systems.builtin.stochastic.continuous.StochasticSIR

systems.builtin.stochastic.continuous.StochasticSIR(*args, **kwargs)

Stochastic SIR epidemic model with demographic noise.

Models disease spread in a finite population with inherent randomness from discrete infection and recovery events. Essential for understanding outbreak extinction, variability in epidemic outcomes, and uncertainty in disease forecasting.

Stochastic Differential Equations

Diffusion approximation of discrete SIR:

dS = -β·S·I/N·dt + σ_S·dW_S
dI = (β·S·I/N - γ·I)·dt + σ_I·dW_I
dR = γ·I·dt + σ_R·dW_R

where: - S(t): Number susceptible (can be infected) - I(t): Number infected (infectious) - R(t): Number recovered (immune) - N = S + I + R: Total population (constant) - β: Transmission rate [1/time] - γ: Recovery rate [1/time] - σ_S, σ_I, σ_R: Noise intensities - W_S, W_I, W_R: Wiener processes

Physical Meaning:

Transmission: β·S·I/N - Rate of new infections - Mass action: Proportional to S·I - β: Contacts per time × transmission probability

Recovery: γ·I - Rate of recoveries - First-order: Proportional to I - γ = 1/(infectious period)

Demographic Noise:

True stochastic SIR has √ diffusion: Diffusion ~ √(rate)

From Poisson statistics of discrete events.

This implementation: Simplified additive noise for tractability.

Physical Interpretation

Susceptibles S:

Decrease only (monotonic): - Start: S(0) ≈ N (nearly all susceptible) - End: S(∞) > 0 (some escape infection) - Never increases (no loss of immunity modeled)

Infected I:

Non-monotonic (rise then fall): - Start: I(0) = small (index cases) - Peak: I_max at t_peak (outbreak peak) - End: I(∞) = 0 (disease dies out)

Recovered R:

Increase only (monotonic): - Start: R(0) = 0 (no immunity initially) - End: R(∞) = final outbreak size - Measure of epidemic impact

Conservation: S + I + R = N (total population constant)

Key Features

Nonlinearity: S·I term creates threshold behavior and epidemic curve.

Positivity: Must have S, I, R ≥ 0 (counts of people).

Conservation: S + I + R = N always (no births/deaths).

Extinction: I → 0 eventually (disease dies out). Time random, final size random.

Threshold (R₀): Probabilistic in stochastic model.

Finite Time: Epidemic is transient (not steady state).

Mathematical Properties

Basic Reproduction Number: R₀ = β/γ

Critical threshold (deterministic): - R₀ < 1: Dies out - R₀ > 1: Epidemic

Stochastic Threshold: Even R₀ > 1: Can die out with probability (1/R₀)^{I₀}

Final Size Relation:

Deterministic: R_∞ satisfies R_∞ = N - S₀·exp(-R₀·R_∞/N)

Stochastic: Distribution around this value.

Epidemic Peak:

Deterministic: I_max ≈ I₀ + S₀ - N/R₀ - (N/R₀)·ln(S₀·R₀/N) t_peak ≈ (1/γ)·ln(R₀·S₀/N)

Stochastic: Random variables (compute via simulation).

Physical Interpretation

Transmission Rate β: - Units: [1/time] - β = contact rate × transmission probability - Typical: 0.2-2.0 per day

Examples: - Influenza: β ≈ 0.5 per day - Measles: β ≈ 1.5 per day - COVID-19: β ≈ 0.3-0.6 per day (varies)

Recovery Rate γ: - Units: [1/time] - γ = 1/(infectious period) - Typical: 0.1-1.0 per day

Examples: - Influenza: γ ≈ 0.5 per day (2 days infectious) - COVID-19: γ ≈ 0.1 per day (10 days) - Measles: γ ≈ 0.1 per day (10 days)

Basic Reproduction Number: R₀ = β/γ

Examples: - Influenza: R₀ ≈ 1-2 - COVID-19: R₀ ≈ 2-5 (variant dependent) - Measles: R₀ ≈ 12-18

Noise Intensity:

From demographic stochasticity (Poisson): σ ≈ √(rate/N)

