We study vaccine control for disease spread on an adaptive network modeling disease avoidance behavior. Control is implemented by adding Poisson distributed vaccination of susceptibles. We show that vaccine control is much more effective in adaptive networks than in static networks due to an interaction between the adaptive network rewiring and the vaccine application. Disease extinction rates using vaccination are computed, and orders of magnitude less vaccine application is needed to drive the disease to extinction in an adaptive network than in a static one. Modeling the spread of epidemics on static networks is a well developed field, but recent studies have begun to account for potentially time-varying network topologies. In particular, people may adjust their social behavior in response to the threat of an epidemic. Both an SIS (susceptible-infected-susceptible) model [1] and SIRS (susceptible-infected-recovered-susceptible) model [2] have been studied on an adaptive network in which non-infected nodes rewire their links adaptively away from infected neighbors and toward other non-infected nodes. Such adaptation typically increases the epidemic threshold and reduces the number of infectious cases, and new disease dynamics and bifurcations are observed. Similar results are seen when the nodes reconnect to randomly selected nodes elsewhere in the network [3, 4] . Because vaccines are available for many diseases, it is desirable to examine the interplay between adaptively fluctuating social contacts and vaccination of susceptible individuals. Almost all diseases exhibit randomness resulting in observed fluctuations, as in [5, 6, 7, 8, 9, 10] . As diseases evolve in large populations, there is the possibility of finite time extinction [10, 11, 12, 13, 14, 15] . Extinction occurs where the number of infectives become so small that there is insufficient transmission to keep the disease in its endemic state [16, 17, 18] . Fluctuations cause the disease free equilibrium (DFE) to be reached in a finite time. Such an extinction process occurs even when the DFE is unstable. Populations based upon adaptive networks further complicate the problem, since social dynamical situations, such as disease avoidance strategies, can cause the endemic and DFE to be bistable [2] . A major characteristic of fluctuation-induced extinction in stochastic models for globally connected large populations is the extinction rate. Viewing disease fadeout as coming from systems far from thermal equilibrium, finite population extinction rate laws have been derived in SIS [15, 19] and SIR [20] models. In contrast to vaccine strategies which stabilize the DFE [21, 22, 23, 24, 25, 26] , periodic pulsed vaccination was generalized to a random Poisson strategy, which exponentially enhances the rate of extinction [27] . Other vaccine strategies have been examined in a variety of static network geometries. Targeting of high degree nodes is widely recognized as more effective than random vaccination, including in scale free networks [28, 29] , small world networks [30] , and more realistic social network geometries [31] . Because targeting the highest degree nodes requires full knowledge of the network geometry, other strategies based on local knowledge have been developed. For example, vaccinating a random acquaintance of a randomly selected node tends to favor high degree nodes and is again more effective than random vaccination [32] . When vaccine is very limited, outbreaks can be minimized by fragmenting the network via a graph partitioning strategy which requires less vaccine than targeting high degree nodes [33] . Here we will use a random vaccination strategy and find that in conjunction with adaptive rewiring, it is extremely effective. We create a new model by modifying an SIS model [1], adding a vaccine class (V). Individuals are connected in a network, with non-infected nodes rewiring adaptively to reduce connections with infected nodes. The transition probabilities are as follows. A susceptible node becomes infected with rate pN I,nbr , where N I,nbr is its number of infected neighbors. An infected node recovers (to susceptible) with rate r. Vaccination occurs in Poisson-distributed pulses with frequency ν, and each pulse vaccinates a fraction A of the susceptible nodes [34] . We assume that the vaccine is not permanent, so a vaccinated node becomes susceptible again with rate q, the resusceptibility rate. While the epidemic spreads, the network is rewired adaptively. Because the vaccine wears off, we assume that vaccinated nodes are uncertain of their infection status and thus rewire in the same way as susceptibles. If a link connects a non-infected node to an infected node, that link is rewired with rate w to connect the noninfected node to another randomly selected non-infected node. Self links and multiple links between nodes are excluded. We simulate the full adaptive network via Monte Carlo simulation in a similar fashion to [2] . Vaccination events occur with average rate ν, and in each event AN S suscep-
Shaw and Schwartz REF examined the effects of vaccination in static versus adaptive networks.
Enhanced vaccine control of epidemics in adaptive networks
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