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

#> [frontmatter]
#> order = "34"
#> title = "6. Uncertainty bithropic model"
#> date = "2025-08-06"
#> tags = ["exercises"]
#> description = "Uncertainty bithropic model"
#> layout = "layout.jlhtml"
#> 
#>     [[frontmatter.author]]
#>     name = "Gauthier Vanhaelewyn"

using Markdown
using InteractiveUtils

# ╔═╡ 7d2ce89c-de3e-4fff-8f23-8756351b377e
using Pkg; Pkg.activate("../../pluto-deployment-environment")

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using Markdown

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using InteractiveUtils

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using Catalyst

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using OrdinaryDiffEq, StatsPlots

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using Measurements

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md"""
### Exercise: Bitrophic model - Uncertainty analysis
"""

# ╔═╡ 8f795d92-b842-4392-8bc3-956080fcc6a8
md"""
In one of the previous practicals we were introduced to a bitrophic model in which the dynamic relationship between a field crop $C$ and a voracious insect population $A$ within an ecosystem was modelled.

$$\begin{eqnarray*}
\frac{dC}{dt} &= \theta C \left(1-\frac{C}{k}\right)-fCA \\
\frac{dA}{dt} &= \phi f CA -(1 + p)\, \mu A
\end{eqnarray*}$$
"""

# ╔═╡ bc2e3724-de5f-4455-a4be-019c6a8accae
md"""
The *reaction network object* for this model could be set-up as:
"""

# ╔═╡ 90e9da81-7576-43bb-b4ce-1293a48bf3ec
bitrophic_model = @reaction_network begin
	θ*(1-C/k), C --> 2C
	f, C+A --> (1+ϕ)*A
    (1+p)*μ, A --> 0
end

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

-  $\theta=0.20 \pm 0.02\;d^{-1}$
-  $\phi=0.20 \pm 0.02$
-  $p=3.0 \pm 0.2$ 

and that the uncertainty in the other parameters values: $k=4000\;kg/ha$, $f=0.001\;ha/(kg\,d)$ and $\mu=0.1\;d^{-1}$ are negligible. Suppose that at the beginning of a season, $100\;kg$ of the crop and $0.5\;kg$ of insects per $ha$ are present. Perform an uncertainty analysis by plotting the uncertainty bands on the simulation results of $C$ and $A$ in a timespan of $[0, 200]\,days$.

Interpret your results.
"""

# ╔═╡ 1129972e-3e3d-4b3b-8c2c-3c57eaa3728a
md"""
Initialize a vector `u0` with the initial conditions, and set the timespan:
"""

# ╔═╡ 79802166-efbd-4944-ae58-fb26a5d9ad4b
# u0 = missing               # Uncomment and complete the instruction

# ╔═╡ 146855a0-5cda-4783-89af-7918c067e952
# tspan = missing            # Uncomment and complete the instruction

# ╔═╡ 4e8fea9e-e16c-4b17-8a6c-52bd8721cfd8
md"""
We initialize a vector `params_uncert` with the parameter values and their corresponding uncertainty:
"""

# ╔═╡ 0588cc16-a4b1-4b3f-8ba9-742680996e11
# params_uncert = missing     # Uncomment and complete the instruction

# ╔═╡ ff76c3e0-396c-4b01-b979-2b95644203a0


# ╔═╡ 6893ccdb-7f6c-4ec9-8d6b-520e784acd03
md"""
We create the corresponding ODE problem and store it in `oprob_uncert`:
"""

# ╔═╡ 60a6fb20-93d2-47d3-8ee1-3087d48c569d
# oprob_uncert = missing       # Uncomment and complete the instruction

# ╔═╡ 0739d816-0456-4cc2-beb7-2e926cf2e229
md"""
We solve the ODE problem. Use `Tsit5()` and `saveat=2.0`. Store the solution in `osol_uncert`:
"""

# ╔═╡ f829fc66-3df8-4ac1-a1bb-8eb2ed722423
# osol_uncert = missing          # Uncomment and complete the instruction

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md"""
Plot the results (simulation of the output variables $C$ and $A$ together with their uncertainty band):
"""

# ╔═╡ 2152a72b-31cd-44b9-90d8-615610e8d261
# missing                        # Uncomment and complete the instruction

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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 $C$ and $A$ and compare with your local sensitivity results of the corresponding parameter.

In order to do that, analyse the effect on the uncertainty bands for $C$ and $A$ by taking one uncertainty on a parameter at a time. In other words, analyse the subsequent cases separately:
- Assume uncertainty only in $\theta$
- Assume uncertainty only in $\phi$
- Assume uncertainty only in $p$
"""

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md"""
!!! question
	Draw your conclusions.
"""

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md"- Answer: missing"

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