Relative noise: 1/√N

State Space

State: X = [S, I, R] ∈ ℝ₊³ - S, I, R ≥ 0 (non-negative counts) - S + I + R = N (conservation) - Bounded: 0 ≤ S, I, R ≤ N

Control: u (optional, interventions) - Reduce β (social distancing) - Increase recovery (treatment) - Vaccination (move S → R)

Noise: w = [w_S, w_I, w_R] ∈ ℝ³ - Demographic stochasticity - Should be correlated (conservation) - This implementation: Simplified independent

Parameters

Name	Type	Description	Default
beta	float	Transmission rate [1/day] - Typical: 0.2-2.0 - Disease and behavior dependent	0.5
gamma	float	Recovery rate [1/day] - 1/γ = infectious period - Typical: 0.1-1.0	0.1
N	float	Total population - Larger N: More deterministic - Smaller N: More stochastic	1000.0
sigma_S	float	Susceptible noise intensity [1/√day] - From √(rate/N) for demographic noise - Typical: √(β·S·I/N²)	0.1
sigma_I	float	Infected noise intensity [1/√day]	0.1
sigma_R	float	Recovered noise intensity [1/√day]	0.1

Stochastic Properties

• System Type: NONLINEAR
• Noise Type: ADDITIVE (simplified)
• SDE Type: Itô
• Noise Dimension: nw = 3
• Stationary: No (epidemic is transient)
• Positive: Should be (may need projection)
• Conserved: S + I + R = N (approximately)

Applications

1. Epidemiology: - COVID-19, influenza, measles modeling - Outbreak prediction with uncertainty - Intervention timing and effectiveness

2. Parameter Estimation: - Estimate β, γ from outbreak data - Bayesian inference with uncertainty - Real-time estimation (particle filter)

3. Public Health Policy: - Vaccination strategies - Social distancing timing - Resource allocation (ICU beds)

4. Extinction Analysis: - Probability small outbreak dies out - Early intervention effectiveness - Import risk assessment

5. Rare Events: - Superspreading events - Large outbreaks in small populations - Timing of peak (ICU planning)

Numerical Integration

Recommended: - Euler-Maruyama: dt = 0.01-0.1 days - Project to non-negative: max(X, 0) - Check conservation: S+I+R ≈ N

Event Detection: - Extinction: I < 0.5 (declare extinct) - Peak: Max I(t) - Duration: Time I > threshold

Monte Carlo Guidelines

Ensemble Analysis: - N_runs = 100-1,000 - Compute: Extinction probability, mean final size - Histogram: Final R (bimodal if near threshold)

Comparison with Deterministic

Deterministic: - Smooth epidemic curve - Single final size - R₀ sharp threshold

Stochastic: - Variable epidemic curves - Distribution of final sizes - R₀ probabilistic threshold - Extinction possible

Limitations

• Additive noise (not √ diffusion)
• Independent noise (should be correlated)
• Homogeneous mixing (no network structure)
• Constant β, γ (no seasonality)
• Closed population (no births/deaths)

Extensions

• SEIR: Add Exposed class
• Age structure: Multiple age groups
• Spatial: Geographic spread
• Network: Contact structure
• Time-varying: Seasonal β(t)
• Vaccination: Control via S → R

See Also

CoxIngersollRoss : Similar √ diffusion structure

Methods

Name	Description
check_conservation	Check conservation constraint S + I + R = N.
define_system	Define stochastic SIR epidemic dynamics.
estimate_extinction_probability	Estimate probability of stochastic extinction (approximation).
get_basic_reproduction_number	Get basic reproduction number R₀ = β/γ.
get_herd_immunity_threshold	Get herd immunity threshold H = 1 - 1/R₀.

check_conservation

systems.builtin.stochastic.continuous.StochasticSIR.check_conservation(x)

Check conservation constraint S + I + R = N.

