evolutionary dynamics on degree heterogeneous graphs the evolution of two species with different fitness is investigated on degree heterogeneous graphs the population evolves either by one individual dying and being replaced by the offspring of a random neighbor voter model dynamics or by an individual giving birth to an offspring that takes over a random neighbor node invasion process dynamics the fixation probability for one species to take over a population of n individuals depends crucially on the dynamics and on the local environment starting with a single fitter mutant at a node of degree k the fixation probability is proportional to k for voter model dynamics and to 1 k for invasion process dynamics

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

on the fixation probability of superstars the moran process models the spread of genetic mutations through populations a mutant with relative fitness r is introduced and the system evolves either reaching fixation an all mutant population or extinction no mutants in a widely cited paper lieberman et al 2005 evolutionary dynamics on graphs nature 433 312 316 generalize the model to populations on the vertices of graphs they describe a class of graphs superstars with a parameter k and state that the fixation probability tends to 1 r k as the graphs get larger we show that this is untrue as stated specifically for k 5 we show that the fixation probability in the limit as graphs get larger cannot exceed 1 1 j r where j r r 4 contrary to the claimed result our proof is fully rigorous though we use a computer algebra package to invert a 31 31 symbolic matrix we do believe the qualitative claim of lieberman et al that superstar fixation probability tends to 1 as k increases and that it can probably be proved similarly to their sketch we were able to run larger simulations than the ones they presented simulations on graphs of around 40 000 vertices do not support their claim but these graphs might be too small to exhibit the limiting behaviour

a first approach to fuzzy differential game problem guarding a territory abstract in this paper the fuzzy differential game problem is proposed and the problem of guarding a territory is considered we reformulate this differential game problem by considering that the locations of the territory and or of one s opponent are imprecisely known to the players thus treating them as fuzzy information a fuzzy control strategy is proposed for this fuzzy differential game problem it will be seen that in the limiting case the result of the fuzzy differential game problem approaches that of the unfuzzified case some computer simulation results are provided in this paper

amplifiers for the moran process the moran process as studied by lieberman hauert and nowak is a randomised algorithm modelling the spread of genetic mutations in populations the algorithm runs on an underlying graph where individuals correspond to vertices initially one vertex chosen uniformly at random possesses a mutation with fitness r 1 all other individuals have fitness 1 during each step of the algorithm an individual is chosen with probability proportional to its fitness and its state mutant or nonmutant is passed on to an out neighbour which is chosen uniformly at random if the underlying graph is strongly connected then the algorithm will eventually reach fixation in which all individuals are mutants or extinction in which no individuals are mutants an infinite family of directed graphs is said to be strongly amplifying if for every r 1 the extinction probability tends to 0 as the number of vertices increases a formal definition is provided in the article strong amplification is a rather surprising property it means that in such graphs the fixation probability of a uniformly placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r 1 independently of n the name strongly amplifying comes from the fact that this selective advantage is amplified strong amplifiers have received quite a bit of attention and lieberman et al proposed two potentially strongly amplifying families superstars and metafunnels heuristic arguments have been published arguing that there are infinite families of superstars that are strongly amplifying the same has been claimed for metafunnels in this article we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying we call the graphs in the family megastars when the algorithm is run on an n vertex graph in this family starting with a uniformly chosen mutant the extinction probability is roughly n 1 2 up to logarithmic factors we prove that all infinite families of superstars and metafunnels have larger extinction probabilities as a function of n finally we prove that our analysis of megastars is fairly tight there is no infinite family of megastars such that the moran algorithm gives a smaller extinction probability up to logarithmic factors also we provide a counterexample which clarifies the literature concerning the isothermal theorem of lieberman et al

a new class of automata networks a new class of automata networks is defined their evolution rules are determined by a probability measure p on the set of all integers z and an indicator function ia on the interval 0 1 it is shown that any cellular automaton rule can be represented by a nonunique rule formulated in terms of a pair p ia this new class of automata networks contains discrete systems which are not cellular automata for a given p a metric can be defined on the space of all rules which induces a metric on the space of all cellular automata rules

