a stochastic model for internal hiv dynamics in this paper we analyse a stochastic model representing hiv internal virus dynamics the stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling we show that the model established in this paper possesses non negative solutions as this is essential in any population dynamics model we also carry out analysis on the asymptotic behaviour of the model we approximate one of the variables by a mean reverting process and find out the mean and variance of this process numerical simulations conclude the paper

a stochastic differential equation sis epidemic model with two independent brownian motions abstract in this paper we introduce two perturbations in the classical deterministic susceptible infected susceptible epidemic model greenhalgh and gray 4 in 2011 use a perturbation on in sis model based on their previous work we consider another perturbation on the parameter and formulate the original model as a stochastic differential equation sde with two independent brownian motions for the number of infected population we then prove that our model has a unique and bounded global solution i t also we establish conditions for extinction and persistence of the infected population i t under the conditions of persistence we show that there is a unique stationary distribution and derive its mean and variance computer simulations illustrate our results and provide evidence to back up our theory

analysis of self organized criticality in ehrenfest s dog flea model the self organized criticality in ehrenfest s historical dog flea model is analyzed by simulating the underlying stochastic process the fluctuations around the thermal equilibrium in the model are treated as avalanches we show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q gaussian the distribution which is obtained naturally in the context of nonextensive statistical mechanics if one avoids the finite size effects by increasing the system size we provide clear numerical evidence that the relation between the exponent ensuremath tau of avalanche size distribution obtained by maximum likelihood estimation and the q value of appropriate q gaussian obeys the analytical result recently introduced by caruso et al phys rev e 75 055101 r 2007 this allows us to determine the value of q parameter a priori from one of the well known exponents of such dynamical systems

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

dynamics in a stochastic heroin model with seasonal variation abstract in this paper a stochastic non autonomous heroin model is formulated we obtain sufficient conditions for extinction and permanence in mean with probability one of the positive solutions in addition we prove that there exists at least one positive periodic solution under some conditions finally numerical simulations are performed to illustrate our theoretical results

a stochastic model of aids and condom use in this paper we introduce stochasticity into a model of aids and condom use via the technique of parameter perturbation which is standard in stochastic population modelling we show that the model established in this paper possesses non negative solutions as desired in any population dynamics we also carry out a detailed analysis on asymptotic stability both in probability one and in pth moment our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system

variance of random walks on cayley trees application to the trapping problem the authors investigate the trapping of particles on regularly multifurcating cayley trees by using random walk methods in order to find a good approximation for the survival probability of the walker they use a cumulant expansion and obtain an exact expression for the variance of the range of nearest neighbour random walks the expressions are corroborated to high accuracy by simulation calculations the method may turn out to be useful for more complex problems such as walks on ultrametric spaces

some thoughts about random walks on figure eight abstract in this paper we discuss the issues of recurrence and transience of random walks in multiply connected spaces and in particular in the presence of knots such problems are of interest in polymer physics since almost all sufficiently long linear polymer chains are quasi knotted at least at theta solvent quality conditions and hence the transience recurrence provides mathematically rigorous definition of the entanglement concept these problems are also of biological interest since they clarify the role of topology in problems involving molecular recognition obtained results extend and generalize the results of earlier published rapid communication kholodenko phys rev e 58 1998 r5213 they also provide some answers to problems which were just mentioned in our earlier published report kholodenko phys reports 298 1998 251

stability of a stochastic sir system abstract we propose a stochastic sir model with or without distributed time delay and we study the stability of disease free equilibrium the numerical simulation of the stochastic sir model shows that the introduction of noise modifies the threshold of system for an epidemic to occur and the threshold stochastic value is found

random walk on a disordered directed bethe lattice the random walk of a particle on a directed bethe lattice of constant coordinancez is examined in the case of random hopping rates as a result the higher the coordinance the narrower the regions of anomalous drift and diffusion the annealed and quenched mean square dispersions are calculated in all dynamical phases in opposition to the one dimensional z 2 case the annealed and quenched mean quadratic dispersions are shown to be identical in all phases

