Abstract:
We study the critical effect of quarantine on the propagation of epidemics on an adaptive network of social contacts. For this purpose, we analyze the susceptible-infected-recovered (SIR) model in the presence of quarantine, where susceptible individuals protect themselves by disconnecting their links to infected neighbors with probability w, and reconnecting them to other susceptible individuals chosen at random. Starting from a single infected individual, we show by an analytical approach and simulations that there is a phase transition at a critical rewiring (quarantine) threshold w_c separating a phase (w= w_c) where the disease does not spread out. We find that in our model the topology of the network strongly affects the size of the propagation, and that w_c increases with the mean degree and heterogeneity of the network. We also find that w_c is reduced if we perform a preferential rewiring, in which the rewiring probability is proportional to the degree of infected nodes.

Abstract:
We obtain conditions for eradication and permanence of the infectives for a nonautonomous SIQR model with time-dependent parameters, that are not assumed to be periodic. The incidence is given by functions of all compartments and the threshold conditions are given by some numbers that play the role of the basic reproduction number. We obtain simple threshold conditions in the autonomous, asymptotically autonomous and periodic settings and show that our thresholds coincide with the ones already established. Additionally, we obtain threshold conditions for the general nonautonomous model with mass-action, standard and quarantine-adjusted incidence.

Abstract:
We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the model in order to model the latent period. The basic reproductive number is obtained. Furthermore, using time delay as a bifurcation parameter, it is proven that there exists a critical value of delay for the stability of diseases prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. The model is extended to assess the impact of some control measures, by reformulating the model as an optimal control problem with vaccination and treatment. The existence of the optimal control is also proved. Finally, some numerical simulations are performed to verify the theoretical analysis. 1. Introduction Mathematical modelling is of considerable importance in the study of epidemiology because it may provide understanding of the underlying mechanisms which influence the spread of disease and may suggest control strategies. The first known mathematical model of epidemiology is formulated and solved by Daniel Bernoulli in 1760. The foundations of the modern mathematical epidemiology based on the compartment models were laid in the early 20th century [1]. Since the middle of the 20th century, mathematical epidemiology has grown exponentially. In particular, the SIR epidemic model is known as one of the most basic epidemic models, in which total host population is divided into three classes called susceptible , infective , and removed . The basic and important research subjects for these systems are the existence of the threshold value which distinguishes whether the infectious disease will die out, the local stability of the disease-free equilibrium and the endemic equilibrium, the Hopf bifurcation, the existence of periodic solutions, optimal control, and so forth. Many models in the literature represent the dynamics of disease by systems of ordinary differential equations without time delay. In order to reflect the real dynamical behaviors of models that depend on the past history of systems, it is reasonable to incorporate time delays into the systems [2]. In fact, inclusion of delays in epidemic models makes them more realistic by allowing the description of the effects of disease latency or immunity [3, 4]. In this paper, we propose the delayed SIR epidemic model governed by the following equations [5]: where is the number of susceptible individuals, is the number of infectious individuals, is the number of recovered individuals, is the specific growth rate, is the environment capacity, is the transmission

Abstract:
A delayed SEIRS epidemic model with pulse vaccination and nonlinear incidence rate is proposed. We analyze the dynamical behaviors of this model and point out that there exists an infection-free periodic solution which is globally attractive if 1<1, 2>1, and the disease is permanent. Our results indicate that a short period of pulse or a large pulse vaccination rate is the sufficient condition for the eradication of the disease. The main feature of this paper is to introduce time delay and impulse into SEIRS model and give pulse vaccination strategies.

Abstract:
We discuss the epidemic network model with infectious force in latent and infected period. We obtain the basic reproduction number and analyze the globally dynamic behaviors of the disease-free equilibrium when the basic reproduction number is less than one. The effects of various immunization schemes are studied. Finally, the final sizes relation is derived for the network epidemic model. The derivation depends on an explicit formula for the basic reproduction number of network of disease transmission models. 1. Introduction Disease spreading has been the subject of intense research since long time ago. With the advent of modern society, fast transportation systems have changed human habits, and some diseases that just a few years ago would have produced local outbreaks are nowadays a global threat for public health systems. A recent example is given by influenza A(H1N1). In order to understand the mechanism of diseases spreading and other similar processes, such as rumors spreading, networks of movie actor collaboration and science collaboration, WWW, and the Internet, it is of great significance to inspect the effect of complex networks features on disease spreading. Therefore, it is of utmost importance to carefully take into account as much details as possible of the structural properties of the network on which the infection dynamics occurs. And in the general case, the epidemic system can be represented as a network where nodes stand for individuals and an edge connecting two nodes denotes the interaction between individuals. The degree of a node is the number of its neighbors, that is, the number of links adjacent to the node. In the past, researchers mainly focused the disease transmission study on the conventional networks [1, 2] such as lattices, regular tree, and ER random graph. Since late 1990s, scientists have presented a series of statistical complex topological characteristics [3–6] such as the small-world (SW) phenomenon [7] and scale-free (SF) property [8] by investigating many real networks. On scale-free networks, it was assumed that the larger the node degree, the greater the infectivity of the node, and the infectivity is just equal to the node degree. Under such an assumption, for instance, Pastor-Satorras et al. concluded that the epidemic threshold for heterogenous networks with sufficiently large size [9]. Subsequently, the studies of dynamical processes on complex networks also have attracted lots of interests with various subjects [10–15], and as one of the typical dynamical processes built on complex networks, epidemic

Abstract:
SEIQR (Susceptible, Exposed, Infectious, Quarantined, and Recovered) models for the transmission of malicious objects with simple mass action incidence and standard incidence rate in computer network are formulated. Threshold, equilibrium, and their stability are discussed for the simple mass action incidence and standard incidence rate. Global stability and asymptotic stability of endemic equilibrium for simple mass action incidence have been shown. With the help of Poincare Bendixson Property, asymptotic stability of endemic equilibrium for standard incidence rate has been shown. Numerical methods have been used to solve and simulate the system of differential equations. The effect of quarantine on recovered nodes is analyzed. We have also analyzed the behavior of the susceptible, exposed, infected, quarantine, and recovered nodes in the computer network.

