Time evolution of non-lethal infectious diseases: a semi-continuous approach

A model describing the dynamics related to the spreading of non-lethal infectious diseases in a fixed-size population is proposed. The model consists of a non-linear delay-differential equation describing the time evolution of the increment in the number of infectious individuals and depends upon a limited number of parameters. Predictions are in good qualitative agreement with data on influenza, which is taken to be a representative type of non-lethal infectious disease.     

Mathematical glimpse on the Y chromosome degeneration

The Y chromosomes are genetically degenerate and do not recombine with their matching partners X. Non-recombination of XY pairs has been pointed out as the key factor for the degeneration of the Y chromosome. The aim here is to show that there is a mathematical asymmetry in sex chromosomes which leads to the degeneration of Y chromosomes even in the absence of XX and XY recombination. A model for sex-chromosome evolution in a stationary regime is proposed. The consequences of their asymmetry are analyzed and lead us to a couple of conclusions. First, Y chromosome degeneration shows up more often than X chromosome degeneration. Second, if nature prohibits female mortalities from beeing exactly , then Y chromosome degeneration is inevitable.     

The effect of spatial structure in adaptive evolution

We study the dynamics of adaptation in a spatially structured population. The model assumes local competition for replication, where each organism interacts only with its nearest neighbors and is inspired by experimental methods that can be used to study the process of adaptive evolution in microbes. In such experiments microbial populations are grown on petri dishes and allowed to adapt by serial passage. We compare the rate of adaptation in a structured population where the structure is maintained intact to those where movement of individuals can occur. We observe that the rate of adaptive evolution is higher and the mean effect of fixed beneficial mutations is lower in intact structures than in structures with mixing.     

Model of evolution with sexual and non-sexual reproduction

Using a previously introduced model (Refs. [9, 10]) of biological evolution, we study the role of the reproduction pattern on the fate of an evolving population. Each individual is under the selectional pressure from the environment and random harmful mutations. The habitat (“climate") is changing periodically. Evolution of populations following three reproduction patterns are compared - an asexual one (without recombination) and two with recombination - asexual (meiotic parthenogenesis) and sexual. We show, via Monte-Carlo simulations, that sexual reproduction leads to a better adaptation to the environment, slightly better survival rates for the individuals and higher probability that the population will not become extinct in difficult external conditions. The benefits of sexual reproduction are enhanced by higher birth rates and lower mutation rates. In the case of low birth rates and high mutation rates there is a small preference for the meiotic parthenogenesis. Received 9 August 1999     

Survival and extinction in cyclic and neutral three-species systems

We study the ABC model ( A + B↦2B, B + C↦2C, C + A↦2A), and its counterpart: the three-component neutral drift model ( A + B↦2A or 2B, B + C↦2B or 2C, C + A↦2C or 2A.) In the former case, the mean-field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account the amplitude of oscillations will drift and eventually one and then two of the three species will become extinct. The second model remains stationary for all initial conditions in the mean-field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the master equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic model exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. The exponential decay is found to be robust against certain changes, such as the three reactions having different rates. Received 14 August 2002 and Received in final form 14 February 2003 / Published online: 1 April 2003 RID="a" ID="a"e-mail: ita@physics.ubc.ca     

Monte Carlo simulations of parapatric speciation

Parapatric speciation is studied using an individual-based model with sexual reproduction. We combine the theory of mutation accumulation for biological ageing with an environmental selection pressure that varies according to the individuals geographical positions and phenotypic traits. Fluctuations and genetic diversity of large populations are crucial ingredients to model the features of evolutionary branching and are intrinsic properties of the model. Its implementation on a spatial lattice gives interesting insights into the population dynamics of speciation on a geographical landscape and the disruptive selection that leads to the divergence of phenotypes. Our results suggest that assortative mating is not an obligatory ingredient to obtain speciation in large populations at low gene flow.     

A history-dependent stochastic predator-prey model: Chaos and its elimination

A non-Markovian stochastic predator-prey model is introduced in which the prey are immobile plants and predators are diffusing herbivors. The model is studied by both mean-field approximation (MFA) and computer simulations. The MFA results a series of bifurcations in the phase space of mean predator and prey densities, leading to a chaotic phase. Because of emerging correlations between the two species distributions, the interaction rate alters and if it is chosen to be the value which is obtained from the simulation, then the chaotic phase disappears. Received 12 July 1999     

Susceptible-infected-recovered epidemics in populations with heterogeneous contact rates

Heterogeneity of contact patterns is recognized as an important feature for realistic modeling of many epidemics. During an outbreak, the frequency of contacts can vary a great deal from person to person and period to period. Contact heterogeneity has been shown to have a large impact on epidemic thresholds and the final size of epidemics. We develop and apply a model which incorporates an arbitrary distribution of contact rates. The model consists of a low-dimensional system of ordinary differential equations which incorporates arbitrary heterogeneity by making use of generating functions of the contact rate distribution. We show further how this model can be applied to the study of simple intervention strategies, such as quarantine of public venues with probability proportional to size. The dynamic model allows us to investigate the effects of gradually implementing such strategies in response to an ongoing epidemic, and we investigate these strategies using data on the contact patterns within a large US city.     

