R₀ Results
Intervention day: 50
R₀
ⓘ
✕
-
Reproduction number (R₀): average secondary cases from a single case in a fully susceptible population.
Herd Immunity Threshold
ⓘ
✕
—
The herd immunity threshold refers to the critical fraction of a population that must acquire immunity in order to interrupt disease transmission and protect susceptible individuals indirectly
Herd immunity
Minimum ≥ — of the population needs vaccination to reach adequate herd immunity.
Day
0
Cattle saved by vaccination
ⓘ
✕
By Day 0: —
—
—
Cattle saved by vaccination
ⓘ
✕
By Day 0: —
—
—
Run the simulation to estimate the vaccination impact.
| Cattle infections — no vaccination | — |
| Cattle infections — with vaccination | — |
| Infections averted (saved) | — |
| Deaths averted | — |
| ↳ from existing vaccinated stock (V_C1+V_C2) | — |
| ↳ from ongoing campaign (α_C) | — |
| Cattle vaccinated (cumulative) | — |
| Vaccinated per infection averted | — |
Model-based estimate: same ODE system run twice over the simulation horizon — current settings vs. a counterfactual with αC=0 and existing vaccinated cattle treated as susceptible. Infections = cumulative EC→IC1 flow (ρ₁·EC); deaths = cumulative DC·IC2.
Preparedness
Day 0
🎲 Stochastic (SDE) Simulation — Itô Diffusion Model
dXi = Fi(X)dt + σiXidWi(t) · Euler–Maruyama · stochastic threshold Rs
Deterministic R₀
—
ρ(FV⁻¹) — deterministic skeleton
mink ak
—
min diagonal transition rate of V
Σ (noise infimum)
—
inf of the Rayleigh quotient
Stochastic threshold Rs
—
R₀ − Σ / (2·min ak)
Extinction in paths
—
% paths with IC2 < 1 at horizon
—
0.080
Noise intensities σi for the 12-dimensional infected subsystem (these enter Rs directly)
Noise intensities for non-infected compartments (S, V, R)
Stochastic Mean E[X(t)]
Deterministic ODE
95% Confidence Band (2.5%–97.5%)
Multi-Color Sample Trajectories
The deterministic overlay is integrated on the same Euler grid as the sample paths, so the
displacement between the black dashed curve and the red mean is attributable purely to the
diffusion term σiXidWi, not to discretisation.
| Metric | Deterministic (ODE) | Stochastic mean | Stochastic 2.5% | Stochastic 97.5% |
|---|
Model & threshold definition
Each of the 25 compartments carries independent multiplicative Itô noise:
dXi = Fi(X)dt + σiXidWi(t),
with the drift Fi identical to the deterministic ODE core used above.
Integrated by Euler–Maruyama on a Δt = 1/n day grid with a projection onto the positive
orthant, consistent with Theorem 1.1 (global existence and positivity).
The stochastic threshold is
Rs = R₀ − (1/(2·minkak))·Σ, where
Σ = infx∈R₊¹² [ Σci²σi²xi² / (Σcixi)² ].
Substituting ui = cixi (admissible because ci > 0)
removes the eigenvector weights and the constrained minimisation of Σσi²ui²
on the simplex gives the closed form
Σ = 1 / Σi=112 σi−2
(attained at ui ∝ σi−2; Σ = 0 if any σi = 0).
The 12-dimensional infected subsystem is
{E_C, I_C1, I_C2, Q_C, q_C, E_B, I_B1, I_B2, Q_B, I_P, I_S, F}, with diagonal transition rates
ak taken from V at the disease-free equilibrium.
Rs < 1 ⇒ ln I(t)/t ≤ (Rs−1)·minkak < 0, so the
infection dies out exponentially almost surely. Rs > 1 ⇒ the system is ergodic
and admits a unique stationary distribution (Has'minskii), i.e. the disease persists.
Auto-filled from the simulation for the selected compartment & day. Editable.
Initial (Day 0)
—
Peak
—
Final
—
Results
Susceptible Cattle (S_C)
1,092
Exposed Cattle (E_C)
179
Asymptomatic Infected Cattle (I_C1)
1,233
Symptomatic Infected Cattle (I_C2)
2,120
Recovered Cattle (R_C)
15,899
Infected Buffalo (I_B)
33
Infected Pig (I_P)
26
Infected Sheep (I_S)
4,584
R₀ summary + derived indicators + Equilibrium
auto
R₀ (results)
—
from model equations (β, γ, μ, D, φ, F_env…)
Herd immunity threshold
—
HIT = 1 − 1/R₀ (if R₀>1)
Peak infected (I_total)
—
Day —
Time to peak
—
days from Day 0
Duration of infection
—
days with I_total > 1
Time endemic starts
—
first stable plateau window
Time epidemic starts
—
first day I_total > 1
Time epidemic ends
—
last day I_total > 1
I_total curve (scaled)
Markers: Peak (red), Endemic start (green), Epidemic start/end (blue/gray)
What-if: vary initial Symptomatic Infected Cattle (I_C20) and compare curves
Threshold-crossing response matrix — —
Rows are generated at each threshold crossing of the I_total curve (rise → peak → decline → endemic).
