Model Structure: A Detailed Walkthrough

This vignette provides a step-by-step technical explanation of how MHASpread is structured—ideal for researchers and modelers.

Overview

MHASpread is a stochastic, multiscale, spatially explicit metapopulation model of FMD transmission. Here we deconstruct its architecture.

Level 1: The Farm (Local Dynamics)

The SEIR Framework

Each farm maintains compartments for each species:

On Farm i, for Species s:

    S_i,s  →  E_i,s  →  I_i,s  →  R_i,s
    (Susceptible) (Exposed) (Infectious) (Recovered)
                ↓
            V_i,s (Vaccinated)

Daily State Transitions

On day $t$, each farm updates:

  1. New infections: $S \xrightarrow{\beta} E$ via mass-action
  2. **Latency: $E \xrightarrow{1/\sigma} I$ after latent period
  3. Recovery: $I \xrightarrow{1/\gamma} R$ after infectious period
  4. Demographic: Birth/death rates applied
  5. Control: Vaccination (if in zone), depopulation (if detected)

Mathematical Formulation

The dynamics per species per farm follow differential equations:

\[\frac{dS}{dt} = u - v - \beta \frac{SI}{N}\] \[\frac{dE}{dt} = \beta \frac{SI}{N} - v \cdot E - \frac{1}{\sigma}E\] \[\frac{dI}{dt} = \frac{1}{\sigma}E - \frac{1}{\gamma}I - v \cdot I\] \[\frac{dR}{dt} = \frac{1}{\gamma}I - v \cdot R\]

where $N = S + E + I + R + V$ (total animals).

Level 2: Transmission Parameters

The β Matrix (Species-Specific Transmission)

Central to MHASpread is the 3×3 transmission matrix:

  Cattle Swine Sheep
Cattle (inf) β_cc β_cs β_cr
Swine (inf) β_sc β_ss β_sr
Sheep (inf) β_rc β_rs β_rr

Example values (per animal per day):

  Cattle Swine Sheep
Cattle (inf) 0.24 0.24 0.24
Swine (inf) 6.14 6.14 6.14
Sheep (inf) 0.105 0.024 0.105

Interpretation: If 1 infected swine is on a farm with 100 cattle, ~0.06 new cattle infections expected per day (6.14 × 1/100).

Disease Duration Parameters

Species-specific to capture biological variation:

Species Latent Period (σ) Infectious Period (γ)
Cattle 3.6 days (range 2–5) 4.4 days (range 3–6)
Swine 3.1 days (range 2–4) 5.7 days (range 5–6)
Sheep 4.8 days (range 3–6) 3.3 days (range 2–4)

These are sampled from distributions (PERT beta-family) for each animal to preserve stochastic variation.

Level 3: Between-Farm Transmission (The Kernel)

The Spatial Kernel Function

Distance-dependent transmission probability for uninfected farm $j$:

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

Breaking this down:

  1. For each infected farm $i$: Calculate transmission contribution
  2. Compute $\frac{I_i(t)}{N_i}$: Infection prevalence (0 to 1)
  3. Multiply by $\phi e^{-\alpha d_{ij}}$: Distance-decay function
  4. Multiply prevalence × decay: Joint probability
  5. Product over all: Combine contributions from all infected farms (exponential model)
  6. 1 - product: Convert to probability farm $j$ is exposed

Kernel Parameters

  • $\phi = 0.044$: Baseline transmission at zero distance
    • Low value reflects that spontaneous spread is rare
    • Most between-farm transmission via animal trade (kernel + events)
  • $\alpha = 0.6$ km$^{-1}$: Decay rate
    • At 1 km: $e^{-0.6 \times 1} = 0.55$ (45% reduction)
    • At 5 km: $e^{-0.6 \times 5} = 0.05$ (95% reduction)
    • At 40 km: $e^{-0.6 \times 40}$ ≈ 0 (transmission negligible)

Kernel Visual 

Transmission Probability (relative to baseline)

1.0 |
    | ███
0.8 |██  ██
    |       ██
0.6 |         ██
    |           ██
0.4 |             ██
    |               ███
0.2 |                   ███
    |                      ███
0.0 |________________________███░░░░░░░
    0        5       10      15      20    40 km
    
██ = High transmission
░░ = Negligible transmission

Level 4: Animal Movements (The Network Layer)

Events File: Trade Network

Each movement event adds stochastic transmission pathway:

date: 2024-01-15
sender: Farm_001 (with 5 infected swine)
receiver: Farm_042 (all susceptible)
species: swine
count: 50 animals

Transmission via Movement

Probability infection transmitted:

\[P(\text{transmission}) = 1 - \left(1 - \frac{I_{\text{sender}}}{N_{\text{sender}}}\right)^n\]

where $n$ = number of animals shipped.

Example: If 5 swine infected out of 100 total (5%), and 50 are shipped: \(P = 1 - (0.95)^{50} \approx 0.92 \text{ (very likely to transmit)}\)

Standstill Effect

When movement standstill active, movements from infected/buffer zones blocked:

  • Cuts network transmission pathway
  • Remains in place 30 days
  • Enforced across simulation for all movements in zones

Level 5: Detection & Diagnosis

Surveillance Zones & Inspection

Active surveillance occurs in farms under surveillance. At iteration $i$:

Number of farms inspected: \(k_i = \max(1, \lfloor P_i / 3 \rfloor)\)

Inspect at least 1 farm, or up to 1/3 of surveillance population.

