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Dynamic Properties of Numerical Solution for Stochastic Epidemic Models with a Fully Implicit Form

Zeyao Jin, Zhiming Li, Shenglong Chen

arXiv:2610.36924v1math.NAcs.NAmath.PRstat.AP

TL;DR

Stochastic epidemic models are needed because random fluctuations can affect transmission and disease extinction, while backward Euler discretization can introduce future information into non-adapted equations. This paper develops a fully implicit stochastic SIS scheme using SSBE and a linear implicit θ-method, and analyzes its numerical and disease dynamics. The results establish extinction and persistence under different conditions, with high environmental noise associated with extinction.

  • Problem

    Backward Euler discretization can place future information in drift and diffusion, making non-adapted stochastic equations difficult to solve numerically.

    Source: §1 · p. 1

  • Method

    The paper combines SSBE with a linear implicit θ-method in the diffusion term to construct fully implicit stochastic SIS models.

    Source: §2 · p. 3, §5 · p. 18

  • Results

    The analysis proves disease extinction or persistence under three scenarios, and simulations verify the theoretical conclusions; high noise is associated with extinction regardless of R_S0.

    Source: §1 · p. 2, §5 · p. 18

  • Takeaways & Limitations

    In the reported scenarios, high environmental noise is linked to disease extinction even regardless of R_S0.

    Source: §1 · p. 2

    The SSBE analysis assumes the solution remains in [0, N] and requires stated coefficient conditions and a step-size restriction.

    Source: §2 · p. 3

Abstract

from arXiv · show

This paper establishes a fully implicit model to investigate the spread of epidemics and studies its dynamic properties, extending the semi-implicit model in Yang et al.(Commun. Nonlinear Sci. Numer. Simul., 152 (2026) 109140 ). We first employ the split-step backward Euler method to address the non-adaptability arising from backward Euler discretization and ensure the global convergence of the numerical solution. By linearizing the model, the computational cost is reduced at the expense of a slight increase in error, effectively avoiding complex iterative solution processes. Then, we derive the existence, uniqueness, positivity, and boundedness of the numerical solution, and obtain its global convergence order. On this basis, the extinction and persistence of the disease are investigated under three different scenarios. The results indicate that environmental noise is a key factor leading to disease extinction. Finally, numerical simulations visually verify the obtained theoretical conclusions.

1. Introduction

The introduction motivates stochastic epidemic modeling by the potential for random fluctuations to affect transmission and extinction, then presents a fully implicit stochastic SIS model to address numerical difficulties with non-adapted equations. It outlines the model’s numerical and disease-dynamics analyses.

1. Introduction

  • Random fluctuations can make transmission trajectories deviate from deterministic predictions and affect whether disease becomes extinct.Source: §1 · p. 1
  • Backward Euler discretization can put future information in both drift and diffusion, complicating numerical solution of non-adapted equations.Source: §1 · p. 1
  • The paper applies SSBE and a linear implicit θ-method to the diffusion term to construct fully implicit stochastic SIS models.The stated aim is to analyze their numerical properties.Source: §1 · p. 2
  • The proposed θ-SIS model includes forward Euler and backward Euler as special cases.Source: §1 · p. 2
  • The paper investigates numerical-solution existence, uniqueness, positivity, and boundedness, and proves disease extinction or persistence under three condition types.Source: §1 · p. 2
  • Linearizing the model reduces computational cost with a small increase in error while retaining global convergence.Source: §1 · p. 3

2. A Fully Implicit Stochastic SIS model

This section constructs a fully implicit stochastic SIS scheme using SSBE and a linear implicit θ-method in the diffusion term. It also establishes conditions for convergence and for positive, bounded numerical solutions.

2. A Fully Implicit Stochastic SIS model

  • The fully implicit scheme addresses the future-information difficulty by applying a linear implicit θ-method to the diffusion term.The preceding approach replaced I_n+1 with I_n; the new method is then simplified to a linear structure.Source: §2 · pp. 3–4
  • Under the stated coefficient conditions and step-size restriction, SSBE is strongly convergent in the mean-square sense.The analysis assumes the solution remains in the bounded domain [0, N].Source: §2 · p. 3
  • Replacing one I* n+1 factor with I* n yields a linear model, and parameter choices recover forward Euler or backward Euler.The nonlinear form otherwise requires iterative solution and is computationally expensive.Source: §2 · p. 4
  • For sufficiently small h, the numerical solution of Eq. (9) is positive and bounded.The proof concludes that I_n+1 lies in (0, N) for h below a specified bound.Source: §2 · p. 4

3. Relevant Properties of Numerical Solutions

This section analyzes the numerical solution’s mean-square convergence and the disease dynamics it generates, including extinction and persistence under differing conditions.

