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The evoSIR model is a modification of the usual SIR process on a graph $G$ in which $S-I$ connections are broken at rate $rho$ and the $S$ connects to a randomly chosen vertex. The evoSI model is the same as evoSIR but recovery is impossible. In an undergraduate project at Duke the critical value for evoSIR was computed and simulations showed that when $G$ is an ErdH os-Renyi graph with mean degree 5, the system has a discontinuous phase transition, i.e., as the infection rate $lambda$ decreases to $lambda_c$, the fraction of individuals infected during the epidemic does not converge to 0. In this paper we study evoSI dynamics on graphs generated by the configuration model. We show that there is a quantity $Delta$ determined by the first three moments of the degree distribution, so that the phase transition is discontinuous if $Delta>0$ and continuous if $Delta<0$.
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In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group.
In this paper, we are concerned with the stochastic susceptible-infectious-susceptible (SIS) epidemic model on the complete graph with $n$ vertices. This model has two parameters, which are the infection rate and the recovery rate. By utilizing the theory of density-dependent Markov chains, we give consistent estimations of the above two parameters as $n$ grows to infinity according to the sample path of the model in a finite time interval. Furthermore, we establish the central limit theorem (CLT) and the moderate deviation principle (MDP) of our estimations. As an application of our CLT, reject regions of hypothesis testings of two parameters are given. As an application of our MDP, confidence intervals with lengths converging to $0$ while confidence levels converging to $1$ are given as $n$ grows to infinity.
The exploration of epidemic dynamics on dynamically evolving (adaptive) networks poses nontrivial challenges to the modeler, such as the determination of a small number of informative statistics of the detailed network state (that is, a few good observables) that usefully summarize the overall (macroscopic, systems level) behavior. Trying to obtain reduced, small size, accurate models in terms of these few statistical observables - that is, coarse-graining the full network epidemic model to a small but useful macroscopic one - is even more daunting. Here we describe a data-based approach to solving the first challenge: the detection of a few informative collective observables of the detailed epidemic dynamics. This will be accomplished through Diffusion Maps, a recently developed data-mining technique. We illustrate the approach through simulations of a simple mathematical model of epidemics on a network: a model known to exhibit complex temporal dynamics. We will discuss potential extensions of the approach, as well as possible shortcomings.
The susceptible--infected--susceptible (SIS) epidemic process on complex networks can show metastability, resembling an endemic equilibrium. In a general setting, the metastable state may involve a large portion of the network, or it can be localized on small subgraphs of the contact network. Localized infections are not interesting because a true outbreak concerns network--wide invasion of the contact graph rather than localized infection of certain sites within the contact network. Existing approaches to localization phenomenon suffer from a major drawback: they fully rely on the steady--state solution of mean--field approximate models in the neighborhood of their phase transition point, where their approximation accuracy is worst; as statistical physics tells us. We propose a dispersion entropy measure that quantifies the localization of infections in a generic contact graph. Formulating a maximum entropy problem, we find an upper bound for the dispersion entropy of the possible metastable state in the exact SIS process. As a result, we find sufficient conditions such that any initial infection over the network either dies out or reaches a localized metastable state. Unlike existing studies relying on the solution of mean--field approximate models, our investigation of epidemic localization is based on characteristics of exact SIS equations. Our proposed method offers a new paradigm in studying spreading processes over complex networks.
In this paper we study the diffusion of an SIS-type epidemics on a network under the presence of a random environment, that enters in the definition of the infection rates of the nodes. Accordingly, we model the infection rates in the form of independent stochastic processes. To analyze the problem, we apply a mean field approximation, which allows to get a stochastic differential equations for the probability of infection in each node, and classical tools about stability, which require to find suitable Lyapunovs functions. Here, we find conditions which guarantee, respectively, extinction and stochastic persistence of the epidemics. We show that there exists 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. These two regions are, unfortunately, not adjacent, as there is a gap between them, whose extension depends on the specific level of noise. In this last region, we perform numerical analysis to suggest the true behavior of the solution.
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