mass action law versus local contagion dynamics a mean field statistical approach with application to the theory of epidemics a mean field approach for epidemic processes with high migration is suggested by analogy with non equilibrium statistical mechanics for large systems a limit of the thermodynamic type is introduced for which both the total size of the system and the total number of individuals tend to infinity but the population density remains constant in the thermodynamic limit the infection rate is proportional to the product of the proportion of individuals susceptible to infection and the average probability of infection the limit form of the average probability of infection is insensitive to the detailed behaviour of the fluctuations of the number of infectious individuals and may belong to two universality classes 1 if the fluctuation of the number of infectives is non intermittent it increases with the increase of the partial density of infectives and approaches exponentially the asymptotic value one for large densities 2 for intermittent fluctuations obeying a power law scaling the average probability of infection also displays a saturation effect for large densities of infectives but the asymptotic value one is approached according to a power law rather than exponentially for low densities of infectives both expressions for the average probability of infection are linear functions of the proportion of infectives and the infection rate is given by the mass action law

automata network sir models for the spread of infectious diseases in populations of moving individuals automata network sir models for the spread of infectious diseases are studied the local rule consists of two subrules the first one applied sequentially describes the motion of the individuals the second is synchronous and models infection and removal or recovery the spatial correlations created by the application of the second subrule are partially destroyed according to the degree of mixing of the population which follows from the application of the first subrule one and two population models are considered in the second case individuals belonging to one population may be infected only by individuals belonging to the other population as is the case for example for the heterosexual propagation of a venereal disease it is shown that the occurrence of the epidemic in one population may be triggered by the occurrence of the epidemic in the other population the emphasis is on the influence of the degree of mixing of the individuals which follows from their diffusive motion in particular the asymptotic behaviours for very small and very large mixing are determined when the degree of mixing tends to infinity the correlations are completely destroyed and the time evolution of the epidemic is then correctly predicted by the mean field approximation

fuzzy controller theory limit theorems for linear fuzzy control rules abstract for a general fuzzy controller employing linear fuzzy control rules it is shown that as the number of rules grow the defuzzified output becomes a linear function of the input in particular for a sufficiently large number of rules the defuzzified output is approximately the same as the pi pid controller

epidemic spread and bifurcation effects in two dimensional network models with viral dynamics we extend a previous network model of viral dynamics to include host populations distributed in two space dimensions the basic dynamical equations for the individual viral and immune effector densities within a host are bilinear with a natural threshold condition in the general model transmission between individuals is governed by three factors a saturating function g i describing emission as a function of originating host virion level a four dimensional array b that determines transmission from each individual to every other individual and a nonlinear function f which describes the absorption of virions by a host for a given net arrival rate a summary of the properties of the viral effector dynamical system in a single host is given in the numerical network studies individuals are placed at the mesh points of a uniform rectangular grid and are connected with an m 2 3n 2 four dimensional array with terms that decay exponentially with distance between hosts g is linear and f has a simple step threshold in a population of n5mn individuals n0 are chosen randomly to be initially infected with the virus we examine the dependence of maximal population viral load on the population dynamical parameters and find threshold effects that can be related to a transcritical bifurcation in the system of equations for individual virus and host effector populations the effects of varying demographic parameters are also examined changes in a which is related to mobility and contact rate b also show threshold effects we also vary the density of randomly chosen initially infected individuals the distribution of final size of the epidemic depends strongly on n0 but is invariably bimodal with mass concentrated mainly near either or both ends of the interval 1 n thus large outbreaks may occur with small probability even with only very few initially infected hosts the effects of immunization of various fractions of the population on the final size of the epidemic are also explored the distribution of the final percentage infected is estimated by simulation the mean of this quantity is obtained as a function of immunization rate and shows an almost linear decline for immunization rates up to about 0 2 when the immunization rate is increased past 0 2 the extra benefit accrues more slowly we include a discussion of some approximations that illuminate threshold effects in demographic parameters and indicate how a mean field approximation and more detailed studies of various geometries and rates of immunization could be a useful direction for future analysis

