We extend the N-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times T(r) and T¯(r) as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance r. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for T¯(r). Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time T¯(r) is almost entirely determined by the basic reproduction number R0 of the network. The value of the upper-transition time T¯(r) around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

The interplay between disease spreading and personal risk perception is of key importance for modelling the spread of infectious diseases. We propose a planar system of ordinary differential equations (ODEs) to describe the co-evolution of a spreading phenomenon and the average link density in the personal contact network. Contrary to standard epidemic models, we assume that the contact network changes based on the current prevalence of the disease in the population, i.e. the network adapts to the current state of the epidemic. We assume that personal risk perception is described using two functional responses: one for link-breaking and one for link-creation. The focus is on applying the model to epidemics, but we also highlight other possible fields of application. We derive an explicit form for the basic reproduction number and guarantee the existence of at least one endemic equilibrium, for all possible functional responses. Moreover, we show that for all functional responses, limit cycles do not exist. This means that our minimal model is not able to reproduce consequent waves of an epidemic, and more complex disease or behavioural dynamics are required to reproduce epidemic waves.

The historical journey of the Benford’s law and its most important definitions and properties are shortly reviewed. Firstly, we define a new class of numbers based on the first-digit law. Secondly, we investigate the relation between Benford’s sequence and normal numbers.

We introduce a Markov Modulated Process (MMP) to describe human mobility. We represent the mobility process as a time-varying graph, where a link specifies a connection between two nodes (humans) at any discrete time step. Each state of the Markov chain encodes a certain modification to the original graph. We show that our MMP model successfully captures the main features of a random mobility simulator, in which nodes moves in a square region. We apply our MMP model to human mobility, measured in a library.

We study an SAIRS-type epidemic model with vaccination, where the role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission patterns of the disease. We provide a global stability analysis for the model. We determine the value of the basic reproduction number R₀ and prove that the disease-free equilibrium is globally asymptotically stable if R₀<1. If R₀>1, the disease free equilibrium is unstable and a unique endemic equilibrium exists. We investigate the global stability of the endemic equilibrium for some variations of the original model under study and answer an open problem proposed in Ansumali et al. (2020). In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when R₀=1. We provide a thorough numerical exploration of our model to illustrate our analytical results.