that at $t=0$ no substrate $S$ is present in the reactor but that there is initially some biomass with a concetration of $0.0005\;g/L$. Perform an uncertainty analysis by plotting the uncertainty bands on the simulation results of $S$ and $X$ in a timespan of $[0, 100]\,h$.

Interpret your results.
"""

# ╔═╡ 562e0006-bb79-4895-98b4-c376ea6ce856
md"""
Initialize a vector `u0` with the initial conditions, and set the timespan:
"""

# ╔═╡ 6afb23f2-3da7-43e0-8a3b-fc4bb5eb9bde
# u0 = missing                 # Uncomment and complete the instruction

# ╔═╡ 1b09c8cb-d3b7-4a15-b6ed-d03019c5f57b
# tspan = missing              # Uncomment and complete the instruction

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md"""
We initialize a vector `params_uncert` with the parameter values and their corresponding uncertainty:
"""

# ╔═╡ 62f4049b-617f-4268-beba-b2c634b46642
# params_uncert = missing       # Uncomment and complete the instruction

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md"""
We create the corresponding ODE problem and store it in `oprob_uncert`:
"""

# ╔═╡ 3cad11d6-c2eb-4865-858a-d407aa25d780
# oprob_uncert = missing        # Uncomment and complete the instruction

# ╔═╡ a7b3b68f-d34e-4e9a-aa44-224919c44fec
md"""
We solve the ODE problem. Use `Tsit5()` and `saveat=2.0`. Store the solution in `osol_uncert`:
"""

# ╔═╡ 40168902-0780-4f92-8eca-23982c80511e
# osol_uncert = missing         # Uncomment and complete the instruction

# ╔═╡ 6b2abf2a-f1f3-4c2d-96e8-cc626848fa68
md"""
Plot the results (simulation of the output variables $S$ and $X$ together with their uncertainty band):
"""

# ╔═╡ 4d753fdb-089d-41dc-89ef-45a1733af786
# missing

# ╔═╡ 4f3900d7-4ff8-4b4b-a59a-f7079ac6c457
md"""
Try to relate the local sensitivity analysis to the uncertainty analysis. Hence, study the effect of the individual parameter uncertainties on the output variables $S$ and $X$ and compare with your local sensitivity results of the corresponding parameter.

In order to do that, analyse the effect on the uncertainty bands for $S$ and $X$ by taking one uncertainty on a parameter at a time. In other words, analyse the subsequent cases separately:
- Assume uncertainty only in $\mu_{max}$
- Assume uncertainty only in $K_s$
- Assume uncertainty only in $S_{in}$
"""

# ╔═╡ ff3e6f75-c02b-4530-8b9a-6ff40915cb5c
md"""
!!! question
	Draw your conclusions.
"""

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md"- Answer: missing"

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