period doubling route to chaos for a global variable of a probabilistic automata network as a function of a parameter characterizing the degree of mixing of site values the density of nonzero sites of some one dimensional cellular automata is shown to exhibit a sequence of period doubling bifurcations and to behave chaotically when the degree of mixing is sufficiently large the automata network rules which are considered appear to be useful to model complex systems as in epidemiology in which the motion of the individuals is believed to play an important role

asymptotically optimal amplifiers for the moran process abstract we study the moran process as adapted by lieberman hauert and nowak this is a model of an evolving population on a graph or digraph where certain individuals called mutants have fitness r and other individuals called non mutants have fitness 1 we focus on the situation where the mutation is advantageous in the sense that r 1 a family of digraphs is said to be strongly amplifying if the extinction probability tends to 0 when the moran process is run on digraphs in this family the most amplifying known family of digraphs is the family of megastars of galanis et al we show that this family is optimal up to logarithmic factors since every strongly connected n vertex digraph has extinction probability n 1 2 next we show that there is an infinite family of undirected graphs called dense incubators whose extinction probability is o n 1 3 we show that this is optimal up to constant factors finally we introduce sparse incubators for varying edge density and show that the extinction probability of these graphs is o n m where m is the number of edges again we show that this is optimal up to constant factors

route to chaos for a global variable of a two dimensional game of life type automata network we consider a two dimensional cellular automaton whose rule consists of two subrules the first applied synchronously is a local rule inspired from the game of life with a larger neighbourhood the second applied sequentially describes the motion of a fraction m of individuals such rules appear to be useful for modelling complex systems in ecology such as natural populations of animals in which the motion of the individuals is believed to play an important role if the motion is long range the density of individuals exhibits a sequence of period doubling bifurcations and behaves chaotically when m is large enough if the motion is short range i e restricted to first neighbours patterns become inhomogeneous spatial correlations decay with a finite correlation length xi of the order of square root m we observe the formation of domains of mean width xi with a chaotic behaviour of the local density of individuals but the collective behaviour is stationary the global density tends to a fixed value when the lattice size is much larger than xi xi

stationary states and spatial patterning in an sis epidemiology model with implicit mobility abstract by means of the asynchronous cellular automata algorithm we study stationary states and spatial patterning in an s i s model in which the individuals are attached to the vertices of a graph and their mobility is mimicked by varying the neighbourhood size q here we consider the following cases q is fixed at certain value and q is taken at random at each step and for each individual the obtained numerical data are then mapped onto the solution of its version corresponding to the limit q this allows for deducing an explicit form of the dependence of the fraction of infected individuals on the curing rate a detailed analysis of the appearance of spatial patterns of infected individuals in the stationary state is performed

effects of distribution of infection rate on epidemic models a goal of many epidemic models is to compute the outcome of the epidemics from the observed infected early dynamics however often the total number of infected individuals at the end of the epidemics is much lower than predicted from the early dynamics this discrepancy is argued to result from human intervention or nonlinear dynamics not incorporated in standard models we show that when variability in infection rates is included in standard susciptible infected susceptible sis and susceptible infected recovered sir models the total number of infected individuals in the late dynamics can be orders lower than predicted from the early dynamics this discrepancy holds for sis and sir models where the assumption that all individuals have the same sensitivity is eliminated in contrast with network models fixed partnerships are not assumed we derive a moment closure scheme capturing the distribution of sensitivities we find that the shape of the sensitivity distribution does not affect r 0 or the number of infected individuals in the early phases of the epidemics however a wide distribution of sensitivities reduces the total number of removed individuals in the sir model and the steady state infected fraction in the sis model the difference between the early and late dynamics implies that in order to extrapolate the expected effect of the epidemics from the initial phase of the epidemics the rate of change in the average infectivity should be computed these results are supported by a comparison of the theoretical model to the ebola epidemics and by numerical simulation

solution of epidemic models with quenched transients we consider a model for single season disease epidemics with a delay latent period in the onset of infectivity and a decay quenching in host susceptibility described by time varying rates of primary and secondary infections the classical susceptible exposed infected sei model of epidemiology is a special case with constant rates the decaying rates force the epidemics to slow down and eventually stop in a quenched transient state that depends on the full history of the epidemic including its initial state this equilibrium state is neutrally stable i e has zero value eigenvalues and cannot be studied using standard equilibrium analysis we introduce a method that gives an approximate analytical solution for the quenched state the method uses an interpolation between two exactly solvable limits and applies to the whole five dimensional parameter space of the model some applications of the solutions for analysis of epidemics are given

