### A Pluto.jl notebook ###
# v0.19.46

#> [frontmatter]
#> order = "13"
#> title = "2. SDE fermentor 2nd order"
#> date = "2025-02-07"
#> tags = ["exercises"]
#> description = "SDE fermentor 2nd order"
#> layout = "layout.jlhtml"
#> 
#>     [[frontmatter.author]]
#>     name = "Gauthier Vanhaelewyn"

using Markdown
using InteractiveUtils

# ╔═╡ 8152f632-af15-4164-a8ff-07c33a9a49b3
begin
	# add this cell if you want the notebook to use the environment from where the Pluto server is launched
	using Pkg
	Pkg.activate("../../pluto-deployment-environment")
end

# ╔═╡ eb142900-1d94-11ef-12ed-6951b45f1817
using Markdown

# ╔═╡ 7139a11b-64db-46b9-a41c-dca83a9eab26
using InteractiveUtils

# ╔═╡ 1ca1d1db-1fdd-4a76-b350-126add7e013c
using Catalyst

# ╔═╡ e04f782d-67da-4e21-a3bf-d2ddff4bba0b
using OrdinaryDiffEq, StochasticDiffEq, StatsPlots

# ╔═╡ a28e7ddf-76e9-4628-888c-e1d838da75ce
md"""
# Exercise: Fermenter - 2nd order kinetics - SDE
"""

# ╔═╡ 959d6307-a30d-4ae7-970d-b2c7584c2c8f
md"""
In a fermenter reactor biomass grows on substrate. The reactor is fed with a inlet flow rate $Q_{in}$ [$L/h$], which consist of a (manipulable) input concentration of substrate $S_{in}$ [$g/L$]. Inside the reactor, biomass, with a concentration of $X$ [$g/L$], is produced through second-order kinetics:

$$\begin{eqnarray*}
%S \xrightarrow[\quad\quad]{\beta} Y \, X
S \xrightarrow[\quad\quad]{r} Y \, X \quad\quad\quad\quad r = k \, S\,X
\end{eqnarray*}$$

with $k$ [$L\,gS^{-1}h^{-1}$] the reaction rate constant, and $Y$ [$gX/gS$] the yield coefficient which is defined here by the amount of produced biomass by consumption of one unit of substrate. Futhermore, the reactor is drained with an outlet flow $Q$ [$L/h$], which consist of the current concentrations of substrate $S$ [$g/L$] and biomass $X$ [$g/L$] inside the reactor. The volume $V$ [$L$] of the reactor content is kept constant by setting $Q_{in} = Q$.
"""

# ╔═╡ 98858b9c-d4f9-451f-a7b9-fcaa012ee28e
md"""
Create a *reaction network object* model for the aforementioned problem in order to simulate the evolution of substrate $S$ and biomass $X$ with time as a Stochastic Differential Equation (SDE) problem with noise scaling. Name it `fermenter_sde_secondorder`.
"""

# ╔═╡ d2e2680c-01c6-449a-bb5f-7472bc1de243
md"""
Assign the following noise scaling values:
- `η = 0.10` for the main *reaction* (default value for `η`)
- noise scaling of `0.05` for the *reaction* describing the inlet $S_{in}$
- noise scaling of `0.0` for the remaining *reactions*
"""

# ╔═╡ 6b627d84-b6a5-444d-8163-40a4cab181bd
# Uncomment and complete the instruction
# fermenter_sde_secondorder = @reaction_network begin
#   @parameters missing
#   missing
#   missing
#   missing
# end

# ╔═╡ 47ca3573-691c-4127-85b9-d5b5a1a23fbb
md"""
Convert the system to a symbolic differential equation model and verify, by analyzing the differential equation, that your model is correctly implemented.
"""

# ╔═╡ e1f11068-e489-4c54-a30d-981f5cb19b47
# osys = missing      # Uncomment and complete the instruction

# ╔═╡ a65c84c8-d2ec-44e4-9a07-cbd59f190c57
md"""
Initialize a vector `u0` with the initial conditions:
"""

# ╔═╡ ef3f7a41-61ab-4449-8ffc-784cf1e5cbe6
# u0 = missing        # Uncomment and complete the instruction

# ╔═╡ 9d113e7f-8499-4b2a-a884-d34ce3da0b82
md"""
Set the timespan for the simulation:
"""

# ╔═╡ 839a624b-ca62-4c57-9511-207b626ce864
# tspan = missing    # Uncomment and complete the instruction

# ╔═╡ a4d28c40-e315-4bb9-87a5-2b45dd633e5f
# params = missing    # Uncomment and complete the instruction

# ╔═╡ aeddc31e-9de2-4792-a2d8-59a14dfc8173
# sprob = missing      # Uncomment and complete the instruction

# ╔═╡ 3a981326-2031-4c63-ad31-c44ddd7a88d5
md"""
Solve the SDE problem. Use `EM()` with `dt=0.1`. Store the solution in `ssol`:
"""

# ╔═╡ 6a69c369-4743-48c1-aed9-4f0ccb095707
# ssol = missing      # Uncomment and complete the instruction

# ╔═╡ 6b6d2229-913b-41a2-8101-00e9fef0945a
md"""
Plot the results with the option `ylim=(0.0, 2.0)`:
"""

# ╔═╡ 593a0e0a-c4d8-4b12-b38b-15b47705f8a7
# missing

# ╔═╡ 08746e97-d794-4261-9ba6-9002cf17e4c1
md"""
Create an `EnsembleProblem` in order to visualize a multiple solutions. Store it in `esprob`.
"""

# ╔═╡ 3d07836a-60b1-4584-a89a-3d8bbc72b8cd
# esprob = missing     # Uncomment and complete the instruction

# ╔═╡ 98c4ee2b-20ef-43a1-b420-7644d568810b
md"""
Solve the `EnsembleProblem` using the same solver (and time step) as before, for $100$ trajectories. Store the solution in `essol`.
"""

# ╔═╡ 683fe575-887b-4bd1-8960-10c04f68354d
# essol = missing      # Uncomment and complete the instruction

# ╔═╡ 26746cab-d3e7-4a01-bbbd-9fcb49ef652f
md"""
Plot the results. Use as option again `ylim=(0.0,2.0)` and also `linealpha=0.5` to modify the line boldness.
"""

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# missing

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