Transmission Dynamics

This page details the specific mechanisms by which foot-and-mouth disease spreads within and between farms in the MHASpread framework.

Overview of Transmission

Transmission of FMD in MHASpread occurs at two interconnected scales:

  1. Within-farm transmission: Direct and indirect contact among housed animals
  2. Between-farm transmission: Distance-driven environmental spread and animal trade

Within-Farm Transmission

Mass-Action Dynamics

Infection spread within farms follows classical mass-action transmission:

\[\text{Force of Infection}_i = \frac{S_i(t) \cdot I_i(t) \cdot \beta_i}{N_i}\]

This formulation assumes:

  • Homogeneous mixing: Any susceptible is equally likely to contact any infectious animal
  • Frequency-dependent contact: Transmission rate scales with density of both susceptible and infectious animals
  • Proportional risk: Risk is $\propto \frac{I}{N}$ (infection prevalence)

Species-Specific Transmission: The Role of Swine

A defining feature of FMD epidemiology is the disproportionate role of swine as disease amplifiers:

Transmission Efficiency Hierarchy

Swine » Cattle » Small Ruminants

Quantitatively:

  • Swine-to-cattle: $\beta \approx 6.14$ animal$^{-1}$ day$^{-1}$ (mode)
  • Cattle-to-cattle: $\beta \approx 0.24$ animal$^{-1}$ day$^{-1}$ (mode)
  • Small ruminant-to-cattle: $\beta \approx 0.105$ animal$^{-1}$ day$^{-1}$ (mode)

Implication: A single infected swine can infect ~25× more cattle individuals per day than an infected cow.

Biological Basis

Differential transmissibility reflects:

  1. Viral shedding rates: Swine shed 10–100× higher viral loads (nasopharyngeal, fecal)
  2. Exposure duration: Swine remain highly infectious for ~6 days vs. cattle ~4 days
  3. Aerosol generation: Swine in confined housing generate more respiratory particles
  4. Fomite transmission: Swine waste contaminates environment more extensively

Temperature and Environmental Persistence

While not explicitly modeled as a time-varying parameter, environmental FMD persistence supports assumption that:

  • Same-farm exposure over hours/days is realistic
  • Within-farm transmission coefficients implicitly include environmental contribution

This is why $\beta$ values are relatively high compared to direct-contact-only models—they represent both direct and indirect (environmental) transmission.

Between-Farm Transmission

The Spatial Transmission Kernel

The exponential transmission kernel models local disease spread:

\[P_{E_j}(t) = 1 - \prod_i \left(1 - \frac{I_i(t)}{N_i} \phi e^{-\alpha d_{ij}}\right)\]

Kernel Components: What Each Parameter Does

Baseline Transmission ($\phi = 0.044$)

At zero distance ($d_{ij} = 0$), probability of between-farm transmission per infectious animal is $\phi \cdot \frac{I_i}{N_i}$.

  • If farm $i$ is at ~10% prevalence and immediately adjacent to farm $j$:
  • $P(\text{transmission}) \approx 1 - (1 - 0.044 \times 0.1) = 0.0044$ per day

This reflects:

  • Rarity of spontaneous spread: FMD typically moves through animal contact/trade, not spontaneously across farms
  • Environmental or fence-line contact: Occasional but not dominant mechanism

Decay Parameter ($\alpha = 0.6$ km$^{-1}$)

Controls how rapidly transmission drops with distance:

\[P(\text{transmission at distance } d) \propto e^{-0.6d}\]
Distance (km) Relative Risk Interpretation
0 1.0 Baseline
1 0.55 ~45% reduction per km
3 0.16 ~84% reduction at 3 km
5 0.05 ~95% reduction at 5 km
10 0.0025 ~99.75% reduction at 10 km
15 0.0001 ~99.99% reduction at 15 km

Maximum Distance Cutoff (40 km)

Transmission negligible beyond 40 km, grounded in:

  • Empirical outbreak data: FMD rarely spreads >40 km without animal movements
  • Mathematical models: Spatial analysis of historic epidemics (Björnham et al., 2020)
  • Biological plausibility: Environmental FMD survival cannot sustain spread over >40 km

Distance Metric

Distances are Euclidean (straight-line) rather than network distance:

\[d_{ij} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}\]

This assumes:

  • Sufficient spatial data resolution (farms mapped at km scale)
  • Landscape homogeneity (terrain doesn’t strongly impede transmission)

Refinement possible: Using road-network distances for fine-scale simulation.

Trade-Network Transmission

Movement Events

Animal movements are the dominant driver of large-scale FMD spread. MHASpread incorporates an events database specifying:

  • Date: Calendar day of shipment
  • Sender: Origin farm
  • Receiver: Destination farm
  • Species: Cattle, swine, or small ruminant
  • Count: Number of animals

Movement-Driven Transmission Probability

When infectious animals are shipped:

\[P(\text{transmission via movement}) = P(\text{shipment includes infectious}) \times P(\text{transmit to receiver})\]

This is implicitly high (often >50–90%) if shipper is infected—hence movements are a critical accelerant of spread.

Control Through Standstill

A 30-day movement restriction prevents all outgoing movements from infected and buffer zones, effectively impeding network-mediated transmission. This is why standstill is a central MHASpread control action.

From Exposure to Disease

Timeline: Latent Period

Once a farm becomes exposed (receives infectious material), animals transition through a latency phase during which:

  • Infection is established but not yet producing virions
  • Animals are not yet infectious to other animals
  • Duration: Species-dependent (mean 2–5 days)

The latent period is sampled from empirically-derived distributions:

\[\sigma \sim \text{Distribution}(\text{species-specific mean, IQR})\]

Impact on dynamics: Latency creates a ~2–5 day delay between exposure and active transmission, allowing time for detection and control action.

