diff --git a/EpiAware/src/EpiAwareUtils/EpiAwareUtils.jl b/EpiAware/src/EpiAwareUtils/EpiAwareUtils.jl
index af0bab2f4..22aceaa91 100644
--- a/EpiAware/src/EpiAwareUtils/EpiAwareUtils.jl
+++ b/EpiAware/src/EpiAwareUtils/EpiAwareUtils.jl
@@ -15,8 +15,8 @@ import Base: eltype
 using Distributions, DocStringExtensions, QuadGK, Statistics, Turing
 
 #Export Structures
-export HalfNormal, DirectSample, SafePoisson, SafeNegativeBinomial, RealValued,
-       RealUnivariateDistribution
+export HalfNormal, DirectSample, SafePoisson, SafeNegativeBinomial, SafeIntValued, SafeInt,
+       SafeDiscreteUnivariateDistribution
 
 #Export functions
 export scan, spread_draws, censored_cdf, censored_pmf, get_param_array, prefix_submodel, ∫F
@@ -34,7 +34,7 @@ include("turing-methods.jl")
 include("DirectSample.jl")
 include("post-inference.jl")
 include("get_param_array.jl")
-include("RealValued.jl")
+include("SafeInt.jl")
 include("SafePoisson.jl")
 include("SafeNegativeBinomial.jl")
 
diff --git a/EpiAware/src/EpiAwareUtils/RealValued.jl b/EpiAware/src/EpiAwareUtils/RealValued.jl
deleted file mode 100644
index 9b5c4c462..000000000
--- a/EpiAware/src/EpiAwareUtils/RealValued.jl
+++ /dev/null
@@ -1,12 +0,0 @@
-"""
-A type to represent real-valued distributions, the purpose of this type is to avoid problems
-with the `eltype` function when having `rand` calls in the model.
-"""
-struct RealValued <: Distributions.ValueSupport end
-Base.eltype(::Type{<:Distributions.Sampleable{F, RealValued}}) where {F} = Real
-
-"""
-A constant alias for `Distribution{Univariate, RealValued}`. This type represents a univariate distribution with real-valued outcomes.
-"""
-const RealUnivariateDistribution = Distributions.Distribution{
-    Distributions.Univariate, RealValued}
diff --git a/EpiAware/src/EpiAwareUtils/SafeInt.jl b/EpiAware/src/EpiAwareUtils/SafeInt.jl
new file mode 100644
index 000000000..a4d26fad4
--- /dev/null
+++ b/EpiAware/src/EpiAwareUtils/SafeInt.jl
@@ -0,0 +1,16 @@
+const SafeInt = Union{Int, BigInt}
+
+"""
+A type to represent real-valued distributions, the purpose of this type is to avoid problems
+with the `eltype` function when having `rand` calls in the model.
+"""
+struct SafeIntValued <: Distributions.ValueSupport end
+function Base.eltype(::Type{<:Distributions.Sampleable{F, SafeIntValued}}) where {F}
+    Union{Int, BigInt}
+end
+
+"""
+A constant alias for `Distribution{Univariate, RealValued}`. This type represents a univariate distribution with real-valued outcomes.
+"""
+const SafeDiscreteUnivariateDistribution = Distributions.Distribution{
+    Distributions.Univariate, SafeIntValued}
diff --git a/EpiAware/src/EpiAwareUtils/SafeNegativeBinomial.jl b/EpiAware/src/EpiAwareUtils/SafeNegativeBinomial.jl
index 010cdb0ac..e57714b80 100644
--- a/EpiAware/src/EpiAwareUtils/SafeNegativeBinomial.jl
+++ b/EpiAware/src/EpiAwareUtils/SafeNegativeBinomial.jl
@@ -65,7 +65,7 @@ var(d)
 2.4617291430060293e40
 ```
 "
-struct SafeNegativeBinomial{T <: Real} <: RealUnivariateDistribution
+struct SafeNegativeBinomial{T <: Real} <: SafeDiscreteUnivariateDistribution
     r::T
     p::T
 
diff --git a/EpiAware/src/EpiAwareUtils/SafePoisson.jl b/EpiAware/src/EpiAwareUtils/SafePoisson.jl
index 0ec1122b9..c85243ebc 100644
--- a/EpiAware/src/EpiAwareUtils/SafePoisson.jl
+++ b/EpiAware/src/EpiAwareUtils/SafePoisson.jl
@@ -45,7 +45,7 @@ var(d)
 7.016735912097631e20
 ```
 "
-struct SafePoisson{T <: Real} <: RealUnivariateDistribution
+struct SafePoisson{T <: Real} <: SafeDiscreteUnivariateDistribution
     λ::T
 
