RADAR5 Hepatitis B Work-Precision Diagrams

RADAR5 Hepatitis B Virus Infection

This is a stiff delay differential equation model of acute hepatitis B virus infection, taken from the RADAR5 test suite by Guglielmi and Hairer. The system has dimension 10 with 5 constant delays ($\tau_1 = 0.6$, $\tau_2 = 0.6$, $\tau_3 = 2.0$, $\tau_4 = 2.0$, $\tau_5 = 3.0$ days).

The model describes the dynamics of the immune response to hepatitis B virus infection, including free virus ($u_1$), infected hepatocytes ($u_2$), antibody-bound virus ($u_3$), CTL effector cells ($u_4$), and various immune cell populations ($u_5$–$u_{10}$). The delays represent the time lags in the immune response cascade.

The problem is extremely stiff due to rate constants spanning over 50 orders of magnitude (from $O(10^{-22})$ to $O(10^{33})$), making it an excellent test case for stiff DDE solvers.

References

Guglielmi, N. and Hairer, E. (2005). Users' Guide for the code RADAR5 - Version 2.1.

using DelayDiffEq, DiffEqDevTools, Plots
using OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK
gr()

Plots.GRBackend()

Problem Definition

function f_hepatitis!(du, u, h, p, t)
    α = p
    xi(z) = 1.0 - z / α[7]

    # Delayed values for 5 delays
    y4_τ1 = h(p, t - α[41]; idxs = 4)
    y4_τ2 = h(p, t - α[42]; idxs = 4)
    y4_τ3 = h(p, t - α[43]; idxs = 4)
    y4_τ4 = h(p, t - α[44]; idxs = 4)
    y4_τ5 = h(p, t - α[45]; idxs = 4)

    y5_τ1 = h(p, t - α[41]; idxs = 5)
    y5_τ3 = h(p, t - α[43]; idxs = 5)

    y6_τ2 = h(p, t - α[42]; idxs = 6)
    y6_τ4 = h(p, t - α[44]; idxs = 6)
    y6_τ5 = h(p, t - α[45]; idxs = 6)

    y7_τ3 = h(p, t - α[43]; idxs = 7)

    y8_τ4 = h(p, t - α[44]; idxs = 8)
    y8_τ5 = h(p, t - α[45]; idxs = 8)

    du[1] = α[1] * u[2] + α[2] * α[3] * u[2] * u[7] - α[4] * u[1] * u[10] -
        α[5] * u[1] - α[6] * u[1] * (α[7] - u[2] - u[3])
    du[2] = α[8] * u[1] * (α[7] - u[2] - u[3]) - α[3] * u[2] * u[7] - α[9] * u[2]
    du[3] = α[3] * u[2] * u[7] + α[9] * u[2] - α[10] * u[3]
    du[4] = α[11] * α[12] * u[1] - α[13] * u[4]
    du[5] = α[14] * (xi(u[3]) * α[15] * y4_τ1 * y5_τ1 - u[4] * u[5]) -
        α[16] * u[4] * u[5] * u[7] + α[17] * (α[18] - u[5])
    du[6] = α[19] * (xi(u[3]) * α[20] * y4_τ2 * y6_τ2 - u[4] * u[6]) -
        α[21] * u[4] * u[6] * u[8] + α[22] * (α[23] - u[6])
    du[7] = α[24] * (
        xi(u[3]) * α[25] * y4_τ3 * y5_τ3 * y7_τ3 -
            u[4] * u[5] * u[7]
    ) - α[26] * u[2] * u[7] + α[27] * (α[28] - u[7])
    du[8] = α[29] * (
        xi(u[3]) * α[30] * y4_τ4 * y6_τ4 * y8_τ4 -
            u[4] * u[6] * u[8]
    ) + α[31] * (α[32] - u[8])
    du[9] = α[33] * xi(u[3]) * α[34] * y4_τ5 * y6_τ5 * y8_τ5 +
        α[35] * (α[36] - u[9])
    du[10] = α[37] * u[9] - α[38] * u[10] * u[1] - α[39] * u[10]

    return nothing
end

α = zeros(45)
# Kinetic parameters
α[1] = 83.0;      α[2] = 5.0;     α[3] = 6.6e14;  α[4] = 3.0e11;  α[5] = 0.4
α[6] = 2.5e7;     α[7] = 0.5e-12; α[8] = 2.3e9;   α[9] = 0.052;   α[10] = 0.15
α[11] = 9.4e9;    α[12] = 1.0e-15; α[13] = 1.2;    α[14] = 2.7e16; α[15] = 2.0
α[16] = 5.3e27;   α[17] = 1.0;    α[18] = 1.0e-18; α[19] = 2.7e16; α[20] = 2.0
α[21] = 8.0e28;   α[22] = 1.0;    α[23] = 1.0e-19; α[24] = 5.3e33; α[25] = 16.0
α[26] = 1.6e14;   α[27] = 0.4;    α[28] = 1.0e-18; α[29] = 8.0e32; α[30] = 16.0
α[31] = 0.1;      α[32] = 1.0e-18; α[33] = 1.7e30; α[34] = 3.0;    α[35] = 0.4
α[36] = 4.3e-22;  α[37] = 0.85e7; α[38] = 8.6e11;  α[39] = 0.043
# Delays
α[41] = 0.6; α[42] = 0.6; α[43] = 2.0; α[44] = 2.0; α[45] = 3.0

function h_hepatitis(p, t; idxs::Union{Nothing, Int} = nothing)
    α = p
    u0 = [
        2.9e-16, 0.0, 0.0, 0.0, α[18], α[23], α[28], α[32], α[36],
        α[37] * α[36] / α[39],
    ]
    return if idxs === nothing
        u0
    else
        u0[idxs]
    end
end

prob = DDEProblem(
    f_hepatitis!, h_hepatitis, (0.0, 130.0), α;
    constant_lags = [0.6, 2.0, 3.0]
)

