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How to prevent the spread of human diseases is a great challenge for thescientific community and so far there are many studies in which immunizationstrategies have been developed. However, these kind of strategies usually donot consider that medical institutes may have limited vaccine resourcesavailable. In this manuscript, we explore the Susceptible-Infected-Recovered(SIR) model with local dynamic vaccination, and considering limited vaccines.In this model, susceptibles in contact with an infected individual, arevaccinated -with probability $\omega$- and then get infected -with probability$\beta$. However, when the fraction of immunized individuals reaches athreshold $V_L$, the vaccination stops, after which only the infection ispossible. In the steady state, besides the critical points $\beta_c$ and$\omega_c$ that separate a non-epidemic from an epidemic phase, we find for arange of $V_L$ another transition points, $\beta^*>\beta_c$ and$\omega^*<\omega_c$, which correspond to a novel discontinuous phasetransition. This critical value separates a phase where the amount of vaccinesis sufficient, from a phase where the disease is strong enough to exhaust allthe vaccination units. For a disease with fixed $\beta$, the vaccinationprobability $\omega$ can be controlled in order to drastically reduce thenumber of infected individuals, using efficiently the available vaccines.  Furthermore, the temporal evolution of the system close to $\beta^*$ or$\omega^*$, shows that after a peak of infection the system enters into aquasi-stationary state, with only a few infected cases. But if there are nomore vaccines, these few infected individuals could originate a secondoutbreak, represented by a second peak of infection. This state of apparentcalm, could be dangerous since it may lead to misleading conclusions and to anabandon of the strategies to control the disease.

Multiple outbreaks in epidemic spreading with local vaccination and limited vaccines