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A Mathematical Study of an SIS Epidemic Model: Global Asymptotic Stability Analysis and Construction of Nonstandard Numerical Schemes
Manh Tuan Hoang
TL;DR
The paper addresses the lack of a theoretical global-stability result for the disease-endemic equilibrium in a saturating-contact SIS model. It establishes global stability results and constructs first- and second-order NSFD schemes; numerical results agree with the theoretical predictions.
Problem
The DEE’s global asymptotic stability had not been established theoretically, although its stability characterizes long-term disease persistence.
Source: §1 · p. 3Method
The authors use Lyapunov and planar-system stability arguments and construct first- and second-order NSFD schemes with weighted nonlocal approximations and renormalized denominator functions.
Source: §1 · pp. 3–4Results
The analysis establishes GAS results for the model’s equilibria, and the proposed schemes preserve positivity and asymptotic stability properties for arbitrary step sizes; numerical results agree with theory.
Source: §5 · p. 22Takeaways & Limitations
The authors suggest the GAS results may provide a basis for extensions to more general saturating contact rates, and the approach may apply to other real-world mathematical models.
Source: §5 · p. 22For first-order schemes of form (3.3), GAS preservation is suggested by simulations when the LAS-preservation condition holds, rather than established theoretically.
Source: §3.1 · p. 10
Abstract
from arXiv · showhide
The aim of this work is to provide a rigorous mathematical analysis of a well-known SIS epidemic model with a saturating contact rate. First, we establish the global asymptotic stability (GAS) of the model's equilibria by employing a suitable Lyapunov function in combination with the Poincaré--Bendixson theorem and the Bendixson--Dulac criterion. The resulting GAS results improve upon previous findings for this SIS model and may also be applicable to its extensions with more general saturating contact rates. Second, we construct families of first- and second-order nonstandard finite difference (NSFD) schemes that preserve the positivity and asymptotic stability properties of the SIS model for arbitrary step sizes. All these schemes are formulated within Mickens' framework; however, compared with their first-order counterparts, the second-order schemes employ a more elaborate construction that combines a weighted nonlocal approximation of the right-hand side with suitably renormalized denominator functions. The weighted nonlocal approximation guarantees dynamic consistency, whereas the denominator functions ensure second-order convergence. Finally, we conduct numerical experiments to validate the theoretical results and demonstrate the advantages of the proposed NSFD schemes. The numerical results are in good agreement with the theoretical results. The approach developed in this work is applicable not only to other epidemiological systems but also, more generally, to mathematical models arising in a wide range of real-world applications.
1 Introduction
The introduction identifies the unproved global stability of the disease-endemic equilibrium as an important gap and sets out a stability analysis alongside the construction of dynamically consistent numerical schemes. The authors also report numerical experiments supporting their theoretical findings.
1 Introduction
- Earlier work established DFE global stability for R0 < 1 and DEE local stability for R0 > 1, but had not theoretically established DEE global stability, which bears on long-term disease persistence.The paper identifies establishing DEE global asymptotic stability as its first objective.Source: §1 · p. 3
- The authors aim to establish DEE global asymptotic stability and give two alternative proofs of DFE global asymptotic stability using a Lyapunov function and an R0-based result.Both DFE proofs differ from the proof in.Source: §1 · p. 3
- The proposed first- and second-order NSFD schemes preserve positivity and asymptotic stability properties of the SIS model for arbitrary step sizes.The introduction contrasts dynamic consistency with the usual first-order accuracy of many NSFD schemes.Source: §1 · p. 3
- The second-order schemes combine a weighted nonlocal approximation, which guarantees dynamic consistency, with renormalized denominator functions that ensure second-order convergence.Their construction follows an approach developed in.Source: §1 · p. 4
- Numerical experiments are reported to support and illustrate the theoretical findings, with results in good agreement with the analysis.Source: §1 · p. 4
2 Complete Global Asymptotic Stability of the SIS Model
This section establishes global asymptotic stability for the model’s endemic equilibrium when it exists and for its disease-free equilibrium when R0 < 1. The proofs combine positivity and planar-system arguments for the endemic case with two approaches to the disease-free case.
