Imitation dynamics in vaccination under partial and temporary immune protection

Paulo Doutor; Ricardo Castelhano; Maria do Céu Soares; Bob Kooi; Paula Patricio; Paulo Doutor; Ricardo Castelhano; Maria do Céu Soares; Bob Kooi; Paula Patricio

doi:10.3934/mbe.2026048

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 5: 1315-1339. doi: 10.3934/mbe.2026048

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Imitation dynamics in vaccination under partial and temporary immune protection

1.
Center for Mathematics and Applications (NOVA Math), NOVA FCT, Caparica, 2829-516, Portugal
2.
Department of Mathematics, NOVA FCT, Caparica, 2829-516, Portugal
3.
Faculty of Science, Vrije Universiteit, Amsterdam, The Netherlands
4.
BCAM - Basque Center for Applied Mathematics, Bilbao, Spain

Received: 31 July 2025 Revised: 19 February 2026 Accepted: 18 March 2026 Published: 02 April 2026

Vaccination has been the most effective strategy to contain and control infectious disease epidemics. Over the past decades, its success and the consequent herd immunity changed public perceptions on the disease and vaccination's expected costs which, in some cases, resulted in vaccine hesitancy. This work extends the classical SIR model by incorporating both temporary and partial immunity, along with individual vaccination decisions driven by an imitation dynamics approach based on perceived expected costs. Our methodology relied on the stability analysis of the equilibria and the investigation of bifurcation phenomena within the model. To this end, we employed classical local asymptotic techniques, complemented by numerical continuation methods and the computation of Lyapunov exponents. The analysis of the model revealed an interplay between incomplete immune protection and imitation parameters, with rich dynamics involving oscillatory and chaotic behavior for low waning immunity rates and/or high protection against reinfection, and for high social-learning rate. Furthermore, exploring temporary immunity led to the conclusion that for diseases with shorter immunity periods and for vaccines with high relative expected cost, people will rapidly decline any vaccination efforts. These findings emphasize the importance of human behavior for a population in an epidemic landscape, and may provide insight on how to adapt and improve public health interventions.
- Epidemiological models,
- suboptimal immunity,
- vaccination strategies,
- imitation dynamics,
- complex dynamics,
- Lyapunov exponents,
- bifurcation analysis
Citation: Paulo Doutor, Ricardo Castelhano, Maria do Céu Soares, Bob Kooi, Paula Patricio. Imitation dynamics in vaccination under partial and temporary immune protection[J]. Mathematical Biosciences and Engineering, 2026, 23(5): 1315-1339. doi: 10.3934/mbe.2026048

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Abstract

Vaccination has been the most effective strategy to contain and control infectious disease epidemics. Over the past decades, its success and the consequent herd immunity changed public perceptions on the disease and vaccination's expected costs which, in some cases, resulted in vaccine hesitancy. This work extends the classical SIR model by incorporating both temporary and partial immunity, along with individual vaccination decisions driven by an imitation dynamics approach based on perceived expected costs. Our methodology relied on the stability analysis of the equilibria and the investigation of bifurcation phenomena within the model. To this end, we employed classical local asymptotic techniques, complemented by numerical continuation methods and the computation of Lyapunov exponents. The analysis of the model revealed an interplay between incomplete immune protection and imitation parameters, with rich dynamics involving oscillatory and chaotic behavior for low waning immunity rates and/or high protection against reinfection, and for high social-learning rate. Furthermore, exploring temporary immunity led to the conclusion that for diseases with shorter immunity periods and for vaccines with high relative expected cost, people will rapidly decline any vaccination efforts. These findings emphasize the importance of human behavior for a population in an epidemic landscape, and may provide insight on how to adapt and improve public health interventions.

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