Abstract

In this thesis we analyze the relationship between epidemiology and network theory, starting from the observation that the viral propagation between interacting agents is determined by intrinsic characteristics of the population contact network. We aim to investigate how a particular network structure can impact on the long-term behavior of epidemics. This field is way too large to be fully discussed; we limit ourselves to consider networks that are partitioned into local communities, in order to incorporate realistic contact structures into the model. 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. We describe the epidemic process as a continuous-time individual-based susceptible–infected–susceptible (SIS) model using a first-order mean-field approximation, both in homogeneous and in heterogeneous setting. For this mean-field model we show that the spectral radius of the smaller quotient graph, in connection with the infecting and curing rates, is related to the epidemic threshold, and it gives conditions in order to decide whether the overall healthy-state defines a globally asymptotically stable or an unstable equilibrium. Moreover we show 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. We also consider the important issue related to the control of the infectious disease. Taking into account the connectivity of the network, we provide a cost-optimal distribution of resources to prevent the disease from persisting indefinitely in the population; for a particular case of two-level immunization problem we report on the construction of a polynomial time complexity algorithm. In the second part of the thesis we include stochasticity in the model, considering the infection rates in the form of independent stochastic processes. This allows us to get stochastic differential equation for the probability of infection in each node. We report on the existence of the solution for all times. Moreover we show that there exist two regions, given in terms of the coefficients of the model, one where the system goes to extinction almost surely, and the other where it is stochastic permanent.