FMD Spatial SPDE Simulator — Stochastic Reaction

1 · The mathematical model click to collapse

transition infection λ_CS contact pressure I_C1 + I_C2 spill-over η = f_env·F (optional) waning-immunity loop ν_kdW stochastic term [G_k(u_k)Ẇ_k](x,t) immobile (D = 0, v⃗ = 0)

Cattle subsystem of the SPDE: two-dose vaccine cascade S_C → V_C1 → V_C2, each dose seroconverting to R_C or waning back to S_C; Q_C is preventive quarantine, q_C clinical isolation (both immobile). Every compartment carries its own Wiener noise (red badges). I_ext is the optional spill-over from buffalo, pigs and sheep/goats (0 for cattle only). Natural mortality μ_C is not drawn. Run the simulation to see ensemble-mean totals in each box.

Governing system (Itô formulation) — densities, animals km⁻²

Each compartment u_k(x,t) is a field over the 2-D landscape Ω. The black terms are the deterministic reaction–diffusion–advection drift; the red terms are the stochastic forcing, one independent Wiener field per compartment. This is the system the solver integrates.

Cattle Subsystem

∂S_C∂t = ∇ · (D_{S_C}∇S_C) − ∇ · (v⃗_CS_C) + Δ_C + α_C3V_C1 + χ₂V_C2 + K₁Q_C + ω_CR_C − (α_C + μ_C + λ_C)S_C + [G_{S_C}(S_C)Ẇ_{S_C}](x, t)

∂E_C∂t = ∇ · (D_{E_C}∇E_C) − ∇ · (v⃗_CE_C) + λ_CS_C − (ρ₁ + μ_C)E_C + [G_{E_C}(E_C)Ẇ_{E_C}](x, t)

∂I_C1∂t = ∇ · (D_{I_C1}∇I_C1) − ∇ · (v⃗_CI_C1) + ρ₁E_C − (μ_C + σ₁)I_C1 + [G_{I_C1}(I_C1)Ẇ_{I_C1}](x, t)

∂I_C2∂t = ∇ · (D_{I_C2}∇I_C2) − ∇ · (v⃗_CI_C2) + σ₁I_C1 + K₂Q_C − (γ₃ + μ_C + φ + 𝒟_C)I_C2 + [G_{I_C2}(I_C2)Ẇ_{I_C2}](x, t)

∂R_C∂t = ∇ · (D_{R_C}∇R_C) − ∇ · (v⃗_CR_C) + α_C1γ₁V_C1 + χ₁γ₂V_C2 + γ₃I_C2 + γ₄q_C − (μ_C + ω_C)R_C + [G_{R_C}(R_C)Ẇ_{R_C}](x, t)

∂V_C1∂t = ∇ · (D_{V_C1}∇V_C1) − ∇ · (v⃗_CV_C1) + α_CS_C − (α_C1γ₁ + α_C2 + α_C3 + μ_C)V_C1 + [G_{V_C1}(V_C1)Ẇ_{V_C1}](x, t)

∂V_C2∂t = ∇ · (D_{V_C2}∇V_C2) − ∇ · (v⃗_CV_C2) + α_C2V_C1 − (χ₁γ₂ + μ_C + χ₂)V_C2 + [G_{V_C2}(V_C2)Ẇ_{V_C2}](x, t)

Quarantine and isolation enforce strict spatial immobility (D_{Q_C} = D_{q_C} = 0, v⃗_{Q_C} = v⃗_{q_C} = 0)

∂Q_C∂t = Δ_QC − (K₁ + K₂ + μ_C)Q_C + [G_{Q_C}(Q_C)Ẇ_{Q_C}](x, t)

∂q_C∂t = φI_C2 − (γ₄ + μ_C)q_C + [G_{q_C}(q_C)Ẇ_{q_C}](x, t)

Regularised force of infection

λ_C(x,t) = β_C · [ I_C1 + I_C2 + f_env(x,t)F(x,t)·(I_B2 + I_P + I_S) ] / max{ N(x,t), ε₀ }

With cattle as the only host, N = N_C (the live sum of the nine compartments) and the other-host term is held as a constant spill-over density I_ext = I_B2 + I_P + I_S (default 0), scaled by η = f_env·F.

Noise operators — the red terms

[G_k(u_k)Ẇ_k](x,t), G_k(u_k) = ν_k · u_k⁺ (g_k(s) = ν_k max{s,0}, g_k(s) = 0 for s ≤ 0)

k ∈ {S_C, E_C, I_C1, I_C2, R_C, V_C1, V_C2, Q_C, q_C}, W_k independent Q-Wiener fields

E[dW_k(x)dW_k(y)] = c(|x−y|) dt, c(r) = exp(−r²/2ℓ²), c(0) = 1

The barrier property (H3) gives almost-sure positivity; ‖G_k(u)‖²_ℒ₂ = ν_k²‖u‖² gives the linear-growth bound (H7), so ν_k is exactly the σ_i of the stochastic threshold theorem.

