# Synopsis: A little vaccine goes a long way

Given that vaccine supplies are often limited, a quantitative understanding of how the number and frequency of vaccinations can affect the growth rate of disease would be useful. Physicists show that even a small number of randomly vaccinated individuals can exponentially increase the extinction rate of a disease.

Most epidemics carry some degree of randomness: both the growth of the population and the rate at which people come into contact can fluctuate in unpredictable ways. As in many nonlinear physical systems far from equilibrium, such fluctuations determine if a disease will continue to spread or become extinct in a finite time.

The delivery of vaccines at random rates into an infected population can similarly be modeled as “noise.” Writing in Physical Review Letters, Mark Dykman of Michigan State University and Ira Schwartz and Alexandra Landsman of the Naval Research Laboratory in Washington D.C. show that even a small number of random vaccinations can lead to an exponential increase in the extinction rate of a disease.

The group adapts an established model in the field of population dynamics known as the SIS model ($S$ and $I$ are variables that define the number of people that are susceptible to an infection and those that are already infected, respectively) and maps the problem to the variational calculus used in classical dynamics. They assume that a small percentage of incoming “susceptibles” receive the vaccination at random times.

The key finding that even weak vaccination can increase the extinction rate of an epidemic exponentially has a physical meaning: as in many dynamical systems, the right frequency of external pulses—in this case, the vaccination rate—can resonate with the system itself. – Jessica Thomas

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