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

#> [frontmatter]
#> order = "4"
#> title = "1. ODE birth rate"
#> date = "2025-02-07"
#> tags = ["exercises"]
#> description = "ODE model of birth rate"
#> layout = "layout.jlhtml"
#> 
#>     [[frontmatter.author]]
#>     name = "Gauthier Vanhaelewyn"

using Markdown
using InteractiveUtils

# ╔═╡ 12fee37d-eef9-4e90-b0af-d892469fad08
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

# ╔═╡ 4a3d066c-f5b1-11ee-0145-2da7c11147a5
using Markdown

# ╔═╡ 0ab4d97a-0c47-4f6e-a76b-9cec00ad410a
using InteractiveUtils

# ╔═╡ 34a2f237-848c-4d07-9cea-fa5505a9e215
using PlutoUI; TableOfContents()

# ╔═╡ b721cc4d-43ba-4221-bbd1-15293aaf54b5
using Catalyst

# ╔═╡ 47e9c791-99cc-4a78-94b0-e0f5d4e0ecc5
using OrdinaryDiffEq, StatsPlots

# ╔═╡ c701d64e-640c-473f-b0fa-688024962f28
md"""
# Exercise: Simple birth-death model for mice

In a simple birth-death model for mice, the birth rate of mice represents the
rate at which new individuals are added to the population through reproduction.
This rate is influenced by factors such as the number of reproductive females,
their fertility, and the frequency of reproduction cycles. Conversely,
the death rate reflects the rate at which individuals are removed from
the population due to mortality factors such as predation, disease,
and environmental stressors. Together, these rates interact dynamically
to shape the population dynamics of mice in their natural habitat.
Denote the number of mice by $X$, the average birth rate by $b$ ($mice / day$), and the average death rate by $d$ ($day^{-1}$). Hence, assume for this overly simplified model, that the birth of mice is a zeroth-order process and that the death of mice is a first-order process.
"""

# ╔═╡ b583efef-dc7d-4447-bf27-e6373f5872c4
md"""
Create a *reaction network object* model for the aforementioned problem in order to simulate the evolution of $X$ with time. Name it `birth_death`.
"""

# ╔═╡ 74c3f7ed-d705-4cb9-b52c-e06c9df5ca13
# Uncomment and complete the instruction
# birth_death = @reaction_network begin
# 	missing
# end

# ╔═╡ c84f2a32-e66b-4a36-a257-7031ce799225
md"""
Convert the system to a symbolic differential equation model and verify, by analyzing the differential equation, that your model is correctly implemented.
"""

# ╔═╡ 124eb20e-6945-4eec-a05a-6b835efbdd2e
# osys  = missing             # Uncomment and complete the instruction

# ╔═╡ 04c645ad-873d-44c1-9bc0-940a412630b9
md"""
## Part 1

Simulate the evolution of the number of mice **per day** during $10$ years starting off with $2$ mice. **Assume that per year 25 pups are born.** Suppose the death rate to be $0.0015\;day^{-1}$.
"""

