Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection

Francesco Salvarani; Gabriel Turinici; Francesco Salvarani; Gabriel Turinici

doi:10.3934/mbe.2018028

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 3: 629-652. doi: 10.3934/mbe.2018028

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Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection

Francesco Salvarani ¹,
Gabriel Turinici ^{2
,
,}

1.
Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France, & Università degli Studi di Pavia, Dipartimento di Matematica, 27100 Pavia, Italy,
2.
Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France, & Institut Universitaire de France, Paris, France

Received: 01 March 2017 Accepted: 29 July 2017 Published: 01 June 2018
MSC : Primary: 92D30; Secondary: 92C42, 60J20, 91A13

We analyze a model of agent based vaccination campaign against influenza with imperfect vaccine efficacy and durability of protection. We prove the existence of a Nash equilibrium by Kakutani's fixed point theorem in the context of non-persistent immunity. Subsequently, we propose and test a novel numerical method to find the equilibrium. Various issues of the model are then discussed, such as the dependence of the optimal policy with respect to the imperfections of the vaccine, as well as the best vaccination timing. The numerical results show that, under specific circumstances, some counter-intuitive behaviors are optimal, such as, for example, an increase of the fraction of vaccinated individuals when the efficacy of the vaccine is decreasing up to a threshold. The possibility of finding optimal strategies at the individual level can help public health decision makers in designing efficient vaccination campaigns and policies.
- Mean Field Games,
- SIR model,
- vaccination persistence,
- durability of protection,
- limited immunity,
- vaccine effectiveness,
- influenza
Citation: Francesco Salvarani, Gabriel Turinici. Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 629-652. doi: 10.3934/mbe.2018028

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Abstract

We analyze a model of agent based vaccination campaign against influenza with imperfect vaccine efficacy and durability of protection. We prove the existence of a Nash equilibrium by Kakutani's fixed point theorem in the context of non-persistent immunity. Subsequently, we propose and test a novel numerical method to find the equilibrium. Various issues of the model are then discussed, such as the dependence of the optimal policy with respect to the imperfections of the vaccine, as well as the best vaccination timing. The numerical results show that, under specific circumstances, some counter-intuitive behaviors are optimal, such as, for example, an increase of the fraction of vaccinated individuals when the efficacy of the vaccine is decreasing up to a threshold. The possibility of finding optimal strategies at the individual level can help public health decision makers in designing efficient vaccination campaigns and policies.

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