Vaccination / immunity / preparedness / alert are derived from R₀, HIT = 1 − 1/R₀, and the current immune fraction.
| Threshold / milestone | Day | I_total | % of peak | Phase | Alert level | Vaccination action | Immunity status | Preparedness | Advisory message |
|---|
Endemic equilibrium point (from equations)
S* (Susceptible cattle)
—
R* (Recovered cattle)
—
I_C2*
—
q_C*
—
📊 District-Level Epidemic Dynamics: I_C2 Cases v/s Susceptible Cattle
Visualizing I_C2 (Symptomatic Infected Cattle) vs Susceptible Cattle with prominent highlights for Peak Day and the High R₀/Rₜ Transmission Window
📍
Peak Day & Shoot-Up Alert
🔥
High R₀ / Rₜ Transmission Phase (Rₜ ≥ 1.0)
Peak I_C2 Cases
—
on Day —
Baseline R₀ (t=0)
—
Transmission pressure
High Rₜ Window
— days
Rₜ ≥ 1.0 duration
Vaccination @ Peak
—
Cattle vaccinated
Susceptible @ Peak
—
Remaining cattle
HIT Requirement
—%
— doses
📈 I_C2 Cases v/s Susceptible Cattle
Red Line: Symptomatic Infected Cattle (I_C2, Left Axis) | Blue Dashed Line: Susceptible Cattle (S_C, Right Axis)
★ Peak Day Highlighted
🔥 High R₀/Rₜ Zone Shaded
▲ Shoot-Up Point Marked
📋 Multi-I_C2 Sensitivity Comparison Table
Shows how Peak Day, Peak Cases & High Rₜ window change as I_C2 seed varies| # | I_C2 Initial | Peak Day | Peak I_C2 Cases | High Rₜ Window (Rₜ≥1) | Max Rₜ (t=0) | Susceptible @ Peak | Vaccinated @ Peak | HIT (%) | Action |
|---|---|---|---|---|---|---|---|---|---|
| Click 📊 Sweep Comparison above to generate multi-I_C2 sensitivity comparison. | |||||||||
📊 Year-wise Simulation Validation — District
How Hindcast Validation Works (RK4 Method):
For each historical year: (1) Year-specific cattle population is used as N. (2) Observed attacks seed I_C2 as the starting condition. (3) A year-specific β_C is back-calculated from observed incidence. (4) The full ODE system is solved using Runge-Kutta 4th order (RK4) for 365 days. (5) Predicted = Average Monthly I_C2 from the RK4 run. (6) Rule: Reported=Yes & model fires (pred > 0) → YES (TP) | Not Reported & model quiet → YES (TN) | Not Reported & model fires → NO (FP) | Reported & model silent → NO (FN).
For each historical year: (1) Year-specific cattle population is used as N. (2) Observed attacks seed I_C2 as the starting condition. (3) A year-specific β_C is back-calculated from observed incidence. (4) The full ODE system is solved using Runge-Kutta 4th order (RK4) for 365 days. (5) Predicted = Average Monthly I_C2 from the RK4 run. (6) Rule: Reported=Yes & model fires (pred > 0) → YES (TP) | Not Reported & model quiet → YES (TN) | Not Reported & model fires → NO (FP) | Reported & model silent → NO (FN).
📂 District Historical Data
Select a district to validate. Cattle population from 2019 Census; FMD attack data 2019–2024.
Required columns: Year,
Observed_Attacks,
Cattle_Population.
Optional: District
Run validation to see hindcast validation against historical outbreaks.
📊 Scenario Comparison — Control Measures Summary
| SCENARIO | VACC RATE (%) | ISOLATION (%) | PEAK R₀ | FINAL R₀ | PEAK INFECTED | PEAK DAY | FINAL INFECTED | ATTACK RATE | EFFICACY VS NO CONTROL | EST. ECONOMIC LOSS |
|---|---|---|---|---|---|---|---|---|---|---|
| Run simulation with different control measures to see scenario comparison | ||||||||||
How to use: Run simulation with different vaccination rates (α_C) and isolation rates (φ) to compare scenarios.
Each scenario is automatically categorized and tracked. The table shows effectiveness metrics and calculates efficacy compared to the "No Control" baseline.
💰 Economic Loss & Cost–Benefit Analysis
Losses are derived from simulated incidence, infectious animal-days and mortality —
Ramanagara District,
horizon 50 days.
