---
title: "Global analysis of a state-dependent impulsive SIRS model with nonlinear incidence"
id: "pubmed-42757765"
canonical_url: "https://medichelpline.com/clinical-feed/pubmed-42757765"
content_type: "clinical_feed_article"
specialty: "Infectious Disease"
source_name: "PubMed / NCBI"
source_url: "https://pubmed.ncbi.nlm.nih.gov/42757765/"
doi: "10.1080/17513758.2026.2732280"
published_at: "2026-09-18T16:30:02.000Z"
evidence_level: "Journal Article"
license: "CC-BY-NC-4.0 / Informational Use"
---
# Global analysis of a state-dependent impulsive SIRS model with nonlinear incidence
## Provenance & Clinical Metadata
- **Canonical URL:** https://medichelpline.com/clinical-feed/pubmed-42757765
- **Specialty:** [Infectious Disease](https://medichelpline.com/clinical-feed/infectious-disease.md)
- **Primary Source:** PubMed / NCBI
- **Source URL:** [Original Journal Publication](https://pubmed.ncbi.nlm.nih.gov/42757765/)
- **DOI:** [10.1080/17513758.2026.2732280](https://doi.org/10.1080%2F17513758.2026.2732280)
- **Published At:** 2026-09-18T16:30:02.000Z
- **Evidence Rating:** Journal Article
## Executive GIST (TL;DR)
- This study formulates a **state-dependent impulsive** control system derived from a nonlinear-incidence **SIRS** infectious-disease model to study regulatory mechanisms of disease control. - Interventions in the model are threshold-triggered: vaccination and isolation impulses occur when the susceptible population exceeds a preset level, producing a state-dependent impulsive differential system. - The authors first analyze the continuous-time ODE subsystem to establish its basic dynamical properties before adding impulses. - They construct and classify a **Poincaré map**, and analyze its domain, range and monotonicity to build a theoretical foundation for periodic solutions of the impulsive system. - Main dynamical outcomes depend on a parameter p tied to the force of infection: for p > 1 the **disease-free periodic solution** is globally asymptotically stable. - For 0 < p < 1 the system can exhibit **bistability** between a disease-free periodic solution and an endemic periodic solution; the long-term outcome depends sensitively on the initial infection size. - The bistability regime is interpreted as representing weak transmissibility and controllable low-level prevalence under threshold-triggered interventions. - The paper situates its analysis within nonlinear dynamics, Poincaré map theory, and impulsive differential equations applied to biological models, and lists relevant mathematical and epidemiological keywords. - MeSH indexing highlights communicable disease epidemiology, computer simulation, nonlinear dynamics, models in biology, incidence, and humans as focus areas.
## Clinical Analysis & Structured Key Points
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Epub 2026 Sep 18. # Global analyses of a state-dependent impulsive system for an SIRS model with a nonlinear incidence rate [Chenxi Huang](https://pubmed.ncbi.nlm.nih.gov/?term=Huang+C&cauthor_id=42757765)[ 1 ](https://pubmed.ncbi.nlm.nih.gov/42757765/#full-view-affiliation-1 "School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China."), [Sanyi Tang](https://pubmed.ncbi.nlm.nih.gov/?term=Tang+S&cauthor_id=42757765)[ 1 ](https://pubmed.ncbi.nlm.nih.gov/42757765/#full-view-affiliation-1 "School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China."), [Qianqian Zhang](https://pubmed.ncbi.nlm.nih.gov/?term=Zhang+Q&cauthor_id=42757765)[ 1 ](https://pubmed.ncbi.nlm.nih.gov/42757765/#full-view-affiliation-1 "School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China.") Affiliations Expand ### Affiliation * 1 School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China. * PMID: **42757765** * DOI: [ 10.1080/17513758.2026.2732280 ](https://doi.org/10.1080/17513758.2026.2732280) Item in Clipboard # Global analyses of a state-dependent impulsive system for an SIRS model with a nonlinear incidence rate Chenxi Huang et al. J Biol Dyn. 2026. Show details Display options Display options Format Abstract PubMed PMID J Biol Dyn Actions * [ Search in PubMed ](https://pubmed.ncbi.nlm.nih.gov/?term=%22J+Biol+Dyn%22%5Bjour%5D&sort=date&sort_order=desc) * [ Search in NLM Catalog ](https://www.ncbi.nlm.nih.gov/nlmcatalog?term=%22J+Biol+Dyn%22%5BTitle+Abbreviation%5D) * [ Add to Search ](https://pubmed.ncbi.nlm.nih.gov/42757765/) . 2026 Dec 31;20(1):2732280. doi: 10.1080/17513758.2026.2732280. Epub 2026 Sep 18. ### Authors [Chenxi Huang](https://pubmed.ncbi.nlm.nih.gov/?term=Huang+C&cauthor_id=42757765)[ 1 ](https://pubmed.ncbi.nlm.nih.gov/42757765/#short-view-affiliation-1 "School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China."), [Sanyi Tang](https://pubmed.ncbi.nlm.nih.gov/?term=Tang+S&cauthor_id=42757765)[ 1 ](https://pubmed.ncbi.nlm.nih.gov/42757765/#short-view-affiliation-1 "School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China."), [Qianqian Zhang](https://pubmed.ncbi.nlm.nih.gov/?term=Zhang+Q&cauthor_id=42757765)[ 1 ](https://pubmed.ncbi.nlm.nih.gov/42757765/#short-view-affiliation-1 "School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China.") ### Affiliation * 1 School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, China. * PMID: **42757765** * DOI: [ 10.1080/17513758.2026.2732280 ](https://doi.org/10.1080/17513758.2026.2732280) Item in Clipboard Cite Display options Display options Format Abstract PubMed PMID ## Abstract This paper establishes a state-feedback control system based on a nonlinear incidence SIRS model to explore the regulatory mechanism of infectious disease control. We develop a state-dependent impulsive model with threshold-triggered vaccination and isolation interventions once the susceptible population exceeds a predefined level. After investigating the basic properties of the ODE subsystem, we define and classify the Poincaré map and analyze its domain, range and monotonicity, which provides a theoretical basis for studying positive periodic solutions of the impulsive system under different ODE dynamics. The results show that the disease-free periodic solution is globally asymptotically stable for p>1, where p is a parameter directly related to the force of the infection. When 0<p<1, however, the model may exhibit bistability between the disease-free periodic solution and an endemic periodic solution, where the final outcome depends sensitively on the initial infection size, which is consistent with the weak transmissibility and controllable low-level prevalence. **Keywords:** 34A37; 34A38; 34C25; 34D20; 92D30; Poincaré map; SIRS model; infectious disease model; state-dependent impulsive differential equations. [PubMed Disclaimer](https://pubmed.ncbi.nlm.nih.gov/disclaimer/) ## Similar articles * [ Analysis of a mathematical model with nonlinear susceptibles-guided interventions. ](https://pubmed.ncbi.nlm.nih.gov/31499725/) Li Q, Xiao YN.Li Q, et al.Math Biosci Eng. 2019 Jun 17;16(5):5551-5583. doi: 10.3934/mbe.2019276.Math Biosci Eng. 2019.PMID: 31499725 * [ Dynamics and bifurcation analysis of a state-dependent impulsive SIS model. ](https://pubmed.ncbi.nlm.nih.gov/34149834/) Wang J.Wang J.Adv Differ Equ. 2021;2021(1):287. doi: 10.1186/s13662-021-03436-3. Epub 2021 Jun 12.Adv Differ Equ. 2021.PMID: 34149834Free PMC article. * [ Vaccination threshold size and backward bifurcation of SIR model with state-dependent pulse control. ](https://pubmed.ncbi.nlm.nih.gov/30017943/) Zhang Q, Tang B, Tang S.Zhang Q, et al.J Theor Biol. 2018 Oct 14;455:75-85. doi: 10.1016/j.jtbi.2018.07.010. 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