SIS Epidemic Model with Ornstein-Uhlenbeck Dynamics

A particle-based simulation of an SIS (Susceptible-Infected-Susceptible) epidemic model where particles move according to an Ornstein-Uhlenbeck process in a 2D periodic domain.

Practical activity for the Physics of Biological Systems minicourse at ICTP-SAIFR

Overview

This simulation combines:

• Spatial dynamics: Particles perform Ornstein-Uhlenbeck (OU) motion, oscillating around equilibrium positions with periodic boundary conditions
• Epidemic dynamics: Distance-dependent infection spreading following an SIS model with spatial interactions

Model Description

Particle Dynamics

Particles move via an Ornstein-Uhlenbeck process:

• Each particle has an equilibrium position x₀
• Position evolves as: dx = -α(x - x₀)dt + σdW
• Periodic boundary conditions (Pac-Man style wrapping)

Epidemic Dynamics

• States: Susceptible (S, blue) or Infected (I, red)
• Recovery: Infected → Susceptible with rate β
• Infection: Probability from neighbor at distance r: p(r) = exp(-λr) dt
• Total infection probability: P(infection) = 1 - ∏(1 - pⱼ) over all infected neighbors

Project Structure

.
├── include/
│   ├── config.h          # System parameters and constants
│   ├── system.h          # System structure and function declarations
│   └── random.h          # Random number generation utilities
├── src/
│   ├── system.c          # Core simulation functions
│   └── random.c          # Random number generators
├── move.c                # OpenGL visualization main
├── main.c                # Simple command-line main
├── run_move.sh           # Compilation script (with OpenGL)
└── run_main.sh           # Compilation script (no OpenGL)

Compilation

With OpenGL Visualization

./run_move.sh [PHI] [RC] [N] [ALPHA] [SIGMA] [DT] [BETA] [LAMBDA]

Command-Line Only

./run_main.sh [PHI] [RC] [N] [ALPHA] [SIGMA] [DT] [BETA] [LAMBDA]

Parameters

• PHI: Particle density (default: 0.9)
• RC: Cutoff radius for interactions (default: 2.5)
• N: Number of particles (default: 1000)
• ALPHA: OU relaxation rate (default: 5.0)
• SIGMA: OU noise strength (default: 0.5)
• DT: Time step (default: 0.01)
• BETA: Recovery rate I→S (default: 0.5)
• LAMBDA: Infection spatial decay (default: 1.0)

Examples

# Use all defaults
./run_move.sh

# Custom parameters
./run_move.sh 0.7 2.5 2500 5.0 0.5 0.01 0.8 1.5

Running the Simulation

Visualization Mode

./move

Controls:

• ESC: Exit
• p: Pause/resume
• g: Toggle cell grid
• +/-: Zoom in/out
• r: Reset zoom
• f/F: Increase/decrease FPS
• Arrow keys: Pan view

Data Collection Mode

./main > output.txt
./main | grep -v "^#" > data.txt

Output format:

step  time    S    I
0     0.0000  999  1
100   1.0000  985  15
200   2.0000  970  30
...

Implementation Details

Spatial Partitioning

• Uses cell lists for efficient neighbor searches
• Cell size = cutoff radius RC
• Each particle searches only in neighboring cells (3×3 grid)

Propagation Models

Three versions available in system.c:

• propagation_v00: Independent transitions (no spatial interaction)
• propagation_v01: Count-based infection (linear in neighbor count)
• propagation_v02: Distance-dependent infection (exponential decay)

Periodic Boundaries

• Minimum image convention for distance calculation
• Position wrapping using modulo arithmetic
• Ensures particles near boundaries interact correctly

Key Functions

System Management:

• makeSystem(): Initialize simulation
• destroySystem(): Free memory
• iteration(): Update particle positions
• propagation_v02(): Update epidemic states

Spatial Partitioning:

• getCellIndex(): Assign particles to cells
• getNeighborList(): Build neighbor cell lists
• minImage(): Compute minimum distance (PBC)

Dependencies

• Standard libraries: stdio.h, stdlib.h, math.h, time.h
• OpenGL (visualization only): GL/gl.h, GL/glu.h, GL/glut.h

Authors

• Matheus L. F. Ribeiro (UFMG)
• Marcelo H. Schwade (IB-USP)
• Vinicius M. Muccillo (IFT-UNESP)
• William G. C. Oropesa (ICTP-SAIFR)