搜索结果：Infection

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate $1$, and infections occur at a rate $λ_{n} \sim \fracλ{n}$ where $n$ is the initial number of healthy individuals with $λ>1$. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically $κ(λ) \log n$ as $n \rightarrow \infty$, where $κ(λ)$ is a continuous function in $λ$ that undergoes a second-order phase transition at $λ_{c}\simeq 1.8038$. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

The main goal of this paper is to model the epidemic and flattening the infection curve of the social networks. Flattening the infection curve implies slowing down the spread of the disease and reducing the infection rate via social-distancing, isolation (quarantine) and vaccination. The nan-pharmaceutical methods are a much simpler and efficient way to control the spread of epidemic and infection rate. By specifying a target group with high centrality for isolation and quarantine one can reach a much flatter infection curve (related to Corona for example) without adding extra costs to health services. The aim of this research is, first, modeling the epidemic and, then, giving strategies and structural algorithms for targeted vaccination or targeted non-pharmaceutical methods for reducing the peak of the viral disease and flattening the infection curve. These methods are more efficient for nan-pharmaceutical interventions as finding the target quarantine group flattens the infection curve much easier. For this purpose, a few number of particular nodes with high centrality are isolated and the infection curve is analyzed. Our research shows meaningful results for flattening the infe

We study the stochastic SIR epidemic model with infection-age dependent infectivity for which a measure-valued process is used to describe the ages of infection for each individual. We establish a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the properly scaled measure-valued processes together with the other epidemic processes to describe the evolution dynamics. In the FLLN, assuming that the hazard rate function of the infection periods is bounded and the ages at time 0 of the infections of the initially infected individuals are bounded, we obtain a PDE limit for the LLN-scaled measure-valued process, for which we characterize its solution explicitly. The PDE is linear with a boundary condition given by the unique solution to a set of Volterra-type nonlinear integral equations. In the FCLT, we obtain an SPDE for the CLT-scaled measure-valued process, driven by two independent white noises coming from the infection and recovery processes. The SPDE is also linear and coupled with the solution to a system of stochastic Volterra-type linear integral equations driven by three independent Gaussian noises, one from the random infection functio

We study a variant of the stochastic SIR model on graphs that has previously been introduced in the physics literature for modelling zombie outbreaks and here referred to as the Zombie Infection Model (ZIM). In this model, initially each node of a graph is either susceptible, infected or removed. As in the SIR model, a susceptible node becomes infected at rate $λ$ times the number of its infected neighbours. Moreover, in the ZIM, an infected node is removed at rate $1$ times the number of its susceptible neighbours. This process exhibits rich and sometimes counterintuitive behaviour. By combining various coupling techniques, we provide a rigorous mathematical analysis of the model, focusing on monotonicity properties and the probability of the infection spreading indefinitely. One of our main results is that this probability is monotone with respect to an increase of $λ$ for the process on trees, but that there are graphs of bounded degree for which it is continuous and yet not monotone. We also establish bounds on this probability for the process on general graphs, and derive more precise results for complete graphs, regular trees, and the $d$-dimensional integer lattice.

We study a dynamic infection spread model, inspired by the discrete time SIR model, where infections are spread via non-isolated infected individuals. While infection keeps spreading over time, a limited capacity testing is performed at each time instance as well. In contrast to the classical, static, group testing problem, the objective in our setup is not to find the minimum number of required tests to identify the infection status of every individual in the population, but to control the infection spread by detecting and isolating the infections over time by using the given, limited number of tests. In order to analyze the performance of the proposed algorithms, we focus on the mean-sense analysis of the number of individuals that remain non-infected throughout the process of controlling the infection. We propose two dynamic algorithms that both use given limited number of tests to identify and isolate the infections over time, while the infection spreads. While the first algorithm is a dynamic randomized individual testing algorithm, in the second algorithm we employ the group testing approach similar to the original work of Dorfman. By considering weak versions of our algorith

Airborne infection risk analysis is usually performed for enclosed spaces where susceptible individuals are exposed to infectious airborne respiratory droplets by inhalation. It is usually based on exponential, dose-response models of which a widely used variant is the Wells-Riley (WR) model. We revisit this infection-risk estimate and extend it to the population level. We use an epidemiological model where the mode of pathogen transmission, either airborne or contact, is explicitly considered. We illustrate the link between epidemiological models and the WR model. We argue that airborne infection quanta are, up to an overall density, airborne infectious respiratory droplets modified by a parameter that depends on biological properties of the pathogen, physical properties of the droplet, and behavioural parameters of the individual. We calculate the time-dependent risk to be infected during the epidemic for two scenarios. We show how the epidemic infection risk depends on the viral latent period and the event time, the time infection occurs. The infection risk follows the dynamics of the infected population. As the latency period decreases, infection risk increases. The longer a su