2 Complete Global Asymptotic Stability of the SIS Model
- For S(0) ≥ 0 and I(0) > 0, both susceptible and infectious populations remain positive for all t > 0.Source: §2 · p. 5
- The DEE is globally asymptotically stable whenever it exists, provided I(0) > 0.The proof uses the Dulac–Bendixson criterion to rule out periodic orbits or graphics in the first quadrant, then excludes the DFE from the omega limit set.Source: §2 · pp. 5–6
- The DFE is globally asymptotically stable if R0 < 1, as established by a Lyapunov function whose derivative is nonpositive and vanishes only at the equilibrium.Source: §2 · pp. 6–7
- A second proof establishes DFE global stability for R0 < 1 by applying a result based on the role of the basic reproduction number in stability analysis.The proof verifies the result’s conditions before applying it.Source: §2 · pp. 7–8
3 Construction of Dynamically Consistent NSFD Schemes
This section constructs first- and second-order NSFD schemes for the SIS model. The second-order schemes combine weighted discrete terms with denominator functions to balance dynamic consistency and convergence order.
3.1 Construction of First-Order NSFD Schemes
- First-order schemes preserve nonnegative approximations for arbitrary step sizes when denominator condition (3.4) holds.This is stated for nonnegative initial data.Source: §3.1 · p. 9
- Under (3.4), first-order schemes retain the model’s equilibria and reproduction number, with local stability preservation governed by parameter-dependent thresholds.The scheme’s basic reproduction number coincides with the model’s.Source: §3.1 · p. 9
- First-order schemes are first-order accurate, while their preservation of global asymptotic stability is suggested by simulations rather than established analytically.The authors describe proving global stability for these schemes as challenging.Source: §3.1 · p. 10
3.2 Construction of Second-Order NSFD Schemes
- Second-order schemes use weighted discrete terms to ensure dynamic consistency and denominator functions chosen to achieve second-order convergence.Their truncation error is O(∆t^3).Source: §3.2 · pp. 10–11
- Second-order schemes preserve nonnegative approximations for arbitrary step sizes under the stated conditions and have the same equilibrium points as the continuous model.The positivity result assumes nonnegative initial data.Source: §3.2 · pp. 10–11
- A parameter-dependent threshold is derived for second-order schemes to preserve local asymptotic stability.The schemes’ denominator condition also gives truncation error O(∆t^3).Source: §3.2 · p. 11
4 Numerical Examples
The numerical examples compare the accuracy and dynamics of first- and second-order NSFD schemes with explicit Euler, and illustrate the model’s global stability results. The NSFD schemes’ numerical behavior agrees with the theoretical analysis, while explicit Euler becomes unstable for the selected step sizes.
4 Numerical Examples
- The numerical examples use φ(∆t) = 1−e^-∆t for the first-order scheme and weights τ1 = τ2 = 1 for the second-order scheme.The schemes are tested with parameters specified for their error analysis and dynamics comparisons.Source: §4 · pp. 12–14
- The first- and second-order NSFD schemes’ error estimates and rates of convergence are consistent with the theoretical analysis.Errors were computed against a reference solution generated by four-stage Runge–Kutta with ∆t = 10^-6 over [0, 1].Source: §4 · p. 13, Table 1 · p. 13, Table 2 · p. 14
- Explicit Euler produces unstable, oscillatory approximations, whereas both NSFD schemes reproduce the continuous-time dynamics for the selected step sizes, including relatively large ones.The parameter set has R0 = 3.7507 > 1, for which the DEE is globally asymptotically stable.Source: §4 · pp. 14–15, Figure 1 · p. 15, Figure 2 · p. 16
- Phase-plane simulations show the DFE globally asymptotically stable when R0 < 1 and the DEE globally asymptotically stable when R0 > 1.The examples use parameter sets for both R0 regimes and plot trajectories from different initial conditions.Source: §4 · p. 19, Table 4 · p. 19, Figure 5 · p. 20
5 Conclusions and Discussions
The paper establishes stability results for the SIS model and constructs first- and second-order NSFD schemes that preserve positivity and asymptotic stability for arbitrary step sizes. Numerical experiments agree with the theoretical predictions, while extensions to other models and higher-order methods are proposed for future work.
5 Conclusions and Discussions
- The analysis establishes global asymptotic stability for the SIS model and may provide a basis for extensions to more general saturating contact rates.It uses a suitable Lyapunov function together with the Poincaré–Bendixson theorem and the Bendixson–Dulac criterion.Source: §5 · p. 22
- First- and second-order NSFD schemes preserve positivity and asymptotic stability for arbitrary step sizes.The second-order schemes combine a weighted nonlocal approximation with renormalized denominator functions; these respectively guarantee dynamic consistency and second-order convergence.Source: §5 · p. 22
- Numerical experiments show good agreement with theoretical predictions and demonstrate advantages of the proposed second-order NSFD schemes.The authors note potential applicability to other epidemiological systems and mathematical models in real-world applications.Source: §5 · p. 22
- Future work will extend the approach to other epidemiological models and develop efficient higher-order methods that preserve continuous-time dynamical properties.Source: §5 · p. 22