FMD Spatial PDE Simulator — Reaction

1 · The mathematical model click to collapse

State variables — densities, animals km⁻²

Every ODE compartment becomes a function of space and time, u(x,y,t). The host population density N(x,y) is reconstructed from your census data: cattle population of each district (or farm) divided by its area, mapped onto the grid.

∂S/∂t = ∇·(D_S∇S) − ∇·(uS) + δN − λS − (α + μ)S + θR + χV

∂E/∂t = ∇·(D_E∇E) − ∇·(uE) + λS − (ρ + μ)E

∂I₁/∂t = ∇·(D₁∇I₁) − ∇·(uI₁) + ρE − (σ + μ)I₁

∂I₂/∂t = ∇·(D₂∇I₂) + σI₁ − (γ + μ + φ + d)I₂

∂Q/∂t = φI₂ − (γ_q + μ)Q

∂V/∂t = ∇·(D_S∇V) + αS − (χ + γ_v + μ)V

∂R/∂t = ∇·(D_S∇R) + γI₂ + γ_qQ + γ_vV − (μ + θ)R

∂W/∂t = D_W∇²W − ∇·(u_wW) + ξ(I₁ + κ_wI₂) − εW

Force of infection — the only non-local term

λ(x,t) = F_env(x) · (1 − m(t)) · [ β_L·Ĩ/N̄ + β_K·(𝒦∗Ĩ)/N̄ + β_W·W ]

with Ĩ = I₁ + κI₂ (κ scales the relative infectiousness of clinical animals), and 𝒦 the dispersal kernel: 𝒦 = w₁·G(σ₁) + w₂·G(σ₂), a short-range plus a fat-tail component representing local contact and long-distance animal trade respectively.

Why a PDE and not just more ODEs

The ODE assumes every animal can contact every other animal. A PDE only lets infection reach a location through its neighbourhood, which is what actually happens in the field.
It produces a travelling wave. The front speed c* = 2√(D·r) is a testable prediction you can compare against the observed spread of outbreak reports across districts.
Ring vaccination and movement bans are inherently spatial — they cannot be represented in a well-mixed model at all.
Clinical animals are lame and under movement control, so D₂ ≪ D₁. The pre-clinical class does the spreading.

Derived quantities computed by the solver

R₀ (next-generation matrix, 4 infected states E, I₁, I₂, W):

R₀ = ρ/(ρ+μ) · [ a₁/(σ+μ) + a₂σ/((σ+μ)(γ+μ+φ+d)) + a₃ξ/ε · ( 1/(σ+μ) + κ_wσ/((σ+μ)(γ+μ+φ+d)) ) ]

r = dominant eigenvalue of the linearised infection Jacobian (day⁻¹)

c* = 2√(D_eff·r) (Fisher–KPP / Kolmogorov front speed, km day⁻¹)

D_eff = Σ v_iD_i / Σ v_i + ½⟨σ²⟩·f_non-local·r

v = dominant eigenvector; the second term is the moment-closure diffusivity equivalent of the dispersal kernel (variance × rate ÷ 2).

Numerical scheme

Method of lines on a uniform Cartesian grid. Diffusion uses the 5-point Laplacian, advection a first-order upwind stencil, reaction a forward-Euler or RK4 update (operator splitting). Boundaries are zero-flux (Neumann) by default, so no animals leak out of the domain. Stability is enforced by the parabolic condition Δt ≤ h²/(4D_max) and the CFL condition Δt ≤ h/(|u_x|+|u_y|); the solver reports both and clamps Δt automatically.

Symbol	Meaning	Unit	Symbol	Meaning	Unit
D₁, D₂	Diffusivity of pre-clinical / clinical infectious animals	km² day⁻¹	ρ	Latent → pre-clinical progression	day⁻¹
D_S, D_E	Diffusivity of susceptible / latent animals	km² day⁻¹	σ	Pre-clinical → clinical progression	day⁻¹
D_W	Environmental virus dispersal (aerosol, fomites)	km² day⁻¹	γ, γ_q, γ_v	Recovery from I₂ / Q / immunity from V	day⁻¹
u	Advection velocity — prevailing trade or wind direction	km day⁻¹	φ	Isolation / quarantine removal rate	day⁻¹
β_L, β_K, β_W	Local, kernel-mediated and environmental transmission	day⁻¹	d	Disease-induced mortality	day⁻¹
σ₁, σ₂	Short-range and long-range kernel widths	km	α, χ, θ	Vaccination, vaccine waning, immunity loss	day⁻¹
ξ, ε	Virus shedding into environment / environmental decay	day⁻¹	F_env	Temperature–humidity–pH survival multiplier	—