Loss — No Control
—
Loss — With Control
—
Losses Averted
—
Incremental Control Cost
—
Net Benefit
—
Benefit–Cost Ratio
—
Cases (No Control)
—
Deaths (No Control)
—
Loss per Case
—
Loss per Animal at Risk
—
Cost & Production Parameters (editable — INR)
Herd composition & price basis
Milk loss — survey, ₹ per affected in-milk animal
Milk loss — yield-based method (used only if selected above)
Draught power loss
Treatment cost — ₹ per affected animal
Opportunity cost of extra labour — ₹ per affected animal
Mortality loss — ₹ per animal died
Distress sale loss — ₹ per animal sold in distress
Other losses (not estimated in the Govindaraj surveys — set to 0 to replicate their framework)
Control & programme costs
Direct-loss defaults: ICAR-NIVEDI field survey, Karnataka 2013-14 (Govindaraj et al. 2020, IJAnS 90:971; 2021, Prev. Vet. Med. 190:105318). Fields marked as assumptions (crossbred/upgraded share, distress-sale rate, buffalo CFR, pig/sheep-goat, weight & reproductive, control costs) are not from these papers — replace with district data.
| LOSS COMPONENT | BASIS | NO CONTROL (₹) | WITH CONTROL (₹) | DIFFERENCE (₹) | % OF TOTAL |
|---|---|---|---|---|---|
| Run the simulation to compute economic losses | |||||
Cumulative Economic Loss (₹)
Loss Composition — No Control (₹)
Method. Cattle and buffalo incidence come from the latent-to-infectious flux
(ρ₁·E_C, ρ₂·E_B); pig and small-ruminant incidence from the infectious out-flux
((γ+μ+D)·I). Morbidity losses scale with infectious animal-days (Σ I·dt); mortality uses the
model's disease-death terms (D_C·I_C₂, D_P·I_P, D_S·I_S) with a user case-fatality for
buffalo, which has no D_B term in the model. Milk loss has an acute component over the
infectious period plus a residual yield depression applied per case. Control costs are driven
by the vaccination flux (α·S), the quarantine compartment (q_C) and the intensity of
m_restrict / λ_surv / b_biosecurity / θ_outbreak / cull_rate.
BCR = averted direct loss ÷ incremental control cost; a BCR > 1 means the control
package pays for itself within the simulated horizon.
Loss framework (Govindaraj et al.). In survey mode, direct losses follow the ICAR-NIVEDI farm-survey framework: milk-yield loss (per affected in-milk animal), draught-power loss (per affected bullock), treatment cost and extra labour (per affected animal), mortality loss (per death, value of the animal in healthy state) and distress-sale loss (per animal sold below pre-disease value). Cattle values are blended by crossbred share, buffalo values by upgraded share, then multiplied by the price index.
Loss framework (Govindaraj et al.). In survey mode, direct losses follow the ICAR-NIVEDI farm-survey framework: milk-yield loss (per affected in-milk animal), draught-power loss (per affected bullock), treatment cost and extra labour (per affected animal), mortality loss (per death, value of the animal in healthy state) and distress-sale loss (per animal sold below pre-disease value). Cattle values are blended by crossbred share, buffalo values by upgraded share, then multiplied by the price index.
| Published benchmark (Govindaraj et al.) | Value |
|---|---|
| FMD loss, cattle + buffalo, Karnataka, 2013-14 (IJAnS 2020) | ₹6,500.93 Cr |
| FMD loss, cattle + buffalo, all India, 2013-14 (IJAnS 2020) | ₹20,896.52 Cr |
| All-India farm-level loss, severe / moderate / mild scenario, 2015-16 prices (PVM 2021) | ₹22,111 / 1,891 / 1,061 Cr |
| Kolar crossbred cattle, per animal: mortality / distress sale / milk / treatment / labour (TBED 2017) | USD 515 / 490 / 327 / 38 / 30 |
| Kolar mean total loss per affected animal, local / upgraded buffalo (TBED 2017) | USD 440 / 513 |
Sources.
Govindaraj G et al. (2020) Economic impact of FMD in cattle and buffaloes in India. Indian J. Anim. Sci. 90(7):971–976. doi:10.56093/ijans.v90i7.106662 ·
Govindaraj G et al. (2021) FMD incidence in cattle and buffaloes and its associated farm-level economic costs in endemic India. Prev. Vet. Med. 190:105318 ·
Govindaraj G et al. (2017) Farm community impacts of FMD outbreaks in cattle and buffaloes in Karnataka. Transbound. Emerg. Dis. 64(3):849–860.
🚨
High Infection Alert — Immediate Action Required!
