Generalization of Pairwise Models to non-Markovian Epidemics on Networks (original) (raw)
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Pairwise approximation for SIR -type network epidemics with non-Markovian recovery
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2018
We present the generalized mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalized model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma- and uniformly distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A ma...
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Complex population structure and the large-scale inter-patch connection human transportation underlie the recent rapid spread of infectious diseases of humans. Furthermore, the fluctuations in the endemicity of the diseases within patch dwelling populations are closely related with the hereditary features of the infectious agent. We present an SIR delayed stochastic dynamic epidemic process in a two-scale dynamic structured population. The disease confers temporary natural or infectionacquired immunity to recovered individuals. The time delay accounts for the timelag during which naturally immune individuals become susceptible. We investigate the stochastic asymptotic stability of the disease free equilibrium of the scale structured mobile population, under environmental fluctuations and the impact on the emergence, propagation and resurgence of the disease. The presented results are demonstrated by numerical simulation results.
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Impact of Non-Markovian Recovery on Network Epidemics
BIOMAT 2015, 2016
We study how the distribution of infectious periods influences the dynamics of epidemics on networks. In our recently developed framework, we use pairwise models for network epidemics with non-Markovian recovery times. It is shown for typical families of distributions (such as gamma, uniform and lognormal) that higher variance in the recovery times generates lower reproduction numbers and different epidemic curves within each distribution family. We also show that knowing the expected value and the variance of the recovery times is not sufficient to determine the key characteristics of the epidemics such as initial growth rate, peak size, peak time and final epidemic size. For accurate predictions, more detailed information on the distribution of the infectious period is required, thus carefully estimating this distribution in the case of real epidemics has paramount public health importance.
Journal of Mathematics in Industry
For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks.
International journal of Computer Networks & Communications, 2014
The SIR model is used extensively in the field of epidemiology, in particular, for the analysis of communal diseases. One problem with SIR and other existing models is that they are tailored to random or Erdos type networks since they do not consider the varying probabilities of infection or immunity per node. In this paper, we present the application and the simulation results of the pSEIRS model that takes into account the probabilities, and is thus suitable for more realistic scale free networks. In the pSEIRS model, the death rate and the excess death rate are constant for infective nodes. Latent and immune periods are assumed to be constant and the infection rate is assumed to be proportional to () () I t N t , where N (t) is the size of the total population and I(t) is the size of the infected population. A node recovers from an infection temporarily with a probability p and dies from the infection with probability (1-p).
Epidemic spread and bifurcation effects in two-dimensional network models with viral dynamics
Physical Review E, 2001
We extend a previous network model of viral dynamics to include host populations distributed in two space dimensions. The basic dynamical equations for the individual viral and immune effector densities within a host are bilinear with a natural threshold condition. In the general model, transmission between individuals is governed by three factors: a saturating function g(•) describing emission as a function of originating host virion level; a four-dimensional array B that determines transmission from each individual to every other individual; and a nonlinear function F, which describes the absorption of virions by a host for a given net arrival rate. A summary of the properties of the viral-effector dynamical system in a single host is given. In the numerical network studies, individuals are placed at the mesh points of a uniform rectangular grid and are connected with an m 2 ϫn 2 four-dimensional array with terms that decay exponentially with distance between hosts; g is linear and F has a simple step threshold. In a population of Nϭmn individuals, N 0 are chosen randomly to be initially infected with the virus. We examine the dependence of maximal population viral load on the population dynamical parameters and find threshold effects that can be related to a transcritical bifurcation in the system of equations for individual virus and host effector populations. The effects of varying demographic parameters are also examined. Changes in ␣, which is related to mobility, and contact rate  also show threshold effects. We also vary the density of ͑randomly chosen͒ initially infected individuals. The distribution of final size of the epidemic depends strongly on N 0 but is invariably bimodal with mass concentrated mainly near either or both ends of the interval ͓1,N͔. Thus large outbreaks may occur, with small probability, even with only very few initially infected hosts. The effects of immunization of various fractions of the population on the final size of the epidemic are also explored. The distribution of the final percentage infected is estimated by simulation. The mean of this quantity is obtained as a function of immunization rate and shows an almost linear decline for immunization rates up to about 0.2. When the immunization rate is increased past 0.2, the extra benefit accrues more slowly. We include a discussion of some approximations that illuminate threshold effects in demographic parameters and indicate how a mean-field approximation and more detailed studies of various geometries and rates of immunization could be a useful direction for future analysis.