Multistability of a Two-Dimensional Map Arising in an Influenza Model

Abstract

In this paper, we propose and analyze a nonsmoothly two-dimensional map arising in a seasonal influenza model. Such map consists of both linear and nonlinear dynamics depending on where the map acts on its domain. The map exhibits a complicated and unpredictable dynamics such as fixed points, period points, chaotic attractors, or multistability depending on the ranges of a certain parameters. Surprisingly, bistable states include not only the coexistence of a stable fixed point and stable period three points but also that of stable period three points and a chaotic attractor. Among other things, we are able to prove rigorously the coexistence of the stable equilibrium and stable period three points for a certain range of the parameters. Our results also indicate that heterogeneity of the population drives the complication and unpredictability of the dynamics. Specifically, the most complex dynamics occur when the underlying basic reproduction number with respect to our model is an intermediate value and the large portion of the population in the same compartment changes in states the following season.

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“…The salient point to note is that mul stability is not an excep on, but rather a prevalent characteris c in complex systems, which o en exhibit high dimensionality and nonlinear dynamics in a diverse span of fields including ecology, neuroscience, and meteorology. While, high dimensionality is not a prerequisite for mul stability, with systems with as few as two variables being able to exhibit this feature 52 , increasing the system's dimensionality tends to enhance the likelihood of observing mul ple stable states, even in systems generated through stochas c processes. Emerging research increasingly supports the idea that the brain, behavior, and their disorders can transi on between dis nct yet stable states, akin to how psychiatric condi ons may represent alterna ve stable states to norma ve behavior 40,53,54 .…”

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“…( 2019 ), Huang et al. ( 2022a ), Yakubu and Franke ( 2006 ) and the works cited therein] are capable of generating complicated and unpredictable dynamics such as period states, bistable periodic states, or chaotic attractors.…”