# ╔═╡ 1cd5136a-f472-4329-980a-b8382d1c04ea
md"""
First, calculate the birth rate in $mice/day$.
"""

# ╔═╡ 6b64ec8b-d85e-4a4f-aa3f-f177f5e34180
# missing              # Uncomment and complete the instruction

# ╔═╡ 52ee74d2-4dca-47ef-8b97-f826ef31ddf9
md"""
Initialize a vector `u0` with the initial conditions:
"""

# ╔═╡ 23f282d8-2eeb-43dd-8ba9-84684daca7a4
# u0 = missing         # Uncomment and complete the instruction

# ╔═╡ 30fa6fc6-54e0-4233-9334-b3d15f630489
md"""
Set the timespan for the simulation:
"""

# ╔═╡ b461a225-bae6-45b8-bb42-54742a72b98f
# tspan = missing      # Uncomment and complete the instruction

# ╔═╡ c753d65f-9ed3-4e60-8e36-6e4bca77c19b
md"""
Initialize a vector `param` with the parameter values:
"""

# ╔═╡ e7942ba9-5434-4b22-b1d4-f18f6227320d
# params = missing     # Uncomment and complete the instruction

# ╔═╡ 2af7f6b9-9d95-4bf6-af67-449803440639
md"""
Create the ODE problem and store it in `oprob`:
"""

# ╔═╡ 9432f90b-7141-4518-b6fd-d55a1389e14a
# oprob = missing      # Uncomment and complete the instruction

# ╔═╡ ab197ac7-413b-4305-8352-2723bf9f2aff
md"""
Solve the ODE problem. Use `Tsit5()` and `saveat=1.0`. Store the solution in `osol`:
"""

# ╔═╡ 8e517114-7c74-4d7c-954c-2787d975a1df
# osol = missing         # Uncomment and complete the instruction

# ╔═╡ 9eaf1ca5-3d65-4263-a307-991bc8bf62d2
md"""
Plot the results:
"""

# ╔═╡ 83e36e63-7e6b-4c17-8f7d-ed0e07f36fc5
# missing           # Uncomment and complete the instruction

# ╔═╡ 5fc64787-795c-4e17-b751-cd3683f8016b
md"""
Interpret the results. Ask yourself the following questions:

1. What is the (approximate) steady state value for $X$?
"""

# ╔═╡ f5718c08-c8aa-4d74-b70f-8c08c6decab2
md"- Answer: missing"

# ╔═╡ fe4526c8-4504-4d91-841a-a83aeef55fc7
md"""
## Part 2

Suppose that at $t = 3\;years$ the death rate of the mice population increases by $50\,\%$ due to a new predator species in the area. Use the same initial condition, timespan and parameter values. Simulate the evolution of the number of mice.
"""

# ╔═╡ 1e869025-c537-4bf1-9c38-93e24b598156
md"""
Create the *condition*. Store it in `condition2`:
"""

# ╔═╡ 1c5d8daa-5c7e-49ef-a6da-6459a1131320
# condition2 = missing            # Uncomment and complete the instruction

# ╔═╡ 4d9ba697-5db3-4879-be88-278765e766f1
md"""
Make a new *reaction system* where the discrete event is included. Name it `birth_death2`.
"""

# ╔═╡ d40c2598-5783-49c5-916d-296e946de3f6
# @named birth_death2 = missing      # Uncomment and complete the instruction

# ╔═╡ afbc4f00-0b4a-427a-9630-3a279ccbc330
md"""
Complete the new *reaction system*. Name it `birth_death2_com`.
"""

# ╔═╡ 3dd3febc-fc3f-477f-a5be-335ae90223e3
# birth_death2_com = missing          # Uncomment and complete the instruction

# ╔═╡ cf05864f-37f0-4fbb-93a5-ef406a2ae014
md"""
Create the ODE problem and store it in `oprob2`:
"""

# ╔═╡ 65e24cee-06c3-4cb7-8f8c-61e02b7dcd29
# oprob2 = missing                    # Uncomment and complete the instruction

# ╔═╡ 9cab94b5-6d2d-4b5a-96f3-3b8eee1c61a1
md"""
Solve the ODE problem. Make a deepcopy and use `Tsit5()` and `saveat=1.0`. Store the solution in `osol2`:
"""

# ╔═╡ 00750aa4-6a00-4d33-a0ab-2d7abe952078
# osol2 = missing                 # Uncomment and complete the instruction

# ╔═╡ 694e5b50-f4c9-4ecc-8860-1b217696e1ea
md"""
Plot the results:
"""

# ╔═╡ c69d0e86-0d15-4915-a890-8abd378da59a
# missing                           # Uncomment and complete the instruction

# ╔═╡ 1f869579-7019-4120-8d14-b38f48d05bdd
md"""
Interpret the results. Ask yourself the following questions:

1. Can you clearly see the effect of the increase in the death rate?
"""

# ╔═╡ 775f2cf3-ffbb-491a-9002-3a39b9c70f3c
md"- Answer: missing"

# ╔═╡ a02b5eac-eacb-422c-be5e-ba6337ed33f8
md"""
2. If the death rate increases at a different timepoint, would you reach the same steady state value for $X$? Explain.
"""

