Dynamical behavior of epidemiological models with nonlinear incidence rates

Area of Science:

• Epidemiology
• Mathematical Biology
• Dynamical Systems Theory

Background:

• Traditional epidemiological models often use bilinear incidence rates (lambda IS).
• Nonlinear incidence rates (lambda IpSq) introduce more complex dynamics.
• Understanding these dynamics is crucial for predicting disease spread.

Purpose of the Study:

• To compare the dynamical behaviors of epidemiological models with nonlinear incidence rates (lambda IpSq) versus bilinear incidence rates (lambda IS).
• To investigate the influence of parameters (p, lambda, q) and initial conditions on disease dynamics.
• To explore the potential for periodic solutions via Hopf bifurcation.

Main Methods:

• Analysis of nonlinear epidemiological models with incidence rate lambda IpSq.
• Comparison with models featuring bilinear incidence rate lambda IS.
• Phase space analysis to identify multiple attractive basins.
• Investigation of Hopf bifurcation for periodic solutions.

Main Results:

• Models with nonlinear incidence rates (lambda IpSq) display a broader range of dynamical behaviors compared to bilinear models (lambda IS).
• Model dynamics are primarily governed by parameters p and lambda, and secondarily by q.
• The eventual extinction or persistence of a disease can be contingent on both model parameters and initial conditions due to multiple attractive basins.
• Periodic solutions can emerge at critical parameter values through Hopf bifurcation.

Conclusions:

• Nonlinear incidence rates significantly enhance the complexity of epidemiological models.
• Initial conditions play a critical role in disease trajectory, alongside model parameters.
• Hopf bifurcation offers a mechanism for the emergence of endemic periodic disease dynamics.