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Return-to-Normality in a Piecewise Deterministic Markov SIR+V Model with Pharmaceutical and Non-pharmaceutical Interventions

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Abstract

This paper studies a two-scales Markov compartmental model (SIRS) with demography, vaccination, reinfections and waning immunity, and with external control inputs of pharmaceutical (vaccination) and non-pharmaceutical (social distancing) type. The demography events are considered to be rare and contribute to a jump mechanism with trajectory depending jump intensity ultimately providing a piecewise deterministic Markov model. We study well-posedness of the system with time-varying total population and the herd immunity zone when intensive care units constraints are enforced. The third contribution describes, via Zubov’s method, the initial configurations that can be driven to the immunity zone. Numeric illustrations are provided using data from INSEE (France) on COVID-19.

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Data Availability

The data that support the findings of this study are openly available from INSEE (Institut national de la statistique et des études économiques) at https://www.insee.fr/fr/statistiques/. under the references: 2381380, 2383440.

Notes

  1. This value is obtained by averaging over 2020–2021 period.

  2. This value is obtained by averaging over 2017–2019 period prior to the epidemics.

  3. This value is obtained by averaging over 2020–2021 period from which the previous \(\mu _-\) is substracted.

  4. The continuity of \(\hat{Q}\) is also true, but this condition on \(\hat{\lambda } \hat{Q}\) suffices for the constructions, see also [21, Remark 1].

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Funding

The authors acknowledge support from the NSF of Shandong Province (NO. ZR202306020015), the National Key R and D Program of China (NO. 2018YFA0703900), and the NSF of P.R. China (NO. 12031009).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University Weihai, Weihai, 264209, People’s Republic of China

    Dan Goreac, Juan Li, Yi Wang & Zhengyang Wang

  2. LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, 77447, Marne-la-Valleée, France

    Dan Goreac

  3. Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, 266237, People’s Republic of China

    Juan Li

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  1. Dan Goreac
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Correspondence to Dan Goreac or Juan Li.

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Goreac, D., Li, J., Wang, Y. et al. Return-to-Normality in a Piecewise Deterministic Markov SIR+V Model with Pharmaceutical and Non-pharmaceutical Interventions. Appl Math Optim 89, 19 (2024). https://doi.org/10.1007/s00245-023-10087-1

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