Global stability of a virus infection model with two delays and two types of target cells

In this paper, incorporating the delay of viral cytopathicity within target cells, we first presented a basic model of viral infection with delay, and then extended it into a model with two delays and two types of target cells. For the models proposed here, both their basic reproduction numbers are found. By constructing Lyapunov functionals, necessary and sufficient conditions ensuring the global stability of the models with delays are given. The obtained results show that, when the basic reproduction number is not greater than one, the infection-free equilibrium is globally stable in the feasible region, which implies that the virus infection goes extinct eventually; when it is greater than one, the infection equilibrium is globally stable in the feasible region, which implies that the virus infection persists in the body of host.     

Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal

In this paper, we investigate the dynamical behavior of two nonlinear models for viral infection with humoral immune response. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The intrinsic growth rate of uninfected cells, incidence rate of infection, removal rate of infected cells, production rate of viruses, neutralization rate of viruses, activation rate of B cells and removal rate of B cells are given by more general nonlinear functions. The second model is a modification of the first one by including an eclipse stage of infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For each model we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the global asymptotic stability of the all equilibria of the models. We perform some numerical simulations for the models with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.     

Generals die in friendly fire,or modeling immune response to HIV

We develop a kinetic model for CD8 T lymphocytes (CTL) whose purpose is to kill cells infected with viruses and intracellular parasites. Using a set of first-order nonlinear differential equations, the model predicts how numbers of different cell types involved in CTL response depend on time. The model postulates that CTL response requires continuous presence of professional antigen-presenting cells (APC) comprised of macrophages and dendritic cells. It assumes that any virus present in excess of a threshold level activates APC that, in turn, activate CTL that expand in number and become armed “effector” cells. In the end, APC are deactivated after virus is cleared. The lack of signal from APC causes effector cells to differentiate, by default, into “transitory cells” that either die, or, in a small part, become long-lived memory cells. Viruses capable of infecting APC will cause premature retirement of effector CTL. If transitory cells encounter virus, which takes place after the premature depletion, CTL become anergic (unresponsive to external stimuli). The model is designed to fit recent experiments on primary CTL response to simian immunodeficiency virus closely related to HIV and lymphocytic choriomeningitis virus. The two viruses are known to infect APC and make them targets for CTL they are supposed to control. Both viruses cause premature depletion and anergy of CTL and persist in the host for life.     

Global stability of delay-distributed viral infection model with two modes of viral transmission and B-cell impairment

This paper formulates a virus dynamics model with impairment of B-cell functions. The model incorporates two modes of viral transmission: cell-free and cell-to-cell. The cell-free and cell-cell incidence rates are modeled by general functions. The model incorporates both, latently and actively, infected cells as well as three distributed time delays. Nonnegativity and boundedness properties of the solutions are proven to show the well-posedness of the model. The model admits two equilibria that are determined by the basic reproduction number R₀. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle's invariance principle. The theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions and time delays on the virus dynamics are studied. We have shown that if the functions of B-cell is impaired, then the concentration of viruses is increased in the plasma. Moreover, we have observed that increasing the time delay will suppress the viral replication.     

Mathematical modelling of acute virus influenza A infections

This article models the immune system and the virus dynamics of acute influenza infection mathematically. We use the model to study the virus dynamics of some well-known and severe and mild types of viruses. Linkages to well-known models in the literature are illustrated. Simulations are compared with experimental results in vivo by comparing with results from infected ferrets where infection closely resembles those in humans. Good agreement is achieved between the model calculations and the experimental values for influenza A viruses. For the Spanish flu virus H1N1 peak virus load is high and virtually all cells are infected in the nostril. In general, the H1N1 viruses show much more prolonged infections than the H3N2 in the nostril. We suggest that the reason is that unspecific immunity attacks H3N2-budded viruses but not H1N1 viruses.     

Using algebraic models of programs for detecting metamorphic malwares

Polymorphic and metamorphic viruses are the most sophisticated malicious programs that give a lot of trouble to virus scanners. Each time when these viruses infect new executables or replicate themselves, they completely modify (obfuscate) their signature to avoid being detected. This contrivance poses a serious threat to antivirus software that relies on classical virus-detection techniques: such viruses do not have any stable specific sequence of instructions that one looks for. In the ultimate case, the only characteristic that remains invariable for all generations of the same virus is their functionality (semantics). To all appearance, the only way to detect for sure a metamorphic malicious code is to look for a pattern that has the same semantics as (i.e., equivalent to) some representative sample of the virus. Thus, metamorphic virus detection is closely related to the equivalence-checking problem for programs. In this paper, we outline some new automata-theoretic framework for the designing of virus detectors. Our approach is based on the equivalence-checking techniques in algebraic models of sequential programs. An algebraic model of programs is an abstract model of computation, where programs are viewed as finite automata operating on Kripke structures. Models of this kind make it possible to focus on those properties of program instructions that are widely used in obfuscating transformations. We give a survey (including the latest results) on the complexity of equivalence-checking problem in various algebraic models of programs and estimate thus the resilience of some obfuscating transformation commonly employed by metamorphic viruses.     

