Some results in Floquet theory, with application to periodic epidemic models

Abstract

In this paper, we present several new results to the classical Floquet theory on the study of differential equations with periodic coefficients. For linear periodic systems, the Floquet exponents can be directly calculated when the coefficient matrices are triangular. Meanwhile, the Floquet exponents are eigenvalues of the integral average of the coefficient matrices when they commute with their antiderivative matrices. For the stability analysis of constant and nontrivial periodic solutions of nonlinear differential equations, we derive a few results based on linearization. We also briefly discuss the properties of Floquet exponents for delay linear periodic systems. To demonstrate the application of these analytical results, we consider a new cholera epidemic model with phage dynamics and seasonality incorporated. We conduct mathematical analysis and numerical simulation to the model with several periodic parameters.

Keywords:

• Floquet exponents
• stability
• epidemic models

AMS Subject Classifications:

• 34C25
• 34D20
• 92D30

Acknowledgements

The authors are grateful to Michael Li at University of Alberta for helpful discussion. The authors would also like to thank the anonymous referees for helpful comments.

Notes

J.P. Tian and J. Wang acknowledge partial support from the National Science Foundation under [grant number 1216907] and [grant number 1216936], respectively.