scholarly journals How Reaction-Diffusion PDEs Approximate the Large-Population Limit of Stochastic Particle Models

2021 ◽  
Vol 81 (6) ◽  
pp. 2622-2657
Author(s):  
Samuel A. Isaacson ◽  
Jingwei Ma ◽  
Konstantinos Spiliopoulos
Keyword(s):  
Large Population ◽  
Reaction Diffusion ◽  
Stochastic Particle ◽  
Large Population Limit

scholarly journals Amplifiers of selection

2015 ◽  
Vol 471 (2181) ◽  
pp. 20150114 ◽  
Author(s):  
B. Adlam ◽  
K. Chatterjee ◽  
M. A. Nowak
Keyword(s):  
Population Structure ◽  
Probability Distribution ◽  
Initial Conditions ◽  
Large Population ◽  
Fixation Probability ◽  
Population Structures ◽  
The Difference ◽  
Large Population Limit

When a new mutant arises in a population, there is a probability it outcompetes the residents and fixes. The structure of the population can affect this fixation probability. Suppressing population structures reduce the difference between two competing variants, while amplifying population structures enhance the difference. Suppressors are ubiquitous and easy to construct, but amplifiers for the large population limit are more elusive and only a few examples have been discovered. Whether or not a population structure is an amplifier of selection depends on the probability distribution for the placement of the invading mutant. First, we prove that there exist only bounded amplifiers for adversarial placement—that is, for arbitrary initial conditions. Next, we show that the Star population structure, which is known to amplify for mutants placed uniformly at random, does not amplify for mutants that arise through reproduction and are therefore placed proportional to the temperatures of the vertices. Finally, we construct population structures that amplify for all mutational events that arise through reproduction, uniformly at random, or through some combination of the two.

scholarly journals Mean field games models of segregation

2017 ◽  
Vol 27 (01) ◽  
pp. 75-113 ◽  
Author(s):  
Yves Achdou ◽  
Martino Bardi ◽  
Marco Cirant
Keyword(s):  
Numerical Methods ◽  
Existence Of Solutions ◽  
Large Population ◽  
Mean Field ◽  
Mean Field Games ◽  
Residential Choice ◽  
Urban Settlements ◽  
Thomas Schelling ◽  
Two Populations ◽  
Large Population Limit

This paper introduces and analyzes some models in the framework of mean field games (MFGs) describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of MFG theory, in the stationary and in the evolutive case. Numerical methods are proposed with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

scholarly journals On the Simpson index for the Wright–Fisher process with random selection and immigration

2020 ◽  
Vol 13 (06) ◽  
pp. 2050046
Author(s):  
Arnaud Guillin ◽  
Franck Jabot ◽  
Arnaud Personne
Keyword(s):  
Large Population ◽  
Fixed Time ◽  
Time Behavior ◽  
Simpson Index ◽  
Weak Selection ◽  
Full Picture ◽  
Long Time ◽  
Object Of Study ◽  
Fisher Process ◽  
Large Population Limit

Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameters for ecologists who study for example, forest dynamics. Following ecological motivations, we will consider, here, the case, where there are various species with fitness and immigration parameters being random processes (and thus time evolving). The Simpson index is difficult to evaluate when the population is large, except in the neutral (no selection) case, because it has no closed formula. Our approach relies on the large population limit in the “weak” selection case, and thus to give a procedure which enables us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. We will also study the long time behavior (invariant measure and convergence speed towards equilibrium) of the Wright–Fisher process in a simplified setting, allowing us to get a full picture for the approximation of the expectation of the Simpson index.

scholarly journals Who is the infector? General multi-type epidemics and real-time susceptibility processes

2019 ◽  
Vol 51 (2) ◽  
pp. 606-631
Author(s):  
Tom Britton ◽  
Ka Yin Leung ◽  
Pieter Trapman
Keyword(s):  
Real Time ◽  
Random Graph ◽  
Branching Processes ◽  
Large Population ◽  
Graph Representation ◽  
Type Model ◽  
Random Weights ◽  
Stochastic Epidemic ◽  
Special Case ◽  
Large Population Limit

AbstractWe couple a multi-type stochastic epidemic process with a directed random graph, where edges have random weights (traversal times). This random graph representation is used to characterise the fractions of individuals infected by the different types of vertices among all infected individuals in the large population limit. For this characterisation, we rely on the theory of multi-type real-time branching processes. We identify a special case of the two-type model in which the fraction of individuals of a certain type infected by individuals of the same type is maximised among all two-type epidemics approximated by branching processes with the same mean offspring matrix.