Abstract:
Worms exploiting zero-day vulnerabilities have drawn significant attention owing to their enormous threats to the Internet. In general, users may immunize their computers with countermeasures in exposed and infectious state, which may take a period of time. Through theoretical analysis, time delay may lead to Hopf bifurcation phenomenon so that the worm propagation system will be unstable and uncontrollable. In view of the above factors, a quarantine strategy is thus proposed in the study. In real network, unknown worms and worm variants may lead to great risks, which misuse detection system fails to detect. However, anomaly detection is of help in detecting these kinds of worm. Consequently, our proposed quarantine strategy is built on the basis of anomaly intrusion detection system. Numerical experiments show that the quarantine strategy can diminish the infectious hosts sharply. In addition, the threshold is much larger after using our quarantine strategy, which implies that people have more time to remove worms so that the system is easier to be stable and controllable without Hopf bifurcation. Finally, simulation results match numerical ones well, which fully supports our analysis. 1. Introduction In recent years, with the rapid development of computer technologies and network applications, Internet has become a powerful mechanism for propagating malicious software programs.Systems running on network computers become more vulnerable to digital threats. In particular,worms that exploit zero-day vulnerabilities have brought severe threats to Internet security. To a certain extent, the propagation of worms in a system of interacting computers could be compared with infectious diseases in a population. Anderson and May discussed the spreading nature of biological viruses, parasites and so forth.leading to infectious diseases in human population through several epidemic models [1, 2]. The action of worms throughout a network can be studied by using epidemiological models for disease propagation [3–8]. Mishra and Saini [4] present a SEIRS model with latent and temporary immune periods, which can reveal common worm propagation. Dong et al. propose a computer virus model with time delay based on an SEIR model and regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation [9]. Ren et al. give a novel computer virus propagation model and study its dynamic behaviors [10]. L.-X. Yang and X. Yang also investigates the propagation behavior of virus programs provided infected

Abstract:
In this paper, a delayed SIR epidemic model with modified saturated incidence rate is proposed. The local stability and the existence of Hopf bifurcation are established. Also some numerical simulations are given to illustrate the theoretical analysis.

Abstract:
We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. We investigate the qualitative behaviour of the model and find the conditions that guarantee the asymptotic stability of corresponding steady states. We present the conditions in the time lag in which the DDE model is stable. Hopf bifurcation analysis is also addressed. Numerical simulations are provided in order to illustrate the theoretical results and gain further insight into the behaviour of this system. 1. Introduction Epidemics have ever been a great concern of human kind, because the impact of infectious diseases on human and animal is enormous, both in terms of suffering and social and economic consequences. Mathematical modeling is an essential tool in studying a diverse range of such diseases to gain a better understanding of transmission mechanisms, and make predictions; determine and evaluate control strategies. Many authors have proposed various kinds of epidemic models to understand the mechanism of disease transmission (see [1–10] and references therein). The basic elements for the description of infectious diseases have been considered by three epidemiological classes: that measures the susceptible portion of population, the infected, and the removed ones. Kermack and McKendrick [11] described the simplest SIR model which computes the theoretical number of people infected with a contagious illness in a closed population over time. Transmission of a disease is a dynamical process driven by the interaction between susceptible and infective. The behaviour of the SIR models are greatly affected by the way in which transmission between infected and susceptible individuals are modelled. The simplest model in which recovery does not give immunity is the SIS model, since individuals move from the susceptible class to the infective class and then back to the susceptible class upon recovery. If individuals recover with permanent immunity, then the simplest model is an SIR model. If individuals recover with temporary immunity so that they eventually become susceptible again, then the simplest model is an SIRS model. If individuals do not recover, then the simplest model is an SI model. In general, SIR (epidemic and endemic) models are appropriate for viral agent diseases such as measles, mumps, and smallpox, while SIS models are appropriate for some bacterial agent diseases such as meningitis, plague, and sexually transmitted diseases, and for protozoan agent diseases such

Abstract:
The customary perspective to reason about epidemic mitigation in temporal networks hinges on the identification of nodes with specific features or network roles. The ensuing individual-based control strategies, however, are difficult to carry out in practice and ignore important correlations between topological and temporal patterns. Here we adopt a mesoscopic perspective and present a principled framework to identify collective features at multiple scales and rank their importance for epidemic spread. We use tensor decomposition techniques to build an additive representation of a temporal network in terms of mesostructures, such as cohesive clusters and temporally-localized mixing patterns. This representation allows to determine the impact of individual mesostructures on epidemic spread and to assess the effect of targeted interventions that remove chosen structures. We illustrate this approach using high-resolution social network data on face-to-face interactions in a school and show that our method affords the design of effective mesoscale interventions.