arXiv subjects

Madeleine Kubasch

Publications and source records attributed to Madeleine Kubasch.

4 recordsLinked to original sources

arXiv2026

Empirical distribution of ancestral lineages in populations with density-dependent interactions

We study a density-dependent Markov jump process describing a population where each individual is characterized by a type, and reproduces at rates depending both on its type and on the population type distribution. We are interested in the empirical distribution of ancestral lineages in the population process. First, we exhibit a time-inhomogeneous Markov process, which allows to capture the behavior of a sampled lineage in the population process. This is achieved through a many-to-one formula, which relates the expected value of a functional evaluated over the lineages in the population process to the expectation of the functional evaluated along this time-inhomogeneous process. This provides a direct interpretation of the underlying survivorship bias, as illustrated on a minimalistic population process. Second, we consider the large population regime, when the population size grows to infinity. Under classical assumptions, the population type distribution converges to a deterministic limit. Here, we focus on the empirical distribution of ancestral lineages in this large population limit, for which we establish a many-to-one formula. Using coupling arguments, we further quantify the approximation error which arises when sampling in this large population approximation instead of the finite-size population process.

math.PR↗

arXiv2026

Large population limit for a multilayer SIR model including households and workplaces

We study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population convergence of the corresponding stochastic process. For this purpose, we use an individual-based model whose state space explicitly takes into account the duration of infectious periods. This allows to deal with the natural correlation of the epidemic states of individuals whose household and workplace share a common infected. In a general setting where a non-exponential distribution of infectious periods may be considered, convergence to the unique deterministic solution of a measurevalued equation is obtained. In the particular case of exponentially distributed infectious periods, we show that it is possible to further reduce the obtained deterministic limit, leading to a closed, finite dimensional dynamical system capturing the epidemic dynamics. This model reduction subsequently is studied from a numerical point of view. We illustrate that the dynamical system derived from the large population approximation is a pertinent model reduction when compared to simulations of the stochastic process or to an alternative edgebased compartmental model, both in terms of accuracy and computational cost.

math.PR↗

arXiv2026

Metacommunity persistence on spatially heterogeneous landscapes

We are interested in the long-time behaviour of the ecological dynamics of two competing species in a spatially heterogeneous environment consisting of two habitat types. Our goal is to provide conditions for the persistence of the two populations. First, we consider a spatially continuous model, formalized as an infinite-dimensional system of integro-differential equations. We show that if each species would persist if it were alone, then mutual invasibility of each other's monospecific equilibrium is a sufficient condition for long time survival of both species. Second, we introduce a finite-dimensional system of ordinary differential equations which approximate the spatial dynamics by averaging over a finite number of habitat types. We derive an analogous sufficient condition for stable coexistence, and show that in this case, there exists a positive coexistence equilibrium. Finally, we complete our theoretical result using a simulation study. Our results indicate that mutual invasibility also is a necessary condition for stable coexistence in both models. In addition, we show that the finite-dimensional model underestimates species' persistance, which indicates that spatial heterogeneity promotes survival.

math.DS↗

arXiv2024

The epidemiological footprint of contact structures in models with two levels of mixing

Models with several levels of mixing (households, workplaces), as well as various corresponding formulations for R0, have been proposed in the literature. However, little attention has been paid to the impact of the distribution of the population size within social structures, effect that can help plan effective interventions. We focus on the influence on the model outcomes of teleworking strategies, consisting in reshaping the distribution of workplace sizes. We consider a stochastic SIR model with two levels of mixing, accounting for a uniformly mixing general population, each individual belonging also to a household and a workplace. The variance of the workplace size distribution appears to be a good proxy for the impact of this distribution on key outcomes of the epidemic, such as epidemic size and peak. In particular, our findings suggest that strategies where the proportion of individuals teleworking depends sublinearly on the size of the workplace outperform the strategy with linear dependence. Besides, one drawback of the model with multiple levels of mixing is its complexity, raising interest in a reduced model. We propose a homogeneously mixing SIR ODE-based model, whose infection rate is chosen as to observe the growth rate of the initial model. This reduced model yields a generally satisfying approximation of the epidemic. These results, robust to various changes in model structure, are very promising from the perspective of implementing effective strategies based on social distancing of specific contacts. Furthermore, they contribute to the effort of building relevant approximations of individual based models at intermediate scales.

physics.soc-ph↗