A mathematical model of combined therapies against cancer using viruses and inhibitors

This paper deals with a procedure for combined therapies against cancer using oncolytic viruses and inhibitors. Replicating genetically modified adenoviruses infect cancer cells, reproduce inside them and eventually cause their death (lysis). As infected cells die, the viruses inside them are released and then proceed to infect other tumor cells. The successful entry of virus into cancer cells is related to the presence of the coxsackie-adenovirus receptor (CAR). Mitogen-activated protein kinase kinase (known as MEK) inhibitors can promote CAR expression, resulting in enhanced adenovirus entry into cancer cells. However, MEK inhibitors can also cause G1 cell-cycle arrest, inhibiting reproduction of the virus. To design an effective synergistic therapy, the promotion of virus infection must be optimally balanced with inhibition of virus production. We introduce a mathematical model to describe the effects of MEK inhibitors and viruses on tumor cells, and use it to explore the reduction of the tumor size that can be achieved by the combined therapies. Furthermore, we find an optimal dose of inhibitor: P _optimal = 1 − μ/δ for a certain initial density of cells (where μ is the removal rate of the dead cells and δ is the death rate of the infected cells). The optimal timing of MEK inhibitors is also numerically studied. This work was supported by the National Natural Science Foundation of China (Grant No. 10571023)     

New conditions for synchronization in complex networks with multiple time-varying delays

In this paper, two kinds of synchronization problems of complex dynamical networks with multiple time-varying delays are investigated, that is, the cases with fixed topology and with switching topology. For the former, different from the commonly used linear matrix inequality (LMI) method, we adopt the approach basing on the scramblingness property of the network’s weighted adjacency matrix. The obtained result implies that the network will achieve exponential synchronization for appropriate communication delays if the network’s weighted adjacency matrix is of scrambling property and the coupling strength is large enough. Note that, our synchronization condition is very new, which would be easy to check in comparison with those previously reported LMIs. Moreover, we extend the result to the case when the interaction topology is switching. The maximal allowable upper bounds of communication delays are obtained in each case. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.     

Dynamics of an HIV Infection Model with Two Infection Routes and Evolutionary Competition between Two Viral Strains

The existing combination therapy of HIV antiretroviral drugs can lead to the emergence of drug-resistant viruses, and cannot effectively block direct cell-to-cell infections, these factors results in incomplete virus suppression and increased risk of disease progression. In this paper, we formulate an HIV model with two strains representing a drug-sensitive virus and a drug-resistant virus to study the joint mechanism of drug resistance. We first reduce the infection-age model to a system of integro-differential equations with infinite delays. Then the stability of the equilibria and the dynamics of competition between two viruses are studied to illuminate the joint effects of infection-age and two infection routes on the evolution of both drug-sensitive and drug-resistant strains before and during drug treatment. Applying a persistence theorem for infinite dimensional systems, we obtain that the disease is always present when the basic reproduction number is larger than unity. Numerical simulations confirm that the basic reproduction numbers and mutation coefficient are the key threshold parameters for determining the competition results of the two viral strains and indicate the cell-to-cell transmission increases the likelihood that HIV breaks out within the host. Finally, sensitivity analyses suggest that the available combination therapy should be taken once symptoms of resistance appear during drug treatment, and demonstrate that the presence of cell-to-cell transmission attenuates the efficacy of the existing antiretroviral drug treatments.     

A sparse hierarchical Bayesian model for detecting relevant antigenic sites in virus evolution

Understanding how viruses offer protection against closely related emerging strains is vital for creating effective vaccines. For many viruses, multiple serotypes often co-circulate and testing large numbers of vaccines can be infeasible. Therefore the development of an in silico predictor of cross-protection between strains is important to help optimise vaccine choice. Here we present a sparse hierarchical Bayesian model for detecting relevant antigenic sites in virus evolution (SABRE) which can account for the experimental variability in the data and predict antigenic variability. The method uses spike and slab priors to identify sites in the viral protein which are important for the neutralisation of the virus. Using the SABRE method we are able to identify a number of key antigenic sites within several viruses, as well as providing estimates of significant changes in the evolutionary history of the serotypes. We show how our method outperforms alternative established methods; standard mixed effects models, the mixed effects LASSO, and the mixed effects elastic nets. We also propose novel proposal mechanisms for the Markov chain Monte Carlo simulations, which improve mixing and convergence over that of the established component-wise Gibbs sampler.     

A propagation model of computer virus with nonlinear vaccination probability

This paper is intended to examine the effect of vaccination on the spread of computer viruses. For that purpose, a novel computer virus propagation model, which incorporates a nonlinear vaccination probability, is proposed. A qualitative analysis of this model reveals that, depending on the value of the basic reproduction number, either the virus-free equilibrium or the viral equilibrium is globally asymptotically stable. The results of simulation experiments not only demonstrate the validity of our model, but also show the effectiveness of nonlinear vaccination strategies. Through parameter analysis, some effective strategies for eradicating viruses are suggested.     

