Spatial cumulant models enable spatially informed treatment strategies and analysis of local interactions in cancer systems

Theoretical and applied cancer studies that use individual-based models (IBMs) havebeen limited by the lack of a mathematical formulation that enables rigorous analysis of these models. However, spatial cumulant models (SCMs), which have arisenfrom theoretical ecology, describe population dynamics generated by a specific family of IBMs, namely spatio-temporal point processes (STPPs). SCMs are spatiallyresolved population models formulated by a system of differential equations thatapproximate the dynamics of two STPP-generated summary statistics: first-order spatial cumulants (densities), and second-order spatial cumulants (spatial covariances).We exemplify how SCMs can be used in mathematical oncology by modelling theoretical cancer cell populations comprising interacting growth factor-producing and non-producing cells. To formulate model equations, we use computational tools that enable the generation of STPPs, SCMs and mean-field population models (MFPMs) from user-defined model descriptions (Cornell et al. Nat Commun 10:4716, 2019).To calculate and compare STPP, SCM and MFPM-generated summary statistics, we develop an application-agnostic computational pipeline. Our results demonstrate that SCMs can capture STPP-generated population density dynamics, even when MFPMs fail to do so. From both MFPM and SCM equations, we derive treatment-induced death rates required to achieve non-growing cell populations. When testing these treatment strategies in STPP-generated cell populations, our results demonstrate that SCM-informed strategies outperform MFPM-informed strategies in terms of inhibiting population growths. We thus demonstrate that SCMs provide a new framework inwhich to study cell-cell interactions, and can be used to describe and perturb STPP-generated cell population dynamics. We, therefore, argue that SCMs can be used to increase IBMs’ applicability in cancer research

An Ikehara-type theorem for functions convergent to zero

We establish an analogue of the Ikehara theorem for positive non-increasing functions convergent to zero. In particular, this provides a complete proof to the results formulated in Diekmann/Kaper (1978) and Carr/Chmaj (2004), which are widely used nowadays to prove the uniqueness of traveling waves for various reaction-diffusion equations

Markov dynamics on the cone of discrete Radon measures

We start with a brief overview of the known facts about the spaces of discrete Radon measures those may be considered as generalizations of configuration spaces. Then we study three Markov dynamics on the spaces of discrete Radon measures: analogues of the contact model, of the Bolker-Dieckmann-Law-Pacala model, and of the Glauber-type dynamics. We show how the results obtained previously for the configuration spaces can be modified for the case of the spaces of discrete Radon measures

Semigroup approach to birth-and-death stochastic dynamics in continuum

We describe a general approach to the construction of a state evolution corresponding to the Markov generator of a spatial birth-and-death dynamics in $\mathbb{R}^d$. We present conditions on the birth-and-death intensities which are sufficient for the existence of an evolution as a strongly continuous semigroup in a proper Banach space of correlation functions satisfying the Ruelle bound. The convergence of a Vlasov-type scaling for the corresponding stochastic dynamics is considered.Comment: 35 page