An SIR epidemic on a weighted network

Disease Spread Model in Structurally Complex Spaces: An Open Markov Chain Approach

Understanding the dynamical behavior of infectious disease propagation within enclosed spaces is crucial for effectively establishing control measures. In this article, we present a modeling approach to analyze the dynamics of individuals in enclosed spaces, where such spaces are comprised of different chambers. Our focus is on capturing the movement of individuals and their infection status using an open Markov chain framework. Unlike ordinary Markov chains, an open Markov chain accounts for individuals entering and leaving the system. We categorize individuals within the system into three different groups: susceptible, carrier, and infected. A discrete-time process is employed to model the behavior of individuals throughout the system. To quantify the risk of infection, we derive a probability function that takes into account the total number of individuals inside the system and the distribution among the different groups. Furthermore, we calculate mathematical expressions for the average number of susceptible, carrier, and infected individuals at each time step. Additionally, we determine mathematical expressions for the mean number and stationary mean populations of these groups. To validate our modeling approach, we compare the theoretical and numerical models proposed in this work.

A Coupled Spatial-Network Model: A Mathematical Framework for Applications in Epidemiology

There is extensive evidence that network structure (e.g., air transport, rivers, or roads) may significantly enhance the spread of epidemics into the surrounding geographical area. A new compartmental modeling framework is proposed which couples well-mixed (ODE in time) population centers at the vertices, 1D travel routes on the graph's edges, and a 2D continuum containing the rest of the population to simulate how an infection spreads through a population. The edge equations are coupled to the vertex ODEs through junction conditions, while the domain equations are coupled to the edges through boundary conditions. A numerical method based on spatial finite differences for the edges and finite elements in the 2D domain is described to approximate the model, and numerical verification of the method is provided. The model is illustrated on two simple and one complex example geometries, and a parameter study example is performed. The observed solutions exhibit exponential decay after a certain time has passed, and the cumulative infected population over the vertices, edges, and domain tends to a constant in time but varying in space, i.e., a steady state solution.

Generating functions for message passing on weighted networks: Directed bond percolation and susceptible, infected, recovered epidemics

We study the SIR (susceptible, infected, removed/recovered) model on directed graphs with heterogeneous transmission probabilities within the message-passing approximation. We characterize the percolation transition, predict cluster size distributions, and suggest vaccination strategies. All predictions are compared to numerical simulations on real networks. The percolation threshold that we predict is a rigorous lower bound to the threshold on real networks. For large, locally treelike networks, our predictions agree very well with the numerical data.

Spreading Properties for SIR Models on Homogeneous Trees

We consider an epidemic model of SIR type set on a homogeneous tree and investigate the spreading properties of the epidemic as a function of the degree of the tree, the intrinsic basic reproduction number and the strength of the interactions between the populations of infected individuals at each node. When the degree is one, the homogeneous tree is nothing but the standard lattice on the integers and our model reduces to a SIR model with discrete diffusion for which the spreading properties are very similar to the continuous case. On the other hand, when the degree is larger than two, we observe some new features in the spreading properties. Most notably, there exists a critical value of the strength of interactions above which spreading of the epidemic in the tree is no longer possible.

Dynamics of epidemic spreading on connected graphs

We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connections between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a collection of numerical simulations for a selection of connected graphs.

Realistic Aspects of Simulation Models for Fake News Epidemics over Social Networks

The pervasiveness of online social networks has reshaped the way people access information. Online social networks make it common for users to inform themselves online and share news among their peers, but also favor the spreading of both reliable and fake news alike. Because fake news may have a profound impact on the society at large, realistically simulating their spreading process helps evaluate the most effective countermeasures to adopt. It is customary to model the spreading of fake news via the same epidemic models used for common diseases; however, these models often miss concepts and dynamics that are peculiar to fake news spreading. In this paper, we fill this gap by enriching typical epidemic models for fake news spreading with network topologies and dynamics that are typical of realistic social networks. Specifically, we introduce agents with the role of influencers and bots in the model and consider the effects of dynamical network access patterns, time-varying engagement, and different degrees of trust in the sources of circulating information. These factors concur with making the simulations more realistic. Among other results, we show that influencers that share fake news help the spreading process reach nodes that would otherwise remain unaffected. Moreover, we emphasize that bots dramatically speed up the spreading process and that time-varying engagement and network access change the effectiveness of fake news spreading.