A virus infection can be initiated with very few or even a single infectious virion, and as such can become extinct, i.e. stochastically fail to take hold or spread significantly. There are many ways that a fully competent infectious virion, having successfully entered a cell, can fail to cause a productive infection, i.e. one that yields infectious virus progeny. Though many discrete, stochastic mathematical models (DSMs) have been developed and used to estimate a virus infection's extinction probability, these typically neglect infection failure post viral entry. The DSM presented herein introduces parameter $γ\in(0,1]$ which corresponds to the probability that a virion's entry into a cell will result in a productive cell infection. We derive an expression for the likelihood of infection extinction in this new DSM, and find that prophylactic therapy with an antiviral acting to reduce $γ$ is best at increasing an infection's extinction probability, compared to antivirals acting on the rates of virus production or virus entry into cells. Using the DSM, we investigate the difference in the fraction of cells consumed by so-called extinct versus established virus infections, and find

Studying the spatiotemporal distribution of SARS-CoV-2 infections among healthcare workers (HCWs) can aid in protecting them from exposure. Existing studies related to HCW infections have emphasized infection rates and protective measures. However, the spatiotemporal patterns and related external environmental factors of HCW infections remain unclear. To fill this gap, an open-source dataset of HCW diagnoses was provided, and the spatiotemporal distributions of SARS-CoV-2 infections among HCWs in Wuhan, China were explored. A geographical detector technique was then used to investigate the impacts of hospital level, type, distance from the infection source, and other external indicators of HCW infections. The results showed that the number of daily HCW infections over time in Wuhan followed a log-normal distribution, with and its mean observed on January 23, 2020 and a standard deviation of 10.8 days. The implementation of high-impact measures, such as the lockdown of the city, may have increased the probability of HCW infections in the short term, especially for HCWs in the outer ring of Wuhan. The infection of HCWs Wuhan exhibited clear spatial heterogeneity. The number of HCW in

We study two simple mathematical models of the epidemic. At first, we study the repetitive infection spreading in a simplified SIRS model including the effect of the decay of the acquired immune. The model is an intermediate model of the SIRS model including the recruitment and death terms and the SIR model in which the recovered population is assumed to be never infected again. When the decay rate δof the immune is sufficiently small, the multiple infection spreading occurs in spikes. The model equation can be reduced to be a map when the decay rate δis sufficiently small, and the spike-like multiple infection spreading is reproduced in the mapping. The period-doubling bifurcation and chaos are found in the simplified SIRS model with seasonal variation. The nonlinear phenomena are reproduced by the map. Next, we study coupled SIRS equations for the directed evolution where the mutation is expressed with a diffusion-type term. A kind of reaction-diffusion equation is derived by the continuum approximation for the infected population I. The reaction-diffusion equation with the linear dependence of infection rate on the type space has an exact Gaussian solution with a time-dependent

Identifying the infection sources in a network, including the index cases that introduce a contagious disease into a population network, the servers that inject a computer virus into a computer network, or the individuals who started a rumor in a social network, plays a critical role in limiting the damage caused by the infection through timely quarantine of the sources. We consider the problem of estimating the infection sources and the infection regions (subsets of nodes infected by each source) in a network, based only on knowledge of which nodes are infected and their connections, and when the number of sources is unknown a priori. We derive estimators for the infection sources and their infection regions based on approximations of the infection sequences count. We prove that if there are at most two infection sources in a geometric tree, our estimator identifies the true source or sources with probability going to one as the number of infected nodes increases. When there are more than two infection sources, and when the maximum possible number of infection sources is known, we propose an algorithm with quadratic complexity to estimate the actual number and identities of the in

We show existence of a non-trivial phase transition for the contact process, a simple model for infection without immunity, on a network which reacts dynamically to the infection trying to prevent an epidemic. This network initially has the distribution of an Erdős-Rényi graph, but is made adaptive via updating in only the infected neighbourhoods, at constant rate. Under these graph dynamics, the presence of infection can help to prevent the spread and so many monotonicity-based techniques fail. Instead, we approximate the infection by an idealised model, the contact process on an evolving forest (CPEF). In this model updating events of a vertex in a tree correspond to the creation of a new tree with that vertex as the root. By seeing the newly created trees as offspring, the CPEF generates a Galton-Watson tree and we use that it survives if and only if its offspring number exceeds one. It turns out that the phase transition in the adaptive model occurs at a different infection rate when compared to the non-adaptive model.

We study a stochastic epidemic model with multiple patches (locations), where individuals in each patch are categorized into three compartments, Susceptible, Infected and Recovered/Removed, and may migrate from one patch to another in any of the compartments. Each individual is associated with a random infectivity function which dictates the force of infectivity while the interactive infection process depends on the age of infection (elapsed time since infection). We prove a functional law of large number for the epidemic evolution dynamics including the aggregate infectivity process, the numbers of susceptible and recovered individuals as well as the number of infected individuals at each time that have been infected for a certain amount of time. From the limits, we derive a PDE model for the density of the number of infected individuals with respect to the infection age, which is a systems of linear PDE equations with a boundary condition that is determined by a set of integral equations.