Integrated (Itô) form — PDE + SDE

du_k = [ ∇·(D_k∇u_k) − ∇·(v⃗_Cu_k) + f_k(u) ] dt + G_k(u_k) dW_k(x,t)

u_k(x,t) = u_k(x,0) + ∫₀^t[ ∇·(D_k∇u_k) − ∇·(v⃗_Cu_k) + f_k(u) ] ds + ∫₀^t ν_ku_k⁺(x,s) dW_k(x,s)

The first integral is the PDE part, the second (red) the SDE part. f_k(u) is the reaction drift of compartment k. The stochastic integral is a martingale, so the ensemble mean stays close to the deterministic skeleton while individual realisations scatter around it.

Why an SPDE and not just the PDE

The PDE gives one smooth answer. In the field, the same incursion can fizzle out in one district and explode in the next; the noise terms reproduce that spread, so outputs come with a 90% band instead of a single curve.
It gives the quantities a PDE cannot: P(invaded) for every district and P(extinction) of the whole epidemic.
Noise lowers the pathwise growth rate (r_S ≈ r − ½ν²), and the stochastic thresholds R₀^S say when extinction is guaranteed almost surely.
The control counterfactual uses the same random seeds, so "cases averted" is not distorted by sampling luck.

Disease-free equilibrium (per unit census density)

Δ_C = π_CN(x), Δ_QC = π_QN(x); Q_C* = Δ_QC/(K₁ + μ_C) (K₂(0) = 0)

θ_C1 = α_C/(α_C1γ₁ + α_C2 + α_C3 + μ_C), θ_C2 = α_C2/(χ₁γ₂ + μ_C + χ₂)

θ_RC = (α_C1γ₁θ_C1 + χ₁γ₂θ_C1θ_C2)/(μ_C + ω_C)

S_C* = (Δ_C + K₁Q_C*) / [α_C + μ_C − (α_C3θ_C1 + χ₂θ_C1θ_C2 + ω_Cθ_RC)]

V_C1* = θ_C1S_C*, V_C2* = θ_C2V_C1*, R_C* = θ_RCS_C*, N_C* = S_C* + V_C1* + V_C2* + R_C* + Q_C*

Basic reproduction number — next-generation matrix

Infected states x = (E_C, I_C1, I_C2, q_C), κ_C = β_CS_C*/N_C*, R₀ = ρ(FV⁻¹)

R₀ = T_CC = (β_CS_C*/N_C*) · ρ₁ / ((ρ₁+μ_C)(μ_C+σ₁)) · [ 1 + σ₁/(γ₃+μ_C+φ+𝒟_C) ]

The multi-host ℛ₀ = max{ℛ_0,CB, ℛ_0,P, ℛ_0,S} reduces to T_CC when cattle are the only host; column q_C of F is zero (isolated animals do not transmit).

Global stochastic threshold: R̄₀ = ρ(F̄V⁻¹) (κ_C → β_C), R₀^S = R̄₀ + max_iσ_i²/(2γ₀)

Local stochastic threshold (R₀ < 1): R₀^S = R₀ + w_maxσ²_max/(2λ_min(Q₀))

r = dominant eigenvalue of F − V; r_S ≈ r − ½⟨σ²⟩; c* = 2√(D_effr)

Numerical scheme — the update applied every step

1. PDE part (drift + transport, Euler or RK4 for f_k):

u_k^* = u_kⁿ + Δt [ D_kΔ_hu_kⁿ − v⃗_C·∇_h^upu_kⁿ + f_k(uⁿ) ]

2. Noise increment: ΔW_kⁿ(x) = √Δt · ξ_kⁿ(x), ξ ~ N(0,1) smoothed to length ℓ

3. SDE part (default, positivity-preserving):

u_kⁿ⁺¹ = (u_k^*)⁺ · exp( ν_kΔW_kⁿ − ½ν_k²Δt )

or Euler–Maruyama: u_kⁿ⁺¹ = max{ 0, u_k^* + ν_k(u_kⁿ)⁺ΔW_kⁿ }

Δ_h is the 5-point Laplacian, ∇_h^up the upwind gradient. Zero-flux (Neumann) boundaries, J·n = 0; Δt ≤ h²/(4D_max) and the CFL limit are enforced automatically. Each realisation has its own seed.

