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112 changes: 66 additions & 46 deletions exercises/ode_model_catalyst_intro.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -168,8 +168,8 @@ The substrates and the products may contain one or more reactants, separated by

# ╔═╡ bb5578b8-40d4-48c9-be87-cfaeadb8c5f9
md"""
!!! hint "Hint"
The Greek letters can be visualized by typing a **backslash** followed by the **name of the Greek letter** and then the **TAB** key. For example `\alpha` followed by the TAB key results in in a list where you can choose `α`.
!!! tip "Tip"
The Greek letters can be visualized by typing a **backslash** followed by the **name of the Greek letter** and then the **TAB** key. For example `\alpha` followed by the TAB key results in in a list where you can choose `α`.
"""

# ╔═╡ af6b169d-55c3-4384-9f57-ad629e02bad6
Expand All @@ -196,7 +196,7 @@ parameters(infection_model)
# ╔═╡ 2bd02577-7321-46b5-9f42-1484ed86d1d3
md"""
!!! important "Important"
You can also get the different species and parameters using `@unpack` followed by comma separated species and/or parameter names followed by the equal sign and the name of the reaction network model. For example:
You can also get the different species and parameters using `@unpack` followed by comma separated species and/or parameter names followed by the equal sign and the name of the reaction network model. For example:
"""

# ╔═╡ 64455b26-4374-435f-b71b-8b9600cab263
Expand All @@ -212,7 +212,7 @@ osys = convert(ODESystem, infection_model)

# ╔═╡ eeb25e50-165b-4e93-8397-5b09fe8e7242
md"""
Note that the model equations are essencially:
Note that the model equations are essentially:

$$\cfrac{dS(t)}{dt} = -\alpha \beta S(t) I(t)$$
$$\cfrac{dI(t)}{dt} = \alpha \beta S(t) I(t) - r I(t)$$
Expand Down Expand Up @@ -248,18 +248,18 @@ parameters(osys)
md"""
### Simulating the system as an ODE-problem

We first need to load the Differential and Plot package, which is required for simulating the system and plotting the results.
We first need to load the `DifferentialEquations` and `Plots` packages, which are required for simulating the system and plotting the results.
"""

# ╔═╡ 3197244f-655b-4dca-80f3-794b30722551
md"""
Now we wish to simulate our model. To do this, we need to provide some the following information:
Now we wish to simulate our model. To do this, we need to provide the following information:

- Initial conditions for the state variables $S$, $I$, $D$ and $R$.
- The parameter values for $\alpha$, $\beta$, $r$ and $m$.
- The timespan, which is the timeframe over which we wish to run the simulation.

Assume in this example that there are $10\,000\,000$ people in the country, and that initially $1\,000$ person are infected. Hence, $I_0 = 1\,000$, $S_0 = 10\,000\,000-I_0 = 9\,999\,000$, $D_0 = 0$ and $R_0 = 0$.\
Assume in this example that there are $10\,000\,000$ people in the country, and that initially $1\,000$ persons are infected. Hence, $I_0 = 1\,000$, $S_0 = 10\,000\,000-I_0 = 9\,999\,000$, $D_0 = 0$ and $R_0 = 0$.\
Furthermore, we take the following values for the parameters: $\alpha = 0.08\;person/contact$, $\beta = 10^{-6}\;contact/(person^2\,day)$, $r = 0.2\;day^{-1}$ (i.e. a person is contagious for an average of $5\;days$) and $m=0.4$. The following table summarizes the above values:

|Initial conditions |Parameters |
Expand All @@ -279,7 +279,7 @@ md"""

# ╔═╡ 1ba859fa-46a5-434f-a99c-e710ba85caf8
md"""
The initial conditions are given as a *Vector*. This is a type which collects several different values. To declare a vector, the values are specific within brackets, `[]`, and separated by `,`. Since we have four species, the vector holds four elements. E.g., we set the value of $I$ using the `:I => 1` syntax. Here, we first denote the name of the species (with a colon `:` pre-appended), next follows a `=>` and then the value of `I`.\
The initial conditions are given as a *Vector*. This is a type which collects several different values. To declare a vector, the values are specified within brackets, `[]`, and separated by `,`. Since we have four species, the vector holds four elements. E.g., we set the value of $I$ using the `:I => 1` syntax. Here, we first denote the name of the species (with a colon `:` pre-appended), next follows a `=>` and then the value of `I`.\
The vector holding the initial conditions for $S$, $I$, $D$ and $R$ can be created in the following way:
"""

Expand All @@ -288,7 +288,7 @@ u0 = [:S => 9_999_000.0, :I => 1_000.0, :D => 0.0, :R => 0.0]

