SIR dynamics in random networks with heterogeneous connectivity

Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE’s. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the theory. Finally, 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.      

Bivariate versus multivariate allometry: A note on a paper by Jungers and German

Jungers and German (1981) found differences when they compared 1) coefficients of allometry from bivariate plots of log measurements versus log body weight with 2) those coefficients from the first principal component of the log measurements excluding body weight. It is argued here that an arbitrary choice of unit for “internal size” is all that separates these coefficients. When the unit is chosen to make internal size isometric with body weight the coefficients agree rather well.     

A note on growth processes in random environment

The purpose of this note is to discuss certain features of population growth models carlier proposed and to construct an alternative diffusion model for regulated growth in random environment. This model is shown to be the analogue of the Malthusian one, although it is a generalization of the latter due to the presence of regulation.     

A note on population growth under random dispersal

An approximation method using a sine function is used to solve the second degree growth equation for the case in which an organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.     

Extinction probabilities in branching processes: A note on holgate and Lakhani's paper

Within the class of offspring distributions with given meanm>1 and probability of no offspringp _o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp ₀<q(m,p₀)≤q<1. A short table illustrates the lower boundq(m,p ₀). Work done at the Mathematics Department, University of Washington, Seattle, U.S.A.     

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

This paper is concerned with a stochastic SIR (susceptible → infective → removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.     

A note on the problem of cycles in random nets

The problem of cycles in random neural nets is considered. An approximation is derived giving the expected numberE(k, a, n) of cycles of lengthk of which a neuron in a random net will be a member. The approximation is further simplifed for the casen> k. In this caseE(k, a, n)=a ^k/n , where (a) is the axone density of the net.     

Effects of predation on host-pathogen dynamics in SIR models

The integration of infectious disease epidemiology with community ecology is an active area of research. Recent studies using SI models without acquired immunity have demonstrated that predation can suppress infectious disease levels. The authors recently showed that incorporating immunity (SIR models) can produce a “hump”-shaped relationship between disease prevalence and predation pressure; thus, low to moderate levels of predation can boost prevalence in hosts with acquired immunity. Here we examine the robustness of this pattern to realistic extensions of a basic SIR model, including density-dependent host regulation, predator saturation, interference, frequency-dependent transmission, predator numerical responses, and explicit resource dynamics. A non-monotonic relationship between disease prevalence and predation pressure holds across all these scenarios. With saturation, there can also be complex responses of mean host abundance to increasing predation, as well as bifurcations leading to unstable cycles (epidemics) and pathogen extinction at larger predator numbers. Firm predictions about the relationship between prevalence and predation thus require one to consider the complex interplay of acquired immunity, host regulation, and foraging behavior of the predator.     

Activity dynamics and propagation of synchronous spiking in locally connected random networks

Random network models have been a popular tool for investigating cortical network dynamics. On the scale of roughly a cubic millimeter of cortex, containing about 100,000 neurons, cortical anatomy suggests a more realistic architecture. In this locally connected random network, the connection probability decreases in a Gaussian fashion with the distance between neurons. Here we present three main results from a simulation study of the activity dynamics in such networks. First, for a broad range of parameters these dynamics exhibit a stationary state of asynchronous network activity with irregular single-neuron spiking. This state can be used as a realistic model of ongoing network activity. Parametric dependence of this state and the nature of the network dynamics in other regimes are described. Second, a synchronous excitatory stimulus to a fraction of the neurons results in a strong activity response that easily dominates the network dynamics. And third, due to that activity response an embedding of a divergent-convergent feed-forward subnetwork (as in synfire chains) does not naturally lead to a stable propagation of synchronous activity in the subnetwork; this is in contrast to our earlier findings in isolated subnetworks of that type. Possible mechanisms for stabilizing the interplay of volleys of synchronous spikes and network dynamics by specific learning rules or generalizations of the subnetworks are discussed.     

A remark on a paper by misra

MISRA (1978) sets confidence intervals for a double linear compound of multivariate normal regression coefficients by using ROY'S maximum root test criterion. The exact test statistic to be used is STUDENT'S t. The t statistic gives narrower confidence bounds than those given by ROY's maximum root statistic. A result given by MORRISON (1975, p. 18, equation 10) for profile analysis is also obtained by using the STUDENT'S t test.     

Spreading dynamics on complex networks: a general stochastic approach

Dynamics on networks is considered from the perspective of Markov stochastic processes. We partially describe the state of the system through network motifs and infer any missing data using the available information. This versatile approach is especially well adapted for modelling spreading processes and/or population dynamics. In particular, the generality of our framework and the fact that its assumptions are explicitly stated suggests that it could be used as a common ground for comparing existing epidemics models too complex for direct comparison, such as agent-based computer simulations. We provide many examples for the special cases of susceptible-infectious-susceptible and susceptible-infectious-removed dynamics (e.g., epidemics propagation) and we observe multiple situations where accurate results may be obtained at low computational cost. Our perspective reveals a subtle balance between the complex requirements of a realistic model and its basic assumptions.