Spatial SIR Model

Overview and Background

The Spatial SIR (Susceptible-Infected-Recovered) model represents a fundamental extension of classical epidemic modeling that incorporates spatial dynamics and individual-level interactions. Unlike traditional compartmental models that treat populations as homogeneous, spatial SIR models explicitly account for the geographic distribution of individuals and their movement patterns.

This agent-based implementation simulates epidemic spread on a two-dimensional torus, where each individual (agent) can move freely and interact with nearby neighbors. The model captures the complex interplay between spatial structure, individual behavior, and epidemic dynamics that characterizes real-world disease transmission.

Mathematical Framework

Agent States and Transitions

Each agent \(i\) at time \(t\) is characterized by:

• Position: \mathbf{x}_i(t) = (x_i(t), y_i(t)) on the unit torus [0,1] \times [0,1]
• Velocity: \mathbf{v}_i(t) = (v_{x,i}(t), v_{y,i}(t)) with magnitude |v_i| and direction \theta_i
• State: s_i(t) ∈ {0, 1, 2} representing Susceptible, Infected, or Recovered
• Infection time: t_{I,i} for tracking recovery probability

Movement Dynamics

Agents undergo continuous random walk with small angular perturbations:

where \mathbf{R}(\xi_i) is a rotation matrix with angle \xi_i \sim \text{Uniform}(-\alpha, \alpha), and v_0 is the target speed.

Transmission Dynamics

Infection occurs when susceptible and infected agents are within interaction radius \(r\):

where d_{ij} is the torus distance between agents i and j, and \beta is the transmission rate.

Recovery Process

Infected agents recover according to an exponential process:

where \(\gamma\) is the recovery rate, independent of infection duration.

Chaos Theory and Lyapunov Analysis

The simulation implements a twin-system approach to measure the finite-time Lyapunov exponent (FTLE), a key indicator of chaotic behavior in dynamical systems. Two nearly identical systems are evolved with identical random seeds, and their divergence is tracked over time.

Twin System Methodology

The primary system and twin system are initialized with identical parameters except for a tiny perturbation in agent positions (\(\delta \sim 10^-5\)). Both systems use the same random number generator to ensure identical stochastic events.

Finite-Time Lyapunov Exponent Calculation

The FTLE is computed from the SIR state vectors rather than individual agent positions:

where \mathbf{S}_i(t) = (S_i(t), I_i(t), R_i(t)) represents the SIR counts of system i.

Interpretation of Lyapunov Exponents

• \(\lambda > 0\): Chaotic behavior - small perturbations grow exponentially
• \(\lambda = 0\): Neutral stability - perturbations neither grow nor decay
• \(\lambda < 0\): Stable behavior - perturbations decay exponentially

Computational Implementation

Spatial Grid Optimization

To efficiently handle the \(O(N^2)\) complexity of proximity detection, the simulation employs a uniform spatial grid:

• Grid cells: Size \(r \times r\) to capture all possible interactions
• Neighbor search: Only check agents in the 9-cell neighborhood (3×3 grid)
• Complexity reduction: From \(O(N^2)\) to \(O(N)\) for sparse systems
• Boundary handling: Torus topology with periodic boundary conditions

Integration Scheme

The simulation uses a semi-implicit Euler integration scheme:

This scheme provides good stability for the stochastic dynamics while maintaining computational efficiency.

Monte Carlo Statistical Analysis

The Monte Carlo framework enables statistical analysis of epidemic outcomes across multiple independent realizations. This approach is essential for understanding the variability inherent in stochastic epidemic processes.

Outbreak Definition

An outbreak is defined as a simulation where the peak infected fraction exceeds a threshold \(\theta\):

Statistical Measures

• Outbreak probability: P_outbreak = (Number of outbreaks) / (Total runs)
• Mean peak infection: ⟨I_peak⟩ = (1/M) × Σ I_peak^(i)
• Final epidemic size: ⟨R_final⟩ = (1/M) × Σ R_final^(i)
• Variance analysis: Confidence intervals and standard deviations for all measures

Epidemiological Applications

Disease Modeling

Spatial SIR models are particularly valuable for modeling diseases where spatial proximity is crucial for transmission, such as:

• Airborne diseases: Influenza, COVID-19, tuberculosis
• Vector-borne diseases: Malaria, dengue fever, Zika virus
• Contact diseases: Measles, chickenpox, norovirus
• Environmental diseases: Cholera, Legionnaires' disease

Public Health Interventions

The model can evaluate various intervention strategies:

• Social distancing: Reducing agent speed and interaction radius
• Quarantine: Removing infected agents from the population
• Vaccination: Converting susceptible agents directly to recovered
• Contact tracing: Targeted isolation of high-risk individuals

Mathematical Properties and Phase Transitions

Basic Reproduction Number

The effective reproduction number \(R_e\) in the spatial model depends on both transmission parameters and spatial structure:

where A is the total area and f_spatial accounts for spatial clustering effects.

Phase Transitions

The model exhibits different phases depending on parameter values:

• Endemic phase (\(R_e < 1\)): Disease dies out quickly
• Epidemic phase (\(R_e > 1\)): Large-scale outbreaks occur
• Critical phase (\(R_e \approx 1\)): Stochastic fluctuations dominate

Research Applications and Extensions

Network Epidemiology

The spatial model can be extended to include explicit contact networks, where agents interact only with their network neighbors. This bridges the gap between spatial and network-based epidemic models.

Multi-Scale Modeling

The agent-based approach naturally incorporates multiple scales:

• Individual level: Agent behavior and decision-making
• Local level: Neighborhood interactions and clustering
• Population level: Global epidemic patterns and statistics
• Temporal level: Short-term dynamics and long-term trends

Machine Learning Integration

The simulation generates rich datasets suitable for machine learning applications:

• Epidemic prediction: Using early-stage data to predict outbreak severity
• Parameter estimation: Inferring transmission parameters from observed data
• Intervention optimization: Finding optimal control strategies
• Pattern recognition: Identifying characteristic epidemic signatures

Limitations and Future Directions

Current Limitations

• Computational complexity: \(O(N^2)\) scaling limits population sizes
• Simplified behavior: Agents lack realistic decision-making processes
• Homogeneous mixing: All agents have identical interaction patterns
• Static parameters: No adaptation or learning mechanisms

Future Research Directions

• Heterogeneous populations: Age-structured models with varying susceptibility
• Adaptive behavior: Agents that modify behavior based on epidemic state
• Multi-pathogen dynamics: Co-circulating diseases with cross-immunity
• Real-world data integration: Incorporating actual mobility and contact patterns
• Machine learning optimization: AI-driven parameter tuning and intervention design