Model Overview

This page provides a comprehensive technical description of the MHASpread model architecture, from compartmental structure through multiscale integration and stochastic processes.

Model Architecture

MHASpread integrates disease dynamics across two coupled spatial scales:

Scale 1: Within-Farm Dynamics (Local)

Individual farms contain multiple host species with homogeneous mixing. Disease progression follows a stochastic SEIR model with species-specific parameters.

Scale 2: Metapopulation Dynamics (Regional)

Farms are embedded in a spatial landscape. Infection spreads between farms via two mechanisms:

  1. Spatial transmission: Distance-dependent kernel (local environmental spread)
  2. Animal movements: Trade shipments and restocking (network-mediated spread)

Compartmental Structure

Each farm maintains disease state tracking for three host species: cattle, swine, and small ruminants.

Core Compartments (Per Species Per Farm)

Compartment Symbol Meaning Transitions
Susceptible $S$ Uninfected, infection-capable animals $\xrightarrow{\beta}$ to Exposed
Exposed $E$ Infected, non-infectious (incubation) $\xrightarrow{1/\sigma}$ to Infectious
Infectious $I$ Infected, infectious animals $\xrightarrow{1/\gamma}$ to Recovered
Recovered $R$ Immune (post-infection) Remains (temporary/permanent immunity)
Vaccinated $V$ Vaccine-protected animals Protected (immunity dependent on efficacy)

Total Population

\[N_i(t) = S_i + E_i + I_i + R_i + V_i\]

Within-Farm Transmission Model

Disease progression within farm $i$ at time $t$ follows a stochastic discrete-time model based on differential equations:

Susceptible Dynamics

\[\frac{dS_i(t)}{dt} = u_i(t) - v_i(t) - \frac{S_i(t)I_i(t)\beta_i}{N_i}\]
  • $u_i(t)$ = animal births entering $S$ compartment
  • $v_i(t)$ = animal deaths exiting all compartments
  • $\frac{S_i(t)I_i(t)\beta_i}{N_i}$ = newly exposed animals (mass-action transmission)
  • $\beta_i$ = within-farm transmission coefficient

Exposed Dynamics

\[\frac{dE_i(t)}{dt} = \frac{S_i(t)I_i(t)\beta_i}{N_i} - v_i(t)E_i(t) - \frac{E_i(t)}{\sigma}\]
  • First term: new exposures from $S$
  • Second term: demographic exit
  • Third term: progression to infectious (latent period = $\sigma$)

Infectious Dynamics

\[\frac{dI_i(t)}{dt} = \frac{E_i(t)}{\sigma} - \frac{I_i(t)}{\gamma} - v_i(t)I_i(t)\]
  • First term: progression from $E$
  • Second term: transition to recovered (infectious period = $\gamma$)
  • Third term: demographic exit

Recovered Dynamics

\[\frac{dR_i(t)}{dt} = \frac{I_i(t)}{\gamma} - v_i(t)R_i(t)\]

Species-Specific Parameters

Transmission Coefficients (β)

The transmission coefficient $\beta$ is a 3×3 matrix reflecting transmission from each infected species to each susceptible species:

Infected Species Susceptible Species $\beta$ Distribution Basis
Cattle Cattle PERT(0.18, 0.24, 0.56) Empirical (Rio Grande do Sul 2000–2001)
Cattle Swine PERT(0.18, 0.24, 0.56) Assumed
Cattle Small ruminants PERT(0.18, 0.24, 0.56) Assumed
Swine Cattle PERT(3.7, 6.14, 10.06) Literature (Eblé et al., 2006)
Swine Swine PERT(3.7, 6.14, 10.06) Literature (Eblé et al., 2006)
Swine Small ruminants PERT(3.7, 6.14, 10.06) Assumed (Eblé et al., 2006)
Small ruminants Cattle PERT(0.044, 0.105, 0.253) Assumed (Orsel et al., 2007)
Small ruminants Swine PERT(0.006, 0.024, 0.09) Literature (Goris et al., 2009)
Small ruminants Small ruminants PERT(0.044, 0.105, 0.253) Literature (Orsel et al., 2007)

Key observation: Swine transmit FMD ~15–25× more efficiently than cattle and ~100× more efficiently than small ruminants, reflecting empirically observed differences in viral shedding and susceptibility.

Disease Duration Parameters

Parameter Species Mean (Median, IQR) Unit Reference
Latent period ($\sigma$) Cattle 3.6 (3, 2–5) days Mardones et al., 2010
  Swine 3.1 (2, 2–4) days Mardones et al., 2010
  Small ruminants 4.8 (5, 3–6) days Mardones et al., 2010
Infectious period ($\gamma$) Cattle 4.4 (4, 3–6) days Mardones et al., 2010
  Swine 5.7 (5, 5–6) days Mardones et al., 2010
  Small ruminants 3.3 (3, 2–4) days Mardones et al., 2010

These parameters are sampled stochastically from species-specific distributions for each simulation replicate, preserving biological uncertainty.

