thresholds for epidemic spreading in networks we study the threshold of epidemic models in quenched networks with degree distribution given by a power law forthesusceptible infected susceptiblemodeltheactivitythresholdc vanishesinthelargesize limitonanynetworkwhosemaximumdegree kmax divergeswiththesystemsize atoddswithheterogeneous mean field hmf theory the vanishing of the threshold has nothing to do with the scale free nature of the networkbutstemsinsteadfromthelargesthubinthesystembeingactiveforanyspreadingrate 1 ffiffiffiffiffiffiffiffiffimax p and playing the role of a self sustained source that spreads the infection to the rest of the system the susceptible infected removedmodel displays insteadagreement with hmftheory and afinite threshold for scale rich networks we conjecture that on quenched scale rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state

a structure preserving method for the distribution of the first hitting time to a moving boundary for some gaussian processes abstract in this work we consider a model for the first hitting time of a moving boundary problem associated to some stochastic processes in addition to a gaussian component the model investigated in this manuscript includes a general deterministic source in either case the coefficients of the deterministic and the stochastic terms are general functions with suitable analytical properties that guarantee that the process under investigation possesses continuous paths as the cornerstone of this manuscript we establish that the cumulative distribution of probability of the first hitting time is governed by a deterministic advection diffusion partial differential equation subject to initial boundary data for which the exact solution is known only in certain specific scenarios motivated by these limitations we propose a crank nicolson discretization of the deterministic model which is capable of preserving the main structural features of probability distributions namely the non negativity the boundedness from above by 1 and the monotonicity to guarantee the preservation of those properties relatively flexible conditions on the parameters need to be imposed additionally we also prove that our scheme is a consistent and unconditionally stable and convergent technique which has second order of convergence some illustrative simulations demonstrate that the order of convergence is indeed quadratic and the comparisons against the known exact solutions establish that the method preserves the same structural properties of the relevant solutions

positivity preserving nonstandard finite difference schemes for cross diffusion equations in biosciences we design nonstandard finite difference nsfd schemes which are unconditionally dynamically consistent with respect to the positivity property of solutions of cross diffusion equations in biosciences this settles a problem that was open for quite some time the study is done in the setting of three concrete and highly relevant cross diffusion systems regarding tumor growth convective predator prey pursuit and evasion model and reaction diffusion chemotaxis model it is shown that nsfd schemes used for classical reaction diffusion equations such as the fisher equation for which the solutions enjoy the positivity property are not appropriate for cross diffusion systems the reliable nsfd schemes are therefore obtained by considering a suitable implementation on the cross diffusive term of mickens rule of nonlocal approximation of nonlinear terms apart from his rule of complex denominator function of discrete derivatives we provide numerical experiments that support the theory as well as the power of the nsfd schemes over the standard ones in the case of the cancer growth model we demonstrate computationally that our nsfd schemes replicate the property of traveling wave solutions of developing shocks observed in marchant et al 2000

langevin approach for the dynamics of the contact process on annealed scale free networks we study the dynamics of the contact process one of the simplest nonequilibrium stochastic processes taking place on a scale free network we consider the network topology as annealed i e all links are rewired at each microscopic time step so that no dynamical correlation can build up this is a practical implementation of the absence of correlations assumed by mean field approaches we present a detailed analysis of the contact process in terms of a langevin equation including explicitly the effects of stochastic fluctuations in the number of particles in finite networks this allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs the fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has a hard upper bound when the upper bound is soft the presence of outliers with huge connectivity perturbs the picture even more inducing an apparent shift of the critical point in light of these findings recent theoretical and numerical results in the literature are critically reviewed

localization transition lifschitz tails and rare region effects in network models effects of heterogeneity in the suspected infected susceptible model on networks are investigated using quenched mean field theory the emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare region effects appearing in simulations and in the lifschitz tails the latter in the linear approximation is related to the spectral density of the laplacian matrix and to the time dependent order parameter i show that these approximations indicate correctly griffiths phases both on regular one dimensional lattices and on small world networks exhibiting purely topological disorder i discuss the localization transition that occurs on scale free networks at 3 degree exponent

