Journal of Mathematical Biology

, Volume 78, Issue 4, pp 1089–1113 | Cite as

Optimal time-profiles of public health intervention to shape voluntary vaccination for childhood diseases

  • Bruno Buonomo
  • Piero Manfredi
  • Alberto d’OnofrioEmail author


In order to seek the optimal time-profiles of public health systems (PHS) Intervention to favor vaccine propensity, we apply optimal control (OC) to a SIR model with voluntary vaccination and PHS intervention. We focus on short-term horizons, and on both continuous control strategies resulting from the forward–backward sweep deterministic algorithm, and piecewise-constant strategies (which are closer to the PHS way of working) investigated by the simulated annealing (SA) stochastic algorithm. For childhood diseases, where disease costs are much larger than vaccination costs, the OC solution sets at its maximum for most of the policy horizon, meaning that the PHS cannot further improve perceptions about the net benefit of immunization. Thus, the subsequent dynamics of vaccine uptake stems entirely from the declining perceived risk of infection (due to declining prevalence) which is communicated by direct contacts among parents, and unavoidably yields a future decline in vaccine uptake. We find that for relatively low communication costs, the piecewise control is close to the continuous control. For large communication costs the SA algorithm converges towards a non-monotone OC that can have oscillations.


Vaccination Human behavior Public health system Communication Optimal control Simulated annealing Forward–backward sweep method 

Mathematics Subject Classification

92D30 49J15 49N90 91A80 91A22 



The work of B. B. has been performed under the auspices of the Italian National Group for the Mathematical Physics (GNFM) of National Institute for Advanced Mathematics (INdAM). We wish to acknowledge the two anonymous referees, whose suggestions helped us to significantly increase the quality and the readability of this work.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and ApplicationsUniversity of Naples Federico IINaplesItaly
  2. 2.Department of Economics and ManagementUniversity of PisaPisaItaly
  3. 3.International Prevention Research InstituteLyonFrance

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