Hypergeometric Sampling

Number of infected farms detected: \(E_i \sim \text{Hypergeometric}(M = P_i, n = I_i, N = k_i)\)

This accounts for:

  • Finite surveillance population
  • Sampling without replacement
  • Realistic detection from limited inspections

Diagnostic Sensitivity

Detected farms subject to test sensitivity: \(E_i^* \sim \text{Binomial}(E_i, s)\)

If $s = 0.95$: 95% of infected farms with clinical signs are confirmed
If $s = 0.80$: Only 80% detected (reflects subclinical cases)

Level 6: Control Actions (The Intervention Layer)

6a. Depopulation

Trigger: Farm detected as infected
Action: Culling of all animals on infected farms
Daily capacity: Scenario-dependent (e.g., 3 farms/day)
Priority: Largest herds (maximize animals removed)

Once depopulated, farm removed from simulation.

6b. Vaccination

Trigger: Farm in infected or buffer zone
Delay: 15 days after zone establishment
Daily capacity: Scenario-dependent (e.g., 5 farms/day)
Species: Cattle only
Efficacy: Scenario-dependent (e.g., 95%)

Progression: Each farm vaccinated over ~15 days: \(\text{Animals vaccinated/day} = \lfloor N \times (1/15) \rfloor\)

Animals transitioned $S \to V$ (protected) following herd proportion.

6c. Movement Standstill

Trigger: Zone establishment
Duration: 30 days
Scope: Outgoing from infected/buffer, incoming to surveillance
Effect: Blocks animal trade movements (cuts network transmission)

6d. Contact Tracing & Traceback

Mechanism: Identify farms that shipped to detected farm in past 30 days
Action: Treat as detected infected, establish zones around traceback farms
Effect: Expands control to connected farms upstream

Level 7: Stochastic Simulation Loop

Daily Iteration Algorithm 

For each day t = 1, 2, ..., max_days:

  1. Within-farm transmission (all farms):
     For each species on each farm:
       New_E = Binomial(S, β × I / N)
       S → E for New_E animals

  2. Disease progression (all farms):
     For each animal, sample latent and infectious periods from distributions
     Apply transitions: E → I → R based on durations

  3. Between-farm transmission:
     Calculate P_E(t) for each uninfected farm
     Farms with exposure become infected (enter E compartment)

  4. Animal movements:
     Apply each movement event
     If sender infected, transmission to receiver with probability P(transmission)

  5. Detection:
     Sample hypergeometric for surveillance farms
     Apply diagnostic sensitivity
     Detected farms trigger control zones

  6. Control actions:
     Depopulate detected farms (if capacity allows, else queue)
     Vaccinate farms in zones (if zone active, lag passed, capacity allows)
     Block movements in standstill zones
     Trace back from detected farms

  7. Record state:
     Store S, E, I, R, V for all farms
     Record control actions taken
     Update zone status

End for

Replicate Runs

Repeat algorithm 100–1000 times with same inputs:

  • Different random seeds
  • Different transmission outcomes
  • Different detection patterns
  • Different stochastic parameter values

Result: Distribution of outcomes (mean ± SD, percentiles)

Level 8: Output Data

Time Series by Farm

For each farm and day:

  • $S_i(t)$, $E_i(t)$, $I_i(t)$, $R_i(t)$, $V_i(t)$ per species
  • Total infected animals across farm
  • Zone assignment (infected, buffer, surveillance, none)

Epidemic Summary Metrics

Across full simulation:

  • Total farms attacked (≥1 infected animal)
  • Total animals infected (cumulative across all)
  • Peak prevalence (max $I(t)$)
  • Outbreak duration (days to elimination)
  • Final size (total farms + animals affected)

Control Action Summary

  • Farms depopulated (timing)
  • Animals vaccinated (number, efficacy)
  • Movement events averted
  • Farms detected via traceback

Cost Inputs

If economic model linked:

  • Depopulation costs (by farm, by animal)
  • Vaccination costs
  • Surveillance costs
  • Lost productivity

Key Assumptions Recap

Assumption Impact Alternative
Homogeneous mixing within farm May overestimate transmission if species segregated Heterogeneous compartment division
Constant β (no season/age variation) Simplified transmission Time-varying or age-structured β
Perfect movement data Assumes all trade recorded Stochastic unobserved movements
Instant control execution No delay in operations Operational delay modeling
Single FMD serotype No cross-strain complexity Multi-strain model
Permanent immunity post-infection No re-infection modeled SEIRS with waning immunity

Computational Complexity

Scalability

  • Small system (50 farms, 1 month): ~seconds per replicate
  • Medium system (500 farms, 3 months): ~minutes per replicate
  • Large system (5000 farms, 1 year): ~hours per replicate × 100 replicates

Parallelization

Replicates run independently → easily parallelized across cores

Validation & Calibration

How Do We Know the Model is Correct?

  1. Historical reconstruction: Simulate known outbreaks (Rio Grande do Sul 2000–2001)
  2. Compare to observed: Do simulated outcomes match field data?
  3. Sensitivity analysis: Vary parameters, check robustness
  4. Expert review: Epidemiologists validate assumptions

Calibration Data

  • FMD experimental transmission studies
  • Field outbreak investigations
  • Livestock census data
  • Trade network analysis

What’s Next?