global stability of endemic equilibrium of an epidemic model with birth and death on complex networks abstract we study global stability of endemic equilibrium of an epidemic model with birth and death on complex networks under some conditions the local asymptotic stability of the endemic equilibrium was established by zhang and jin 2011 for correlated networks and the global asymptotic stability was obtained by chen and sun 2014 for uncorrelated networks in this work we remove those conditions and prove by constructing a lyapunov function that the endemic equilibrium is globally asymptotically stable numerical simulations are also presented to illustrate the feasibility of the result

two energy conserving numerical schemes for the sine gordon equation two explicit conservative numerical schemes for the sine gordon equation are proposed their stability and convergence are proved numerical simulation of a sine gordon soliton shows that the new schemes are very accurate and fast

contact processes on random graphs with power law degree distributions have critical value 0 if we consider the contact process with infection rate on a random graph on n vertices with power law degree distributions mean field calculations suggest that the critical value c of the infection rate is positive if the power 3 physicists seem to regard this as an established fact since the result has recently been generalized to bipartite graphs by gomez gardenes et al proc natl acad sci usa 105 2008 1399 1404 here we show that the critical value c is zero for any value of 3 and the contact process starting from all vertices infected with a probability tending to 1 as n maintains a positive density of infected sites for time at least exp n 1 for any 0 using the last result together with the contact process duality we can establish the existence of a quasi stationary distribution in which a randomly chosen vertex is occupied with probability it is expected that c as 0 here we show that 1 2 3 and so 2 for 3 thus even though the graph is locally tree like does not take the mean field critical value 1

a deterministic model for the distribution of the stopping time in a stochastic equation and its numerical solution in this work we consider a stochastic differential equation that generalizes the well known paris equation from fracture of materials the model describes the propagation of cracks on solids and it includes a deterministic summand and a stochastic component in terms of a brownian motion the use of ito s stochastic integral gives an equivalent stochastic integral equation that is further generalized here we note that the probability distribution of the stopping time of the general model satisfies a deterministic diffusion advection partial differential equation for which the solution is known only in a reduced number of particular cases motivated by these analytical results we develop a fast finite difference method to approximate the distribution of the stopping time the method is an explicit exponential like technique that preserves the main features of a probability distribution namely the non negativity the boundedness from above by 1 as well as the spatial monotonicity moreover the method is a monotone technique that is also capable of preserving the temporal monotonicity of the approximations these properties of the proposed methodology are thoroughly established in the present manuscript a continuity condition of the numerical solutions in terms of the initial conditions and the temporal computational parameter is established also together with a limiting property of the methodology when the free parameter tends to infinity for comparison purposes we are providing an implicit and stable discretization of the mathematical model which has a second order of convergence but for which conditions that guarantee the positivity the boundedness and the monotonicity of approximations are not available the numerical simulations obtained with implementations of our techniques show that the explicit method is an efficient scheme that preserves the characteristics of interest non negativity boundedness from above by 1 and monotonicity and that the numerical approximations are in good agreement with the known exact solutions

the tangent approximation to one sided brownian exit densities let p t be the density of the first exit time of a brownian motion over a one sided moving boundary and let p1 t be the density at t of the first exit time over the tangent to the boundary at t when is p1 t a good approximation to p t we investigate this question by means of a new integral equation for p t which makes possible explicit estimates for the error of the approximation

epidemic thresholds of the susceptible infected susceptible model on networks a comparison of numerical and theoretical results recent work has shown that different theoretical approaches to the dynamics of the susceptible infectedsusceptible sis model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks here we present large scale numerical simulations of the sis dynamics on various types of networks allowing the precise determination of the effective threshold for systems of finite size n we compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean field theory and of the quenched mean field theory we show that the latter is in general more accurate scaling with n with the correct exponent but often failing to capture the correct prefactor

