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Periodicity in an epidemic model with a generalized non-linear incidence
1Institute for Biodiagnostics, National Research Council Canada, 435 Ellice Avenue, Winnipeg, Man., Canada R3B 1Y6. murray.alexander@nrc-cnrc.gc.ca
This study introduces a simple SIV epidemic model with a non-linear incidence rate, revealing complex dynamics like bistability and periodicity. The basic reproductive number remains constant, and Hopf bifurcations are analyzed for disease control insights.
Area of Science:
- Mathematical epidemiology
- Dynamical systems theory
- Infectious disease modeling
Background:
- Existing epidemic models often use simplified incidence rates.
- Non-linear incidence can introduce complex behaviors not captured by linear models.
- Understanding these dynamics is crucial for predicting and controlling disease spread.
Purpose of the Study:
- To develop and analyze a simple Susceptible-Infected-Vaccinated (SIV) epidemic model.
- To investigate the impact of a generalized non-linear incidence rate on epidemic dynamics.
- To explore phenomena like bistability, periodicity, and Hopf bifurcations.
Main Methods:
- Development of a compartmental SIV model with a generalized non-linear incidence rate.
- Application of Poincaré index theory for detailed dynamical analysis.
- Computation of the first Lyapunov coefficient to classify Hopf bifurcations.
- Numerical simulations with realistic parameters for infectious diseases of childhood.
Main Results:
- Non-linearity in the incidence rate generates vital dynamics (bistability, periodicity) without external forcing.
- The basic reproductive number is independent of the specific functional form of the non-linear incidence.
- Hopf bifurcations (forward, backward, subcritical) were identified under specific conditions, leading to limit cycles.
Conclusions:
- Non-linear incidence rates significantly enrich epidemic model dynamics, leading to complex behaviors.
- The model provides a framework for understanding disease persistence and oscillations.
- Findings are applicable to real-world infectious diseases, aiding in control strategy development.
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