a probabilistic automata network epidemic model with births and deaths exhibiting cyclic behaviour a probabilistic automata network model for the spread of an infectious disease in a population of moving individuals is studied the local rule consists of two subrules the first one applied synchronously models infection birth and death processes it is a probabilistic cellular automaton rule the second applied sequentially describes the motion of the individuals the model contains six parameters the probabilities p for a susceptible to become infected by contact with an infective the respective birth rates bs and bi of the susceptibles from either a susceptible or an infective parent the respective death rates ds and di of susceptibles and infectives and a parameter m characterizing the motion of the individuals the model has three fixed points the first is trivial it describes a stationary state with no living individuals the second corresponds to a disease free state with no infectives the third and last one characterizes an endemic state with non zero densities of susceptibles and infectives moreover the model may exhibit oscillatory behaviour of the susceptible and infective densities as functions of time through a hopf type bifurcation the influence of the different parameters on the stability of all these states is studied with a particular emphasis on the influence of motion which has been found to be a stabilizing factor of the cyclic behaviour

dimer automata and cellular automata abstract we define a class of discrete dynamical systems which we call dimer automata whereas in a cellular automaton the new state of one cell is a function of the states in the neighborhood in a dimer automaton the new states of two neighboring cells are functions of the states of these two cells dimer automata with synchronous dynamics seem artificial but with asynchronous dynamics such systems are very natural they are as simple as cellular automata they have some advantages in modeling spatial spread we present the definition some easy consequences a classification of one dimensional dimer automata a first approach to determine a characteristic equation and a formula for an approximate asymptotic density as well as a comparison to computer simulations finally we compare synchronous and asynchronous cellular automata

mean field type equations for spread of epidemics the small world model abstract in the paper we study a cellular automata ca model of epidemic dynamics the effects of local spatial correlations on a temporal aggregated spread of single epidemics are studied as a function of increasing proportion of global contacts small world model we conjecture that even in the presence of high local correlations the aggregated mean field type models can be quite successful if the contact rate is treated as a free parameter the dependence of the estimated contact rate on the mixing parameter can be understood in terms of a simple probabilistic model the contact rate reflects not only a microscopic and epidemiological situation but also a complicated social pattern including short and long range contacts as well as a possibly hierarchical structure of human society

migration in cellular automata abstract we investigate the effect of spatial migration of cells in cellular automata that have been used in the study of excitable media and epidemiology with an infinite range migration while the periods of oscillation of the densities of cells in each state are mostly unaffected by migration the amplitudes of these oscillations behave very differently when migration is present spatial structures which develop in the absence of migration are shown to be unstable to even the lowest possible level of migration decay of these structures is shown to scale with system size demonstrating the absence of structure in the asymptotic regime for any system size decay of structure is also shown to occur on a time scale t scaling roughly as t t m where m is the migration rate and t is time elapsed the exact shape of the decay of structure however is not self similar overall structure decays almost entirely after each cell has migrated twice on an average we conclude from this that the asymptotic behavior after the structure has subsided is governed by the mean field approximation

walds martingale and the conditional distributions of absorption time in the moran process many models of evolution are stochastic processes where some quantity of interest fluctuates randomly in time one classic example is the moranbirthdeath process where that quantity is the number

an analysis of the fixation probability of a mutant on special classes of non directed graphs there is a growing interest in the study of evolutionary dynamics on populations with some non homogeneous structure in this paper we follow the model of lieberman et al lieberman et al 2005 na