# ╔═╡ 95c1c0ea-51d0-47ee-8948-a5b87c42a70d
md"
Note that the order of the vector elements doesn't matter, because the initial values of each of the species is indicated using its variable name.
Note that the order of the vector elements doesn't matter here, since the initial values of each of the species is indicated using its variable name.
"

# ╔═╡ 57036f49-2f1f-4327-89ed-2c96098a1c22
Expand All @@ -298,7 +298,7 @@ md"""

# ╔═╡ 56ebb72d-2351-4e67-b268-1f48bbb77cb3
md"""
The timespan sets the time point at which we start the simulation (typically `0.0` is used) and the final time point of the simulation. These are combined into a two-valued Tuple. Tuples are similar to vectors, but are enclosed by `()` and not `[]`. Again, we will let both time points be decimal valued.
The timespan sets the time point at which we start the simulation (typically `0.0` is used) and the final time point of the simulation. These are combined into a two-valued *Tuple*. Tuples are similar to vectors, but are enclosed by `()` and not `[]`. Again, we will let both time points be decimal valued.
"""

# ╔═╡ a25d5925-a254-488b-b782-d29cff4470a2
Expand All @@ -311,7 +311,7 @@ md"""

# ╔═╡ a235c7ce-f14a-4c7c-86a1-08aa5f2d9c85
md"""
Similarly, the parameters values are also given as a vector. We have four parameters, hence, the parameter vector will also contain four elements. We use a similar notation for setting the parameter values as the initial condition (first the colon, then the parameter name, then an arrow, then the value).
Similarly, the parameter values are also given as a vector. We have four parameters, hence, the parameter vector will also contain four elements. We use a similar notation for setting the parameter values as the initial condition (first the colon, then the parameter name, then an arrow, then the value).
"""

# ╔═╡ 9a9440fa-d8a3-44bc-8037-4bf1f8af40b0
Expand Down Expand Up @@ -358,7 +358,7 @@ md"""

# ╔═╡ 0af3c166-46b6-455d-af6b-a72c4d2a5ce4
md"""
Finally, we can plot the solution through the plot function.
Finally, we can plot the solution through the `plot` function.
"""

# ╔═╡ 513c037b-c54c-47fa-b97a-06f69a983386
Expand All @@ -374,7 +374,7 @@ plot(osol, idxs=[:S, :I]) # brackets [ ]

# ╔═╡ f3cf01a1-3c65-4a37-bf9a-cd2233cb470b
md"""
If you want a fase plot of for example just $I$ versus $S$, you can specify this with the option `idxs=(:S, :I)` (*notice the parentheses*) in the plot function. You can indicate the $S$ and $I$ axes with the additional options `xlab="S"` and `ylab="I"`.
If you want a phase plot of, for example, just $I$ versus $S$, you can specify this with the option `idxs=(:S, :I)` (*notice the parentheses*) in the plot function. You can indicate the $S$ and $I$ axes with the additional options `xlab="S"` and `ylab="I"`.
"""

# ╔═╡ ff8f4c23-5695-48aa-9965-02677103f2c9
Expand Down Expand Up @@ -437,7 +437,7 @@ You may have noticed that while using the Pluto notebooks, when you change the v
md"""
### Example 1 - Influence of $r$

Influence of the duration of infection $1/r$ for average infection periods of between $10$, days and $1$ day contagious ($r$ between $0.1$ and $1.0$, step $0.1$, default value $0.1$).
Influence of the duration of infection $1/r$ for average infection periods of between $10$ days and $1$ day of being contagious ($r$ between $0.1$ and $1.0$, step $0.1$, default value $0.1$).
"""

# ╔═╡ b6baafc2-6d5e-43c3-8ef9-845961cdd20b
Expand All @@ -450,14 +450,14 @@ We will create a slider for the $r$-values between $0.1$ and $1.0$, stepsize $0.

# ╔═╡ ae38c663-0ee4-409e-bfca-5f13ed88b67d
md"""
We will create een new parameter value vector, ODE problem and solution object by putting `1` at the end of the corresponding variable names. In that way, the previous simulation results will be unaffected! The model, the initial conditions and the timespan are identical as before. In there we also use the variable `r` coupled to the slider.
We will create a new parameter value vector, ODE problem and solution object by putting `1` at the end of the corresponding variable names. In that way, the previous simulation results will be unaffected! The model, the initial conditions and the timespan are identical as before. In there we also use the variable `r` coupled to the slider.
"""

# ╔═╡ d44da6c6-c93d-4c61-8125-9eee464c897e
params1 = [:α => 0.08, :β => 1.0e-6, :r => r, :m => 0.4]

# ╔═╡ 00a72697-d36a-41cc-9eec-8e821829ce0e
# put semi-colon at end of instruction to avoid seeing its output.
# type a semi-colon at end of an instruction to avoid seeing its output
oprob1 = ODEProblem(infection_model, u0, tspan, params1);