epidemic spreading in correlated complex networks we study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node s connectivities for the case of markovian complex networks showing only correlations between pairs of nodes we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links which from a node with connectivity k go to nodes with connectivity k ensuremath numerical simulations on a correlated growing network model provide support for our conclusions

predicting epidemic threshold of correlated networks a comparison of methods abstract being able to theoretically predict the outbreak threshold of an epidemic is essential in disease control real world correlated networks are ubiquitous and their topological structure strongly affects the prediction accuracy of present day theoretical methods quantifying their accuracy and fitness in predicting outbreak thresholds of correlated networks is thus essential we use a susceptible infected removed sir model to examine four widely used theoretical methods the heterogeneous mean field h m f quenched mean field q m f dynamical message passing d m p and connectivity matrix c m methods to predict the outbreak threshold of a correlated network the potential topological structure of correlated network impacts on prediction accuracy of these four methods we emphasize that the quantitative changes of degree correlation degree distribution exponent and network size strongly affect the outbreak of sir spreading dynamics and compare the simulation results with the theoretical ones obtained from these four methods the extensive experiments in synthetic networks show that a the increasing degree correlation coefficients reduce outbreak threshold and suppress outbreak size b the increasing degree distribution exponents raise outbreak threshold but suppress outbreak size c the increasing network sizes decrease outbreak threshold but do not affect outbreak size d as for four theoretical methods c m and d m p are more likely to precisely predict outbreak threshold because they to some extent incorporate network topology with dynamical correlations the experimental results in 50 real world networks also prove the above conclusions

how to simulate the quasistationary state for a large class of processes with an absorbing state statistical properties of the surviving sample attain time independent values in the quasistationary qs regime we propose a practical simulation method for studying quasistationary properties based on the equation of motion governing the qs distribution in applications to the contact process the method is shown to reproduce exact results for the process on a complete graph and known scaling behavior to high precision at the critical point our method is about an order of magnitude more efficient than conventional simulation

spectral analysis and slow spreading dynamics on complex networks the susceptible infected susceptible sis model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous annihilation reactions the effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean field qmf approximations in scale free networks qmf can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix therefore it can provide predictions on possible rare region effects thus on the occurrence of slow dynamics i compare qmf results of sis with simulations on various large dimensional graphs in particular i show that for erd s renyi graphs this method predicts correctly the occurrence of rare region effects it also provides a good estimate for the epidemic threshold in case of percolating graphs griffiths phases emerge if the graph is fragmented or if we apply a strong exponentially suppressing weighting scheme on the edges the latter model describes the connection time distributions in the face to face experiments in case of a generalized barabasi albert type of network with aging connections strong rare region effects and numerical evidence for griffiths phase dynamics are shown the dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices

quasistationary analysis of the contact process on annealed scale free networks we present an analysis of the quasistationary qs state of the contact process cp on annealed scale free networks using a mapping of the cp dynamics in a one step process and analyzing numerically and analytically the corresponding master equation the relevant qs quantities determined via the master equation exhibit an excellent agreement with direct qs stochastic simulations of the cp the high accuracy of the resulting data allows a probe of the strong corrections to scaling present in both the critical and supercritical regions corrections that mask the correct finite size scaling obtained analytically by applying an exact heterogeneous mean field approach our results represent a promising starting point for a deeper understanding of the contact process and absorbing phase transitions on real quenched complex networks

