Mean field at distance one

  title={Mean field at distance one},
  author={Ka Yin Leung and Mirjam E E Kretzschmar and Odo Diekmann},
To be able to understand how infectious diseases spread on networks, it is important to understand the network structure itself in the absence of infection. In this text we consider dynamic network models that are inspired by the (static) configuration network. The networks are described by population-level averages such as the fraction of the population with k partners, k = 0, 1, 2, … This means that the bookkeeping contains information about individuals and their partners, but no information… 
Stochastic epidemics on random networks
This thesis considers stochastic epidemic models for the spread of epidemics in structured populations. The asymptotic behaviour of the models is analysed by using branching process approximations.


Dangerous connections: on binding site models of infectious disease dynamics
This work forms models for the spread of infection on networks that are amenable to analysis in the large population limit and is able to characterize population-level epidemiological quantities, such as $$R_0$$R0, r, the final size, and the endemic equilibrium, in terms of the corresponding variables.
SIR dynamics in random networks with heterogeneous connectivity
  • E. Volz
  • Mathematics
    Journal of mathematical biology
  • 2008
The dynamic equations provide an alternative way of determining the epidemic threshold where large-scale epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.
Dynamic Random Networks in Dynamic Populations
We consider a random network evolving in continuous time in which new nodes are born and old may die, and where undirected edges between nodes are created randomly and may also disappear. The node
Mathematics of Epidemics on Networks: From Exact to Approximate Models
This textbook provides an exciting new addition to the area of network science featuring a stronger and more methodical link of models to their mathematical origin and explains how these relate to
A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon
The main findings are that clustering tends to decrease the spread of disease, and the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and disease spread broadly increases with degree correlation.
Large graph limit for an SIR process in random network with heterogeneous connectivity
We consider an SIR epidemic model propagating on a Configuration Model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the
A Dynamic Network in a Dynamic Population: Asymptotic Properties
A criterion for when a giant connected component exists after the process has evolved for a long period of time is derived, assuming that the node population grows to infinity.
Dynamic concurrent partnership networks incorporating demography.
Law of large numbers for the SIR epidemic on a random graph with given degrees
The main result is that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time.
Random Graphs and Complex Networks: Volume 1
This rigorous introduction to network science presents random graphs as models for real-world networks and investigates key properties, such as the connectivity of nodes, in several important models for complex networks.