Symbol	Meaning	Unit	Symbol	Meaning	Unit
β_C, ε₀	Cattle transmission rate, regularisation threshold	day⁻¹, km⁻²	ρ₁, σ₁	Latent → pre-clinical, pre-clinical → clinical	day⁻¹
γ₃, γ₄	Recovery from I_C2 / from isolation q_C	day⁻¹	φ, 𝒟_C	Isolation rate, disease-induced mortality	day⁻¹
α_C, α_C1γ₁	Vaccination S_C → V_C1; V_C1 → R_C	day⁻¹	α_C2, α_C3, χ₁γ₂, χ₂	Booster, failure, V_C2 → R_C, V_C2 → S_C	day⁻¹
K₁, K₂	Quarantine release to S_C / progression to I_C2	day⁻¹	ω_C, μ_C	Loss of immunity, natural mortality	day⁻¹
Δ_C, Δ_QC	Recruitment into S_C and into Q_C (π·N(x))	km⁻² day⁻¹	D_k, v⃗_C	Diffusivity of class k, advection velocity	km² day⁻¹, km day⁻¹
G_k, ν_k	Noise operator and intensity of compartment k (red terms)	day^−½	Ẇ_k, ℓ	Space–time Wiener noise, its correlation length	km

Symbol

Meaning

Unit

Symbol

Meaning

Unit

β_C, ε₀

Cattle transmission rate, regularisation threshold

day⁻¹, km⁻²

ρ₁, σ₁

Latent → pre-clinical, pre-clinical → clinical

day⁻¹

γ₃, γ₄

Recovery from I_C2 / from isolation q_C

day⁻¹

φ, 𝒟_C

Isolation rate, disease-induced mortality

day⁻¹

α_C, α_C1γ₁

Vaccination S_C → V_C1; V_C1 → R_C

day⁻¹

α_C2, α_C3, χ₁γ₂, χ₂

Booster, failure, V_C2 → R_C, V_C2 → S_C

day⁻¹

K₁, K₂

Quarantine release to S_C / progression to I_C2

day⁻¹

ω_C, μ_C

Loss of immunity, natural mortality

day⁻¹

Δ_C, Δ_QC

Recruitment into S_C and into Q_C (π·N(x))

km⁻² day⁻¹

D_k, v⃗_C

Diffusivity of class k, advection velocity

km² day⁻¹, km day⁻¹

G_k, ν_k

Noise operator and intensity of compartment k (red terms)

day^−½

Ẇ_k, ℓ

Space–time Wiener noise, its correlation length

Set inputs here, then open Results & graphs.

2 · Spatial domain & data source

The scale you pick determines the grid extent, the cell size and how the host density field N(x,y) is built from your FMD data.

Data provenance. District centroids and geographic areas in the built-in Karnataka table are real. The cattle-population and reported-attack series bundled with this file are a demonstration series anchored on the Ramanagara figures used by the ODE dashboard and interpolated across districts and years — they are placeholders for wiring and teaching, and must not be quoted as surveillance results. For any analysis that leaves this screen, load your own NADRES / Livestock Census extract through Upload custom domain (CSV) (columns: name, lat, lon, area_km2, population, infected); the uploaded table replaces the built-in one completely.

Farm level

2 km × 2 km domain, 20 m cells. Herd spread over sheds/paddocks; index case in one pen. Diffusion = animal movement within the holding. Only meaningful when the herd is spatially structured (separate sheds, paddocks, calf pens); a freely mixing herd is better served by the ODE model.

District level

Square domain matching the selected district's area, cattle density from the census column, seeded with that year's observed attacks at a chosen entry point.

State level — Karnataka mosaic

Real lat/lon grid over all 30 districts. Density and seeding come from the population and attack columns of the loaded table — the bundled one is a demonstration series; upload your own for real work. Required for the district risk map.

Extend the domain across state borders cross-border spread

Adds border districts of the six neighbouring states — Maharashtra, Goa, Telangana, Andhra Pradesh, Tamil Nadu and Kerala — to the Karnataka mosaic. These start with zero reported cases, so anything that lights up there arrived by spreading across the state boundary. State borders are drawn on the field, and every unit is tagged with its state in the outcome table. District centroids and areas are approximate real values; the populations are demonstration figures — for real work upload your own CSV (an optional state column colours the boundaries).

Outbreak year (seeding)

Reported attacks of this year become the initial I_C2 field.

Initial infected (index cases)

0 = use the reported attacks from the data.

