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Bistability of Evolutionary Stable Vaccination Strategies in the Reinfection SIRI Model

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Abstract

We use the reinfection SIRI epidemiological model to analyze the impact of education programs and vaccine scares on individuals decisions to vaccinate or not. The presence of the reinfection provokes the novelty of the existence of three Nash equilibria for the same level of the morbidity relative risk instead of a single Nash equilibrium as occurs in the SIR model studied by Bauch and Earn (PNAS 101:13391–13394, 2004). The existence of three Nash equilibria, with two of them being evolutionary stable, introduces two scenarios with relevant and opposite features for the same level of the morbidity relative risk: the low-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a low probability; and the high-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a high probability. We introduce the evolutionary vaccination dynamics for the SIRI model and we prove that it is bistable. The bistability of the evolutionary dynamics indicates that the damage provoked by false scares on the vaccination perceived morbidity risks can be much higher and much more persistent than in the SIR model. Furthermore, the vaccination education programs to be efficient they need to implement a mechanism to suddenly increase the vaccination coverage level.

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Acknowledgements

The authors thank the referees for their comments and suggestions. The authors also thank the financial support of LIAAD-INESC TEC and FCT Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) within project UID/EEA/50014/2013 and ERDF (European Regional Development Fund) through the COMPETE Program (operational program for competitiveness) and by National Funds through the FCT within Project “Dynamics, optimization and modelling”, with reference PTDC/MAT-NAN/6890/2014 and Project “NanoSTIMA: Macro-to-Nano Human Sensing: Towards Integrated Multimodal Health Monitoring and Analytics/NORTE-01-0145-FEDER-000016” financed by the North Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, and through the European Regional Development Fund (ERDF).

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Authors and Affiliations

  1. LIAAD-INESC TEC, R Dr Roberto Frias, 4200-465, Porto, Portugal

    José Martins & Alberto Pinto

  2. School of Technology and Management, Polytechnic Institute of Leiria, Campus 2, Morro do Lena - Alto do Vieiro, 2411-901, Leiria, Portugal

    José Martins

  3. Faculty of Sciences, University of Porto, R Campo Alegre, 4169-007, Porto, Portugal

    Alberto Pinto

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  1. José Martins
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Correspondence to José Martins.

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Martins, J., Pinto, A. Bistability of Evolutionary Stable Vaccination Strategies in the Reinfection SIRI Model. Bull Math Biol 79, 853–883 (2017). https://doi.org/10.1007/s11538-017-0257-6

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  • DOI: https://doi.org/10.1007/s11538-017-0257-6

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