Stability and Hopf Bifurcation for a Delayed SLBRS Computer Virus Model

By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results

Modeling and Analysis of Bifurcation in a Delayed Worm Propagation Model

A delayed worm propagation model with birth and death rates is formulated. The stability of the positive equilibrium is studied. Through theoretical analysis, a critical value τ0 of Hopf bifurcation is derived. The worm propagation system is locally asymptotically stable when time delay is less than τ0. However, Hopf bifurcation appears when time delay τ passes the threshold τ0, which means that the worm propagation system is unstable and out of control. Consequently, time delay should be adjusted to be less than τ0 to ensure the stability of the system stable and better prediction of the scale and speed of Internet worm spreading. Finally, numerical and simulation experiments are presented to simulate the system, which fully support our analysis

A general class of spreading processes with non-Markovian dynamics

In this paper we propose a general class of models for spreading processes we call the $SI^*V^*$ model. Unlike many works that consider a fixed number of compartmental states, we allow an arbitrary number of states on arbitrary graphs with heterogeneous parameters for all nodes and edges. As a result, this generalizes an extremely large number of models studied in the literature including the MSEIV, MSEIR, MSEIS, SEIV, SEIR, SEIS, SIV, SIRS, SIR, and SIS models. Furthermore, we show how the $SI^*V^*$ model allows us to model non-Poisson spreading processes letting us capture much more complicated dynamics than existing works such as information spreading through social networks or the delayed incubation period of a disease like Ebola. This is in contrast to the overwhelming majority of works in the literature that only consider spreading processes that can be captured by a Markov process. After developing the stochastic model, we analyze its deterministic mean-field approximation and provide conditions for when the disease-free equilibrium is stable. Simulations illustrate our results

bifurcation analysis of a delayed worm propagation model with saturated incidence

This paper is concerned with a delayed SVEIR worm propagation model with saturated incidence. The main objective is to investigate the effect of the time delay on the model. Sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation are obtained by choosing the time delay as the bifurcation parameter. Particularly, explicit formulas determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold theorem. Numerical simulations for a set of parameter values are carried out to illustrate the analytical results

A stochastic SIRI epidemic model with relapse and media coverage

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Faculty of Sciences (Ibn Tofail University

Epidemiological models and optimal control theory - Applications to marketing and computer viruses transmission

Dissertação de Mestrado em Engenharia de SistemasEpidemiological models and Optimal Control Theory are closely interrelated, inasmuch as the development of control interventions that could help to perceive and minimize the spread of infection diseases is a pressing need. However, the application of these two subjects can be extended to other scientific domains apart from health. In this respect, this dissertation takes advantage of the foundations of Mathematical Epidemiology and Optimal Control Theory to study the dynamics of Viral Marketing within a certain population and, with less detail, Computer Viruses Transmission into network systems. Due to the fierce marketing competition, not all the marketing campaigns become viral, and formulate strategies that could leverage traditional marketing campaigns is not a trivial task. In this regard, since the process of diffusing Viral Marketing campaigns through social networks can be modeled under concepts of Mathematical Epidemiology and being strategy the keyword, the benefits of Optimal Control Theory on the diffusion of a real viral advertisement are studied and control strategies to maximize the spread of viral messages with low cost are investigated and proposed in optimal time windows. Numerical methods based on the Pontryagin’s Maximum Principle (indirect methods) and methods that treat the Optimal Control problem as a nonlinear constrained optimization problem (direct methods) are tested and compared, using different numerical solvers. In an introductory way, Computer Viruses Transmission and its dynamics are briefly studied by using epidemiological modeling over networks. In addition, an R software package related to the mathematical modeling of infectious diseases is discussed and the propagation of computer viruses within a network system is illustrated.O desenvolvimento de políticas de controlo é crucial para perceber e minimizar a propagação de doenças infeciosas. Sob este pressuposto, os modelos epidemiológicos e a Teoria do Controlo Ótimo surgem, cada vez mais, intimamente relacionados. Todavia, a aplicação destes domínios científicos pode ser estendida a outras áreas do conhecimento. Neste contexto, a presente dissertação estuda não só as dinâmicas do Marketing Viral no seio de uma população, recorrendo a fundamentos de Epidemiologia Matemática e à Teoria do Controlo Ótimo, mas também, com menor enfoque, a Transmissão de Vírus Informáticos em sistemas em rede. Atendendo à crescente competitividade dos mercados, nem todas as campanhas de marketing se tornam virais. Para além disso, a conceção de estratégias de marketing que, por sua vez, permitam colmatar as carências das campanhas tradicionais pode revelar-se uma tarefa bastante complexa. Sob evidências científicas de que o processo de difusão de campanhas de Marketing Viral pode ser modelado sob conceitos de Epidemiologia Matemática, são estudadas as potencialidades da Teoria do Controlo Ótimo na disseminação de um campanha real de marketing. Mais ainda, são discutidas e propostas estratégias de controlo para maximizar a difusão de mensagens virais a um baixo custo. São, igualmente, testados e comparados métodos númericos baseados no Princípio do Máximo de Pontryagin (métodos indirectos) e, por outro lado, métodos que tratam um problema de Controlo Ótimo como um problema de otimização não linear com restrições (métodos diretos). No que concerne à Transmissão de Vírus Informáticos, a sua dinâmica de propagação é analisada, de uma forma puramente introdutória, sob pressupostos relativos à modelação epidemiológica em redes. Para isso, é explorado um pacote do software R dedicado à modelação de doenças infeciosas, permitindo ilustrar a propagação de vírus informáticos num dado sistema

Global Stability of Multigroup SIRS Epidemic Model with Varying Population Sizes and Stochastic Perturbation around Equilibrium

We discuss multigroup SIRS (susceptible, infectious, and recovered) epidemic models with random perturbations. We carry out a detailed analysis on the asymptotic behavior of the stochastic model; when reproduction number ℛ0>1, we deduce the globally asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium of the deterministic model in time average. Numerical methods are employed to illustrate the dynamic behavior of the model and simulate the system of equations developed. The effect of the rate of immunity loss on susceptible and recovered individuals is also analyzed in the deterministic model

A novel dynamics model of fault propagation and equilibrium analysis in complex dynamical communication network

International audienceTo describe failure propagation dynamics in complex dynamical communication networks, we propose an efficient and compartmental standard-exception-failure propagation dynamics model based on the method of modeling disease propagation in social networks. Mathematical formulas are derived and differential equations are solved to analyze the equilibrium of the propagation dynamics. Stability is evaluated in terms of the balance factor G and it is shown that equilibrium where the number of nodes in different states does not change, is globally asymptotically stable if G≥1. The theoretical results derived are verified by numerical simulations. We also investigate the effect of some network parameters, e.g. node density and node movement speed, on the failure propagation dynamics in complex dynamical communication networks to gain insights for effective measures of control of the scale and duration of the failure propagation in complex dynamical communication networks