early appraisal of the fixation probability in directed networks in evolutionary dynamics the probability that a mutation spreads through the whole population having arisen from a single individual is known as the fixation probability in general it is not possible to find the fixation probability analytically given the mutant s fitness and the topological constraints that govern the spread of the mutation so one resorts to simulations instead depending on the topology in use a great number of evolutionary steps may be needed in each of the simulation events particularly in those that end with the population containing mutants only we introduce two techniques to accelerate the determination of the fixation probability the first one skips all evolutionary steps in which the number of mutants does not change and thereby reduces the number of steps per simulation event considerably this technique is computationally advantageous for some of the so called layered networks the second technique which is not restricted to layered networks consists of aborting any simulation event in which the number of mutants has grown beyond a certain threshold value and counting that event as having led to a total spread of the mutation for advantageous mutations in large populations and regardless of the network s topology we demonstrate both analytically and by means of simulations that using a threshold of about n r ensuremath 1 1 4 mutants where n is the number of simulation events and r is the ratio of the mutants fitness to that of the remainder of the population leads to an estimate of the fixation probability that deviates in no significant way from that obtained from the full fledged simulations we have observed speedups of two orders of magnitude for layered networks with 10 text 000 nodes

critical behaviour of a probabilistic automata network sis model for the spread of an infectious disease in a population of moving individuals a probabilistic automata network sis susceptible infective susceptible model for the spread of an infectious disease in a population of moving individuals is studied the local rule consists of two subrules the first one applied synchronously models infection and recovery it is a probabilistic cellular automaton rule the second applied sequentially describes the motion of the individuals the model contains three parameters the probabilities pi to get infected and pr to recover and the average number of tentative moves per individual m depending upon the values of these parameters in the infinite time limit the system is either in the disease free state or in the endemic state it goes from one state to the other through a transcritical bifurcation similar to a second order phase transition characterized by a non negative order parameter whose role is played in this model by the stationary density of infected individuals the pi pr phase diagram and the critical behaviour of the stationary density of infectives in the neighbourhood of the phase transition are studied as a function of m according to whether the individuals perform short or long range moves it is found that the parameters characterizing the transition have a qualitatively different behaviour as m varies when m is very large the correlations created by the application of the subrule modelling infection and recovery are destroyed and as expected the behaviour of the system is then correctly predicted by a mean field type approximation which assumes a homogeneous mixing of the individuals when m is not large this assumption is no longer correct

computational complexity of ecological and evolutionary spatial dynamics there are deep yet largely unexplored connections between computer science and biology both disciplines examine how information proliferates in time and space central results in computer science describe the complexity of algorithms that solve certain classes of problems an algorithm is deemed efficient if it can solve a problem in polynomial time which means the running time of the algorithm is a polynomial function of the length of the input there are classes of harder problems for which the fastest possible algorithm requires exponential time another criterion is the space requirement of the algorithm there is a crucial distinction between algorithms that can find a solution verify a solution or list several distinct solutions in given time and space the complexity hierarchy that is generated in this way is the foundation of theoretical computer science precise complexity results can be notoriously difficult the famous question whether polynomial time equals nondeterministic polynomial time i e p np is one of the hardest open problems in computer science and all of mathematics here we consider simple processes of ecological and evolutionary spatial dynamics the basic question is what is the probability that a new invader or a new mutant will take over a resident population we derive precise complexity results for a variety of scenarios we therefore show that some fundamental questions in this area cannot be answered by simple equations assuming that p is not equal to np

discrete stochastic modeling for epidemics in networks in this paper we construct a microscopic mechanism for the epidemic processes in heterogeneous populations which contains two basic assumptions all edges of the network are broken and reconnected at every time step among the epidemic process the probability of randomly chosen two half edges to make a pair is identical we define the stochastic epidemic process and get the epidemic distributions numerically according to the transmission probability two different phases are observed which means the onset of phase transition and the threshold value is very small under the thermodynamic limit the process can be approximated by a deterministic dynamical system about this deterministic system we get the analytical threshold value which is consistent with the numerical results of the epidemic distributions

dynamical phases in a cellular automaton model for epidemic propagation abstract a directed epidemic propagation process is modeled by a deterministic cellular automaton with three local states infected immunized and susceptible the model is characterized by the choice of the lifetimes of the infected and immunized states as external parameters and by the existence of a continuous control parameter determining the fraction of synchronized infection vectors the various dynamical regimes observed in the fully synchronized state are described in a region of parameter space where a statistically stationary disordered regime is observed evidence is given of a phase transition between a localized damage and a spreading damage regime