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Modelling the impact of precaution on disease dynamics and its evolution

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In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public’s level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease’s eventual vanishing for the framework model under the assumption that the basic reproduction number \(R_0<1\). For \(R_0>1\), we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing “adopting the best response” with “adapting toward the best response”, we also explore the adaptive long-term dynamics.

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The authors would like to thank (a) The Mathematical Research Center (MRC) of the University of Pittsburgh, which is funded by the Dietrich School of Arts and Sciences and (b) NSF conference Grant DMS- 2304988 for the support to the authors for their participation in the 2023 Mathematical Ecology Conference in which the authors presented this work and received feedback, which helped improve the paper.

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Correspondence to Xingfu Zou.

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Dedicated to Professor Glenn Webb in the occasion of his 80th birthday. Partially supported by NSERC of Canada (RGPIN-2022-04744).

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Cheng, T., Zou, X. Modelling the impact of precaution on disease dynamics and its evolution. J. Math. Biol. 89, 1 (2024).

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