Multiple outbreaks in epidemic spreading with local vaccination and limited vaccines

M. A. Di Muro, L. G. Alvarez-Zuzek, S. Havlin, L. A. Braunstein

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19 Scopus citations


How to prevent the spread of human diseases is a great challenge for the scientific community and so far there are many studies in which immunization strategies have been developed. However, these kind of strategies usually do not consider that medical institutes may have limited vaccine resources available. In this manuscript, we explore the susceptible-infected-recovered model with local dynamic vaccination, and considering limited vaccines. In this model, susceptibles in contact with an infected individual, are vaccinated-with probability ω-and then get infected-with probability β. However, when the fraction of immunized individuals reaches a threshold V L, the vaccination stops, after which only the infection is possible. In the steady state, besides the critical points β c and ω c that separate a non-epidemic from an epidemic phase, we find for a range of V L another transition points, β∗ > β c and ω∗ < ω c, which correspond to a novel discontinuous phase transition. This critical value separates a phase where the amount of vaccines is sufficient, from a phase where the disease is strong enough to exhaust all the vaccination units. For a disease with fixed β, the vaccination probability ω can be controlled in order to drastically reduce the number of infected individuals, using efficiently the available vaccines. Furthermore, the temporal evolution of the system close to β∗ or ω∗, shows that after a peak of infection the system enters into a quasi-stationary state, with only a few infected cases. But if there are no more vaccines, these few infected individuals could originate a second outbreak, represented by a second peak of infection. This state of apparent calm, could be dangerous since it may lead to misleading conclusions and to an abandon of the strategies to control the disease.

Original languageEnglish
Article number083025
JournalNew Journal of Physics
Issue number8
StatePublished - Aug 2018

Bibliographical note

Publisher Copyright:
© 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft.


SH thanks the Israel Science Foundation, ONR, the Israel Ministry of Science and Technology (MOST) with the Italy Ministry of Foreign Affairs, BSF-NSF, MOST with the Japan Science and Technology Agency, the BIU Center for Research in Applied Cryptography and Cyber Security, and DTRA (Grant no. HDTRA-1-10-1-0014) for financial support. LAB wish to thank DTRA (Grant no. HDTRA-1-10-1-0014) for financial support. MAD, LGAZ and LAB wish to thank to UNMdP and CONICET (PIP 00443/2014) for financial support.

FundersFunder number
Israel Ministry of Science and Technology
Office of Naval Research
Japan Science and Technology AgencyHDTRA-1-10-1-0014
Consejo Nacional de Investigaciones Científicas y TécnicasPIP 00443/2014
Ministry of Science and Technology
Israel Science Foundation
Ministry for Foreign Affairs
Universidad Nacional de Mar del Plata


    • SIR model
    • complex networks
    • epidemic modeling
    • percolation


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