Modified SIR Model with Vaccination and Waning Immunity

Model Parameters

Disease Parameters

Basic Reproduction Number (R₀) ⓘ

4.0

Range: 2.0 (mild) to 8.0 (very contagious)

Immunity Decay Rate (α) ⓘ

0.014

Range: 0.008 (slow decay) to 0.02 (fast decay)

Recovery Rate (α) ⓘ

0.25

Range: 0.05 (20 days) to 0.5 (2 days)

Infectious period: 4.0 days

Initial Infected Population ⓘ

0.01%

Range: 0.01% to 1%

Vaccination Parameters

Vaccination Rate (γ₀) ⓘ

0.005

Range: 0.001 (low) to 0.02 (high)

Population vaccinated daily: 0.5%

Vaccination Strategy

Model Structure

Maximum Immunity Levels (m) ⓘ

365

Range: 100 to 365 levels

Simulation Duration (days) ⓘ

1500

Range: 500 to 3000 days

Key Results

Immunity Threshold (k₀):

Critical Vaccination Rate:

Current Vaccination Rate:

Population Dynamics

Infection Detail

R Values

Model Explanation

This interactive simulation implements the modified SIR model with waning immunity and vaccination as described in the paper by Flavius Guiaș. The model extends the standard SIR approach with:

Multiple immunity levels: Immunity wanes gradually over time, with level 0 being no immunity and level m being maximum immunity.
Exponential decay of immunity: R_k = R₀·e^(-α·k) where R_k is the reproduction number at immunity level k.
Targeted vaccination: Vaccination can be applied to specific immunity levels based on strategy.
Critical vaccination rate: The minimum vaccination rate needed to eventually eradicate the epidemic.

The simulation shows long-term epidemic dynamics including possible recurring waves and endemic states.

Modified SIR Model with Vaccination and Several Levels of Immunity

Model Parameters

Disease Parameters

Vaccination Parameters

Model Structure

Model Explanation