Enable functional/time dependent parameters

[TorkelE]

Adds docs and tests. Also modifies the main code to enable these to be used within Catalyst.

Example:

using Catalyst, DataInterpolations, OrdinaryDiffEqDefault, Plots

# Create input function/data
tend = 120.0
ts = collect(0.0:1.0:tend)
light = sin.(ts/6) .+ 1
light = [max(0.0, l - rand()) for l in light]
interpolated_light = LinearInterpolation(light, ts)
plot(interpolated_light)

# Create model
t = default_t()
in_type = typeof(interpolated_light)
@parameters kA kD (light_in::in_type)(..)
@species Pₐ(t) Pᵢ(t)
rxs = [
    Reaction(kA*light_in(t), [Pᵢ], [Pₐ]),
    Reaction(kD, [Pₐ], [Pᵢ])
]
@named rs = ReactionSystem(rxs, t)
rs = complete(rs)

# Simulate the model.
u0 = [Pᵢ => 1.0, Pₐ => 0.0]
ps = [kA => 1.5, kD => 1.0, light_in => interpolated_light]
oprob = ODEProblem(rs, u0, tend, ps)
sol = solve(oprob)
plot(sol)

[TorkelE]

Builds on the change in #1192, however, but otherwise this is now ready.

[isaacsas]

It would be nicer if we could make something like

Suggested change

	k1*$inf_rate, S --> I
	k1*($I_rate)(I), S --> I

work.

[isaacsas]

Having to declare t and/or I(t) symbolically is a bit clunky.

[TorkelE]

Agreed. had wanted to do this but get a

ERROR: LoadError: TypeError: in isdefined, expected Symbol, got a value of type Expr
Stacktrace:
  [1] recursive_escape_functions!(expr::Expr, syms_skip::Vector{Any})
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:980
  [2] (::Catalyst.var"#232#233"{Expr, Vector{Any}})(i::Int64)
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:978
  [3] foreach
    @ ./abstractarray.jl:3187 [inlined]
  [4] recursive_escape_functions!(expr::Expr, syms_skip::Vector{Any})
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:978
  [5] recursive_escape_functions!
    @ ~/.julia/dev/Catalyst/src/dsl.jl:977 [inlined]
  [6] get_rxexpr(rx::Catalyst.DSLReaction)
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:555
  [7] #219
    @ ~/.julia/dev/Catalyst/src/dsl.jl:546 [inlined]
  [8] foreach(f::Catalyst.var"#219#221"{Expr}, itr::Vector{Catalyst.DSLReaction})
    @ Base ./abstractarray.jl:3187
  [9] get_rxexprs(reactions::Vector{Catalyst.DSLReaction}, equations::Vector{Expr}, all_syms::Vector{Union{Expr, Symbol}})
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:546
 [10] make_reaction_system(ex::Expr, name::QuoteNode)
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:323
 [11] make_rs_expr(name::QuoteNode, network_expr::Expr; complete::Bool)
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:153
 [12] make_rs_expr(name::QuoteNode, network_expr::Expr)
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:152
 [13] var"@reaction_network"(__source__::LineNumberNode, __module__::Module, name::Symbol, network_expr::Expr)
    @ Catalyst ~/.julia/dev/Catalyst/src/dsl.jl:92
in expression starting at /home/loman/Desktop/Julia Playground/Environment - Catalyst Docs (full)/catalyst_docs_playground.jl:71

[isaacsas]

Let’s get an issue to follow up and fix this then. (Post V15.)

[isaacsas]

I still think the initial example is likely to just confuse readers as to what a functional parameter is, and would be better off just using a basic function instead of bringing in DataInterpolations (or else add an explanation that you want to show how one can use functional parameters to represent time-dependent parameters that are only measured at a discrete set of points via building an interpolant and setting it to be the functional parameter). But that said, feel free to merge if you want.

[TorkelE]

I added a note at the end of that example to clarify things a bit, but I have now expanded further to

!!! note
    For this simple example, $(2 + t)/(1 + t)$ could have been used directly as a reaction rate (or written as a normal function), technically making the functional parameter approach unnecessary. However, here we used this function as a simple example of how discrete data can be made continuous using DataInterpolations, and then have its values inserted using a (functional) parameter.

Do you think that is helpful?

[isaacsas]

Sure, seems helpful. In any case, finish it as you see fit and then merge. (I should move on to other V15 blocker PRs.)

[TorkelE]

Sounds good, I will merge like this (once things pass). Happy to get back to this though. It might even make sense to have a dedicated example tutorial to the various ways of modelling functions.