docs: add discrete system tutorial page

Author: AayushSabharwal

docs/pages.jl

@@ -8,6 +8,7 @@ pages = [
         "tutorials/modelingtoolkitize.md",
         "tutorials/programmatically_generating.md",
         "tutorials/stochastic_diffeq.md",
+        "tutorials/discrete_system.md",
         "tutorials/parameter_identifiability.md",
         "tutorials/bifurcation_diagram_computation.md",
         "tutorials/domain_connections.md"],

docs/src/tutorials/discrete_system.md

@@ -0,0 +1,36 @@
+# Modeling Discrete Systems
+
+In this example, we will use the new [`DiscreteSystem`](@ref) API
+to create an SIR model.
+
+```@example discrete
+using ModelingToolkit
+using ModelingToolkit: t_nounits as t
+using OrdinaryDiffEq: FunctionMap
+
+@inline function rate_to_proportion(r, t)
+    1 - exp(-r * t)
+end
+@parameters c δt β γ
+@constants h = 1
+@variables S(t) I(t) R(t)
+k = ShiftIndex(t)
+infection = rate_to_proportion(β * c * I / (S * h + I + R), δt * h) * S
+recovery = rate_to_proportion(γ * h, δt) * I
+
+# Equations
+eqs = [S(k + 1) ~ S - infection * h,
+    I(k + 1) ~ I + infection - recovery,
+    R(k + 1) ~ R + recovery]
+@mtkbuild sys = DiscreteSystem(eqs, t)
+
+u0 = [S => 990.0, I => 10.0, R => 0.0]
+p = [β => 0.05, c => 10.0, γ => 0.25, δt => 0.1]
+tspan = (0.0, 100.0) # value function (from Symbolics) is used to convert a Num to Float64
+prob = DiscreteProblem(sys, u0, tspan, p)
+sol = solve(prob, FunctionMap())
+```
+
+Currently, all updates happen at integer intervals (`t = 1, 2, 3...`). All shifts
+in the equations must be positive. For example, instead of specifying the update equation
+for the Fibonacci series as `x ~ x(k-1) + x(k-2)` use `x(k+2) ~ x(k+1) + x`.