# Speeding up disease extinction with a limited amount of vaccine 

M. Khasin $^{1}$, M.I. Dykman1 and B. Meerson 2 

$^{1}$Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824 USA and $^{2}$Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel 

We consider optimal vaccination protocol where the vaccine is in short supply. In this case, disease extinction results from a large and rare fluctuation. We show that the probability of such fluctuation can be exponentially increased by vaccination. For periodic vaccination with fixed average rate, the optimal vaccination protocol is model independent and presents a sequence of short pulses. The effect of vaccination can be resonantly enhanced if the pulse period coincides with the characteristic period of the disease dynamics or its multiples. This resonant effect is illustrated using a simple epidemic model. If the system is periodically modulated, the pulses must be synchronized with the modulation, whereas in the case of a wrong phase the vaccination can lead to a negative result. The analysis is based on the theory of fluctuation-induced population extinction in periodically modulated systems that we develop. 

Spreading of an infectious disease is a random pro- cess. An important source of the randomness is demo- graphic noise associated with the stochastic character of the events of infection, recovery, birth, death, etc. In a large population demographic noise is small on aver- age, and the infection spread leads to an endemic state where a certain fraction of the population stays infected for a long time. The disease, however, can disappear as a result of a large rare fluctuation, an unlikely chain of events where, for example, susceptible individuals hap- pen to avoid getting infected while infected ones recover [1–3]. Then, if there is no influx of infected individu- als from the outside, the population will be disease-free. Such spontaneous disappearance of a disease is an exam- ple of population extinction studied in stochastic popu- lation dynamics. 

Spontaneous extinction is also important for various physical and chemical reaction systems. This is a conse- quence of the underlying similarity of the dynamics that result from short random events, like collisions between molecules that lead to chemical reactions and interac- tions between individuals that lead to the disease spread. Extinction can be understood for different types of sys- tems within the same general formalism, which provides a broader scope for the present paper. Moreover, the method of optimal control of extinction that we propose can be applied to systems of various types. 

A conventional way of fighting epidemics is via vac- cination. If there is enough vaccine, the infection can be eradicated “deterministically” by eliminating the en- demic state [4]. The amount of available vaccine, how- ever, is often insufficient. The vaccine may be expensive, or it may be dangerous to store in large amounts, as in the case of anthrax, or it may be effectively short-lived due to mutations of the infection agent, as for HIV [5] and influenza [6]. 

Even where the endemic state may not be eliminated deterministically, vaccination can dramatically affect the 

PACS numbers: 87.23.Cc, 02.50.Ga, 05.40.-a 

## I. INTRODUCTION 

stochastic dynamics of the epidemics. The underlying mechanism is the change of the rate of large fluctuations leading to disease extinction. For a well mixed popu- lation, this rate W$_{e}$ is usually exponentially small for a large total population size N ≫ 1, W$_{e}$ ∝ exp( −Q ) with Q ∝ N , [3, 7–14]. We call Q the disease extinc- tion barrier. Vaccination changes the value of Q /N . In turn, this changes the disease extinction rate exponen- tially strongly. This effect was previously discussed for vaccination applied at random [13]. 

The goal of this paper is to find an optimal way of administering a limited amount of vaccine which would maximally increase the disease extinction rate. We find a vaccination protocol that applies for a broad class of epidemic models. Our approach is based on the obser- vation that, in a large fluctuation that leads to disease extinction, the population is most likely to evolve in a well-defined way. It moves along the most probable path in the space of the dynamical variables which characterize different sub-populations [13–15]. Vaccination perturbs the system as it moves along the optimal path. One can think of vaccination as “force” and its effect as “work” done on the system. This work reduces the barrier Q . The problem then is to maximize the work for given con- straints on the vaccine. 

Optimization of the effect of vaccination resembles an- other problem of optimal control of random systems: controlling large fluctuations in noise-driven dynamical systems by applying an external field with a given av- erage intensity [16, 17]. There are, however, important differences, which come from the very nature of the con- trol field. Indeed, vaccination only reduces the number of susceptible individuals. In other words, as a control field, vaccination never changes sign. Then, as we find, if the available amount of vaccine is constrained by a given mean vaccination rate, the optimal vaccination protocol turns out to be model-independent. This applies also to using a limited amount of medications and other situa- tions where the control field drives the system only in one direction. 

