Threshold analysis of the susceptible-infected-susceptible model on overlay networks

In this paper, we study epidemic spreading on overlay networks in which n multiple sets of links interconnect among the same nodes. By using the microscopic Markov-chain approximation (MMA) approach, we establish the conditions of epidemic outbreak for two kinds of spreading mechanisms in such an overlay network: the concatenation case and the switching case. When a uniform infection rate is set in all the subnetworks, we find the epidemic threshold for the switching case is just n times as large as that of concatenation case. We also find that the overlay network with a uniform infection rate can be considered as an equivalent (in the sense of epidemic dynamics and epidemic threshold) weighted network. To be specific, the concatenation case corresponds to the integer weighted network, while the switching case corresponds to the fractional weighted network. Interestingly, the time-varying unweighted network can be mapped into the static weighted network. Our analytic results exhibit good agreement with numerical simulations.     

两类带有确定潜伏期的SEIS传染病模型的分析

通过研究两类带有确定潜伏期的SEIS传染病模型,发现对种群的常数输入和指数输入会使疾病的传播过程产生本质的差异．对于带有常数输入的情形,找到了地方病平衡点存在及局部渐近稳定的阈值,证明了地方病平衡点存在时一定局部渐近稳定,并且疾病一致持续存在．对于带有指数输入的情形,发现地方病平衡点当潜伏期充分小时是局部渐近稳定的,当潜伏期充分大时是不稳定的．     

Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogenous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school. Supplementary material for this article is available online.     

Dynamics of an epidemic model with host migration

Host migration among discrete geographical regions is demonstrated as an important factor that brings about the diffusion and outbreak of many vector-host diseases. In the paper, we develop a mathematical model to explore the effect of host migration between two patches on the spread of a vector-host disease. Analytical results show that the reproduction number R₀ provides a threshold condition that determines the uniform persistence and extinction of the disease. If both the patches are identical, it is shown that an endemic equilibrium is locally stable. It is also shown that a unique endemic equilibrium, which exists when the disease cannot induce the death of the host, is globally asymptotically stable. Finally, two examples are given to illustrate the effect of host migration on the spread of the vector-host disease.     

Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient

A diffusive epidemic model for H1N1 influenza is formulated with a view to gain basic understanding of the virus behavior. All newborns are assumed to be susceptible. Mortality rate for infective individuals in the population is assumed to be greater than natural mortality rate. Latent, infectious and immune periods are assumed to be constants throughout this study. The numerical solutions of this model are carried out under three different initial populations distribution. In order to investigate the effect of the disease transmission coefficient on the spread of disease, β is taken to be constant as well as a function of seasonally varying time t and a function of spatial variable x  . The threshold quantity (_R0)$({\mathfrak{R}}_0)$ that governs the disease dynamics is derived. Numerical simulation shows that the system supports the existence of sustained and damped oscillations depending on initial populations distribution, the disease transmission rate and diffusion.     

Complex dynamics of epidemic models on adaptive networks

There has been a substantial amount of well mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this paper we study epidemic dynamics in an adaptive network proposed by Gross et al., where the susceptibles are able to avoid contact with the infectious by rewiring their network connections. Such rewiring of the local connections changes the topology of the network, and inevitably has a profound effect on the transmission of the disease, which in turn influences the rewiring process. We rigorously prove that the adaptive epidemic model investigated in this paper exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation. Our study shows that adaptive behaviors during an epidemic may induce complex dynamics of disease transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classical network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks.     

一类病毒自身发生变异的传染病模型的近似解

针对一类病毒变异前后传染病患者具有不同传染率的情形,利用同伦映射方法,得到其相应分阶段传播的动力学生态模型的渐近解.     

Time-varying human mobility patterns with metapopulation epidemic dynamics

In this paper, explicitly considering the influences of an epidemic outbreak on human travel, a time-varying human mobility pattern is introduced to model the time variation of global human travel. The impacts of the pattern on epidemic dynamics in heterogeneous metapopulation networks, wherein each node represents a subpopulation with any number of individuals, are investigated by using a mean-field approach. The results show that the pattern does not alter the epidemic threshold, but can slightly lower the final average density of infected individuals as a whole. More importantly, we also find that the pattern produces different impacts on nodes with different degree, and that there exists a critical degree _kc$k_c$. For nodes with degree smaller than _kc$k_c$, the pattern produces a positive impact on epidemic mitigation; conversely, for nodes with degree larger than _kc$k_c$, the pattern produces a negative impact on epidemic mitigation.     

Reducing mosquito-borne disease outbreak size: The relative importance of contact and transmissibility in a network model

The complex biological and environmental factors involved in the transmission of mosquito-borne diseases in humans have made their control elusive in many instances. Conceptual models contribute to gain insight and help to reduce the risk of taking poor managerial decisions. The focus of this paper is to compare, using a contact network model, the impact that perturbation of the number infectious contacts and of transmissibility have on the size of an outbreak. We illustrate the analysis on a contact network parametrized with data that associates humans and the mosquito Culex quinquefasciatus, a vector for lymphatic filariasis. The model suggests that, if the values corresponding to transmissibility and number of infectious contacts is relatively large, variations in the size of an outbreak are significantly in favor of control measures to reduce infectious contacts.