Seed placement

Upload custom domain (CSV)

Columns: name, lat, lon, area_km2, population, infected

Karnataka mosaic · 30 districts

3 · Parameters

Transmission

β_C transmission (day⁻¹)

ε₀ regularisation

F fomite scaling (0–1)

I_ext spill-over (km⁻²)

Progression & isolation

ρ₁ latent→precl. (day⁻¹)

σ₁ precl.→clinical

γ₃ recovery from I₂

φ isolation (day⁻¹)

𝒟_C disease death

γ₄ recovery in q_C

μ_C natural death

Vaccination & immunity

α_C vaccination S→V₁

α_C1 efficacy (V₁)

γ₁ seroconversion V₁

α_C2 booster V₁→V₂

α_C3 failure V₁→S

χ₁ efficacy (V₂)

γ₂ seroconversion V₂

χ₂ waning V₂→S

ω_C immunity loss R→S

Quarantine & recruitment

π_C recruitment Λ_C/N

π_Q quarantine Λ_QC/N

K₁ quarantine release

K₂ quarantine→I₂

K₂ form

K₂ half-prevalence

Movement — diffusion & advection

D_S susceptible

D_E latent

D_I1 pre-clinical

D_I2 clinical

D_R recovered

D_V1 vaccinated (1)

D_V2 vaccinated (2)

v_x drift (km d⁻¹)

v_y drift (km d⁻¹)

Boundary

Environment f_env

Temp (°C)

Humidity (%)

f_env = — · η = f_env·F = — — Briere-type response, clamped to [0, 1]

Stochastic forcing — red noise terms (dW)

ν_S

ν_E

ν_I1

ν_I2

ν_R

ν_V1

ν_V2

ν_Q

ν_q

Noise multiplier

Noise structure

ℓ correlation (km)

Noise step

Realisations M

Random seed

Deterministic skeleton

Thresholds (live)

DFE: S*/N* = — · V*/N* = — · R*/N* = — · Q*/N* = —

N* ÷ census density = —

R₀ = T_CC = — · R̄₀ = —

R₀^S (global) = — · R₀^S (local) = —

Numerics

Grid Nx

Grid Ny

Horizon T (days)

Δt (days)

Drift scheme

Frames saved

Initial condition

Stability check

cell size h = — km · domain —

diffusion limit Δt ≤ — d

CFL limit Δt ≤ — d

noise: max ν√Δt = —

Spatial control measures

Trigger — detection threshold (clinical prevalence, %)

Response delay (days)

Ring radius (km)

Ring vaccination α_ring

Ring culling rate

Movement ban m (0–1)

Ban applies to

Baseline run for comparison

Idle.

4 · Spatial field

Day 0

low

high

max = —

Run the simulation to render the field.

Thresholds & invasion metrics

R₀ = T_CC

—

R₀^S global · local

—

Growth r · r_S

—

Predicted c*

—

Simulated front speed

—

P(extinction by T)

—

Peak infectious

—

Peak day

—

Final attack rate

—

Area invaded (P ≥ ½)

—

Infectious I₁+I₂ — ensemble band, sample paths, deterministic skeleton

Compartments — ensemble mean, domain totals

Front position vs time

Radius of the invaded region from the seed centroid: sample path #1 with its linear fit, and the ensemble-mean radius. The fitted slope is the simulated front speed.

Distribution of final attack rate across realisations

4b · Geographic spread — district risk map

Every district polygon is filled by its simulated risk class (ensemble mean); Karnataka is highlighted.

Risk basis

High ≥

Moderate ≥

Fill

Labels

Highlight Karnataka

Waiting for a run.

Each district is filled by a risk class: High, Moderate or Low. The score is 0–1. Count-type bases (attack rate, cases, peak) are divided by the highest district; timing bases are scored by earliness, (T − day) / T. The composite index is 0.5·(attack ÷ max attack) + 0.3·arrival earliness + 0.2·reached (1 if the front arrived, else 0). Districts with no infected animals are Low. High ≥ and Moderate ≥ set the class cut-offs. With Highlight Karnataka on, the state carries a bold outer boundary and full colour while neighbour-state districts are dimmed, so colour outside the line is cross-border spread. Boundaries are Census 2011 districts: Narayanpet and Vikarabad (created later) have no polygon.

5 · Unit-wise outcome (ensemble)

Unit	State	Population	P(reached)	Arrival day (if reached)	Peak day	Peak infectious (mean)	Cumulative cases (mean)	90% interval	Attack rate	Δ vs no control
Run the simulation.

Unit

State

Population

P(reached)

Arrival day (if reached)

Peak day

Peak infectious (mean)

Cumulative cases (mean)

90% interval

Attack rate

Δ vs no control

Run the simulation.

Every number is computed over the M realisations of the SPDE. P(reached) is the fraction of realisations in which the unit's infectious prevalence crossed 10⁻⁴; Arrival day is the mean arrival over those realisations only. 90% interval is the 5th–95th percentile of cumulative cases. Δ vs no control compares ensemble means against the uncontrolled counterfactual run with the same noise seeds, so the difference is due to control and not to sampling luck. Cumulative cases count new infections λ_CS_C plus quarantine breakdowns K₂Q_C; on long horizons waning (ω_C, α_C3, χ₂) allows reinfection, so attack rates can exceed 100%.