# ╔═╡ ce733400-9bf4-42b3-8f77-351ef9b1cd92
md"- Answer: missing"

# ╔═╡ Cell order:
# ╠═4a3d066c-f5b1-11ee-0145-2da7c11147a5
# ╠═0ab4d97a-0c47-4f6e-a76b-9cec00ad410a
# ╠═12fee37d-eef9-4e90-b0af-d892469fad08
# ╠═34a2f237-848c-4d07-9cea-fa5505a9e215
# ╟─c701d64e-640c-473f-b0fa-688024962f28
# ╠═b721cc4d-43ba-4221-bbd1-15293aaf54b5
# ╟─b583efef-dc7d-4447-bf27-e6373f5872c4
# ╠═74c3f7ed-d705-4cb9-b52c-e06c9df5ca13
# ╟─c84f2a32-e66b-4a36-a257-7031ce799225
# ╠═124eb20e-6945-4eec-a05a-6b835efbdd2e
# ╠═47e9c791-99cc-4a78-94b0-e0f5d4e0ecc5
# ╟─04c645ad-873d-44c1-9bc0-940a412630b9
# ╟─1cd5136a-f472-4329-980a-b8382d1c04ea
# ╠═6b64ec8b-d85e-4a4f-aa3f-f177f5e34180
# ╟─52ee74d2-4dca-47ef-8b97-f826ef31ddf9
# ╠═23f282d8-2eeb-43dd-8ba9-84684daca7a4
# ╟─30fa6fc6-54e0-4233-9334-b3d15f630489
# ╠═b461a225-bae6-45b8-bb42-54742a72b98f
# ╟─c753d65f-9ed3-4e60-8e36-6e4bca77c19b
# ╠═e7942ba9-5434-4b22-b1d4-f18f6227320d
# ╟─2af7f6b9-9d95-4bf6-af67-449803440639
# ╠═9432f90b-7141-4518-b6fd-d55a1389e14a
# ╟─ab197ac7-413b-4305-8352-2723bf9f2aff
# ╠═8e517114-7c74-4d7c-954c-2787d975a1df
# ╟─9eaf1ca5-3d65-4263-a307-991bc8bf62d2
# ╠═83e36e63-7e6b-4c17-8f7d-ed0e07f36fc5
# ╟─5fc64787-795c-4e17-b751-cd3683f8016b
# ╟─f5718c08-c8aa-4d74-b70f-8c08c6decab2
# ╟─fe4526c8-4504-4d91-841a-a83aeef55fc7
# ╟─1e869025-c537-4bf1-9c38-93e24b598156
# ╠═1c5d8daa-5c7e-49ef-a6da-6459a1131320
# ╟─4d9ba697-5db3-4879-be88-278765e766f1
# ╠═d40c2598-5783-49c5-916d-296e946de3f6
# ╟─afbc4f00-0b4a-427a-9630-3a279ccbc330
# ╠═3dd3febc-fc3f-477f-a5be-335ae90223e3
# ╟─cf05864f-37f0-4fbb-93a5-ef406a2ae014
# ╠═65e24cee-06c3-4cb7-8f8c-61e02b7dcd29
# ╟─9cab94b5-6d2d-4b5a-96f3-3b8eee1c61a1
# ╠═00750aa4-6a00-4d33-a0ab-2d7abe952078
# ╟─694e5b50-f4c9-4ecc-8860-1b217696e1ea
# ╠═c69d0e86-0d15-4915-a890-8abd378da59a
# ╟─1f869579-7019-4120-8d14-b38f48d05bdd
# ╟─775f2cf3-ffbb-491a-9002-3a39b9c70f3c
# ╟─a02b5eac-eacb-422c-be5e-ba6337ed33f8
# ╟─ce733400-9bf4-42b3-8f77-351ef9b1cd92