The Susceptible–Vaccinated–Infectious–Recovered–Deceased (SIRDV) model extends the classical SIR framework by incorporating prophylactic intervention through vaccination and disease-induced mortality. It improves realism by distinguishing between individuals who leave the susceptible compartment through infection or vaccination, and differentiates between those who recover and those who die from infection.
🔄 Compartmental Structure and Dynamics
In the SIRDV model, the total population N is divided into five mutually exclusive compartments:
| Compartment | Symbol | Description |
|---|---|---|
| Susceptible | S | Individuals at risk of infection. They transition to Infected (S → I) or to Vaccinated (S → V) at a rate ω. |
| Vaccinated | V | Individuals vaccinated from the susceptible pool. They have partial protection (efficacy ε) but can still get infected. |
| Infectious | I | Infected individuals capable of spreading the disease. They recover (I → R) or die (I → D) at rates γ and μ respectively. |
| Recovered | R | Individuals who recover and gain immunity. |
| Deceased | D | Individuals who die due to the infection. |
The compartmental transitions are governed by the following equations:
- dS/dt = – (β Ă— S Ă— I) / N – ω Ă— S
- dV/dt = ω Ă— S – (1 – ε) Ă— β Ă— V Ă— I / N
- dI/dt = (β Ă— S Ă— I) / N + (1 – ε) Ă— β Ă— V Ă— I / N – Îł Ă— I – ÎĽ Ă— I
- dR/dt = Îł Ă— I
- dD/dt = ÎĽ Ă— I
đź“Š Parameter Table (General Viral Disease Context)
| Parameter | Meaning | Typical Range |
|---|---|---|
| β | Transmission rate | 0.1 – 1.0 per day |
| γ | Recovery rate | 0.1 – 0.3 per day |
| μ | Disease-induced death rate | 0.0001 – 0.01 per day |
| ω | Vaccination rate | 0.001 – 0.01 per day |
| ε | Vaccine efficacy | 0.5 – 0.9 |
| δ | Waning immunity rate | 0 – 0.005 per day |
⚙️ Model Assumptions and Extensions
- Partial Vaccine Efficacy: Vaccinated individuals can still get infected but at a reduced rate based on ε.
- Waning Immunity: Immunity from vaccination can wane over time, with rate δ returning individuals from V to S.
- Delayed Protection: Immunity may not be instant after vaccination; delay can be represented in extended models.
- Stratification: The model can be extended to multiple population groups (e.g., by age or risk level) to reflect targeted vaccination.
🎯 Applications
- Optimal Control: Determine best vaccination schedules to reduce infections or deaths.
- Threshold Estimation: Evaluate vaccination coverage needed to bring effective reproduction number below 1.
- Public Health Forecasting: Estimate peak infections, death tolls, and the long-term impact of vaccination campaigns.
đź“š Reference Papers
- Liu, X., Takeuchi, Y., & Iwami, S. (2008). SVIR epidemic models with vaccination strategies. Journal of Theoretical Biology, 253(1), 1–11.
- Tuong, T. D., Nguyen, D. H., & Nguyen, N. N. (2024). Stochastic multi-group epidemic SVIR models: Degenerate case. Communications in Nonlinear Science and Numerical Simulation, 128, 107588.
- Saad-Roy, C. M., Wagner, C. E., Baker, R. E., et al. (2020). Immune life history, vaccination, and the dynamics of SARS-CoV-2 over the next 5 years. Science, 370, 811–818.
- Turkyilmazoglu, M. (2022). An extended epidemic model with vaccination: Weak-immune SIRVI. Physica A, 598, 127429.
- Wang, Y., Ullah, S., Khan, I. U., AlQahtani, S. A., & Hassan, A. M. (2023). Numerical assessment of multiple vaccinations to mitigate the transmission of COVID-19 via a new epidemiological modeling approach. Results in Physics, 52, 106889.