stationary distribution and extinction of sir model with nonlinear incident rate under markovian switching abstract taking into account of both white and colored noises a stochastic epidemic model with nonlinear incident rate under regime switching is formulated based on this model we investigate the dynamic behaviors such as ergodicity and extinction of the sir model with beddington deangelis incidence rate and markov switching first we study the existence of the unique positive solution of system 1 3 secondly by using lyapunov functions we prove that the system has a ergodic stationary distribution under certain sufficient conditions then we obtain the conditions for extinction finally numerical simulations are employed to illustrate our theoretical analysis

on polya random walks lattice green functions and the bond percolation threshold based on numerical evidence the authors conjecture a connection between the bond percolation threshold of bravais lattices in three or more dimensions and the value at the origin of a lattice green function related to the probability of return to the origin for a polya random walk

unraveling the structure of treelike networks from first passage times of lazy random walkers we study the problem of random search in finite networks with a tree topology where it is expected that the distribution of the first passage time f t decays exponentially we show that the slope of the exponential tail is independent of the initial conditions of entering the tree in general and scales exponentially or as a power law with the extent of the tree l depending on the tendency p to jump toward the target node it is unfeasible to uniquely determine l and p from measuring the tail slope or the mean first passage time mfpt of an ordinary diffusion along the tree to unravel the structure we consider lazy random walkers that take steps with probability m when jumping on the nodes and return with probability q from the leaves by deriving an exact analytical expression for the mfpt of the intermittent random walk we verify that the structural information of the tree can be uniquely extracted by measuring the mfpt for two randomly chosen types of tracer particles with distinct experimental parameters m and q we also address the applicability of our approach in the presence of disorder in the structure of the tree or statistical uncertainty in the experimental parameters

an analysis of stability of milstein method for stochastic differential equations with delay this paper deals with the adapted milstein method for solving linear stochastic delay differential equations it is proved that the numerical method is mean square ms stable under suitable conditions the obtained result shows that the method preserves the stability property of a class of linear constant coefficient problems this is also verified by several numerical examples

the kernel of the rost invariant serre s conjecture ii and the hasse principle for quasi split groups 3 6d4 e6 e7 we prove that for a simple simply connected quasi split group of type 3 6d4 e6 e7 defined over a perfect field f of characteristic 2 3 the rost invariant has trivial kernel in certain cases we give a formula for the rost invariant it follows immediately from the result above that if cd f 2 resp vcd f 2 then serre s conjecture ii resp the hasse principle holds for such a group for a c2 field in particular x y we prove the stronger result that serre s conjecture ii holds for all not necessary quasi split exceptional groups of type 3 6d4 e6 e7

strongly inner anisotropic forms of simple algebraic groups the definition of the index of a semisimple algebraic group over a field will be recalled in 1 55 more precisely we show in proposition 2 b cf 3 1 that if over a given field k there exists a central division algebra of degree 4 i e of dimension 16 and of order 4 in the brauer group then there exists a group of type ei t over a purely transcendental extension of degree 1 k t of k this will be deduced in two different ways from theorem 1 of section 2 whose precise statement is too technical to be given in this introduction but whose spirit is that it enables one from the existence of groups of a given type with preassigned properties of a geometric and cohomological nature over a field k to infer the existence of groups of a more complicated type meeting similar requirements over k t that theorem has other interesting applications to classification problems for instance we also deduce from it again in two different ways proposition 3 b in 3 2 that the existence of a division algebra of degree 3 over k implies the existence of groups with index

