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A vertically transmitted epidemic model with two state-dependent pulse controls.

Xunyang Wang1,2, Canyun Huang1, Yuanjie Liu1

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This study introduces an SIR epidemic model incorporating vertical transmission and pulse controls. Mathematical analysis and simulations demonstrate the potential to control epidemics by reducing vertical transmission and applying timely interventions.

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SIR epidemic modelorbital stabilityperiodic solutionstate-dependent pulsesvertical transmission

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Infectious Disease Dynamics

Background:

  • Vertical transmission, the mother-to-offspring spread of pathogens, is a significant factor in disease dynamics.
  • Existing epidemic models often do not fully account for the complexities of vertical transmission and external control strategies.

Purpose of the Study:

  • To develop and analyze a mathematical SIR (Susceptible-Infectious-Recovered) model that includes vertical transmission.
  • To investigate the impact of two state-dependent pulse controls on epidemic dynamics.
  • To determine the conditions for the existence and stability of periodic solutions in the model.

Main Methods:

  • Formulation of an SIR epidemic model with vertical transmission.
  • Application of the Poincare map and qualitative analysis.
  • Utilizing a Poincare criterion analogue for stability analysis.
  • Numerical simulations and phase diagram generation.

Main Results:

  • Proof of the existence and stability of positive order-1 and order-2 periodic solutions.
  • Numerical simulations confirm the analytical findings and demonstrate model behavior.
  • Validation of the effectiveness of state-dependent pulse controls in managing epidemic spread.

Conclusions:

  • The proposed SIR model with vertical transmission and pulse controls provides a framework for understanding epidemic dynamics.
  • State-dependent pulse controls are effective in mitigating epidemic spread.
  • Reducing vertical transmission is crucial for comprehensive epidemic control strategies.