scholarly journals On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2103
Author(s):  
Giacomo Ascione
Keyword(s):  
Deterministic Model ◽  
Large Population ◽  
Epidemic Models ◽  
Fluid Limit ◽  
Sir Epidemic ◽  
Generalized Fractional Calculus ◽  
Stochastic Epidemic ◽  
Non Local ◽  
Stochastic Analog ◽  
Large Population Limit

Fractional-order epidemic models have become widely studied in the literature. Here, we consider the generalization of a simple SIR model in the context of generalized fractional calculus and we study the main features of such model. Moreover, we construct semi-Markov stochastic epidemic models by using time changed continuous time Markov chains, where the parent process is the stochastic analog of a simple SIR epidemic. In particular, we show that, differently from what happens in the classic case, the deterministic model does not coincide with the large population limit of the stochastic one. This loss of fluid limit is then stressed in terms of numerical examples.

scholarly journals Nonparametric adaptive inference of birth and death models in a large population limit

2021 ◽  
Vol 3 (1) ◽  
pp. 1-69
Author(s):  
Alexandre Boumezoued ◽  
Marc Hoffmann ◽  
Paulien Jeunesse
Keyword(s):  
Large Population ◽  
Adaptive Inference ◽  
Large Population Limit

scholarly journals The emergence of a birth-dependent mutation rate in asexuals: causes and consequences

2021 ◽  
Author(s):  
Florian Patout ◽  
Raphael Forien ◽  
Matthieu Alfaro ◽  
Julien Papaix ◽  
Lionel Roques
Keyword(s):  
Mutation Rate ◽  
Birth Rate ◽  
Fitness Landscape ◽  
Large Population ◽  
Reaction Diffusion ◽  
Weak Selection ◽  
Positive Dependence ◽  
Evolutionary Trajectories ◽  
Complex Landscape ◽  
Phenotype Distribution

In unicellular organisms such as bacteria and in most viruses, mutations mainly occur during reproduction. Thus, genotypes with a high birth rate should have a higher mutation rate. However, standard models of asexual adaptation such as the 'replicator-mutator equation' often neglect this generation-time effect. In this study, we investigate the emergence of a positive dependence between the birth rate and the mutation rate in models of asexual adaptation and the consequences of this dependence. We show that it emerges naturally at the population scale, based on a large population limit of a stochastic time-continuous individual-based model with elementary assumptions. We derive a reaction-diffusion framework that describes the evolutionary trajectories and steady states in the presence of this dependence. When this model is coupled with a phenotype to fitness landscape with two optima, one for birth, the other one for survival, a new trade-off arises in the population. Compared to the standard approach with a constant mutation rate, the symmetry between birth and survival is broken. Our analytical results and numerical simulations show that the trajectories of mean phenotype, mean fitness and the stationary phenotype distribution are in sharp contrast with those displayed for the standard model. The reason for this is that the usual weak selection limit does not hold in a complex landscape with several optima associated with different values of the birth rate. Here, we obtain trajectories of adaptation where the mean phenotype of the population is initially attracted by the birth optimum, but eventually converges to the survival optimum, following a hook-shaped curve which illustrates the antagonistic effects of mutation on adaptation.

scholarly journals Coalescent models for populations with time-varying population sizes and arbitrary offspring distributions

2017 ◽  
Author(s):  
Christophe Fraser ◽  
Lucy M Li
Keyword(s):  
Population Dynamics ◽  
Infectious Diseases ◽  
Large Population ◽  
Disease Outbreaks ◽  
Time Varying ◽  
Offspring Distribution ◽  
Population Sizes ◽  
The Relationship ◽  
Original Derivation ◽  
Large Population Limit

AbstractThe coalescent has been used to infer from gene genealogies the population dynamics of biological systems, such as the prevalence of an infectious disease. The offspring distribution affects the relationship between population dynamics and the genealogy, and for infectious diseases, the offspring distribution is often highly overdispersed. Here, we provide a general formula for the coalescent rate for populations with time-varying sizes and any offspring distribution. The formula is valid in the same large population limit as Kingman’s original derivation. By relating our derivation to existing formulations of the coalescent, we show that differences in the coalescent rate derived for many population models may be explained by differences in the offspring distribution. The coalescent derivations presented here could be used to quantify the overdispersion in the offspring distribution of infectious diseases, which is useful for accurate modelling disease outbreaks.

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