Intrinsic fluctuations in discrete and continuous time models

  • C├ęsar Parra Rojas

    Student thesis: Phd


    This thesis explores the stochastic features of models of ecological systems in discrete and in continuous time. Our interest lies in models formulated at the microscale, from which a mesoscopic description can be derived. The stochasticity present in the models, constructed in this way, is intrinsic to the systems under consideration and stems from their finite size. We start by exploring a susceptible-infectious-recovered model for epidemic spread on a network. We are interested in the case where the connectivity, or degree, of the individuals is characterised by a very broad, or heterogeneous, distribution, and in the effects of stochasticity on the dynamics, which may depart wildly from that of a homogeneous population. The model at the mesoscale corresponds to a system of stochastic differential equations with a very large number of degrees of freedom which can be reduced to a two-dimensional model in its deterministic limit. We show how this reduction can be carried over to the stochastic case by exploiting a time-scale separation in the deterministic system and carrying out a fast-variable elimination. We use simulations to show that the temporal behaviour of the epidemic obtained from the reduced stochastic model yields reasonably good agreement with the microscopic model under the condition that the maximum allowed degree that individuals can have is not too close to the population size. This is illustrated using time series, phase diagrams and the distribution of epidemic sizes.The general mesoscopic theory used in continuous-time models has only very recently been developed for discrete-time systems in one variable. Here, we explore this one-dimensional theory and find that, in contrast to the continuous-time case, large jumps can occur between successive iterates of the process, and this translates at the mesoscale into the need for specifying `boundary' conditions everywhere outside of the system. We discuss these and how to implement them in the stochastic difference equation in order to obtain results which are consistent with the microscopic model. We then extend the theoretical framework to make it applicable to systems containing an arbitrary number of degrees of freedom. In addition, we extend a number of analytical results from the one-dimensional stochastic difference equation to arbitrary dimension, for the distribution of fluctuations around fixed points, cycles and quasi-periodic attractors of the corresponding deterministic map. We also derive new expressions, describing the autocorrelation functions of the fluctuations, as well as their power spectrum. From the latter, we characterise the appearance of noise-induced oscillations in systems of dimension greater than one, which have been previously observed in continuous-time systems and are known as quasi-cycles. Finally, we explore the ability of intrinsic noise to induce chaotic behaviour in the system for parameter values for which the deterministic map presents a non-chaotic attractor; we find that this is possible for periodic, but not for quasi-periodic, states.
    Date of Award1 Aug 2017
    Original languageEnglish
    Awarding Institution
    • The University of Manchester
    SupervisorAlan Mckane (Supervisor) & Tobias Galla (Supervisor)


    • Stochastic processes
    • Population dynamics
    • Mathematical epidemiology

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