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Pasa Analytics | Epidemiological Modeling - Advanced SIR Model

Epidemiological Modeling - SIR Model

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Home › Epidemiological Modeling - SIR Model

1. The Basic SIR Model [February 2025]

Methodology

This study presents a comprehensive analysis of the SIR model (Susceptible-Infectious-Removed) with numerical resolution of differential equations and nullcline analysis.

The Three Compartments

The classic SIR model divides the population into three compartments:

  • S (Susceptible): non-immune individuals who can be contaminated
  • I (Infectious): disease carriers who contaminate susceptibles
  • R (Removed): immune (after recovery) or deceased
$$ S = N_S/N, \quad I = N_I/N, \quad R = N_R/N $$

where N is the total population, and N_S, N_I, N_R are the sizes of each compartment.

Model Equations

Disease transmission is modeled by a mass action term:

$$ \beta S I $$

where β is the transmission rate. The force of infection is defined as λ = β I.

Let τ be the infectious period, then the recovery rate is γ = 1/τ.

dS/dt = μ - β S I - δ S
dI/dt = β S I - γ I - δ I
dR/dt = γ I - δ R

with:

  • μ: birth rate
  • δ: death rate (often δ = μ for constant population)
  • γ: recovery rate
  • β: transmission rate

Basic Reproduction Number R₀

An infectious individual contaminates on average β people per time unit during a period τ:

$$ \mathcal{R}_0 = \beta \tau = \frac{\beta}{\gamma}$$

This fundamental parameter determines epidemic behavior:

  • R₀ < 1: epidemic dies out
  • R₀ > 1: epidemic spreads
  • R₀ = 1: critical threshold
Removed and susceptibles as a fraction as a function of $ R_0$.

Impact du nombre de reproduction de base R₀ sur la sévérité de l'épidémie

Removed fraction as a function of $.

Évolution temporelle des trois compartiments : Susceptibles (S), Infectés (I), Retirés (R)

When the epidemic starts, the fraction of infected I(t) grows exponentially with an initial growth rate given by r = γ (R₀ - 1). The higher R₀, the faster the initial growth and the more severe the epidemic. For typical R₀ values (2 to 3 for influenza, 3 to 5 for COVID-19), we observe a rapid growth in the number of infected, highlighting the importance of control measures to reduce R₀ below 1. We also note that the final fraction of the population affected by the epidemic (R*) is an increasing function of R₀, with a rapid transition around R₀ = 1. This fraction does not necessarily equal 1 even for high R₀ values, due to herd immunity that develops as individuals move into the R compartment. When the number of susceptibles S(t) falls below the critical threshold 1/R₀, the growth of I(t) slows down and eventually declines, even if R₀ > 1. This explains the characteristic bell-shaped curve of the epidemic: rapid growth followed by a gradual decline as herd immunity develops.

2. Nullclines and Stationary Points [February 2025]

Nullclines

Nullclines are curves where derivatives vanish. For the SIR model with δ = μ:

I nullcline (dI/dt = 0):

$$ \beta S - \gamma - \delta = 0 \quad \Rightarrow \quad \boxed{S_{nc} = \frac{\gamma + \delta}{\beta}} $$

S nullcline (dS/dt = 0):

$$ \mu - \beta S_{nc} I - \delta S_{nc} = 0 \quad \Rightarrow \quad \boxed{I_{nc} = \frac{\mu - \delta S_{nc}}{\beta S_{nc}} = \frac{\mu}{\gamma + \delta} - \frac{\delta}{\beta}} $$
Removed fraction as a function of $.

Portrait de phase S-I : évolution de la fraction d'infectés en fonction des susceptibles

Stationary Points

Two types of fixed points:

  • Disease-free fixed point: (S*, I*, R*) = (1, 0, 0) (or adapted according to μ, δ)
  • Endemic fixed point: (S*, I*, R*) with I* > 0 (appears when R₀ > 1)
Removed fraction as a function of $.