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

rare regions of the susceptible infected susceptible model on barabasi albert networks i extend a previous work to susceptible infected susceptible sis models on weighted barab asi albert scale free networks numerical evidence is provided that phases with slow power law dynamics emerge as the consequence of quenched disorder and tree topologies studied previously with the contact process i compare simulation results with spectral analysis of the networks and show that the quenched mean field qmf approximation provides a reliable relatively fast method to explore activity clustering this suggests that qmf can be used for describing rare region effects due to network inhomogeneities finite size study of the qmf shows the expected disappearance of the epidemic threshold ensuremath lambda c in the thermodynamic limit and an inverse participation ratio ensuremath sim 0 25 meaning localization in case of disassortative weight scheme contrarily for the multiplicative weights and the unweighted trees this value vanishes in the thermodynamic limit suggesting only weak rare region effects in agreement with the dynamical simulations strong corrections to the mean field behavior in case of disassortative weights explains the concave shape of the order parameter ensuremath rho ensuremath lambda at the transition point application of this method to other models may reveal interesting rare region effects griffiths phases as the consequence of quenched topological heterogeneities

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

global stability of endemic equilibrium for a siqrs epidemic model on complex networks abstract we study the global stability of endemic equilibrium of a siqrs epidemic model on complex networks this model was proposed by li et al 2014 the local asymptotic stability of disease free equilibrium and the permanence of the disease were proved by them by use of a combination of lyapunov function method and a monotone iterative technique we prove that the endemic equilibrium is globally asymptotically stable numerical simulations indicate that quarantine strategy is an effective measure in preventing and controlling epidemic spreading

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

effects of heterogeneous self protection awareness on resource epidemic coevolution dynamics recent studies have demonstrated that the allocation of individual resources has a significant influence on the dynamics of epidemic spreading in the real scenario individuals have a different level of awareness for self protection when facing the outbreak of an epidemic to investigate the effects of the heterogeneous self awareness distribution on the epidemic dynamics we propose a resource epidemic coevolution model in this paper we first study the effects of the heterogeneous distributions of node degree and self awareness on the epidemic dynamics on artificial networks through extensive simulations we find that the heterogeneity of self awareness distribution suppresses the outbreak of an epidemic and the heterogeneity of degree distribution enhances the epidemic spreading next we study how the correlation between node degree and self awareness affects the epidemic dynamics the results reveal that when the correlation is positive the heterogeneity of self awareness restrains the epidemic spreading while when there is a significant negative correlation strong heterogeneous or strong homogeneous distribution of the self awareness is not conducive for disease suppression we find an optimal heterogeneity of self awareness at which the disease can be suppressed to the most extent further research shows that the epidemic threshold increases monotonously when the correlation changes from most negative to most positive and a critical value of the correlation coefficient is found when the coefficient is below the critical value an optimal heterogeneity of self awareness exists otherwise the epidemic threshold decreases monotonously with the decline of the self awareness heterogeneity at last we verify the results on four typical real world networks and find that the results on the real world networks are consistent with those on the artificial network

sampling methods for the quasistationary regime of epidemic processes on regular and complex networks a major hurdle in the simulation of the steady state of epidemic processes is that the system will unavoidably visit an absorbing disease free state at sufficiently long times due to the finite size of the networks where epidemics evolves in the present work we compare different quasistationary qs simulation methods where the absorbing states are suitably handled and the thermodynamical limit of the original dynamics can be achieved we analyze the standard qs sqs method where the sampling is constrained to active configurations the reflecting boundary condition rbc where the dynamics returns to the pre absorbing configuration and hub reactivation hr where the most connected vertex of the network is reactivated after a visit to an absorbing state we apply the methods to the contact process cp and susceptible infected susceptible sis models on regular and scale free networks the investigated methods yield the same epidemic threshold for both models for cp that undergoes a standard collective phase transition the methods are equivalent for sis whose phase transition is ruled by the hub mutual reactivation the sqs and hr methods are able to capture localized epidemic phases while rbc is not we also apply the autocorrelation time as a tool to characterize the phase transition and observe that this analysis provides the same finite size scaling exponents for the critical relaxation time for the investigated methods finally we verify the equivalence between rbc method and a weak external field for epidemics on networks

