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Floquet Stability Analysis of Periodic Epidemic Models

Author(s) Neeraj Kumar
Country India
Abstract Seasonally forced epidemic models are nonautonomous dynamical systems, in which the contact rate and the transmission process change in time. For these models, the constant-coefficient eigenvalue analysis of autonomous theory no longer applies, and stability should be read off from the behaviour of the solutions in one forcing period. In this article we develop Floquet theory, the spectral theory of linear periodic differential equations, as the natural language for stability analysis of periodic epidemic models. We introduce the fundamental matrix, the monodromy matrix, and the Floquet multipliers, together with the classical criterion that every Floquet multiplier must remain inside the unit circle. Linearizing the infected subsystem in terms of the disease-free state leads to a periodic linear system whose monodromy matrix determines the invasion. We now describe the periodic next-generation operator and the associated periodic basic reproduction number R_0, and we state the central correspondence. : R_0 < 1"," = 1"," > 1 exactly when the dominant Floquet multiplier ρ satisfies ρ < 1"," = 1"," > 1, respectively. This yields the threshold theorem: the disease-free periodic state is locally asymptotically stable for R_0 < 1 and unstable for R_0 > 1, in which case the disease is uniformly persistent. We also discuss the Floquet stability of positive periodic (endemic) solutions and the resonance phenomena that periodic forcing can produce. A seasonally forced SIR model is worked out in closed form to illustrate the theory. The results surveyed are classical; detailed proofs may be found in the references.
Keywords Floquet theory; Floquet multipliers; monodromy matrix; periodic epidemic models; seasonal forcing; basic reproduction number; threshold dynamics; stability analysis.
Field Mathematics
Published In Volume 8, Issue 4, April 2017
Published On 2017-04-06
DOI https://doi.org/10.71097/IJTAS.v8.i4.1439
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Crossref DOI prefix of IJTAS is 10.71097/IJTAS

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