Champ de vecteurs montrant la direction de l'évolution du système

Nullcline Interpretation
  • Nullcline intersection gives equilibrium points
  • I nullcline is vertical: critical susceptible threshold
  • Below threshold (S < S_nc), I decreases
  • Above threshold (S > S_nc), I increases (if I > 0)

3. SIR Model without Birth/Death [February 2025]

Without population dynamics (μ = δ = 0), the model simplifies:

dS/dt = -β S I
dI/dt = β S I - γ I
dR/dt = γ I

Proliferation Condition

For disease spread, we need dI/dt > 0:

$$ I(β S - γ) > 0 \quad \Rightarrow \quad S > \frac{\gamma}{\beta} = \frac{1}{\mathcal{R}_0}$$

The disease can only proliferate if the susceptible fraction is high enough.

Implicit S-R Relation

Dividing the S equation by the R equation:

$$ \frac{dS}{dR} = -\mathcal{R}_0 S $$

This equation is readily solved:

$$ S(R) = S_0 e^{-\mathcal{R}_0 R} $$

with S(0) = 1 (initially, no one is immune).

Removed fraction as a function of $.

Relation implicite entre susceptibles et retirés : S = exp(-R₀·R)

Stationary Solution

At equilibrium (t → ∞), I* = 0 and S* + R* = 1, giving the implicit equation:

$$ \boxed{1 - e^{-\mathcal{R}_0 R^*} - R^* = 0} $$

Solving this equation gives the final fraction of the population affected by the epidemic.

Removed fraction as a function of $ R_0 $.

Fraction finale de la population touchée par l'épidémie en fonction de R₀

Numerical Example
  • R₀ = 3.5
  • τ = 5 days → γ = 0.2 day⁻¹
  • β = R₀ × γ = 0.7 day⁻¹
  • Stationary state: S* = 0.034, I* = 0, R* = 0.966
  • Interpretation: 96.6% of the population will be affected
Removed fraction as a function of $.

Trajectoire du système dans le plan infectés-retirés

4. SIR Model with Birth and Death [February 2025]

Introducing population dynamics (μ > 0, δ = μ) maintains constant total population but allows susceptible renewal through births.

Parameters

  • R₀ = 3.5
  • γ = 0.2 day⁻¹ (τ = 5 days)
  • β = 0.7 day⁻¹
  • μ = δ = 0.02 day⁻¹ (birth/death rate)

New Endemic Fixed Point

Unlike the case without birth/death, the system converges to an endemic fixed point with I* > 0:

$$ S^* = \frac{\gamma + \delta}{\beta} = \frac{0.2 + 0.02}{0.7} = 0.314 $$
$$ I^* = \frac{\mu}{\gamma + \delta} - \frac{\delta}{\beta} = \frac{0.02}{0.22} - \frac{0.02}{0.7} = 0.0623 $$
$$ R^* = 1 - S^* - I^* = 0.6234 $$
Removed fraction as a function of $.

Modèle avec dynamique de population : apparition d'un point fixe endémique non nul

Behavior Comparison

Without birth/death
I* = 0
S* = 0.034
R* = 0.966

Disease extinction

With birth/death
I* = 0.062
S* = 0.314
R* = 0.623

Stable endemic state

Births maintain a constant flow of new susceptibles, allowing the disease to persist in the population. This is typical of endemic diseases like seasonal influenza.

Numerical Verification

After simulation up to t = 200 days, values converge to the theoretical fixed point:

S ≈ 0.315, I ≈ 0.062, R ≈ 0.623

5. Conclusion and Perspectives [February 2025]

Key Results

SIR model analysis highlights several fundamental results in mathematical epidemiology:

  • Epidemic threshold: R₀ > 1 is necessary for epidemic spread
  • Herd immunity: when susceptible fraction drops below 1/R₀, epidemic recedes
  • Final size: given by implicit equation 1 - exp(-R₀ R*) - R* = 0
  • Endemicity: with population renewal, disease can persist indefinitely

Model Extensions

The basic SIR model can be enriched for more complex situations:

SEIR

Adds E (Exposed) compartment for incubation period

SIRS

Immunity loss: recovered return to susceptible compartment

Age-structured

Age-dependent contact and mortality rates

Spatial mobility

Geographic disease spread (meta-population models)

Practical Applications

These models are used daily by public health agencies for:

  • Forecasting epidemic evolution (influenza, COVID-19, Ebola...)
  • Evaluating control measures impact (lockdown, vaccination)
  • Optimizing resource allocation (hospital beds, vaccines)
  • Determining vaccination thresholds for herd immunity
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