Dynamics and bifurcation analysis of a state-dependent impulsive SIS model

Area of Science:

• Mathematical epidemiology
• Dynamical systems theory
• Public health modeling

Background:

• Previous research focused on SIR models with state-dependent impulses.
• Understanding disease dynamics with saturated treatment is crucial for effective control.
• Bifurcation analysis helps identify critical parameter values influencing disease spread.

Purpose of the Study:

• To propose and analyze a novel state-dependent impulsive SIS model.
• To investigate the existence and stability of periodic solutions.
• To explore bifurcation phenomena related to control parameters.

Main Methods:

• Utilizing ordinary differential equations (ODEs) to model disease dynamics.
• Recalling and analyzing the complex dynamics of ODE systems with saturated treatment.
• Applying Poincaré map definition and properties to study bifurcations.
• Systematically investigating bifurcations near semi-trivial periodic solutions.

Main Results:

• Established the existence and stability of the semi-trivial periodic solution.
• Demonstrated the existence and stability of positive periodic solutions through bifurcation analysis.
• Identified key control parameters influencing disease dynamics.

Conclusions:

• The proposed state-dependent impulsive SIS model offers insights into disease dynamics.
• Bifurcation analysis is a powerful tool for understanding disease control strategies.
• The findings contribute to the development of more effective public health interventions.