Time-varying parameter specification in DiscreteProblem?

Specific Domains Modelling & Simulations
3

Here are some ideas:

  1. I would simply pass the vector v as a parameter, e.g.,
    v = exp.(.- (0:0.1:40) ./ 10)  # Note: Covers same times as your simulation
p = (0.5,0.25,1//10,v)  # β₀,γ,δt,v
  2. In the function, access v at the right place, i.e.,
    • either converting time to index
    • or running the simulation in discrete steps in the first place

Here is code illustrating both options:

@inline function rate_to_proportion(r::Real,t::Real)  # Float not general enough for Rational time
    1-exp(-r*t)
end

function sir_map_v1!(unp1,un,p,t)
     (S,I,C) = un
     (β₀,γ,δt,v) = p
     β = β₀ * v[Int(t // δt) + 1]  # Note: Rational to prevent rounding issues
     ... # cut some lines here
end

function sir_map_v2!(unp1,un,p,t)
     (S,I,C) = un
     (β₀,γ,δt,v) = p
     β = β₀ * v[t]  # Note: t is actually steps here
     ... # cut some lines here
end

# Run option 1
tspan = (0//1, 40//1)
p = (0.5,0.25,1//10,v)  # Rational δt
prob = DiscreteProblem(sir_map_v1!,u0,tspan,p)
sol = solve(prob,FunctionMap(),dt=p[3])

# Run option 2
tspan = (1, 401)  # Discrete steps matching indices of `v`
p = (0.5,0.25,0.1,v)
prob = DiscreteProblem(sir_map_v2!,u0,tspan,p)
sol = solve(prob,FunctionMap(),dt=1)  # discrete stepping, i.e, dt isa Int