laplacian spectra of and random walks on complex networks are scale free architectures really important we study the laplacian operator of an uncorrelated random network and as an application consider hopping processes diffusion random walks signal propagation etc on networks we develop a strict approach to these problems we derive an exact closed set of integral equations which provide the averages of the laplacian operator s resolvent this enables us to describe the propagation of a signal and random walks on the network we show that the determining parameter in this problem is the minimum degree q m of vertices in the network and that the high degree part of the degree distribution is not that essential the position of the lower edge of the laplacian spectrum c appears to be the same as in the regular bethe lattice with the coordination number q m namely c 0 if q m 2 and c 0 if q m 2 in both of these cases the density of eigenvalues p 0 as c 0 but the limiting behaviors near c are very different in terms of a distance from a starting vertex the hopping propagator is a steady moving gaussian broadening with time this picture qualitatively coincides with that for a regular bethe lattice our analytical results include the spectral density p near c and the long time asymptotics of the autocorrelator and the propagator

stochastic asymptotic analysis of a multi host model with vector transmission abstract we present a stochastic epidemic model with vectorial transmission and multi host structure to include environmental noise we stochastically perturb biting rates with general state functional intensities so we derive a stochastic differential equation sde which describes a vector disease with two types of hosts humans and animals and give conditions to assure disease extinction and persistence finally via numerical experiments we extend and illustrate our results using literature parameters for chagas

anomalous biased diffusion in networks we study diffusion with a bias toward a target node in networks this problem is relevant to efficient routing strategies in emerging communication networks like optical networks bias is represented by a probability p of the packet or particle to travel at every hop toward a site that is along the shortest path to the target node we investigate the scaling of the mean first passage time mfpt with the size of the network we find by using theoretical analysis and computer simulations that for random regular rr and erd ifmmode mbox h o else h o fi s r enyi networks there exists a threshold probability p mathrm th such that for pl p mathrm th the mfpt scales anomalously as n ensuremath alpha where n is the number of nodes and ensuremath alpha depends on p for pg p mathrm th the mfpt scales logarithmically with n the threshold value p mathrm th of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes an exact solution for every value of p is given for the scaling of the mfpt in rr networks the regime transition is also observed for the second moment of the probability distribution function the standard deviation for the case of scale free sf networks we present analytical bounds and simulations results showing that the mfpt scales at most as mathrm ln n to a positive power for any finite bias which means that in sf networks even a very small bias is considerably more efficient in comparison to unbiased walk

asymptotic behavior and stability of a stochastic model for aids transmission a stochastic model for aids transmission is presented in this paper the existence and uniqueness of its solution is proved and the solution belongs to an interval satisfying the actual condition furthermore the disease equilibrium can be demonstrated to be existent and both its almost surely exponential stability and its pth moment exponential stability can be concluded based on parameter analysis the first quadrant can be integrated into several parts according to the comparison between almost surely exponential stability and pth moment exponential stability and the potential meaning of the stability is further demonstrated finally based on the above theoretical analysis some conditions for aids to decrease and then to die out are obtained by computer simulation

brownian bridge on hyperbolic spaces and on homogeneous trees let b be the brownian motion on a noncompact non euclidean rank one symmetric space h a typical examples is an hyperbolic space hn n 2 for 0 the brownian bridge b of length on h is the process bt 0 t conditioned by b0 b o where o is an origin in h it is proved that the process converges weakly to the brownian excursion when the brownian excursion is the radial part of the brownian bridge on 3 the same result holds for the simple random walk on an homogeneous tree

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

polymer statistics on a cayley tree a study is made of linear and branched polymers on a cayley tree values of the critical monomer fugacity are found in closed form for any value of the branching point fugacity the correlation length exponent nu is found to be 1 2 through finite size scaling arguments this gives independent support to the idea that mean field and cayley tree approximations are not equivalent a proposal is made for the behaviour of nu against branching point fugacity

a stochastic differential equation sis epidemic model in this paper we extend the classical susceptible infected susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation sde for the number of infectious individuals i t we then prove that this sde has a unique global positive solution i t and establish conditions for extinction and persistence of i t we discuss perturbation by stochastic noise in the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance the results are illustrated by computer simulations including two examples based on real life diseases

