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The Analysis of Epidemic Network Model with Infectious Force in Latent and Infected Period  [PDF]
Juping Zhang,Zhen Jin
Discrete Dynamics in Nature and Society , 2010, DOI: 10.1155/2010/604329
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
Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment  [PDF]
Erika Rivero-Esquivel,Eric ávila-Vales,Gerardo E. García-Almeida
Mathematics , 2015,
Abstract: We study the dynamics of a SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment, that takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions we prove the existence of backward bifurcation, the stability and the direction of Hopf bifurcation. We also explore how the mechanism of backward bifurcation affects the control of the infectious disease. Numerical simulations are presented to illustrate the theoretical findings.
Bifurcation Analysis of an SIR Epidemic Model with the Contact Transmission Function  [PDF]
Guihua Li,Gaofeng Li
Abstract and Applied Analysis , 2014, DOI: 10.1155/2014/930541
Abstract: We consider an SIR endemic model in which the contact transmission function is related to the number of infected population. By theoretical analysis, it is shown that the model exhibits the bistability and undergoes saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. Furthermore, we find that the threshold value of disease spreading will be increased, when the half-saturation coefficient is more than zero, which means that it is an effective intervention policy adopted for disease spreading. However, when the endemic equilibria exist, we find that the disease can be controlled as long as we let the initial values lie in the certain range by intervention policy. This will provide a theoretical basis for the prevention and control of disease. 1. Introduction The classical SIR model for disease transmission has been widely studied. It is one of the most important issues that the dynamical behaviors are changed by the different incidence rate in epidemic system. For the incidence rate, we divided into two categories: one is that Capasso and Serio [1] proposed the infection force which is a saturated curve, described “crowding effect” or “protection measures;” the other is the infection force that describes the effect of “intervention policy,” for example, closing schools and restaurants and postponing conferences (see Figure 1). For the model with the saturated infection force, , which is one of the typical infection forces, the rich dynamical behaviors were founded by Ruan and Wang [2] and Tang et al. [3]. The model with the incidence rate can be suited for many infectious diseases, including measles, mumps, rubella, chickenpox, and influenza. For more research literatures about nonlinear infection rate see [4–9]. However, for some parasite-host models, by observing macro- and microparasitic infections, one finds that the infection rate is an increasing function of the parasite dose, usually sigmoidal in shape [10, 11]. So we will build a model with sigmoidal incidence rate which is taken into account “crowding effect” and “saturated effect.” According to the parasite-host model which is proposed by Anderson and May (1979) [12, 13], the model is as follows: where are susceptible hosts, infected hosts, and removed hosts, respectively. is the birth rate of susceptible host, is the natural death rate of a population, is the removal rate, and is the per capita infection-related death rate. If we denote infection force , can be explained as the rate of valid contact. At the beginning of disease, most people have poor awareness of
Complex Dynamics of Discrete SEIS Models with Simple Demography  [PDF]
Hui Cao,Yicang Zhou,Baojun Song
Discrete Dynamics in Nature and Society , 2011, DOI: 10.1155/2011/653937
Abstract: We investigate bifurcations and dynamical behaviors of discrete SEIS models with exogenous reinfections and a variety of treatment strategies. Bifurcations identified from the models include period doubling, backward, forward-backward, and multiple backward bifurcations. Multiple attractors, such as bistability and tristability, are observed. We also estimate the ultimate boundary of the infected regardless of initial status. Our rigorously mathematical analysis together with numerical simulations show that epidemiological factors alone can generate complex dynamics, though demographic factors only support simple equilibrium dynamics. Our model analysis supports and urges to treat a fixed percentage of exposed individuals. 1. Introduction Pure demographic (or ecological) discrete models can have very complicated dynamics. It is well known that the logistic model , Ricker's model [1–3], and Hassell model [4] all have complex dynamics through the mechanics of period-doubling bifurcation. When adding epidemiological effects to these demographic models to describe the dynamics of an infectious disease, it is not surprising at all that the complex dynamics still retain [5, 6]. Yet an interesting question is that if adding epidemiological effects to a simple demographic model that has only trivial dynamics, for example, , is it possible to observe complex dynamics? Our work in this paper finds a quite positive answer by bifurcation analysis. It is shown that the chaotic dynamics is possible. The bifurcation approach has been extensively utilized in theoretical epidemiology. The use of equilibrium bifurcation to investigate the dynamical behavior of continuous epidemic models has been successful, for instance, the work in [7–15]. Meanwhile, discrete epidemic models have gained more popularity [5, 6, 16–25], since epidemiological data are usually collected at discrete times and it becomes easier to compare data with models. We will use the bifurcation approach to analyze discrete SEIS epidemic models in this paper. The appearance of multiple attractors enhances uncertainty of outcomes because of the lack of information to the initial epidemiological status, even though the domain of attractions of each attractor is fully depicted. In this case, it would be necessary to estimate the prevalence regardless of the initial data. The management measures to completely wipe out an infectious disease may be too costly, far beyond the capability of any society. A practical and feasible way is to have an infectious disease under control, that is, to keep the number of the
Epidemic dynamics on an adaptive network  [PDF]
Thilo Gross,Carlos Dommar D'Lima,Bernd Blasius
Physics , 2005, DOI: 10.1103/PhysRevLett.96.208701
Abstract: Many real world networks are characterized by adaptive changes in their topology depending on the dynamic state of their nodes. Here we study epidemic dynamics in an adaptive network, where susceptibles are able to avoid contact with infected by rewiring their network connections. We demonstrate that adaptive rewiring has profound consequences for the emerging network structure, giving rise to assortative degree correlation and a separation into two loosely connected sub-compartments. This leads to dynamics such as oscillations, hysteresis and 1st order transitions. We describe the system in terms of a simple model using a pair-approximation and present a full local bifurcation analysis. Our results indicate that the interplay between dynamics and topology can have important consequences for the spreading of infectious diseases and related applications.