second order mean field susceptible infected susceptible epidemic threshold given the adjacency matrix a of a network we present a second order mean field expansion that improves on the first order n intertwined susceptible infected susceptible sis epidemic model unexpectedly we found that in contrast to first order second order mean field theory is not always possible the network size n should be large enough under the assumption of large n we show that the crucial and characterizing quantity the sis epidemic threshold ensuremath tau c obeys an eigenvalue equation more complex than the one in the first order n intertwined model however the resulting epidemic threshold is more accurate ensuremath tau c 2 ensuremath tau c 1 o frac ensuremath tau c 1 n where the first order epidemic threshold is ensuremath tau c 1 frac 1 ensuremath lambda 1 a and where ensuremath lambda 1 a is the spectral radius of the adjacency matrix a

parallel activation of memories in an oscillatory neural network we describe a feedback neural network whose elements possess dynamic thresholds this network has an oscillatory mode that we investigate by measuring the activities of memory patterns as functions of time we observe spontaneous and induced transitions between the different oscillating memories moreover the network exhibits pattern segmentation by oscillating between different memories that are included as a mixture in a constant input the efficiency of pattern segmentation decreases strongly as the number of the input memories is increased using oscillatory inputs we observe resonance behavior

high prevalence regimes in the pair quenched mean field theory for the susceptible infected susceptible model on networks reckoning of pairwise dynamical correlations significantly improves the accuracy of mean field theories and plays an important role in the investigation of dynamical processes in complex networks in this work we perform a nonperturbative numerical analysis of the quenched mean field theory qmf and the inclusion of dynamical correlations by means of the pair quenched mean field pqmf theory for the susceptible infected susceptible model on synthetic and real networks we show that the pqmf considerably outperforms the standard qmf theory on synthetic networks of distinct levels of heterogeneity and degree correlations providing extremely accurate predictions when the system is not too close to the epidemic threshold while the qmf theory deviates substantially from simulations for networks with a degree exponent 2 5 the scenario for real networks is more complicated still with pqmf significantly outperforming the qmf theory however despite its high accuracy for most investigated networks in a few cases pqmf deviations from simulations are not negligible we found correlations between accuracy and average shortest path while other basic network metrics seem to be uncorrelated with the theory accuracy our results show the viability of the pqmf theory to investigate the high prevalence regimes of recurrent state epidemic processes in networks a regime of high applicability

effect of the interconnected network structure on the epidemic threshold most real world networks are not isolated in order to function fully they are interconnected with other networks and this interconnection influences their dynamic processes for example when the spread of a disease involves two species the dynamics of the spread within each species the contact network differs from that of the spread between the two species the interconnected network we model two generic interconnected networks using two adjacency matrices a and b in which a is a 2n 2n matrix that depicts the connectivity within each of two networks of size n and b a 2n 2n matrix that depicts the interconnections between the two using an n intertwined mean field approximation we determine that a critical susceptible infected susceptible sis epidemic threshold in two interconnected networks is 1 1 a b where the infection rate is within each of the two individual networks and in the interconnected links between the two networks and 1 a b is the largest eigenvalue of the matrix a b in order to determine how the epidemic threshold is dependent upon the structure of interconnected networks we analytically derive 1 a b using a perturbation approximation for small and large the lower and upper bound for any as a function of the adjacency matrix of the two individual networks and the interconnections between the two and their largest eigenvalues and eigenvectors we verify these approximation and boundary values for 1 a b using numerical simulations and determine how component network features affect 1 a b we note that given two isolated networks g1 and g2 with principal eigenvectors x and y respectively 1 a b tends to be higher when nodes i and j with a higher eigenvector component product xiyj are interconnected this finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics

an implicit numerical method for the two dimensional fractional percolation equation in this paper an implicit numerical method for the two dimensional fractional percolation equation without the assumption of continued and rigid body motion is proposed consistency stability and convergence of the implicit finite difference method are established finally some numerical examples are given the numerical results demonstrate the effectiveness of the theoretical analysis