# ╔═╡ cf39b4cf-9cd0-4755-80db-ca4aea7c1084
Expand All @@ -476,44 +476,52 @@ md"""
Try to interpret the results yourself.
Ask yourself the following questions:

1. What are the trends in the results obtained?
!!! question
1. What are the trends in the results obtained?
"""

# ╔═╡ ff9ce0c5-931f-44f4-aa0d-4aead0284160
md"- Answer: missing"
# ╔═╡ 4242ec68-b36d-4f7f-9001-2c90c66d5b9b
md"""
- Answer:
"""

# ╔═╡ f5f98168-119c-4b49-a8e3-a3cc775faeb0
md"""2. How can this be explained from the model structure?"""
md"""
!!! question
2. How can this be explained from the model structure?
"""

# ╔═╡ a308de53-80a2-4b37-a752-cb2c383cf7a4
md"- Answer: missing"
# ╔═╡ 30b0e2b3-2b8d-43a5-9587-9fcb0b89b258
md"""
- Answer:
"""

# ╔═╡ ade413c2-d7d9-4250-8490-75534900a389
md"""
### Example 2 - Discrete Event

Suppose that regulations are such that on day 14, people need to reduce their contacts by 50%. Hence, this means that the parameter value $\beta$ needs to be divided by a factor of 2 at timepoint 14. In order to realize that we need to now the order of the parameters in the model because we will need to address the value of $\beta$ by means of an index.
Suppose that regulations are such that on day 14, people need to reduce their contacts by 50%. Hence, this means that the parameter value $\beta$ needs to be divided by a factor of 2 at timepoint 14.
"""

# ╔═╡ 58730ac6-d83b-420d-a000-2f50545f0d39
md"""
We need to state that the parameter $\beta$ needs to be reduce by $50\%$ at time $t=14\,days$. We put this in a condition named `condition2`.
We need to state that the parameter $\beta$ needs to be reduced by $50\%$ at time $t=14$. We include this condition in a variable named `condition2` in the following way:
"""

# ╔═╡ 7411474c-fac7-4b7c-8ded-4c2df5956fb0
condition2 = [14.0] => [infection_model.β ~ infection_model.β/2]

# ╔═╡ e6949052-6d12-4414-ac67-cb9435b46290
md"""
The discrete time event needs to be included in our model.
The discrete time event needs to be now included in our model.
"""

# ╔═╡ 8752373f-a602-437b-9b3a-2602c8babf87
@named infection_model2 = ReactionSystem(equations(infection_model), discrete_events=condition2)

# ╔═╡ ee596a6c-29bf-4d18-b687-be943c128aa7
md"""
After that, we need to *complete* our *reaction network model*.
After that, we need to *complete* our *reaction network model*, so that the model can be simulated.
"""

# ╔═╡ 40f849cb-20f9-4bbe-806e-512abd6f3210
Expand Down Expand Up @@ -555,19 +563,25 @@ osol2.u[end]
md"""
Try to interpret the results yourself. Ask yourself the following questions:

1. What are the trends in the results obtained?
!!! question
1. What are the trends in the results obtained?
"""

# ╔═╡ a736a17b-ee8e-491b-840a-6319619a8dab
md"- Answer: missing"
# ╔═╡ 6eb368f5-8d43-4fb8-b6eb-c783675d1dd9
md"""
- Answer:
"""

# ╔═╡ 2ba54805-e901-4281-9b07-cc7137b8c809
md"""
2. How much less casualties are there compared to not altering the contact rate?
!!! question
2. How much less casualties are there compared to not altering the contact rate?
"""

# ╔═╡ 20d56d2f-2532-4286-88cc-b42ccb22d246
md"- Answer: missing"
# ╔═╡ fec6e165-bd1a-4e57-b7f3-ae712e03276c
md"""
- Answer:
"""

# ╔═╡ cb8a6f77-f08c-4fc8-9445-bd1c17521fcc
md"""
Expand All @@ -578,7 +592,7 @@ Suppose that when the number of infected individuals reaches $1\,000\,000$, then

# ╔═╡ 5005a09d-a844-4f9d-a058-0d93583d5bab
md"""
Normally in a continuous event the value of one or more species can be changed when a certain condition is met. In our specific case we want the change in the species happening only once! So, if you want that the continuous event: ''when $I$ reaches $10^6$ then $999000$ is subtrated from $I$'' happens only once, then we need to include a ficticious new *species* in our *reaction network model*. We will call this ficticious *species* `pwc` (a short for _**p**roceed **w**ith **c**ondition_) and we set it default to `true`.
Normally in a continuous event the value of one or more species can be changed when a certain condition is met. In our specific case we want the change in the species happening only once! So, if you want that the continuous event "when $I$ reaches $10^6$ then $999000$ is subtracted from $I$" happens only once, then we need to include a ficticious new *species* in our *reaction network model*. We will call this ficticious species `pwc` (a short for _**p**roceed **w**ith **c**ondition_) and we set its default value to `true`.
"""