Parameters

Name	Type	Description	Default
x	np.ndarray	State [S, I, R]	required

Returns

Name	Type	Description
	float	Absolute error |S+I+R - N|

Examples

>>> sir = StochasticSIR(N=1000)
>>> x = np.array([500, 200, 300])
>>> error = sir.check_conservation(x)
>>> print(f"Conservation error: {error:.2f}")

define_system

systems.builtin.stochastic.continuous.StochasticSIR.define_system(
    beta=0.5,
    gamma=0.1,
    N=1000.0,
    sigma_S=0.1,
    sigma_I=0.1,
    sigma_R=0.1,
)

Define stochastic SIR epidemic dynamics.

Parameters

Name	Type	Description	Default
beta	float	Transmission rate [1/day] - β = contact rate × transmission probability - Typical: 0.2-2.0 per day - Higher → faster spread	0.5
gamma	float	Recovery rate [1/day] - γ = 1/(infectious period) - Typical: 0.1-1.0 per day - Higher → faster recovery	0.1
N	float	Total population (constant) - Larger N: More deterministic - Smaller N: More stochastic - Typical: 100-1,000,000	1000.0
sigma_S	float	Susceptible noise intensity [1/√day] - From demographic noise: ~ √(β·S·I/N²) - Typical: 0.01-1.0	0.1
sigma_I	float	Infected noise intensity [1/√day] - From demographic noise: ~ √(β·S·I/N² + γ·I/N) - Most critical (affects outbreak dynamics)	0.1
sigma_R	float	Recovered noise intensity [1/√day] - From demographic noise: ~ √(γ·I/N)	0.1

Notes

Basic Reproduction Number: R₀ = β/γ

Determines epidemic threshold: - R₀ < 1: Dies out (subcritical) - R₀ = 1: Critical (boundary) - R₀ > 1: Epidemic (supercritical)

Stochastic Extinction:

Even if R₀ > 1, can randomly die out: P(extinction | I₀) ≈ (1/R₀)^{I₀}

Example: R₀ = 2, I₀ = 1 → P_ext = 50%

Noise Scaling:

Physical demographic noise: σ_physical ~ √(rate/N)

For N = 1000, typical rates ~ 0.1-1: σ ~ 0.01-0.1

Larger population → smaller relative noise.

Infectious Period: T_inf = 1/γ

Examples: - Influenza: 2-3 days (γ ≈ 0.4) - COVID-19: 7-14 days (γ ≈ 0.1) - Measles: 7-10 days (γ ≈ 0.12)

Herd Immunity: H = 1 - 1/R₀

Fraction immune needed to prevent epidemic.

Validation:

Check conservation: S + I + R should equal N. If drifts: Renormalize periodically.

estimate_extinction_probability

systems.builtin.stochastic.continuous.StochasticSIR.estimate_extinction_probability(
    I0,
)

Estimate probability of stochastic extinction (approximation).

For R₀ > 1: P_ext ≈ (1/R₀)^{I₀}

Parameters

Name	Type	Description	Default
I0	float	Initial number infected	required

Returns

Name	Type	Description
	float	Extinction probability

Examples

>>> sir = StochasticSIR(beta=0.6, gamma=0.2)  # R₀=3
>>> P_ext_1 = sir.estimate_extinction_probability(I0=1)
>>> P_ext_10 = sir.estimate_extinction_probability(I0=10)
>>> print(f"1 infected: P_ext = {P_ext_1:.2%}")
>>> print(f"10 infected: P_ext = {P_ext_10:.2%}")

get_basic_reproduction_number

systems.builtin.stochastic.continuous.StochasticSIR.get_basic_reproduction_number(
)

Get basic reproduction number R₀ = β/γ.

Returns

Name	Type	Description
	float	R₀

Examples

>>> sir = StochasticSIR(beta=0.5, gamma=0.1)
>>> R0 = sir.get_basic_reproduction_number()
>>> print(f"R₀ = {R0:.2f}")

get_herd_immunity_threshold

systems.builtin.stochastic.continuous.StochasticSIR.get_herd_immunity_threshold(
)

Get herd immunity threshold H = 1 - 1/R₀.

Returns

Name	Type	Description
	float	Herd immunity threshold (fraction)

Examples

>>> sir = StochasticSIR(beta=0.5, gamma=0.1)
>>> H = sir.get_herd_immunity_threshold()
>>> print(f"Herd immunity: {H:.1%}")