diff --git a/src/systems/alias_elimination.jl b/src/systems/alias_elimination.jl
index da3d71db8a..4142f14240 100644
--- a/src/systems/alias_elimination.jl
+++ b/src/systems/alias_elimination.jl
@@ -732,6 +732,7 @@ function alias_eliminate_graph!(graph, var_to_diff, mm_orig::SparseMatrixCLIL)
                     c, dr = reach₌[idx]
                     @assert c == 1
                 end
+                dr in stem_set && break
                 var_to_diff[prev_r] = dr
                 push!(updated_diff_vars, prev_r)
                 prev_r = dr
@@ -901,10 +902,14 @@ function alias_eliminate_graph!(graph, var_to_diff, mm_orig::SparseMatrixCLIL)
     # RHS or its derivaitves must still exist in the system to be valid aliases.
     needs_update = false
     function contains_v_or_dv(var_to_diff, graph, v)
+        counter = 0
         while true
             isempty(𝑑neighbors(graph, v)) || return true
             v = var_to_diff[v]
             v === nothing && return false
+            counter += 1
+            counter > 10_000 &&
+                error("Internal error: there's an infinite loop in the `var_to_diff` graph.")
         end
     end
     for (v, (c, a)) in ag
diff --git a/test/reduction.jl b/test/reduction.jl
index c0f22df73c..e534c7aa30 100644
--- a/test/reduction.jl
+++ b/test/reduction.jl
@@ -295,3 +295,10 @@ eqs = [x ~ 0
 ss = alias_elimination(sys)
 @test isempty(equations(ss))
 @test sort(observed(ss), by = string) == ([D(x), x, y] .~ 0)
+
+eqs = [D(D(x)) ~ -x]
+@named sys = ODESystem(eqs, t, [x], [])
+ss = alias_elimination(sys)
+@test length(equations(ss)) == length(states(ss)) == 1
+ss = structural_simplify(sys)
+@test length(equations(ss)) == length(states(ss)) == 2