Infectious Period

Once animals develop clinical or subclinical infection, they shed virus and transmit:

  • Duration: Species-dependent (mean 3–6 days)
  • Peak shedding: Days 2–4 of clinical disease
  • Transmission: Active throughout (no distinction between clinical/subclinical)

Recovery

After infectious period, animals recover and are assumed to develop durable immunity:

\[I \xrightarrow{\gamma} R\]

Assumption of durability: In reality, reinfection with heterologous serotypes possible but not modeled here.

From Import to Farm-Level Spread

Example: How a Single Infected Shipment Cascades

Day 0 (Import): 50 infected swine arrive at recipient farm with 200 cattle, 100 swine, 50 sheep.

Day 1–3 (Latency): Swine are infected but not yet transmitting. Recipient farm appears seronegative.

Day 3–4 (Infectiousness emerges): Swine begin shedding. Within-farm transmission kernel activates:

  • Swine-to-cattle transmission: $\beta \approx 6.14$
  • Many cattle become exposed

Day 6–7 (Cattle infectious): First cattle become infectious. Between-farm kernel activates:

  • Neighboring farms within 3 km have elevated exposure probability
  • Movements to other locations may occur before detection

Day 10–15 (Exponential phase): If undetected, neighboring farms become infected, spatial wave progresses.

Heterogeneity in Susceptibility

Farm-Level Differences

Transmission dynamics vary by farm context:

Farm Type Transmission Rate Reason
Single species (cattle) Baseline Homogeneous, low amplification
Mixed cattle/swine 5–10× faster Swine amplification
Large swine farm Highest Density + shedding
Smaller operations Lower Reduced density, limited contact

MHASpread captures this through within-farm size ($N$) and species composition but assumes homogeneous mixing once animals are on-site.

Detection Heterogeneity

Infection detection varies by:

  • Farm inspection intensity (more inspections → higher detection probability)
  • Diagnostic sensitivity (clinical disease easier to detect than subclinical)
  • Farmer perceptiveness (farm owner’s ability to recognize disease)

Modeled via hypergeometric sampling and diagnostic sensitivity parameter $s$.

Transmission in the Context of Control Zones

Phase 1: Uncontrolled Spread (Pre-Detection)

20-day silent spread phase simulates:

  • Unrestricted transmission
  • No control actions
  • Virus establishes across unknown number of farms
  • Movement still occurs (no standstill yet)

Output: Realistic distribution of epidemic sizes and farm networks affected before awareness.

Phase 2: Detection and Zone Establishment

Once infected farms are detected:

  1. Infected zone (3 km) centered on detected farm
  2. Buffer zone (7 km) surrounding infected zone
  3. Surveillance zone (15 km) outer perimeter

Transmission continues but is progressively restricted:

  • Movements blocked: Standstill prevents shipment-mediated transmission
  • Depopulation: Reduces $I$ (infectious animals) in zone
  • Vaccination: Converts cattle from $S \to V$ (protected)

Phase 3: Containment and Resolution

If control actions are effective:

  • $I$ decreases (from depopulation and recovery)
  • $E$ decreases (latent animals mature but infectious pool reduced)
  • Spatial kernel indicates no new exposures (few infectious farms remain)

Containment success depends on depopulation speed, vaccination coverage, and detection sensitivity.

Stochasticity in Transmission

Why Randomness Matters

Three sources of stochasticity in transmission:

  1. Binomial sampling: Actual number of exposed individuals from force of infection (sampling error)
  2. Parameter variation: $\beta$, $\sigma$, $\gamma$ drawn from distributions, reflecting biological uncertainty
  3. Spatial randomness: Which farms are nearby (subset of population randomly contacted)

Consequence: Outbreak Variability

Starting from identical initial conditions, replicate simulations can yield:

  • Complete fadeout: Epidemic dies before large spread (5–10% of replicates in some scenarios)
  • Rapid expansion: Exponential spread to dozens of farms
  • Variable duration: 30–300 days depending on control effectiveness

Implication: Policy evaluation must use ensemble statistics (median, percentiles) rather than single deterministic simulation.

Comparison to Deterministic Models

Aspect Deterministic MHASpread (Stochastic)
Transmission Force of infection computes exact new infections Binomial sampling introduces chance
Parameters Fixed to mean values Sampled from distributions
Replicates Single simulation 100–1000 replicates
Outcomes All identical Distribution of outcomes
Extinction risk Only from explicit control May occur by chance
Computational cost Very fast Moderate (minutes to hours per scenario)

Trade-off: Stochasticity adds realism but requires ensemble analysis.

Real-World Calibration

Data Sources for Transmission Parameters

  1. Experimental infections (lab studies): Direct measurement of $\beta$, recovery kinetics
  2. Field outbreak data (active surveillance): Documented secondary attack rates, spread patterns
  3. Retrospective reconstruction: Analysis of known epidemics (Rio Grande do Sul 2000–2001)
  4. Comparative epidemiology: Meta-analyses across global FMD events

Uncertainty Quantification

MHASpread explicitly represents parameter uncertainty through:

  • PERT (Program Evaluation and Review Technique) distributions: Beta-family distributions with specified mode and range
  • Sensitivity analysis: Outer and inner parameter bounds tested
  • Ensemble inference: Bayesian approaches integrating simulation and field data

This ensures that model outputs reflect not just simulation uncertainty but also fundamental epidemiological uncertainty.

Next Steps