The Hopf Bifurcation Analysis and Optimal Control of a Delayed SIR Epidemic Model  [PDF]
Abdelhadi Abta,Hassan Laarabi,Hamad Talibi Alaoui
International Journal of Analysis , 2014, DOI: 10.1155/2014/940819
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
Backward Bifurcation and Control in Transmission Dynamics of Arboviral Diseases  [PDF]
Hamadjam Abboubakar,Jean Claude Kamgang,Daniel Tieudjo
Quantitative Biology , 2015,
Abstract: In this paper, we derive and analyze a compartmental model for the control of arboviral diseases which takes into account an imperfect vaccine combined with individual protection and some vector control strategies already studied in the literature. After the formulation of the model, a qualitative study based on stability analysis and bifurcation theory reveals that the phenomenon of backward bifurcation may occur. The stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the reproduction number, R 0 , is less than unity. Using Lyapunov function theory, we prove that the trivial equilibrium is globally asymptotically stable; When the disease-- induced death is not considered, or/and, when the standard incidence is replaced by the mass action incidence, the backward bifurcation does not occur. Under a certain condition , we establish the global asymptotic stability of the disease--free equilibrium of the full model. Through sensitivity analysis, we determine the relative importance of model parameters for disease transmission. Numerical simulations show that the combination of several control mechanisms would significantly reduce the spread of the disease, if we maintain the level of each control high, and this, over a long period.
Nonlinear Pulse Vaccination in an SIR Epidemic Model with Resource Limitation  [PDF]
Wenjie Qin,Sanyi Tang,Robert A. Cheke
Abstract and Applied Analysis , 2013, DOI: 10.1155/2013/670263
Abstract: Mathematical models can assist in the design and understanding of vaccination strategies when resources are limited. Here we propose and analyse an SIR epidemic model with a nonlinear pulse vaccination to examine how a limited vaccine resource affects the transmission and control of infectious diseases, in particular emerging infectious diseases. The threshold condition for the stability of the disease free steady state is given. Latin Hypercube Sampling/Partial Rank Correlation Coefficient uncertainty and sensitivity analysis techniques were employed to determine the key factors which are most significantly related to the threshold value. Comparing this threshold value with that without resource limitation, our results indicate that if resources become limited pulse vaccination should be carried out more frequently than when sufficient resources are available to eradicate an infectious disease. Once the threshold value exceeds a critical level, both susceptible and infected populations can oscillate periodically. Furthermore, when the pulse vaccination period is chosen as a bifurcation parameter, the SIR model with nonlinear pulse vaccination reveals complex dynamics including period doubling, chaotic solutions, and coexistence of multiple attractors. The implications of our findings with respect to disease control are discussed. 1. Introduction Epidemiology is the study of the spread of diseases with the objective of tracing factors that are responsible for or contribute to their occurrence and serves as the foundation and logic of interventions made in the interest of public health and preventive medicine. Mathematical models describing the population dynamics of infectious diseases have played an important role in better understanding epidemiological patterns and disease control for a long time. Various epidemic models have been proposed and explored extensively and considerable progress has been achieved in the studies of disease control and prevention (see [1–3] and the references therein). Outbreaks of infectious diseases have not only caused the loss of billions of lives but have often also rapidly damaged social economic systems, bringing about much human misery. Consequently, the focus of our research has been on how to prevent and cure infectious diseases effectively. It is well known that one of the most important concerns in the analysis of epidemic logical models is the efficacy of vaccination programmes. This subject gained prominence as a result of highly successful application of vaccinations for the worldwide eradication of smallpox
Forward Hysteresis and Backward Bifurcation Caused by Culling in an Avian Influenza Model  [PDF]
Hayriye Gulbudak,Maia Martcheva
Quantitative Biology , 2013,
Abstract: The emerging threat of a human pandemic caused by the H5N1 avian influenza virus strain magnifies the need for controlling the incidence of H5N1 infection in domestic bird populations. Culling is one of the most widely used control measures and has proved effective for isolated outbreaks. However, the socio-economic impacts of mass culling, in the face of a disease which has become endemic in many regions of the world, can affect the implementation and success of culling as a control measure. We use mathematical modeling to understand the dynamics of avian influenza under different culling approaches. We incorporate culling into an SI model by considering the per capita culling rates to be general functions of the number of infected birds. Complex dynamics of the system, such as backward bifurcation and forward hysteresis, along with bi-stability, are detected and analyzed for two distinct culling scenarios. In these cases, employing other control measures temporarily can drastically change the dynamics of the solutions to a more favorable outcome for disease control.
Hopf Bifurcation in an SEIDQV Worm Propagation Model with Quarantine Strategy  [PDF]
Yu Yao,Wenlong Xiang,Andong Qu,Ge Yu,Fuxiang Gao
Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/304868
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
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