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A Unified Stochastic SIR Model Driven By Lévy Noise With Time-Dependency

Terry Easlick, Wei Sun

Abstract

We propose a unified stochastic SIR model driven by Lévy noise. The model is structural enough to allow for time-dependency, nonlinearity, discontinuity, demography and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.

A Unified Stochastic SIR Model Driven By Lévy Noise With Time-Dependency

Abstract

We propose a unified stochastic SIR model driven by Lévy noise. The model is structural enough to allow for time-dependency, nonlinearity, discontinuity, demography and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.
Paper Structure (5 sections, 4 theorems, 98 equations, 7 figures)

This paper contains 5 sections, 4 theorems, 98 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Model for population numbers
  3. Model for population proportions
  4. Simulations
  5. Conclusion

Key Result

Theorem 2.1

Suppose that Assumptions assm21--assm24 hold. Then, for any given initial value $(X_0,Y_0, Z_0) \in \mathbb{R}^3_+$, the system (p1_eq2) has a unique strong solution taking values in $\mathbb{R}^3_+$.

Figures (7)

  • Figure 1: Simulation 1 using E-M scheme of system (\ref{['XC']}) and displaying the average and three randomly-selected sample paths.
  • Figure 2: Simulation 2 using E-M scheme of system (\ref{['ex1b']}) and displaying the average and three randomly-selected sample paths.
  • Figure 3: Simulation 3 using E-M scheme of system (\ref{['XC']}) and displaying the average and three randomly-selected sample paths.
  • Figure 4: Simulation using E-M scheme of system (\ref{['ex34a']}) and displaying the average and three randomly-selected sample paths.
  • Figure 5: Simulation using E-M scheme of system (\ref{['ex34b']}) and displaying the average and three randomly-selected sample paths.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Example 3.3
  • Remark 4.1