5.1 SIR model formulation and analysis

The SIR model is a cornerstone of mathematical epidemiology. It divides a population into Susceptible, Infected, and Recovered groups, using differential equations to track disease spread over time. This powerful tool helps predict outbreaks and evaluate control strategies.

Key concepts include the basic reproduction number (R0), which determines epidemic potential, and equilibrium points that represent long-term disease states. Numerical solutions allow for practical applications, such as forecasting healthcare needs and comparing intervention effectiveness.

SIR Model Fundamentals

Formulation of SIR model

Top images from around the web for Formulation of SIR model

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

• 

  Identifying Critical Parameters in SIR Model for Spread of Disease View original

  Is this image relevant?

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Formulation of SIR model

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

• 

  Identifying Critical Parameters in SIR Model for Spread of Disease View original

  Is this image relevant?

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

• 

  The SIR dynamic model of infectious disease transmission and its analogy with chemical kinetics ... View original

  Is this image relevant?

1 of 3

• SIR model components divide population into distinct compartments
  • S: Susceptible population susceptible to infection
  • I: Infected population actively spreading disease
  • R: Recovered population immune to reinfection
• Differential equations describe population flow between compartments
  • Rate of change for susceptible: $\frac{dS}{dt} = -\beta SI$ decreases as infections occur
  • Rate of change for infected: $\frac{dI}{dt} = \beta SI - \gamma I$ increases with new infections, decreases with recovery
  • Rate of change for recovered: $\frac{dR}{dt} = \gamma I$ increases as infected individuals recover
• Parameters control disease dynamics
  • $\beta$: Transmission rate determines infection spread (contacts per time)
  • $\gamma$: Recovery rate governs transition from infected to recovered (1/infectious period)
• Assumptions simplify model but limit realism
  • Closed population neglects demographic changes (births, deaths, migration)
  • Homogeneous mixing assumes equal probability of contact between individuals
  • Immunity after recovery prevents reinfection (may not hold for all diseases)

Stability of equilibrium points

• Disease-free equilibrium (DFE) represents absence of infection
  • Occurs when I = 0, no infected individuals in population
  • Stable when R0 < 1, disease unable to spread effectively
• Endemic equilibrium indicates persistent disease presence
  • Exists when R0 > 1, disease spreads efficiently
  • Represents long-term disease state in population
• Local stability analysis examines behavior near equilibrium
  1. Linearize system of equations around equilibrium point
  2. Evaluate Jacobian matrix to determine stability properties
  3. Analyze eigenvalues to classify equilibrium stability
• Global stability considerations extend beyond local analysis
  • Lyapunov functions prove global stability for some SIR models
  • Phase plane analysis visualizes system behavior across all initial conditions
• Epidemiological interpretation guides public health decisions
  • DFE stability suggests possibility of disease elimination (smallpox)
  • Endemic equilibrium informs long-term management strategies (influenza)

SIR Model Analysis and Application

Basic reproduction number

• R0 quantifies disease transmissibility
  • Average number of secondary infections caused by one infected individual in fully susceptible population
• Calculation method relates transmission and recovery rates
  • $R_0 = \frac{\beta S_0}{\gamma}$ where S0 is initial susceptible population
  • Higher β or lower γ increases R0, indicating more efficient spread
• Threshold behavior determines epidemic potential
  • R0 < 1: Disease dies out, unable to sustain transmission
  • R0 > 1: Epidemic occurs, exponential growth in early stages
• Factors influencing R0 guide intervention strategies
  • Contact rate affected by population density and social behavior
  • Transmission probability linked to pathogen characteristics and host susceptibility
  • Duration of infectiousness varies by disease (influenza vs HIV)
• Implications for disease control inform public health policy
  • Vaccination strategies aim to reduce susceptible population below threshold
  • Social distancing measures decrease contact rates to lower effective R0

Numerical solutions for disease dynamics

• Numerical methods approximate solutions when analytical methods fail
  • Euler's method uses simple linear approximation (less accurate but easy to implement)
  • Runge-Kutta methods provide higher-order accuracy (commonly used in practice)
• Initial conditions define starting point for simulation
  • S(0), I(0), R(0) specify initial population in each compartment
  • Often set I(0) small to model disease introduction
• Time series plots visualize compartment dynamics
  • Susceptible curve shows depletion of vulnerable population
  • Infected curve displays epidemic trajectory (outbreak size and duration)
  • Recovered curve indicates cumulative cases over time
• Epidemic curve analysis reveals key outbreak characteristics
  • Peak infection time helps predict maximum healthcare demand
  • Final epidemic size estimates total impact on population
• Parameter sensitivity analysis explores model behavior
  • Varying β examines effect of transmission control measures
  • Changing γ models impact of treatment or virulence changes
• Practical applications support evidence-based decision making
  • Predicting outbreak trajectories guides resource allocation (hospital beds)
  • Evaluating intervention strategies compares effectiveness of control measures (masks vs lockdowns)