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docs: update discrete system docs
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docs/src/tutorials/discrete_system.md

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@@ -15,13 +15,14 @@ end
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@constants h = 1
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@variables S(t) I(t) R(t)
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k = ShiftIndex(t)
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infection = rate_to_proportion(β * c * I(k-1) / (S(k-1) * h + I(k-1) + R(k-1)), δt * h) * S(k-1)
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recovery = rate_to_proportion(γ * h, δt) * I(k-1)
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infection = rate_to_proportion(
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β * c * I(k - 1) / (S(k - 1) * h + I(k - 1) + R(k - 1)), δt * h) * S(k - 1)
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recovery = rate_to_proportion(γ * h, δt) * I(k - 1)
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# Equations
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eqs = [S(k) ~ S(k-1) - infection * h,
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I(k) ~ I(k-1) + infection - recovery,
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R(k) ~ R(k-1) + recovery]
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eqs = [S(k) ~ S(k - 1) - infection * h,
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I(k) ~ I(k - 1) + infection - recovery,
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R(k) ~ R(k - 1) + recovery]
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@mtkbuild sys = DiscreteSystem(eqs, t)
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u0 = [S(k - 1) => 990.0, I(k - 1) => 10.0, R(k - 1) => 0.0]
@@ -31,13 +32,14 @@ prob = DiscreteProblem(sys, u0, tspan, p)
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sol = solve(prob, FunctionMap())
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```
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All shifts must be negative. If default values are provided, they are treated as the value
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for the variable at the previous timestep. For example, consider the following system to
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generate the Fibonacci series:
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All shifts must be non-positive, i.e., discrete-time variables may only be indexed at index
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`k, k-1, k-2, ...`. If default values are provided, they are treated as the value of the
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variable at the previous timestep. For example, consider the following system to generate
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the Fibonacci series:
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```@example discrete
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@variables x(t) = 1.0
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@mtkbuild sys = DiscreteSystem([x ~ x(k-1) + x(k-2)], t)
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@mtkbuild sys = DiscreteSystem([x ~ x(k - 1) + x(k - 2)], t)
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```
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Note that the default value is treated as the initial value of `x(k-1)`. The value for

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