Abstract. We study the diffusion of epidemics on networks that are partitioned into local communities. The gross structure of hierarchical networks of this kind can be described by a quotient graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. In community models the nodal infection probability is thus expected to depend mainly on the interaction of a few, large interconnected clusters. In this work, we describe the epidemic process as a continuous-time individual-based susceptible-infected-susceptible (SIS) model using a first-order mean-field approximation.A key feature of our model is that the spectral radius of this smaller quotient graph (which only captures the macroscopic structure of the community network) is all we need to know in order to decide whether the overall healthy-state defines a globally asymptotically stable or an unstable equilibrium. Indeed, the spectral radius is related to the epidemic threshold of the system.Moreover we prove that, above the threshold, another steady-state exists that can be computed using a lower-dimensional dynamical system associated with the evolution of the process on the quotient graph. Our investigations are based on the graph-theoretical notion of equitable partition and of its recent and rather flexible generalization, that of almost equitable partition.Key words. susceptible-infected-susceptible model, hierarchical networks, graph spectra, equitable and almost equitable partitions AMS subject classifications.1. Introduction. Metapopulation models of epidemics consider the entire population partitioned into communities (also called households, clusters or subgraphs). Such models assume that each community shares a common environment or is defined by a specific relationship (see, e.g., [1,2,3]).Several authors also account for the effect of migration between communities [4,5]. Conversely, the model we are interested in suits better the diffusion of computer viruses or stable social communities, which do not change during the infection period; hence we do not consider migration.In this work, we study the diffusion of epidemics over an undirected graph G = (V, E) with edge set E and node set V . The order of G, denoted N , is the cardinality of V , whereas the size of G is the cardinality of E, denoted L. Connectivity of the graph G is conveniently encoded in the N × N adjacency matrix A. We are interested in the case of networks that can be naturally partitioned into n communities: they are represented by a node set partition π = {V 1 , ..., V n }, i.e., a sequence of mutually disjoint nonempty subsets of V , called cells, whose union is V .The epidemic model adopted in the rest of the paper is a continuous-time Markovian individual-based susceptible-infected-susceptible (SIS) model. In the SIS model a node can be repeatedly infected, recover and yet be infected again. The viral state of a node i, at time t, is thus described by a Bernoulli random var...

No abstract

Graphene has recently emerged as a novel material in the biomedical field owing to its optical properties, biocompatibility, large specific surface area and low cost. In this paper, we provide the first demonstration of the possibility of using light to remotely trigger the release of drugs from graphene in a highly controlled manner. Different drugs including chemotherapeutics and proteins are firmly adsorbed onto reduced graphene oxide (rGO) nanosheets dispersed in a biopolymer film and then released by individual millisecond-long light pulses generated by a near infrared (NIR) laser. Here graphene plays the dual role of a versatile substrate for temporary storage of drugs and an effective transducer of NIR-light into heat. Drug release appears to be narrowly confined within the size of the laser spot under noninvasive conditions and can be precisely dosed depending on the number of pulses. The approach proposed paves the way for tailor-made pharmacological treatments of chronic diseases, including cancer, anaemia and diabetes.

We consider a model for the diffusion of epidemics in a population that is partitioned into local communities. In particular, assuming a mean-field approximation, we analyze a continuous-time susceptible-infected-susceptible (SIS) model that has appeared recently in the literature. The probability by which an individual infects individuals in its own community is different from the probability of infecting individuals in other communities. The aim of the model, compared to the standard, nonclustered one, is to provide a compact description for the presence of communities of local infection where the epidemic process is faster compared to the rate at which it spreads across communities. Ultimately, it provides a tool to express the probability of epidemic outbreaks in the form of a metastable infection probability. In the proposed model, the spatial structure of the network is encoded by the adjacency matrix of clusters, i.e., the connections between local communities, and by the vector of the sizes of local communities. Thus, the existence of a nontrivial metastable occupancy probability is determined by an epidemic threshold which depends on the clusters' size and on the intercommunity network structure.

The design of an efficient curing policy, able to stem an epidemic process at an affordable cost, has to account for the structure of the population contact network supporting the contagious process. Thus, we tackle the problem of allocating recovery resources among the population, at the lowest cost possible to prevent the epidemic from persisting indefinitely in the network. Specifically, we analyze a susceptible-infected-susceptible epidemic process spreading over a weighted graph, by means of a firstorder mean-field approximation. First, we describe the influence of the contact network on the dynamics of the epidemics among a heterogeneous population, that is possibly divided into communities. For the case of a community network, our investigation relies on the graph-theoretical notion of equitable partition; we show that the epidemic threshold, a key measure of the network robustness against epidemic spreading, can be determined using a lower-dimensional dynamical system. Exploiting the computation of the epidemic threshold, we determine a cost-optimal curing policy by solving a convex minimization problem, which possesses a reduced dimension in the case of a community network. Lastly, we consider a two-level optimal curing problem, for which an algorithm is designed with a polynomial time complexity in the network size.In the case of a uniform curing policy we have that the value of δ such that λ = 0 is the largest eigenvalue is (β + β 2 + 4β 2 ε 2 k)/2 and the value of the total cost is U u =c 0 β+ β 2 +4β 2 ε 2 k +c 1 k β+ β 2 +4β 2 ε 2 k .