# ╔═╡ f77d2d75-468f-4746-b81e-0bbbf33fc8d7
Expand Down Expand Up @@ -638,7 +652,7 @@ Now we can plot the results.
"""

# ╔═╡ 02173eaa-1178-4383-b57a-03e53ce38af8
plot(osol3)
plot(osol3, idxs=[:S, :I, :D, :R])

# ╔═╡ 6d1aa79b-8614-4a77-a327-f7e6962d1944
md"""
Expand All @@ -652,19 +666,25 @@ osol3.u[end]
md"""
Try to interpret the results yourself. Ask yourself the following questions:

1. What are the trends in the results obtained?
!!! question
1. What are the trends in the results obtained?
"""

# ╔═╡ 7872701b-8506-4303-b615-32d3c1afa50d
md"- Answer: missing"
# ╔═╡ 33ff901b-6c7e-4b8c-a163-663d9443a213
md"""
- Answer:
"""

# ╔═╡ 0d6c2233-35c5-4257-9dfd-398704080ba3
md"""
2. How much less casualties are there compared to not putting $999\,000$ individuals into isolation? (Hint: you also need to take into account the casualties in the $999\,000$ individuals that had been put into isolation.)
!!! question
2. How much less casualties are there compared to not putting $999\,000$ individuals into isolation? (Hint: you also need to take into account the casualties in the $999\,000$ individuals that had been put into isolation.)
"""

# ╔═╡ 029a2024-1ec7-4af9-9e73-717f13483468
md"- Answer: missing"
# ╔═╡ f5d23dba-02a9-4d27-8eb1-ff89d2902cdf
md"""
- Answer:
"""

# ╔═╡ Cell order:
# ╠═be326550-25ea-4c5f-ac4a-dd74d12bc89a
Expand Down Expand Up @@ -755,9 +775,9 @@ md"- Answer: missing"
# ╠═52901bbf-e47b-4da1-95c4-f0869812398c
# ╟─102b4fbc-23b1-46ed-bb72-124eb88517ce
# ╟─f6cbafbf-98b4-4286-86d2-fe95821a5ff4
# ╟─ff9ce0c5-931f-44f4-aa0d-4aead0284160
# ╟─4242ec68-b36d-4f7f-9001-2c90c66d5b9b
# ╟─f5f98168-119c-4b49-a8e3-a3cc775faeb0
# ╟─a308de53-80a2-4b37-a752-cb2c383cf7a4
# ╟─30b0e2b3-2b8d-43a5-9587-9fcb0b89b258
# ╟─ade413c2-d7d9-4250-8490-75534900a389
# ╟─58730ac6-d83b-420d-a000-2f50545f0d39
# ╠═7411474c-fac7-4b7c-8ded-4c2df5956fb0
Expand All @@ -774,9 +794,9 @@ md"- Answer: missing"
# ╟─18151ab1-1e4e-48d8-be70-4fa4c8a43af1
# ╠═7d7ed974-7c9c-4c30-ab2a-e3028ce702dd
# ╟─7375edbc-24a4-4300-bdef-2686cb377cfc
# ╟─a736a17b-ee8e-491b-840a-6319619a8dab
# ╟─6eb368f5-8d43-4fb8-b6eb-c783675d1dd9
# ╟─2ba54805-e901-4281-9b07-cc7137b8c809
# ╟─20d56d2f-2532-4286-88cc-b42ccb22d246
# ╟─fec6e165-bd1a-4e57-b7f3-ae712e03276c
# ╟─cb8a6f77-f08c-4fc8-9445-bd1c17521fcc
# ╟─5005a09d-a844-4f9d-a058-0d93583d5bab
# ╠═f77d2d75-468f-4746-b81e-0bbbf33fc8d7
Expand All @@ -796,6 +816,6 @@ md"- Answer: missing"
# ╟─6d1aa79b-8614-4a77-a327-f7e6962d1944
# ╠═d04b8d97-e9d3-4428-a695-bfde4b44a291
# ╟─70fd136e-5ded-4c30-818e-a5de61fbbf86
# ╟─7872701b-8506-4303-b615-32d3c1afa50d
# ╟─33ff901b-6c7e-4b8c-a163-663d9443a213
# ╟─0d6c2233-35c5-4257-9dfd-398704080ba3
# ╟─029a2024-1ec7-4af9-9e73-717f13483468
# ╟─f5d23dba-02a9-4d27-8eb1-ff89d2902cdf
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