This work develops a control-oriented mathematical model based on a nonlinear-incidence SIRS framework to examine regulatory mechanisms for infectious disease control. The model couples a standard continuous-time ODE SIRS subsystem with state-dependent impulses: when the susceptible population surpasses a predefined threshold, instantaneous interventions are applied in the form of vaccination and isolation pulses. The impulses are triggered by the state of the system rather than by fixed time schedules, producing a state-dependent impulsive differential equation system driven by a nonlinear incidence rate.
The formulation emphasizes how threshold-triggered public-health actions can be represented within dynamical systems and control theory. The source describes the inclusion of a parameter p that is directly related to the force of infection and plays a central role in the subsequent dynamical analysis.
Before analyzing the full impulsive system, the authors study the basic properties of the continuous ODE subsystem. This step establishes the existence and qualitative behavior of flows between impulses, which is necessary for constructing return maps and for understanding how impulses interact with the continuous dynamics.
The ODE analysis provides the groundwork for later classification of periodic orbits and for determining invariant sets of the flow. Specific technical properties of the ODE subsystem are used to characterize how trajectories evolve up to the moment when the susceptible population reaches the intervention threshold.
A principal methodological element is the definition and classification of a Poincaré map (return map) associated with the impulsive system. The paper analyzes the Poincaré map's domain and range and studies its monotonicity properties. These map characteristics form the theoretical basis for proving the existence and stability of positive periodic solutions of the impulsive system under different regimes of the underlying ODE dynamics.
By examining the Poincaré map, the authors translate the continuous-plus-impulsive dynamics into discrete iterations, enabling the application of fixed-point and monotone-map techniques to determine long-term behaviors such as periodic solutions and their stability.
The analysis identifies a clear threshold condition in terms of the parameter p, which is linked to the force of infection. For p > 1, the study shows that the disease-free periodic solution of the impulsive system is globally asymptotically stable. In this regime, regardless of initial conditions (within the biologically relevant domain reported in the source), trajectories converge to the disease-free periodic orbit under the model's threshold-triggered vaccination and isolation interventions.
This global stability result indicates that when the effective transmission parameter p exceeds unity under the model assumptions, the combined continuous dynamics and state-dependent impulses reliably drive the system toward eradication of infection in a periodic fashion determined by the impulse mechanism.
When 0 < p < 1, the model can exhibit bistability: both the disease-free periodic solution and an endemic periodic solution may coexist. In this parameter regime the eventual outcome depends sensitively on the initial size of the infected population. Small initial infections may decay and lead to the disease-free periodic orbit, while larger initial infections can push the system toward a sustained endemic periodic state.
The paper interprets this bistability as consistent with situations of weak transmissibility (lower p) where low-level prevalence is controllable but outcomes remain sensitive to initial conditions and the timing or magnitude of state-triggered interventions.
The modeled threshold-triggered vaccination and isolation represent a class of interventions that are responsive to the susceptible population level rather than scheduled pulses. The results suggest that the effectiveness of such state-dependent control depends strongly on the transmission-related parameter p. In regimes where p > 1 the interventions ensure global convergence to a disease-free periodic regime, whereas lower p values admit multiple attractors and therefore require attention to initial conditions and possibly to the design of thresholds or pulse strengths to avoid undesirable endemic outcomes.
The conclusions point toward the importance of coupling nonlinear incidence dynamics with realistic, state-dependent intervention rules when using mathematical models to inform control policy. The model provides a theoretical framework to examine how threshold-based vaccination and isolation may produce qualitatively different long-term epidemiological outcomes depending on transmissibility.
The study is framed within nonlinear dynamics and impulsive differential equation theory, using tools such as the Poincaré map to analyze periodic behavior. Relevant mathematical classifications and keywords given in the source include Poincaré map, state-dependent impulsive differential equations, and SIRS model with nonlinear incidence. MeSH terms and indexing noted for the article emphasize communicable disease epidemiology, computer simulation, nonlinear dynamics, models in biology, and incidence. Specific numerical details, proofs, and technical lemmas used in the paper are reported in the original article but are not duplicated verbatim here.