Sexually transmitted infections and the marriage problem

We study an SIS epidemiological model for a sexually transmitted infection in a monogamous population where the formation and breaking of couples is governed by individual preferences. The mechanism of couple recombination is based on the so-called bar dynamics for the marriage problem. We compare the results with those of random recombination – where no individual preferences exist – for which we calculate analytically the infection incidence and the endemic threshold. We find that individual preferences give rise to a large dispersion in the average duration of different couples, causing substantial changes in the incidence of the infection and in the endemic threshold. Our analysis yields also new results on the bar dynamics, that may be of interest beyond the field of epidemiological models.     

Nonlinear voter models: the transition from invasion to coexistence

In nonlinear voter models the transitions between two states depend in a nonlinear manner on the frequencies of these states in the neighborhood. We investigate the role of these nonlinearities on the global outcome of the dynamics for a homogeneous network where each node is connected to m = 4 neighbors. The paper unfolds in two directions. We first develop a general stochastic framework for frequency dependent processes from which we derive the macroscopic dynamics for key variables, such as global frequencies and correlations. Explicit expressions for both the mean-field limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter space that distinguishes between different dynamic regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) complete invasion; (ii) random coexistence; and – most interestingly – (iii) correlated coexistence. These findings are contrasted with predictions from the mean-field phase diagram and are confirmed by extensive computer simulations of the microscopic dynamics.     

Complex spatial organization in a simple model of resource allocation

A dynamical model for the distribution of resources among competing agents is studied. The model is exactly solvable in the case of global competition, which leads to the accumulation of all the resources by the agent with the highest performance. On the other hand, local competition allows for a wider resource distribution, with a much weaker correlation with individual performances. Multiplicative processes give rise to almost-ordered spatial structures, through the enhancement of random fluctuations. Received 17 August 2000 and Received in final form 1st November 2000     

Bounding the quality of stochastic oscillations in population models

We analyze general two-species stochastic models, of the kind generally used for the study of population dynamics. Although usually defined a priori, the deterministic version of these models can be obtained as the infinite volume limit of many stochastic models (which are necessarily defined by more parameters than the deterministic one). It is known that damped oscillations in a deterministic model usually correspond to oscillatory-like fluctuations in their deterministic counterparts. The quality of these “oscillations" depends on details of each stochastic model. We show, however, that the parameters of the deterministic system are generally enough to obtain very good bounds for the quality of “oscillations" in any of its stochastic counterparts. These bounds are shown to depend on only one dimensionless parameter.     

Replicators with Hebb interactions

We do an analytical study of the statistical properties of an ecosystem composed of species that are coupled via pairwise interactions that are given by the Hebb rule and have deterministic self-interactions u. In the model each species is characterized by an infinite set of p = αN traits. As one of our main results, we observe that the ecosystem becomes less cooperative as the complexity of species (number of traits) is increased. Received 23 September 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: viviane@if.sc.usp.br     

Moment equations for a spatially extended system of two competing species

The dynamics of a spatially extended system of two competing species in the presence of two noise sources is studied. A correlated dichotomous noise acts on the interaction parameter and a multiplicative white noise affects directly the dynamics of the two species. To describe the spatial distribution of the species we use a model based on Lotka-Volterra (LV) equations. By writing them in a mean field form, the corresponding moment equations for the species concentrations are obtained in Gaussian approximation. In this formalism the system dynamics is analyzed for different values of the multiplicative noise intensity. Finally by comparing these results with those obtained by direct simulations of the time discrete version of LV equations, that is coupled map lattice (CML) model, we conclude that the anticorrelated oscillations of the species densities are strictly related to non-overlapping spatial patterns.     