Symbol

Meaning

Unit

Symbol

Meaning

Unit

D₁, D₂

Diffusivity of pre-clinical / clinical infectious animals

km² day⁻¹

Latent → pre-clinical progression

day⁻¹

D_S, D_E

Diffusivity of susceptible / latent animals

km² day⁻¹

Pre-clinical → clinical progression

day⁻¹

D_W

Environmental virus dispersal (aerosol, fomites)

km² day⁻¹

γ, γ_q, γ_v

Recovery from I₂ / Q / immunity from V

day⁻¹

Advection velocity — prevailing trade or wind direction

km day⁻¹

Isolation / quarantine removal rate

day⁻¹

β_L, β_K, β_W

Local, kernel-mediated and environmental transmission

day⁻¹

Disease-induced mortality

day⁻¹

σ₁, σ₂

Short-range and long-range kernel widths

α, χ, θ

Vaccination, vaccine waning, immunity loss

day⁻¹

ξ, ε

Virus shedding into environment / environmental decay

day⁻¹

F_env

Temperature–humidity–pH survival multiplier

—

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 I₂ 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

Epidemiology

β_L local (day⁻¹)

β_K kernel (day⁻¹)

β_W environment

κ clinical infectivity

ρ latent→precl. (day⁻¹)

σ precl.→clinical

γ recovery (day⁻¹)

φ isolation (day⁻¹)

γ_q recovery in Q

d disease death

μ natural death

δ recruitment

α vaccination (day⁻¹)

χ vaccine waning

γ_v V→R

θ immunity loss

Environmental virus

ξ shedding (day⁻¹)

ε decay (day⁻¹)

κ_w clinical shedding

D_W (km² day⁻¹)

Movement & dispersal

D_S susceptible

D_E latent

D₁ pre-clinical

D₂ clinical

σ₁ short kernel (km)

σ₂ long kernel (km)

w₂ long-range weight

Mixing

u_x trade/wind (km d⁻¹)

u_y trade/wind (km d⁻¹)

Wind factor on W

Boundary

Environment F_env

Temp (°C)

Humidity (%)

F_env = — — same Briere-type response used in your ODE dashboard

Numerics

Grid Nx

Grid Ny

Horizon T (days)

Δt (days)

Reaction scheme

Frames saved

Stability check

cell size h = — km · domain —

diffusion limit Δt ≤ — d

CFL limit Δt ≤ — d

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.

Invasion metrics

R₀ (local)

—

Growth rate r

—

Predicted c*

—

Simulated front speed

—

Peak prevalence

—

Peak day

—

Final attack rate

—

Area invaded

—

Epidemic curve — domain totals

Front position vs time

Radius of the invaded region measured from the seed centroid. A straight line confirms a constant-speed travelling wave; its slope is the simulated front speed.

4b · Geographic spread — district risk map

Every district polygon is filled by its simulated risk class; 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

Unit	State	Population	Arrival day	Peak day	Peak infectious	Cumulative cases	Attack rate	Δ vs no control
Run the simulation.

Unit

State

Population

Arrival day

Peak day

Peak infectious

Cumulative cases

Attack rate

Δ vs no control

Run the simulation.

Arrival day is the first day the infectious density in that unit crosses the detection threshold — the quantity to compare against your dated outbreak reports, since it is the one output a well-mixed ODE cannot produce. Δ vs no control is the reduction in cumulative cases attributable to the ring vaccination and movement ban. Attack rate is cumulative infections divided by population, so on long horizons it can exceed 100% — waning immunity (θ) and vaccine waning (χ) return animals to S and they can be infected again. Shorten the horizon or set θ = χ = 0 if you want a single-epidemic attack rate.