DDEProblem with uType Vector{Float64} and tType Float64. In-place: true
Non-trivial mass matrix: false
timespan: (0.0, 130.0)
u0: 10-element Vector{Float64}:
 2.9e-16
 0.0
 0.0
 0.0
 1.0e-18
 1.0e-19
 1.0e-18
 1.0e-18
 4.3e-22
 8.500000000000002e-14

Reference Solution

sol = solve(
    prob, MethodOfSteps(Rodas5P());
    reltol = 1.0e-12, abstol = 1.0e-20, dt = 1.0e-6
)
test_sol = TestSolution(sol)
plot(sol; title = "Hepatitis B Solution")

We also plot the first three components (virus dynamics) separately:

plot(sol; idxs = [1, 2, 3], title = "Hepatitis B: Virus Dynamics")

High Tolerances

Rosenbrock methods

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [
    Dict(:alg => MethodOfSteps(Rosenbrock23())),
    Dict(:alg => MethodOfSteps(Rodas4())),
    Dict(:alg => MethodOfSteps(Rodas5())),
    Dict(:alg => MethodOfSteps(Rodas5P())),
]
names = ["Rosenbrock23", "Rodas4", "Rodas5", "Rodas5P"]
wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1.0e5), error_estimate = :final,
    dt = 1.0e-6
)
plot(wp; title = "Hepatitis B: Rosenbrock Methods (final error)")

wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1.0e5), error_estimate = :L2,
    dt = 1.0e-6
)
plot(wp; title = "Hepatitis B: Rosenbrock Methods (L2 error)")

SDIRK methods

setups = [
    Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(KenCarp4())),
    Dict(:alg => MethodOfSteps(Kvaerno4())),
]
names = ["TRBDF2", "KenCarp4", "Kvaerno4"]
wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1.0e5), error_estimate = :final,
    dt = 1.0e-6
)
plot(wp; title = "Hepatitis B: SDIRK Methods (final error)")

wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1.0e5), error_estimate = :L2,
    dt = 1.0e-6
)
plot(wp; title = "Hepatitis B: SDIRK Methods (L2 error)")

All stiff methods comparison

setups = [
    Dict(:alg => MethodOfSteps(Rosenbrock23())),
    Dict(:alg => MethodOfSteps(Rodas5P())),
    Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(KenCarp4())),
]
names = ["Rosenbrock23", "Rodas5P", "TRBDF2", "KenCarp4"]
wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1.0e5), error_estimate = :final,
    dt = 1.0e-6
)
plot(wp; title = "Hepatitis B: All Stiff Methods (final error)")

Low Tolerances

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [
    Dict(:alg => MethodOfSteps(Rosenbrock23())),
    Dict(:alg => MethodOfSteps(Rodas4())),
    Dict(:alg => MethodOfSteps(Rodas5P())),
    Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(KenCarp4())),
]
names = ["Rosenbrock23", "Rodas4", "Rodas5P", "TRBDF2", "KenCarp4"]
wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1.0e5), error_estimate = :final,
    dt = 1.0e-6
)
plot(wp; title = "Hepatitis B: Low Tolerances (final error)")

wp = WorkPrecisionSet(
    prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1.0e5), error_estimate = :L2,
    dt = 1.0e-6
)
plot(wp; title = "Hepatitis B: Low Tolerances (L2 error)")

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/StiffDDE","Hepatitis_wpd.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/StiffDDE/Project.toml`
⌃ [f42792ee] DDEProblemLibrary v0.1.5
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⌃ [1344f307] OrdinaryDiffEqLowOrderRK v2.0.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.0.0
⌅ [2d112036] OrdinaryDiffEqSDIRK v2.0.0
⌃ [b1df2697] OrdinaryDiffEqTsit5 v2.0.0
⌃ [79d7bb75] OrdinaryDiffEqVerner v2.0.0
⌃ [91a5bcdd] Plots v1.41.6
⌃ [31c91b34] SciMLBenchmarks v0.1.3
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

And the full manifest:

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  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.1.0+0
⌃ [1d63c593] LLVMOpenMP_jll v18.1.8+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.42.0+0
⌃ [89763e89] Libtiff_jll v4.7.2+0
  [38a345b3] Libuuid_jll v2.42.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
⌃ [9bd350c2] OpenSSH_jll v10.3.1+0
⌃ [458c3c95] OpenSSL_jll v3.5.6+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
⌃ [36c8627f] Pango_jll v1.57.1+0
⌅ [30392449] Pixman_jll v0.44.2+0
⌃ [c0090381] Qt6Base_jll v6.10.2+1
⌃ [629bc702] Qt6Declarative_jll v6.10.2+1
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
⌃ [f50d1b31] Rmath_jll v0.5.1+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.3+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
⌃ [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
⌃ [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [8f1865be] ZeroMQ_jll v4.3.6+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
⌃ [a4ae2306] libaom_jll v3.13.3+0
  [0ac62f75] libass_jll v0.17.4+0
  [1183f4f0] libdecor_jll v0.2.2+0
⌃ [8e53e030] libdrm_jll v2.4.125+1
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.58+0
  [a9144af2] libsodium_jll v1.0.21+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
⌃ [1317d2d5] oneTBB_jll v2022.0.0+1
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`