Spatial Transmission Model

Transmission Kernel

Between-farm transmission is modeled as a distance-dependent process. The probability that farm $j$ becomes exposed during time step $t$ is:

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

where:

  • $i$ = index of all presently infected farms
  • $j$ = target (uninfected) farm
  • $I_i(t)$ = number of infectious animals at farm $i$ at time $t$
  • $N_i$ = total population at farm $i$
  • $d_{ij}$ = Euclidean distance between farm $i$ and $j$ (kilometers)
  • $\phi$ = baseline transmission probability at zero distance = 0.044
  • $\alpha$ = kernel decay parameter = 0.6 (determines steepness of distance decay)
  • Maximum distance cutoff: 40 km (beyond which transmission considered negligible)

Kernel Parameterization

The parameters $\phi$ and $\alpha$ are derived from:

  • Livestock disease epidemiology literature: Documented patterns of FMD spread via environmental transmission and animal movements
  • Brazilian FMD outbreak data: Spatial analysis of 2000–2001 Rio Grande do Sul epidemics
  • International validation: Consistent with spatial transmission estimates from European and African FMD outbreaks

Interpretation

The kernel captures that:

  1. Local transmission dominates: ~80–90% of between-farm spread occurs within 10 km
  2. Distance remains protective: Risk drops 50% by ~3 km, 90% by ~12 km
  3. Rare long-distance events: Elevated risk beyond 20 km reflects occasional long-distance movement or environmental transmission

Stochastic Simulation Framework

MHASpread operates as a discrete-time stochastic simulation. At each time step $t$ (1 day):

Step 1: Within-Farm Transmission

For each farm and each $(S, I)$ species pair:

\[\text{New exposures}_i \sim \text{Binomial}\left(S_i, \frac{I_i \beta_i}{N_i}\right)\]

Step 2: Disease Progression

  • Sampled transitions: $E \to I$, $I \to R$ from stochastic distributions of $\sigma$ and $\gamma$
  • Demographic events: Births and deaths applied per farm

Step 3: Spatial Transmission

For each uninfected farm:

\[P(\text{exposure to infection}) = P_{E_j}(t)\]

If infected, farm transitions from susceptible to exposed state.

Step 4: Control Actions

  • Detection: Stochastic inspection and diagnostic testing
  • Depopulation: Removal of detected infected farms
  • Vaccination: Progressive immunization of animals in control zones
  • Movement restrictions: Prevention of outgoing/incoming shipments

Output

Each simulation produces:

  • Daily time series of $S, E, I, R, V$ for each farm and species
  • Spatial attack patterns (which farms attacked)
  • Control action outcomes (farms depopulated, animals vaccinated)
  • Epidemic metrics (final size, duration, peak prevalence)

Integration with Animal Movements

Trade Network Transmission

Animal movements between farms are a critical transmission pathway. The model integrates an events file specifying:

  • Date: When animals were moved
  • Sender, Receiver: Movement endpoints
  • Species, Count: Number and type of animals

If an infectious animal is moved from an infected to a susceptible farm, transmission occurs with probability dependent on:

  • Number of infectious animals in shipment
  • Farm-level infection prevalence at origin
  • Species susceptibility at destination

Standstill Implementation

During control zone activation, a 30-day movement standstill restricts:

  • Outgoing movements: From infected + buffer zones (prevents propagation)
  • Incoming movements: To surveillance zone (reduces exposure)

Once movement is blocked, that farm cannot receive animals for 30 days or until all control zones are dissolved.

Assumptions and Simplifications

Key Assumptions

  1. Homogeneous within-farm mixing: Contact rates uniform across species and farms
  2. Panmictic animal populations: No structural subdivision within farms (silos, barns treated as single unit)
  3. Constant transmission parameters: $\beta$ and $\sigma$, $\gamma$ do not vary with season or epidemic stage
  4. Maximum distance cutoff: Transmission negligible >40 km
  5. Instant control action execution: Assumes perfect compliance and logistics (realistic with scenario sensitivity)
  6. Single-serotype pathogen: Model does not represent multi-strain dynamics

Where These Matter

  • Within-farm heterogeneity: Species may be physically separated (reduce transmission)
  • Seasonality: FMD temporal patterns (winter peaks in some regions)
  • Compliance variability: Real-world standstills, vaccination uptake <100%
  • Coinfection/cross-immunity: Other pathogens not modeled

Deterministic vs. Stochastic Behavior

Stochastic Elements

  • Transmission (binomial sampling)
  • Disease progression timing (distributions, not fixed)
  • Detection (hypergeometric + binomial processes)
  • Animal movements (probabilistic routing, if applicable)

Deterministic Elements

  • Control zone radii (fixed: 3, 7, 15 km)
  • Disease duration means (fixed but distributions sampled)
  • Kernel function form (exponential, fixed decay)
  • Computational operations (arithmetic)

Implication: Replicate simulations with identical inputs will produce variable but statistically consistent outcomes. Ensemble analysis (100s of replicates) is standard practice.

Mathematical Notation Summary

Symbol Meaning
$S, E, I, R, V$ Compartment sizes (animals)
$N$ Total population
$\beta$ Transmission coefficient (per animal per day)
$\sigma$ 1/latent period (days$^{-1}$)
$\gamma$ 1/infectious period (days$^{-1}$)
$u$ Birth rate
$v$ Death rate
$d_{ij}$ Distance between farm $i$ and $j$ (km)
$\phi$ Baseline transmission probability ($d=0$)
$\alpha$ Kernel decay rate (km$^{-1}$)
$P_E$ Probability of farm exposure
$t$ Time (days)

Next Steps