We study the problem of identifying a single infection source in a network under the susceptible-infected-recovered-infected (SIRI) model. We describe the infection model via a state-space model, and utilizing a state propagation approach, we derive an algorithm known as the heterogeneous infection spreading source (HISS) estimator, to infer the infection source. The HISS estimator uses the observations of node states at a particular time, where the elapsed time from the start of the infection is unknown. It is able to incorporate side information (if any) of the observed states of a subset of nodes at different times, and of the prior probability of each infected or recovered node to be the infection source. Simulation results suggest that the HISS estimator outperforms the dynamic message pass- ing and Jordan center estimators over a wide range of infection and reinfection rates.

We study an individual-based stochastic spatial epidemic model where the number of locations and the number of individuals at each location both grow to infinity. Each individual is associated with a random infection-age dependent infectivity function. Individuals are infected through interactions across the locations with heterogeneous effects. The epidemic dynamics can be described using a time-space representation for the the total force of infection, the number of susceptible individuals, the number of infected individuals that are infected at each time and have been infected for a certain amount of time, as well as the number of recovered individuals. We prove a functional law of large numbers for these time-space processes, and in the limit, we obtain a set of time-space integral equations. We then derive the PDE models from the limiting time-space integral equations, in particular, the density (with respect to the infection age) of the time-age-space integral equation for the number of infected individuals tracking the age of infection satisfies a linear PDE in time and age with an integral boundary condition. These integral equation and PDE limits can be regarded as dynamic

Algorithms for identifying the infection states of nodes in a network are crucial for understanding and containing infections. Often, however, only a relatively small set of nodes have a known infection state. Moreover, the length of time that each node has been infected is also unknown. This missing data -- infection state of most nodes and infection time of the unobserved infected nodes -- poses a challenge to the study of real-world cascades. In this work, we develop techniques to identify the latent infected nodes in the presence of missing infection time-and-state data. Based on the likely epidemic paths predicted by the simple susceptible-infected epidemic model, we propose a measure (Infection Betweenness) for uncovering these unknown infection states. Our experimental results using machine learning algorithms show that Infection Betweenness is the most effective feature for identifying latent infected nodes.

We consider real-time timely tracking of infection status (e.g., covid-19) of individuals in a population. In this work, a health care provider wants to detect infected people as well as people who have recovered from the disease as quickly as possible. In order to measure the timeliness of the tracking process, we use the long-term average difference between the actual infection status of the people and their real-time estimate by the health care provider based on the most recent test results. We first find an analytical expression for this average difference for given test rates, infection rates and recovery rates of people. Next, we propose an alternating minimization based algorithm to find the test rates that minimize the average difference. We observe that if the total test rate is limited, instead of testing all members of the population equally, only a portion of the population may be tested in unequal rates calculated based on their infection and recovery rates. Next, we characterize the average difference when the test measurements are erroneous (i.e., noisy). Further, we consider the case where the infection status of individuals may be dependent, which happens when an i

In this work, we aim to understand the influence of the heterogeneity of infection rates on the Susceptible-Infected-Susceptible (SIS) epidemic spreading. Employing the classic SIS model as the benchmark, we study the influence of the independently identically distributed infection rates on the average fraction of infected nodes in the metastable state. The log-normal, gamma and a newly designed distributions are considered for infection rates. We find that, when the recovery rate is small, i.e.\ the epidemic spreads out in both homogeneous and heterogeneous cases: 1) the heterogeneity of infection rates on average retards the virus spreading, and 2) a larger even-order moment of the infection rates leads to a smaller average fraction of infected nodes, but the odd-order moments contribute in the opposite way; when the recovery rate is large, i.e.\ the epidemic may die out or infect a small fraction of the population, the heterogeneity of infection rates may enhance the probability that the epidemic spreads out. Finally, we verify our conclusions via real-world networks with their heterogeneous infection rates. Our results suggest that, in reality the epidemic spread may not be so

The SIR infection theory initiated by Kermack-Mckendrick in 1927 discusses the infection in an isolated population with uniform properties such as the uniform population distribution. In the infection, there exist two aspects: (1) The quantitative aspect and (2) the temporal aspect. Since the SIR theory is a mean-field theory, it can't match both aspects simultaneously. If the quantitative aspect is matched, the temporal aspect can't be matched, versa. The infection starts from a cluster, and it spreads to different places increasing the size of the infection. In general, even in the case of the infection in a big city, the infection grows within a limited population. Namiki found and named this kind of population as an effective population. He proposes that if the hypothesis is adopted, the quantitative and temporal aspects can be matched simultaneously.

Internet worm infection continues to be one of top security threats and has been widely used by botnets to recruit new bots. In this work, we attempt to quantify the infection ability of individual hosts and reveal the key characteristics of the underlying topology formed by worm infection, i.e., the number of children and the generation of the worm infection family tree. Specifically, we first apply probabilistic modeling methods and a sequential growth model to analyze the infection tree of a wide class of worms. We analytically and empirically find that the number of children has asymptotically a geometric distribution with parameter 0.5. As a result, on average half of infected hosts never compromise any vulnerable host, over 98% of infected hosts have no more than five children, and a small portion of infected hosts have a large number of children. We also discover that the generation follows closely a Poisson distribution and the average path length of the worm infection family tree increases approximately logarithmically with the total number of infected hosts. Next, we empirically study the infection structure of localized-scanning worms and surprisingly find that most of t