3.1. Convergence in Mean-square

  • The numerical solution converges in mean square to the analytic solution on [0, T] with global order 1/2.The proof uses positivity and boundedness of the numerical solution.Source: §3.1 · pp. 5, 7

3.2. Persistence and Extinction

  • Extinction is shown when the threshold is below one with small noise and in a separate high-noise regime.The section considers these alongside the case where the threshold exceeds one.Source: §3.2 · pp. 11–12
  • For the persistence case, the long-run lower and upper limits of the numerical solution bracket the unique positive root ξ+ in (0, N).The root is identified through a strictly increasing function.Source: §3.2 · pp. 12–13

4. Numerical Simulation

The simulations illustrate positivity preservation, computational efficiency, and the predicted extinction or persistence behavior under three conditions. They show that Wiener truncation removes negative solutions, the linearized scheme is faster with comparable error, and the disease outcomes vary with noise and parameter regimes.

Example 1. Effect of Wiener truncation on positivity preservation

  • Wiener truncation eliminates negative numerical solutions observed without truncation, particularly at larger noise intensity or step size.The comparison tests two parameter combinations and indicates improved robustness in long-time simulations.Source: §4 · p. 14, Figure 1 · p. 14

Example 2. Computational efficiency and accuracy of the linearized scheme

  • Eq. (9) reduces computation time by approximately 25% across tested step sizes, while its mean-square error remains very close to Eq. (8)'s.The ratio between the schemes’ mean-square errors is close to 1.Source: §4 · p. 14, Figure 2 · p. 14, Table 1 · p. 15

Example 3. Disease extinction under low noise

  • With R^S_0 < 1 and sufficiently small noise, all five tested trajectories decay to zero, and all 100 simulated paths converge to zero.Extinction occurs for all 25 tested combinations of the implicit parameters θ and ϑ.Source: §4 · pp. 15–16, Figure 3 · p. 15, Figure 4 · p. 15

Example 4. Disease extinction under high noise

  • Under high noise, the disease becomes extinct across five initial conditions and 100 paths, faster than in the low-noise case and independently of θ and ϑ.The tested noise intensity is σ = 0.1, satisfying the stated high-noise condition.Source: §4 · pp. 16–17, Figure 5 · p. 16, Figure 6 · p. 16

Example 5. Disease persistence

  • When R^S_0 > 1 under the tested regime, trajectories from five initial conditions converge to a positive endemic level, with the 100-path confidence interval remaining above zero after the initial transient.Varying θ and ϑ changes the endemic level slightly but preserves the persistence behavior.Source: §4 · pp. 17–18, Figure 7 · p. 17, Figure 8 · p. 17

5. Conclusion

The conclusion presents a fully implicit stochastic SIS scheme built from SSBE discretization, Wiener truncation, and linear implicit θ-methods in the diffusion term. It reports positivity and boundedness, half-order global convergence, and extinction or persistence across three conditions, with simulations illustrating these results.

Contributions

  • The model combines SSBE discretization, Wiener truncation, and linear implicit θ-methods in the diffusion term to produce the linear fully implicit scheme Eq. (9).The conclusion describes this as simplifying the model’s structure.Source: §5 · p. 18

Theoretical results

  • The numerical solution of Eq. (9) is positive and bounded, and its global convergence order is 1/2.The positivity and boundedness result follows after introducing Wiener truncation.Source: §5 · p. 18
  • Theorems 4–6 address disease extinction and persistence across three scenarios; when R^S_0 < 1 in a low-noise environment, the disease eventually goes extinct.The examples report extinction under the conditions of Theorems 4 and 5, and persistence under the conditions of Theorem 6.Source: §5 · p. 18

Numerical simulations

  • The simulations illustrate positivity under Wiener truncation, lower computation time with a small error difference for Eq. (9) versus Eq. (8), and the predicted extinction and persistence outcomes.Examples 3–5 correspond to the conditions of Theorems 4–6.Source: §5 · p. 18

Funding

The work was supported by three funding sources.

Funding

  • The Xinjiang Talent Development Fund supported the work under grant XJRC-2025-KJ-PY-KJLJ-108.Source: Funding · p. 18
  • The National Natural Science Foundation of China supported the work under grant 12561047.Source: Funding · p. 18
  • The 2025 Central Guidance for Local Science and Technology Development Fund supported the work under grant ZYYD2025ZY20.Source: Funding · p. 18

CRediT authorship contribution statement

The statement assigns drafting, methodology, and conceptualization to Zeyao Jin, and review and editing to Zhiming Li and Shenglong Chen.

Author contributions

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