Global dynamics of tick-borne diseases

Area of Science:

• Epidemiology
• Mathematical Biology
• Disease Modeling

Background:

• Tick-borne diseases exhibit seasonal peaks due to tick life cycle dynamics.
• Disease transmission is primarily by adult ticks, necessitating time-delayed models.
• Existing models may not fully capture the impact of tick life stages on disease spread.

Purpose of the Study:

• To develop and analyze a tick-borne disease model incorporating a nonlinear incidence rate.
• To implement a piecewise constant delay representing the adult tick biting period.
• To investigate the global asymptotic stability of disease-free and endemic equilibria.

Main Methods:

• Development of a mathematical model for tick-borne disease transmission.
• Application of a nonlinear incidence rate to reflect realistic transmission dynamics.
• Utilizing Lyapunov functions and the Lyapunov-LaSalle technique for stability analysis.
• Incorporation of a piecewise constant delay to simulate tick maturation and biting periods.

Main Results:

• The global asymptotic stability of the disease-free equilibrium was mathematically proven.
• The global asymptotic stability of the endemic equilibrium was also demonstrated.
• Theoretical stability results were corroborated through numerical simulations.

Conclusions:

• The developed tick-borne disease model accurately captures disease dynamics influenced by tick life cycles.
• The model predicts stable disease-free and endemic states under specific conditions.
• Mathematical stability analysis provides a robust framework for understanding disease persistence and eradication.