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Studies on the basic reproduction number in stochastic epidemic models with random perturbations
Andrés Ríos-Gutiérrez1, Soledad Torres2, Viswanathan Arunachalam1
1Department of Statistics, Universidad Nacional de Colombia, Bogotá, Colombia.
Advances in Difference Equations
|June 21, 2021
Summary
This study defines the basic reproduction number for stochastic epidemic models, analyzing SIR, SIS, and SEIR models using stochastic differential equations to determine stability conditions for disease-free equilibrium.
Area of Science:
- Mathematical epidemiology
- Stochastic modeling
- Dynamical systems
Background:
- Epidemic models are crucial for understanding disease spread.
- Stochasticity and random perturbations play a significant role in real-world epidemics.
- Existing deterministic models may not fully capture the nuances of disease dynamics.
Purpose of the Study:
- To define and analyze the basic reproduction number for stochastic epidemic models.
- To investigate the stability of disease-free equilibrium in SIR, SIS, and SEIR models under random perturbations.
- To extend findings to deterministic epidemic models.
Main Methods:
- Definition of the basic reproduction number using integral or survival functions.
- Analysis of systems of stochastic differential equations for SIR, SIS, and SEIR models.
- Stability analysis of equilibrium points.
Main Results:
- The basic reproduction number is defined for stochastic epidemic models.
- Stability conditions for the disease-free equilibrium point are established.
- Numerical conditions for asymptotic stability are provided.
Conclusions:
- Stochastic differential equations provide a robust framework for epidemic modeling.
- The defined basic reproduction number is a key indicator for disease persistence.
- Understanding stability is vital for public health interventions.
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