Peak Infected
—
Peak Day
—
Outbreak Days
—
Risk Level
—
💡 Recommended Actions:
🔬 With vs Without Control Measures — Comparison
Side-by-side Itotal curves showing impact of movement restriction, biosecurity, surveillance, culling & quarantine
No Control Peak
—
With Control Peak
—
Peak Reduction
—
% fewer infected
No Ctrl Final I
—
end-of-sim
With Ctrl Final I
—
end-of-sim
Duration Saved
—
fewer outbreak days
| Metric | Without Control measures | With Control measures | Change |
|---|---|---|---|
| Run simulation to see control measures comparison | |||
📐 View Control Measure Equations
Force of Infection (Cattle):
λ_C = β_C · (1 - m_restrict) · (I_C1 + I_C2 + f_env·F·(1-b_biosec)·I_B2 + f_env·F·(1-b_biosec)·I_P + f_env·F·(1-b_biosec)·I_S) / N
dI_C2/dt = σ₁·I_C1 + K₂·Q_C − [γ₃ + μ_C + φ_base·(1+λ_surv)·(1+θ_outbreak) + D_C + cull_rate]·I_C2
dQ_C/dt = Δ_QC·(1−s_efficacy) − (K₁ + K₂ + μ_C + φ_quarantine)·Q_C
dS_C/dt = Δ_C + α_C3·ε_vac·V_C1 + χ₂·V_C2 + K₁·Q_C·(1−s_efficacy) + ∅·R_C − [α_C + μ_C]·S_C − λ_C·S_C
λ_C = β_C · (1 - m_restrict) · (I_C1 + I_C2 + f_env·F·(1-b_biosec)·I_B2 + f_env·F·(1-b_biosec)·I_P + f_env·F·(1-b_biosec)·I_S) / N
dI_C2/dt = σ₁·I_C1 + K₂·Q_C − [γ₃ + μ_C + φ_base·(1+λ_surv)·(1+θ_outbreak) + D_C + cull_rate]·I_C2
dQ_C/dt = Δ_QC·(1−s_efficacy) − (K₁ + K₂ + μ_C + φ_quarantine)·Q_C
dS_C/dt = Δ_C + α_C3·ε_vac·V_C1 + χ₂·V_C2 + K₁·Q_C·(1−s_efficacy) + ∅·R_C − [α_C + μ_C]·S_C − λ_C·S_C
🎛️ Parameter Zero-Impact Analysis
Shows what happens when each selected control parameter is individually set to zero — all other params stay at current values
Baseline (all controls ON):
Peak = —
Final I = —
No Control (all zero):
Peak = —
Final I = —
| Parameter Set to Zero | Peak Infected Cases | Peak Day infected cases | Final Infected at end day of simulation | Outbreak Days | Impact vs With-Control | Risk Verdict |
|---|---|---|---|---|---|---|
| Select parameters above and click "Run Zero-Impact Analysis" | ||||||
How to read: Each row shows what happens if that one control parameter is zeroed while all other controls stay at their current set values.
"Impact vs With-Control" = extra infections caused by removing that parameter. Higher impact = that control is more critical.
📊 3-Scenario Control Measure Comparison
Uniform distribution Monte Carlo · Baseline + Minimal / Moderate / Maximum · Exportable charts & tables
Baseline
No Control (all = 0)
Minimal
U(0.05, 0.20)
Moderate
U(0.30, 0.55)
Maximum
U(0.80, 0.95)
| Metric | Baseline | Minimal (mean±SD) | Moderate (mean±SD) | Maximum (mean±SD) |
|---|---|---|---|---|
| Click ▶ Run to generate Monte Carlo results | ||||
📋 Uniform Distribution Parameter Ranges
| Parameter | ODE Effect | Baseline | Minimal | Moderate | Maximum |
|---|
💉 VACCINATION IMPACT ANALYSIS
Vaccination Coverage
—%
R₀ After Vaccination
—
Current Re
—
Required Control %
—%
⚠️ Run simulation to see analysis
📊 How Much Vaccination Is Needed to Control FMD?
X-axis = Vaccination Coverage %, Y-axis = Effective Reproduction Re. Push the blue dot below the green dashed line to eliminate FMD.
Run the simulation to see the analysis here.
📌 How to read this graph:
🔵 Blue curve: Re for YOUR simulation R₀ as vaccination % rises
🔴 Red curve: Re if a high-risk outbreak strain (R₀=4.5) hits
🟢 Green dashed line: Control threshold — anything below this means FMD will die out
🔵 Filled dot: YOUR current vaccination coverage and Re
Dashed vertical lines show the minimum vaccination % needed to cross the threshold.
🔵 Blue curve: Re for YOUR simulation R₀ as vaccination % rises
🔴 Red curve: Re if a high-risk outbreak strain (R₀=4.5) hits
🟢 Green dashed line: Control threshold — anything below this means FMD will die out
🔵 Filled dot: YOUR current vaccination coverage and Re
Dashed vertical lines show the minimum vaccination % needed to cross the threshold.