sde sis epidemic model with demographic stochasticity and varying population size an sde sis epidemic model with demographic stochasticity is developed existence of a unique non negative solution is shown an upper bound is found for the infectives in terms of the population size the infectives and susceptibles go extinct in finite time a s results are supported numerically some with real examples in this paper we look at the two dimensional stochastic differential equation sde susceptible infected susceptible sis epidemic model with demographic stochasticity where births and deaths are regarded as stochastic processes with per capita disease contact rate depending on the population size first we look at the sde model for the total population size and show that there exists a unique non negative solution then we look at the two dimensional sde sis model and show that there exists a unique non negative solution which is bounded above given the total population size furthermore we show that the number of infecteds and the number of susceptibles become extinct in finite time almost surely lastly we support our analytical results with numerical simulations using theoretical and realistic disease parameter values

probability measures and hamiltonian models on bethe lattices i properties and construction of mrt probability measures the properties of one step markov rotationally and m step m 1 or 2 translationally invariant mrt probability measures on q state site qss bethe lattices are studied a theorem is proven which completely defines such measures in terms of m q2 q fundamental probabilities these are explicitly calculated for any mrt qss hamiltonian model as a consequence of our approach the dychotomy between alternative solutions of hamiltonian models on bethe lattices is solved

site percolation and random walks on d dimensional kagome lattices the site percolation problem is studied on d dimensional generalizations of the kagome lattice these lattices are isotropic and have the same coordination number q as the hyper cubic lattices in d dimensions namely q 2d the site percolation thresholds are calculated numerically for d 3 4 5 and 6 the scaling of these thresholds as a function of dimension d or alternatively q is different than for hypercubic lattices instead of the latter is the bethe approximation which is usually assumed to hold for all lattices in high dimensions a series expansion is calculated in order to understand the different behaviour of the kagome lattice the return probability of a random walker on these lattices is also shown to scale as for bond percolation on d dimensional diamond lattices these results imply

random walks on the bethe lattice we obtain random walk statistics for a nearest neighbor polya walk on a bethe lattice infinite cayley tree of coordination numberz and show how a random walk problem for a particular inhomogeneous bethe lattice may be solved exactly we question the common assertion that the bethe lattice is an infinite dimensional system

the effect of media coverage on threshold dynamics for a stochastic sis epidemic model abstract media coverage is one of the important measures for controlling infectious diseases but the effect of media coverage on diseases spreading in a stochastic environment still needs to be further investigated here we present a stochastic susceptible infected susceptible sis epidemic model incorporating media coverage and environmental fluctuations by using feller s test and stochastic comparison principle we establish the stochastic basic reproduction number r 0 s which completely determines whether the disease is persistent or not in the population if r 0 s 1 the disease will go to extinction if r 0 s 1 the disease will also go to extinction in probability which has not been reported in the known literatures and if r 0 s 1 the disease will be stochastically persistent in addition the existence of the stationary distribution of the model and its ergodicity are obtained numerical simulations based on real examples support the theoretical results the interesting findings are that i the environmental fluctuation may significantly affect the threshold dynamical behavior of the disease and the fluctuations in different size scale population and ii the media coverage plays an important role in affecting the stationary distribution of disease under a low intensity noise environment

random walks on pseudo lattices abstract the problem of a polya unbiased nearest neighbor stepping random walk is solved for several pseudo lattices related to the bethe lattice cayley tree in each case the solution is derived by projecting the walk on a pseudo lattice onto a walk on a linear chain with internal states and a point defect random walk statistics exhibited explicitly include the probability of return to the starting point the mean time to return if return does occur and the asymptotic behavior of the expected number of distinct sites visited in a walk of long duration conjectured relationships between random walk statistics and percolation theory are discussed in the context of pseudo lattices

grothendieck s theorem on non abelian h 2 and local global principles a theorem of grothendieck asserts that over a perfect field k of cohomological dimension one all non abelian h 2 cohomology sets of algebraic groups are trivial the purpose of this paper is to establish a formally real generalization of this theorem the generalization to the context of perfect fields of virtual cohomological dimension one takes the form of a local global principle for the h 2 sets with respect to the orderings of the field this principle asserts in particular that an element in h 2 is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k our techniques provide a new proof of grothendieck s original theorem an application to homogeneous spaces over k is also given

