We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range of biological applications. Our primary motivation is modelling of immunotherapy of malignant tumours. In this context the different actors, T-cells, cytokines or cancer cells, are modelled as single particles (individuals) in the stochastic system. The main expansions of the model are distinguishing cancer cells by phenotype and genotype, including environment-dependent phenotypic plasticity that does not affect the genotype, taking into account the effects of therapy and introducing a competition term which lowers the reproduction rate of an individual in addition to the usual term that increases its death rate. We illustrate the new setup by using it to model various phenomena arising in immunotherapy. Our aim is twofold: on the one hand, we show that the interplay of genetic mutations and phenotypic switches on different timescales as well as the occurrence of metastability phenomena raise new mathematical challenges. On the other hand, we argue why understanding purely stochastic events (which cannot be obtained with deterministic models) may help to understand the resistance of tumours to therapeutic approaches and may have non-trivial consequences on tumour treatment protocols. This is supported through numerical simulations.
We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0, 1, ..., L} and has a natural reproduction rate, a logistic death rate due to age or competition, and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when the mutation rate is low enough, metastability occurs: the exit time of the valley is an exponentially distributed random variable.2010 Mathematics Subject Classification. 92D25, 60J80, 60J27, 92D15, 60F15, 37N25.
In the late 1970s, in two celebrated papers, Aizenman and Higuchi independantly established that all infinitevolume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature β ≥ 0 are of the form αµ + β + (1 − α)µ − β , where µ + β and µ − β are the two pure phases and 0 ≤ α ≤ 1. We present here a new approach to this result, with a number of advantages:1. We obtain a finite-volume, quantitative analogue (implying the classical claim);2. the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon;3. this new approach seems substantially more robust.
Known results for Gibbs measures and the 2d Ising model
We prove that all Gibbs states of the q-state nearest neighbor Potts model on Z 2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature β > β c (q) = log 1 + √ q .
We study the discrete massless Gaussian Free Field on Z d , d ≥ 2, in the presence of a disordered square-well potential supported on a finite strip around zero. The disorder is introduced by reward/penalty interaction coefficients, which are given by i.i.d. random variables. Under minimal assumptions on the law of the environment, we prove that the quenched free energy associated to this model exists in R + , is deterministic, and strictly smaller than the annealed free energy whenever the latter is strictly positive.
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