robustness and fragility of the susceptible infected susceptible epidemic models on complex networks we analyze two alterations of the standard susceptible infected susceptible sis dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree all models have the same epidemic thresholds in mean field theories but depending on the network properties simulations yield a dual scenario in which the epidemic thresholds of the modified sis models can be either dramatically altered or remain unchanged in comparison with the standard dynamics for uncorrelated synthetic networks having a power law degree distribution with exponent 5 2 differences are more remarkable for 3 where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one this duality is elucidated in terms of epidemic lifespan on star graphs we verify that the activation of the modified sis models is triggered in the innermost component of the network given by a k core decomposition for 3 the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard sis the duality also appears in the finite size scaling of the critical quantities where mean field behaviors are observed for the modified but not for the original dynamics our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models

nature of the epidemic threshold for the susceptible infected susceptible dynamics in networks we develop an analytical approach to the susceptible infected susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks we find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold in particular small world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit

weighted curvature preserving pde image filtering method the tensor driven curvature preserving partial differential equation is an outstanding anisotropic diffusion filtering model to effectively preserve image edge well a weighted curvature preserving pde based filtering method is proposed which employ local image directional information to design weight coefficient for different vector fields experimental results indicate that new approach shows superior performance on preserving image edge and curvature geometric structure

transcritical bifurcation yielding global stability for network processes abstract several network processes exhibiting transcritical bifurcation have globally stable steady states their dynamical behaviour is captured by a simple property of the right hand side of the corresponding system of odes based on this property a class of dynamical systems is introduced for which the local stability of the trivial steady state determines the global behaviour of the system it is shown that this condition is satisfied by three network models namely the individual based and degree based ode approximations of sis epidemic propagation on networks and the hopfield model with non negative weights the general result enables us to describe the global behaviour of these systems that was not available for the first and third models and was proved in a significantly more complicated way for the second

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

the numerical solution of a generalized burgers huxley equation through a conditionally bounded and symmetry preserving method in this article we propose a non standard finite difference scheme to approximate the solutions of a generalized burgers huxley equation from fluid dynamics our numerical method preserves the skew symmetry of the partial differential equation under study and under some analytical constraints of the model constants and the computational parameters involved it is capable of preserving the boundedness and the positivity of the solutions in the linear regime the scheme is consistent to first order in time due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative and to second order in space we compare the results of our computational technique against the exact solutions of some particular initial boundary value problems our simulations indicate that the method presented in this work approximates well the theoretical solutions and moreover that the method preserves the boundedness of solutions within the analytical constraints derived here in the problem of approximating solitary wave solutions of the model under consideration we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique for which a minimum relative error is achieved finally we linearly perturb a steady state solution of the partial differential equation under investigation and show that our simulations still converge to the same constant solution establishing thus robustness of our method in this sense

phase transition of the susceptible infected susceptible dynamics on time varying configuration model networks we present a degree based theoretical framework to study the susceptible infected susceptible sis dynamics on time varying rewired configuration model networks using this framework on a given degree distribution we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the sis process this analysis is suitable for the characterization of the phase transition and leads to three main contributions 1 we obtain a self consistent expression for the absorbing state threshold able to capture both collective and hub activation 2 we recover the predictions of a number of existing approaches as limiting cases of our analysis providing thereby a unifying point of view for the sis dynamics on random networks 3 we obtain bounds for the critical exponents of a number of quantities in the stationary state this allows us to reinterpret the concept of hub dominated phase transition within our framework it appears as a heterogeneous critical phenomenon observables for different degree classes have a different scaling with the infection rate this phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold

phase transitions with infinitely many absorbing states in complex networks we investigate the properties of the threshold contact process tcp a process showing an absorbing state phase transition with infinitely many absorbing states on random complex networks the finite size scaling exponents characterizing the transition are obtained in a heterogeneous mean field hmf approximation and compared with extensive simulations particularly in the case of heterogeneous scale free networks we observe that the tcp exhibits the same critical properties as the contact process which undergoes an absorbing state phase transition to a single absorbing state the accordance among the critical exponents of different models and networks leads to conjecture that the critical behavior of the contact process in a hmf theory is a universal feature of absorbing state phase transitions in complex networks depending only on the locality of the interactions and independent of the number of absorbing states the conditions for the applicability of the conjecture are discussed considering a parallel with the susceptible infected susceptible epidemic spreading model which in fact belongs to a different universality class in complex networks

boundary element method for a free boundary problem modeling three dimensional tumor growth abstract in the mathematical model of tumor growth the initial boundary surface of the three dimensional tumor domain is known but the tumor domain and its boundary surface change over time in a way that is unknown in advance people are concerned whether the tumor is likely to spread or shrink and how the tumor will change in this article by the boundary element method the free boundary problem in the three dimensional tumor domain will be solved via the integration on a two dimensional boundary surfaces at the expense of singularity in the green s functions we will numerically compute and graphically show the changing boundary surfaces of the three dimensional tumor domain over time we will numerically analyze the trend of tumor growth with varying proliferation rate our numerical approach professor bei hu proposed is new and our numerical experiments contribute to the prediction of tumor growth in clinical medicine

spread of epidemic like failures in telecommunication networks abstract we study epidemic like failures in telecommunication networks a mean field model taking two levels of failure into account is introduced where infection recovery and transition rates are node link specific regarding the short term epidemic outbreak an epidemic threshold is stated in terms of the basic reproduction number computed as the largest eigenvalue of a weighted adjacency matrix of the network as to the long term endemic situation we have proved the existence and uniqueness of a steady state we check the accuracy of the model by means of monte carlo simulations to improve the level of accuracy we propose a slight modification of the mean field equations which changes the way we compute the probability for a node of acquiring the infection from one of its neighbors as a consequence correlations between probabilities of different states are implicitly incorporated into the model giving improved predictions and being very close to simulation based data

two structure preserving doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation in this paper we present two structure preserving doubling like algorithms for obtaining the positive definite solution of the nonlinear matrix equation x a h x 1 a q where x c n i n is an unknown matrix and q c n i n is a hermitian positive definite matrix we prove that the sequences generated by the algorithms converge to the positive definite solution of the considered matrix equation r quadratically in addition we also present some numerical results to illustrate the behavior of the considered algorithm

susceptible infected susceptible model a comparison of n intertwined and heterogeneous mean field approximations we introduce the susceptible infected susceptible sis spreading model which is taken as a benchmark for the comparison between the n intertwined approximation and the pastor satorras and vespignani heterogeneous mean field hmf approximation of the sis model the n intertwined approximation the hmf approximation and the sis spreading model are compared for different graph types we focus on the epidemic threshold and the steady state fraction of infected nodes in networks with different degree distributions overall the n intertwined approximation is superior to the hmf approximation the n intertwined approximation is exactly the same as the hmf approximation in regular graphs however for some special graph types such as the square lattice graph and the path graph the two mean field approximations are both very different from the sis spreading model

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

effects of local population structure in a reaction diffusion model of a contact process on metapopulation networks we investigate the effects of local population structure in reaction diffusion processes representing a contact process cp on metapopulations represented as complex networks considering a model in which the nodes of a large scale network represent local populations defined in terms of a homogeneous graph we show by means of extensive numerical simulations that the critical properties of the reaction diffusion system are independent of the local population structure even when this one is given by a ordered linear chain this independence is confirmed by the perfect matching between numerical critical exponents and the results from a heterogeneous mean field theory suited in principle to describe situations of local homogeneous mixing the analysis of several variations of the reaction diffusion process allows us to conclude the independence from population structure of the critical properties of cp like models on metapopulations and thus of the universality of the reaction diffusion description of this kind of models