Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

Stochastic epidemics in a homogeneous community

These notes describe stochastic epidemics in a homogenous community. Our main concern is stochastic compartmental models (i.e. models where each individual belongs to a compartment, which stands for its status regarding the epidemic under study : S for susceptible, E for exposed, I for infectious, R for recovered) for the spread of an infectious disease. In the present notes we restrict ourselves to homogeneously mixed communities. We present our general model and study the early stage of the epidemic in chapter 1. Chapter 2 studies the particular case of Markov models, especially in the asymptotic of a large population, which leads to a law of large numbers and a central limit theorem. Chapter 3 considers the case of a closed population, and describes the final size of the epidemic (i.e. the total number of individuals who ever get infected). Chapter 4 considers models with a constant influx of susceptibles (either by birth, immigration of loss of immunity of recovered individuals), and exploits the CLT and Large Deviations to study how long it takes for the stochastic disturbances to stop an endemic situation which is stable for the deterministic epidemic model. The document ends with an Appendix which presents several mathematical notions which are used in these notes, as well as solutions to many of the exercises which are proposed in the various chapters.Comment: Part I of "Stochastic Epidemic Models with Inference", T. Britton & E. Pardoux eds., Lecture Notes in Mathematics 2255, Springer 201

Quasi-stationary distributions

This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth-death processes and related models. Results on the question of whether a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, are reviewed as well. The paper is concluded with a summary of recent developments in numerical and approximation methods

A phase transition for measure-valued SIR epidemic processes

We consider measure-valued processes $X=(X_t)$ that solve the following martingale problem: for a given initial measure $X_0$, and for all smooth, compactly supported test functions $\varphi$, \begin{eqnarray*}X_t(\varphi )=X_0(\varphi)+\frac{1}{2}\int _0^tX_s(\Delta \varphi )\,ds+\theta \int_0^tX_s(\varphi )\,ds\\{}-\int_0^tX_s(L_s\varphi )\,ds+M_t(\varphi ).\end{eqnarray*} Here $L_s(x)$ is the local time density process associated with $X$, and $M_t(\varphi )$ is a martingale with quadratic variation $[M(\varphi )]_t=\int_0^tX_s(\varphi ^2)\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $\theta_c(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $\theta>\theta_c(d)$, then the solution survives forever with positive probability, but if $\theta<\theta_c(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $\theta$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $\theta$; that is, for any compact set $K$, the process $X_t(K)=0$ eventually.Comment: Published in at http://dx.doi.org/10.1214/13-AOP846 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Co-evolution of Content Popularity and Delivery in Mobile P2P Networks

Mobile P2P technology provides a scalable approach to content delivery to a large number of users on their mobile devices. In this work, we study the dissemination of a \emph{single} content (e.g., an item of news, a song or a video clip) among a population of mobile nodes. Each node in the population is either a \emph{destination} (interested in the content) or a potential \emph{relay} (not yet interested in the content). There is an interest evolution process by which nodes not yet interested in the content (i.e., relays) can become interested (i.e., become destinations) on learning about the popularity of the content (i.e., the number of already interested nodes). In our work, the interest in the content evolves under the \emph{linear threshold model}. The content is copied between nodes when they make random contact. For this we employ a controlled epidemic spread model. We model the joint evolution of the copying process and the interest evolution process, and derive the joint fluid limit ordinary differential equations. We then study the selection of the parameters under the content provider's control, for the optimization of various objective functions that aim at maximizing content popularity and efficient content delivery.Comment: 21 pages, 16 figure

Nested Sequential Monte Carlo Methods

We propose nested sequential Monte Carlo (NSMC), a methodology to sample from sequences of probability distributions, even where the random variables are high-dimensional. NSMC generalises the SMC framework by requiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correct SMC algorithm. Furthermore, NSMC can in itself be used to produce such properly weighted samples. Consequently, one NSMC sampler can be used to construct an efficient high-dimensional proposal distribution for another NSMC sampler, and this nesting of the algorithm can be done to an arbitrary degree. This allows us to consider complex and high-dimensional models using SMC. We show results that motivate the efficacy of our approach on several filtering problems with dimensions in the order of 100 to 1 000.Comment: Extended version of paper published in Proceedings of the 32nd International Conference on Machine Learning (ICML), Lille, France, 201

Evolutionary dynamics on any population structure

Evolution occurs in populations of reproducing individuals. The structure of a biological population affects which traits evolve. Understanding evolutionary game dynamics in structured populations is difficult. Precise results have been absent for a long time, but have recently emerged for special structures where all individuals have the same number of neighbors. But the problem of determining which trait is favored by selection in the natural case where the number of neighbors can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class which suggests there is no efficient algorithm. Whether there exists a simple solution for weak selection was unanswered. Here we provide, surprisingly, a general formula for weak selection that applies to any graph or social network. Our method uses coalescent theory and relies on calculating the meeting times of random walks. We can now evaluate large numbers of diverse and heterogeneous population structures for their propensity to favor cooperation. We can also study how small changes in population structure---graph surgery---affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.Comment: 68 pages, 10 figure