the average number of distinct sites visited by a random walker on random graphs we study the linear large n behaviour of the average number of distinct sites s n visited by a random walker after n steps on a large random graph an expression for the graph topology dependent prefactor b in s n bn is proposed we use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message passing equations to calculate its value numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs scaling with system size and average graph connectivity are also analysed

exact closed form of the return probability on the bethe lattice an exact closed form solution for the return probability of a random walk on the bethe lattice is given the long time asymptotic form confirms a previously known expression it is however shown that this exact result reduces to the proper expression when the bethe lattice degenerates on a line unlike the asymptotic result which is singular this is shown to be an artefact of the asymptotic expansion the density of states is also calculated

asymptotic properties of switching diffusion epidemic model with varying population size the purpose of this work is to investigate the asymptotic properties of a stochastic version of the classical sis epidemic model with standard incidence and varying population size the stochastic model studied here includes white vector noise and telegraph noise modeled by markovian switching we established conditions for extinction both in probability one and in pth moment we also established the persistence of disease under different conditions on the intensities of noises and the parameters of the model furthermore we showed the existence of a stationary distribution and derive expressions for its mean and variance the presented results are demonstrated by numerical simulations

random walk on the bethe lattice and hyperbolic brownian motion we give the exact solution to the problem of a random walk on the bethe lattice through a mapping on an asymmetric random walk on the half line we also study the continuous limit of this model and discuss in detail the relation between the random walk on the bethe lattice and brownian motion on a space of constant negative curvature

stability of sirs system with random perturbations epidemiological models with bilinear incidence rate si usually have an asymptotically stable trivial equilibrium corresponding to the disease free state or an asymptotically stable non trivial equilibrium i e interior equilibrium corresponding to the endemic state in this paper we consider an epidemiological model which is an sirs model with or without distributed time delay influenced by random perturbations we present the stability conditions of the disease free equilibrium of the associated stochastic sirs system

evolution of predator prey systems described by a lotka volterra equation under random environment abstract in this paper we consider the evolution of a system composed of two predator prey deterministic systems described by lotka volterra equations in random environment it is proved that under the influence of telegraph noise all positive trajectories of such a system always go out from any compact set of int r 2 with probability one if two rest points of the two systems do not coincide in case where they have the rest point in common the trajectory either leaves from any compact set of int r 2 or converges to the rest point the escape of the trajectories from any compact set means that the system is neither permanent nor dissipative

continuum diffusion on networks trees with hyperbranched trunks and fractal branches the probability that a random walker returns to its origin for large times scales as t d over 2 where d over is the spectral dimension we calculate d over for a class of tree structures using a renormalization technique on an infinite continued fraction we consider a wide range of homogeneous networks based on replacing the branches of a self similar tree with arbitrary fractals and composite fractals we also consider a new class of inhomogeneous hyperbranched trees

r equivalence and special unitary groups abstract a norm homomorphism for the group of r equivalence classes of all simply connected semisimple classical algebraic groups is constructed the group of r equivalence classes for special unitary groups su b is computed it is proved that the variety of su b is rational if ind b 3 and the stable birational type of su b depends only on the brauer class of b and does not depend on the involution

global behavior of a multi group sis epidemic model with age structure abstract we study global dynamics of a system of partial differential equations the system is motivated by modelling the transmission dynamics of infectious diseases in a population with multiple groups and age dependent transition rates existence and uniqueness of a positive endemic equilibrium are established under the quasi irreducibility assumption which is weaker than irreducibility on the function representing the force of infection we give a classification of initial values from which corresponding solutions converge to either the disease free or the endemic equilibrium the stability of each equilibrium is linked to the dominant eigenvalue s a where a is the infinitesimal generator of a quasi irreducible semigroup generated by the model equations in particular we show that if s a 0 then the disease free equilibrium is globally stable if s a 0 then the unique endemic equilibrium is globally stable