A particle system approach to aggregation phenomena

Inspired by a PDE-ODE system of aggregation developed in the biomathematical literature, an interacting particle system representing aggregation at the level of individuals is investigated. It is proved that the empirical density of the individual converges to solution of the PDE-ODE system

Habits and demand changes after COVID-19

In this paper, we investigate how a transitory lockdown of a sector of the economy may have changed our habits and, therefore, altered the goods’ demand permanently. In a two-sector infinite horizon economy, we show that the demand of the goods produced by the sector closed during the lockdown could shrink or expand with respect to their pre-pandemic level depending on the lockdown’s duration and the habits’ strength. We also show that the end of a lockdown may be characterized by a price surge due to a combination of strong demand of both goods and rigidities in production

A Novel Averaging Principle Provides Insights in the Impact of Intratumoral Heterogeneity on Tumor Progression

From MDPI via Jisc Publications RouterHistory: accepted 2021-09-14, pub-electronic 2021-10-09Publication status: PublishedFunder: Mic2Mode-I2T; Grant(s): 01ZX1710B, 01ZX1308D, 01ZX1707C, 031L0085B, ZT-I- 392 0010, 96 732Typically stochastic differential equations (SDEs) involve an additive or multiplicative noise term. Here, we are interested in stochastic differential equations for which the white noise is nonlinearly integrated into the corresponding evolution term, typically termed as random ordinary differential equations (RODEs). The classical averaging methods fail to treat such RODEs. Therefore, we introduce a novel averaging method appropriate to be applied to a specific class of RODEs. To exemplify the importance of our method, we apply it to an important biomedical problem, in particular, we implement the method to the assessment of intratumoral heterogeneity impact on tumor dynamics. Precisely, we model gliomas according to a well-known Go or Grow (GoG) model, and tumor heterogeneity is modeled as a stochastic process. It has been shown that the corresponding deterministic GoG model exhibits an emerging Allee effect (bistability). In contrast, we analytically and computationally show that the introduction of white noise, as a model of intratumoral heterogeneity, leads to monostable tumor growth. This monostability behavior is also derived even when spatial cell diffusion is taken into account

Network of interacting neurons with random synaptic weights

Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N

Some perspectives on mathematical modeling for aggregation phenomena

The main theme of the thesis here proposed is given by the bio-physical phenomenon of aggregation. More specifically we propose some developments in the direction of scaling limits. More specifically, given some PDE (or a system of PDEs), that describe macroscopically a specific phenomena, we propose a microscopic view through a proper system of interacting particles system (in a continuous in time and space setting). In the first chapter we work on a microscopic approach to a PDE which describes cell-cell adhesion (transport-diffusion equation). In the second chapter we focus on system of particles interacting with themselves and with an incompressible fluid. In particular we work on a scaling limit for the Vlasov Fokker Planck Navier Stokes system. Finally in the third chapter we present an averaging result, applied to the study of evolution of a cancerous population, which present a peculiar type of aggregation

A Stochastic Model of Economic Growth in Time-Space

2noreservedWe deal with an infinite horizon, infinite dimensional stochastic optimal control problem arising in the study of economic growth in time-space. Such a problem has been the object of various papers in deterministic cases when the possible presence of stochastic disturbances is ignored (see, e.g., [P. Brito, The Dynamics of Growth and Distribution in a Spatially Heterogeneous World, working paper 2004/14, ISEG-Lisbon School of Economics and Management, University of Lisbon, 2004], [R. Boucekkine, C. Camacho, and G. Fabbri, J. Econom. Theory, 148 (2013), pp. 2719--2736], [G. Fabbri, J. Econom. Theory, 162 (2016), pp. 114--136], and [R. Boucekkine, G. Fabbri, S. Federico, and F. Gozzi, J. Econom. Geography, 19 (2019), pp. 1287--1318]). Here we propose and solve a stochastic generalization of such models where the stochastic term, in line with the standard stochastic economic growth models (see, e.g., the books [A. G. Malliaris and W. A. Brock, Stochastic Methods in Economics and Finance, Advanced Textbooks in Economics 17, North Holland, 1982, Chapter 3] and [H. Morimoto, Stochastic Control and Mathematical Modeling: Applications in Economics, Cambridge Books, 2010, Chapter 9]), is a multiplicative one, driven by a cylindrical Wiener process. The problem is studied using the dynamic programming approach. We find an explicit solution of the associated HJB equation, use a verification type result to prove that such a solution is the value function, and find the optimal feedback strategies. Finally, we use this result to study the asymptotic behavior of the optimal trajectories.mixedGozzi, Fausto; Leocata, MartaGozzi, Fausto; Leocata, Mart