Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Seasonal forcing and exponential threshold incidence in cholera dynamics

Pages: 2261 - 2290, Volume 22, Issue 6, August 2017      doi:10.3934/dcdsb.2017095

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Jinhuo Luo - College of Information Technology, Shanghai Ocean University, Shanghai 201306, China (email)
Jin Wang - Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States (email)
Hao Wang - Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada (email)

Abstract: We propose a seasonal forcing iSIR (indirectly transmitted SIR) model with a modified incidence function, due to the fact that the seasonal fluctuations can be the main culprit for cholera outbreaks. For this nonautonomous system, we provide a sufficient condition for the persistence and the existence of a periodic solution. Furthermore, we provide a sufficient condition for the global stability of the periodic solution. Finally, we present some simulation examples for both autonomous and nonautonomous systems. Simulation results exhibit dynamical complexities, including the bistability of the autonomous system, an unexpected outbreak of cholera for the nonautonomous system, and possible outcomes induced by sudden weather events. Comparatively the nonautonomous system is more realistic in describing the indirect transmission of cholera. Our study reveals that the relative difference between the value of immunological threshold and the peak value of bacterial biomass is critical in determining the dynamical behaviors of the system.

Keywords:  Cholera, nonautonomous, stability, seasonal forcing, immunological threshold, persistence, periodic solution, exponential incidence, sudden events.
Mathematics Subject Classification:  93A30, 37B55, 34D20, 34D23, 97M10, 34C25, 37D35, 34C60.

Received: March 2015;      Revised: January 2017;      Available Online: March 2017.