Effects of disease characteristics and population distribution on dynamics of epidemic spreading among residential sites

We investigate the spreading processes of epidemic diseases among many residential sites for different disease characteristics and different population distributions by constructing and solving a set of integrodifferential equations for the evolutions of position-dependent infected and infective rates, taking into account the infection processes both within a single site and among different sites. In a spreading process the states of an individual include susceptible (S), incubative (I), active (A) and recovered (R) states. Although the transition from S to I mainly depends on the active rate, the susceptible rate and the connectivity among individuals, the transitions from I to A and from A to R are determined by intrinsic characteristics of disease development in individuals. We adopt incubation and infection periods to describe the intrinsic features of the disease. By numerically solving the equations we find that the asymptotic behavior of the spreading crucially depends on the infection period and the population under affection of an active individual. Other factors, such as the structure of network and the popular distribution, play less important roles. The study may provide useful information for analyzing the key parameters affecting the dynamics and the asymptotic behavior.     

A model based on <Emphasis Type="Italic">TTC</Emphasis> to describe how drivers controltheir vehicles

We have used the Penna ageing model to analyze how the differences in evolution of sex chromosomes depend on the strategy of reproduction. In panmictic populations, when females (XX) can freely choose the male partner (XY) for reproduction from the whole population, the Y chromosome accumulates defects and eventually the only information it brings is a male sex determination. As a result of shrinking Y chromosome the male genomes de facto loose one copy of the X chromosome information and, as a result, males are characterized by higher mortality, observed also in the human populations. If it is assumed in the model that the presence of the male is indispensable at least during the pregnancy of his female partner and he cannot be seduced by another female at least during the one reproduction cycle-the Y chromosome preserves its content, does not shrink and the lifespan of females and males is the same. Thus, Y chromosome shrinks not because of existing in one copy, without the possibility of recombination, but because it stays under weaker selection pressure; in panmictic populations without the necessity of being faithful, a considerable fraction of males is dispensable and they can be eliminated from the population without reducing its reproduction potential.     

SIR model of epidemic spread with accumulated exposure

We study an extended and modified SIR model of epidemic spread in which susceptible agents during interactions with infectious neighbors are exposed to the disease and can consequently become infectious. The studied model is extended to include heterogeneity of interactions which is modelled assuming random character of the dose accumulated by susceptible agents in every interaction with infectious neighbors. When the accumulated exposure is larger than the individual’s resistance, an agent becomes infectious and consequently introduces a new source of an epidemic which is capable of passing the disease further. We study statistical properties characterizing the course of an epidemic. The examination of the modified SIR model reveals a possible “resonant activation”-like behavior of the system in the duration of the epidemic outbreak and a possible bistable behavior of the model with accumulated exposure. Furthermore, the linear scaling of the duration of the epidemic with the system size for a wide range of the model parameters is recorded.     

Editorial: Ecological complex systems

Main aim of this topical issue is to report recent advances in noisy nonequilibrium processes useful to describe the dynamics of ecological systems and to address the mechanisms of spatio-temporal pattern formation in ecology both from the experimental and theoretical points of view. This is in order to understand the dynamical behaviour of ecological complex systems through the interplay between nonlinearity, noise, random and periodic environmental interactions. Discovering the microscopic rules and the local interactions which lead to the emergence of specific global patterns or global dynamical behaviour and the noise’s role in the nonlinear dynamics is an important, key aspect to understand and then to model ecological complex systems.     

Asymptotic regime in N random interacting species

The asymptotic regime of a complex ecosystem with N random interacting species and in the presence of an external multiplicative noise is analyzed. We find the role of the external noise on the long time probability distribution of the ith density species, the extinction of species and the local field acting on the ith population. We analyze in detail the transient dynamics of this field and the cavity field, which is the field acting on the ith species when this is absent. We find that the presence or the absence of some population give different asymptotic distributions of these fields.     

Pattern formation,social forces,and diffusion instability in games with success-driven motion

A local agglomeration of cooperators can support the survival or spreading of cooperation, even when cooperation is predicted to die out according to the replicator equation, which is often used in evolutionary game theory to study the spreading and disappearance of strategies. In this paper, it is shown that success-driven motion can trigger such local agglomeration and may, therefore, be used to supplement other mechanisms supporting cooperation, like reputation or punishment. Success-driven motion is formulated here as a function of the game-theoretical payoffs. It can change the outcome and dynamics of spatial games dramatically, in particular as it causes attractive or repulsive interaction forces. These forces act when the spatial distributions of strategies are inhomogeneous. However, even when starting with homogeneous initial conditions, small perturbations can trigger large inhomogeneities by a pattern-formation instability, when certain conditions are fulfilled. Here, these instability conditions are studied for the prisoner’s dilemma and the snowdrift game. Furthermore, it is demonstrated that asymmetrical diffusion can drive social, economic, and biological systems into the unstable regime, if these would be stable without diffusion.