     SafePoisson{T}(λ::Real) where {T <: Real} = new{T}(λ)
@@ -142,12 +142,12 @@ ad_rand(λ) = ad_rand(Random.GLOBAL_RNG, λ)
 function ad_rand(rng::AbstractRNG, λ)
     s = sqrt(λ)
     d = 6.0 * λ^2
-    L = floor(λ - 1.1484)
+    L = _safe_int_floor(λ - 1.1484)
     # Step N
     G = λ + s * randn(rng)
 
     if G >= 0.0
-        K = floor(G)
+        K = _safe_int_floor(G)
         # Step I
         if K >= L
             return K
@@ -177,7 +177,7 @@ function ad_rand(rng::AbstractRNG, λ)
             continue
         end
 
-        K = floor(λ + s * T)
+        K = _safe_int_floor(λ + s * T)
         px, py, fx, fy = procf(λ, K, s)
         c = 0.1069 / λ
 
@@ -229,7 +229,7 @@ function log1pmx(x::Float64)
 end
 
 # Procedure F
-function procf(λ, K, s::Float64)
+function procf(λ, K::SafeInt, s::Float64)
     # can be pre-computed, but does not seem to affect performance
     ω = 0.3989422804014327 / s
     b1 = 0.041666666666666664 / λ
@@ -241,7 +241,7 @@ function procf(λ, K, s::Float64)
 
     if K < 10
         px = -float(λ)
-        py = λ^K / factorial(floor(Int, K))
+        py = λ^K / factorial(K)
     else
         δ = 0.08333333333333333 / K
         δ -= 4.8 * δ^3
diff --git a/EpiAware/test/EpiAwareUtils/RealValued.jl b/EpiAware/test/EpiAwareUtils/RealValued.jl
deleted file mode 100644
index 13205084d..000000000
--- a/EpiAware/test/EpiAwareUtils/RealValued.jl
+++ /dev/null
@@ -1,8 +0,0 @@
-@testitem "RealValued Type Tests" begin
-    using Distributions
-    struct DummySampleable <: Sampleable{Univariate, RealValued} end
-
-    @test RealValued <: Distributions.ValueSupport
-    @test eltype(DummySampleable) == Real
-    @test RealUnivariateDistribution == Distribution{Univariate, RealValued}
-end
diff --git a/EpiAware/test/EpiAwareUtils/SafeInt.jl b/EpiAware/test/EpiAwareUtils/SafeInt.jl
new file mode 100644
index 000000000..d1a59dfab
--- /dev/null
+++ b/EpiAware/test/EpiAwareUtils/SafeInt.jl
@@ -0,0 +1,8 @@
+@testitem "SafeInt Type Tests" begin
+    using Distributions
+    struct DummySampleable <: Sampleable{Univariate, SafeIntValued} end
+
+    @test SafeIntValued <: Distributions.ValueSupport
+    @test eltype(DummySampleable) <: Union{Int, BigInt}
+    @test SafeDiscreteUnivariateDistribution == Distribution{Univariate, SafeIntValued}
+end
diff --git a/EpiAware/test/EpiAwareUtils/SafeNegativeBinomial.jl b/EpiAware/test/EpiAwareUtils/SafeNegativeBinomial.jl
index 7e5bb98cd..4e1ee09db 100644
--- a/EpiAware/test/EpiAwareUtils/SafeNegativeBinomial.jl
+++ b/EpiAware/test/EpiAwareUtils/SafeNegativeBinomial.jl
@@ -70,7 +70,7 @@ end
 
     dist = SafeNegativeBinomial(r, p)
     @testset "Large value of mean samples a BigInt with SafePoisson" begin
-        @test rand(dist) isa Real
+        @test rand(dist) isa Union{Int, BigInt}
     end
     @testset "Large value of mean sample failure with Poisson" begin
         _dist = EpiAware.EpiAwareUtils._negbin(dist)
diff --git a/EpiAware/test/EpiAwareUtils/SafePoisson.jl b/EpiAware/test/EpiAwareUtils/SafePoisson.jl
index e4f889254..874cf3423 100644
--- a/EpiAware/test/EpiAwareUtils/SafePoisson.jl
+++ b/EpiAware/test/EpiAwareUtils/SafePoisson.jl
@@ -2,9 +2,9 @@
     λ = 10.0
     dist = SafePoisson(λ)
     @test typeof(dist) <: SafePoisson
-    @test rand(dist) isa Real
-    @test rand(dist, 10) isa Vector{Real}
-    @test rand(dist, 10, 10) isa Array{Real}
+    @test rand(dist) isa SafeInt
+    @test rand(dist, 10) isa Vector{SafeInt}
+    @test rand(dist, 10, 10) isa Array{SafeInt}
 end
 
 @testitem "Check distribution properties of SafePoisson" begin
@@ -54,7 +54,7 @@ end
     bigλ = exp(48.0) #Large value of λ
     dist = SafePoisson(bigλ)
     @testset "Large value of mean samples a BigInt with SafePoisson" begin
-        @test rand(dist) isa Real
+        @test rand(dist) isa SafeInt
     end
     @testset "Large value of mean sample failure with Poisson" begin
         _dist = Poisson(dist.λ)