Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An unconditional boundary and dynamics preserving scheme for the stochastic epidemic model

Ruishu Liu, Xiaojie Wang, Lei Dai

TL;DR

A logarithm transformation based Milstein-type method for the stochastic susceptible-infected-susceptible (SIS) epidemic model evolving in the domain, which unconditionally reproduces the extinction and persistence behavior of the original model, with no additional requirements imposed on the model parameters.

Abstract

In the present article, we construct a logarithm transformation based Milstein-type method for the stochastic susceptible-infected-susceptible (SIS) epidemic model evolving in the domain (0,N). The new scheme is explicit and unconditionally boundary and dynamics preserving, when used to solve the stochastic SIS epidemic model. Also, it is proved that the scheme has a strong convergence rate of order one. Different from existing time discretization schemes, the newly proposed scheme for any time step size h > 0, not only produces numerical approximations living in the entire domain (0,N), but also unconditionally reproduces the extinction and persistence behavior of the original model, with no additional requirements imposed on the model parameters. Numerical experiments are presented to verify our theoretical findings.

An unconditional boundary and dynamics preserving scheme for the stochastic epidemic model

TL;DR

A logarithm transformation based Milstein-type method for the stochastic susceptible-infected-susceptible (SIS) epidemic model evolving in the domain, which unconditionally reproduces the extinction and persistence behavior of the original model, with no additional requirements imposed on the model parameters.

Abstract

In the present article, we construct a logarithm transformation based Milstein-type method for the stochastic susceptible-infected-susceptible (SIS) epidemic model evolving in the domain (0,N). The new scheme is explicit and unconditionally boundary and dynamics preserving, when used to solve the stochastic SIS epidemic model. Also, it is proved that the scheme has a strong convergence rate of order one. Different from existing time discretization schemes, the newly proposed scheme for any time step size h > 0, not only produces numerical approximations living in the entire domain (0,N), but also unconditionally reproduces the extinction and persistence behavior of the original model, with no additional requirements imposed on the model parameters. Numerical experiments are presented to verify our theoretical findings.
Paper Structure (8 sections, 11 theorems, 87 equations, 8 figures, 4 tables)

This paper contains 8 sections, 11 theorems, 87 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The stochastic epidemic model and the proposed scheme
  3. The first-order strong convergence of the scheme
  4. Dynamics behavior of numerical approximations
  5. Unconditional extinction preserving
  6. Unconditional persistence preserving
  7. Numerical experiments
  8. Conclusion

Key Result

Lemma 2.1

For any initial value $I_0 \in (0,N)$, the SDE 2023SIS-eq:SIS_original has a unique global positive solution $I_t\in (0,N)$ for all $t \geq 0$ with probability one, namely,

Figures (8)

  • Figure 1: Example \ref{['2023SIS-eg:convergence_1']}
  • Figure 2: Example \ref{['2023SIS-eg:convergence_2']}
  • Figure 3: Example \ref{['2023SIS-eg:extinction_2']}: $h_{\text{exact}} = 2^{-14}$
  • Figure 4: Example \ref{['2023SIS-eg:extinction_2']}: $h = 2^{-4}$
  • Figure 5: Example \ref{['2023SIS-eg:extinction_2']}: $h = 2^{-2}$
  • ...and 3 more figures

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Exponential moment bounds
  • Lemma 2.4: Moment bounds
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition 4.1: Extinction
  • Theorem 4.2
  • Theorem 4.3
  • ...and 8 more