Interventions in GARCE Branching Processes with Application to Ebola Virus Data

In hindsight, even a cursory look may have revealed substantial growth of the 2014 Ebola infection and death cases in West Africa before drastic interventions showed an effect in late 2014. Yet a timely assessment as to whether an intervention has a sufficient impact to stabilize and eventually end an outbreak is equally important as early detection and accurate prediction of the magnitude of the outbreak several months before it spins out of control. To this aim, we consider an intervention effect in the GARCE branching process model, proposed by Hueter, that was successful to early detect the magnitude of the outbreak when data became available in early 2014. This model provides a novel and simple approach to branching processes that allows for time-varying random environments and instances of peak growth and near extinction-type rates as seen in Ebola viruses, tuberculosis infections, and infectious diseases. We present results on the survival and extinction behaviours, characterization of the phase transition between the subcritical and supercritical phases, and a sufficient condition for escape from supercriticality upon a level shift intervention. Intervention analysis of the Ebola outbreak data are presented and findings on the outbreak’s estimated phase and intervention effect are discussed.     

Two classes of piecewise-linear difference equations with eventual periodicity three

In this paper, we will study two classes of difference equations which are piecewise-linear and of similar forms. We will show that all nontrivial solutions of one equation are eventually periodic with prime period three. We will show this result for one case of the second equation.     

Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus

Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function.     

具有垂直传染和接触传染的传染病模型的稳定性研究

本文研究了一类具有垂直传染和接触传染的传染病模型.利用常微分方程定性与稳定性方法,分析了该模型非负平衡点的存在性及其局部稳定性.同时,利用LaSalle不变性原理和通过构造适当的Lyapunov函数,获得了平凡平衡点、无病平衡点和地方病平衡点全局渐近稳定的充分条件.结果表明当基本再生数小于等于1时,所有种群趋于灭绝;当基本再生数大于1和病毒主导再生数小于1时,病毒很快被清除;当基本再生数大于1和病毒主导再生数大于1以及满足一定条件时,病毒持续流行并将成为一种地方病.     

Two crucial examples related to properness of weakly relaxed delayed controls

In this paper, we deal with the problem of finding a proper relaxation procedure (in the sense that relaxed minimizers can be approximated with ordinary controls) for optimal control problems involving delays in the control variables. For systems involving commensurate delays it is well known that a ‘strong’ model provides a proper relaxation procedure. For the noncommensurate case, however, the question of how to properly relax delayed controls has remained open and a natural candidate has been a modified version of the strong model, which we call the ‘weak’ procedure, first introduced by Warga. In this paper, we show by means of two examples that this procedure may fail to be proper for systems with either commensurate or noncommensurate delays. It is expected that these examples will provide some insight into the problem of finding a proper relaxation procedure applicable to the general case.     

Multi-period lot-sizing with supplier selection: Structural results,complexity and algorithms

We consider a multi-period lot-sizing problem with multiple products and multiple suppliers. Demand is deterministic and time-varying. The objective is to determine order quantities to minimize the total cost over a finite planning horizon. This problem is strongly NP-hard. For a special case, we extend the classical zero-inventory-ordering principle and solve it by dynamic programming. Based on this new extension, we also develop a heuristic algorithm for the general problem and computationally show that it works well.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

Flocking dynamics for a multiagent system involving task strategy

Collective behavior sometimes requires forming a particular formation or reaching a certain velocity to accomplish a specific task, such as bird migration. In this paper, we investigate the collective migration model, which consists of two parts: Cucker–Smale type interaction and target velocity. Each agent has a strategy to allocate limited energy to group interaction and velocity tracking. In this case, if the system achieves monocluster flocking then the final velocity is equal to target velocity. When the strategy is invariant, we show that 1/2 is a critical threshold which is consistent with the classical Cucker–Smale model. When the strategy is time varying, we provide a time-varying strategy named threshold strategy to ensure that for any initial state the system achieves monocluster flocking and the final velocity reaches target velocity. In addition, the case of multiple target velocities is considered. According to the theory of bicluster flocking, we obtain a sufficient framework to guarantee that the system achieves bicluster flocking and two groups would reach their target velocities, respectively.     

一个在肿瘤细胞中可复制的病毒模型的数学分析

作者考虑一种向肿瘤注射可复制病毒的癌症治疗方法.病毒感染肿瘤细胞,在其中复制,最终引起肿瘤细胞死亡(溶解).一旦肿瘤细胞死亡,其中的病毒释放并感染邻近的肿瘤细胞.上述过程可用一个(一阶)双曲偏微分方程系统的自由边界问题来刻画,其中自由边界是肿瘤的表面.未知变量包含未被感染的细胞、感染的细胞、坏死的细胞密度、自由病毒密度、肿瘤中细胞的速度以及自由边界r=R(t).该文的目的是对上述数学模型进行分析并找一个使得肿瘤体积收缩到零的条件.