Stability of a non-local kinetic model for cell migration with density dependent orientation bias

Loy, Nadia; Preziosi, Luigi

doi:10.3934/krm.2020035

Cited by 8 publications

(13 citation statements)

References 24 publications

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“…In scenario S1, the endemic components ρ E I1 S1 , ρ E I2 S1 are computed through the expressions in (26) and Table 1, respectively. (22). In particular, in our numerical set, equation ( 23) admits three positive roots, but just one of them makes all the other endemic components (22) positive, hence a unique endemic equilibrium exists.…”

Section: Viral Load-dependent Vs Constant Isolation Controlmentioning

confidence: 87%

An SIR-like kinetic model tracking individuals' viral load

Della Marca,

Loy,

Tosin

2021

Preprint

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Mathematical models are formal and simplified representations of the knowledge related to a phenomenon. In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infectious individual. Here, we attempt to investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. In the framework of multiagents systems, we propose a particle stochastic model describing the infection transmission trough interactions among agents and the individual physiological course of the disease. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong and switches in consequence of a Markovian process, and a microscopic trait, representing a normalized measure of their viral load, that changes in consequence of binary interactions or interactions with a background. Specifically, we consider Susceptible-Infected-Removed-like dynamics where infectious individuals may be isolated from the general population and the isolation rate may depend on the viral load sensitivity and frequency of tests. We derive kinetic evolution equations for the distribution functions of the viral load of the individuals in each compartment, whence, via suitable upscaling procedures, we obtain a macroscopic model for the densities and viral load momentum. We perform then a qualitative analysis of the ensuing macroscopic model, and we present numerical tests in the case of both constant and viral load-dependent isolation control. Also, the matching between the aggregate trends obtained from the macroscopic descriptions and the original particle dynamics simulated by a Monte Carlo approach is investigated.

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Section: Viral Load-dependent Vs Constant Isolation Controlmentioning

confidence: 87%

An SIR-like kinetic model tracking individuals' viral load

Della Marca,

Loy,

Tosin

2021

Preprint

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“…Since we consider the agents labelled with i = 2 as inert, we implicitly mean that they do not interact either with one another or with the agents labelled with i = 1. Therefore, the reference equation for this application is (15)…”

Section: Kinetic Description Of the Label Switchingmentioning

confidence: 99%

“…with 0 ≤ β(v, t) ≤ 1 for all t > 0 and all v ∈ R + . The kinetic equations describing the evolution of f 1 and f 2 can be deduced from (15), considering that the agents of the population i = 2 do not interact. Therefore, we have…”

Section: Kinetic Description Of the Label Switchingmentioning

confidence: 99%

“…Conversely, quarantined people (x = 2) do not interact due to isolation and possibly switch back to label x = 1 when their viral load has decreased enough. To describe these dynamics, the reference framework is provided by the kinetic equation (15), which takes into account interactions within the same labels only. In particular, we model the transition probabilities as…”

Section: A Kinetic Model Of the Contagion Of Infectious Diseases With Quarantinementioning

confidence: 99%

“…Boltzmann-type kinetic equations are a valuable tool to model multi-agent systems, their success being confirmed by a great variety of modern applications which take advantage of the formalism of the collisional kinetic theory. The literature in the field is constantly growing as witnessed by very recent contributions to econophysics [9], human ecology [27], vehicular traffic with autonomous vehicles [23,28,29], opinion formation [22], biology [14,15,24]. Kinetic models of multi-agent systems are based on a revisitation of the methods of the classical kinetic theory, however with remarkable differences due to the different nature of the physical systems at hand.…”

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Boltzmann-type equations for multi-agent systems with label switching

Loy¹,

Tosin²

2021

KRM

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<p style='text-indent:20px;'>In this paper, we propose a Boltzmann-type kinetic description of mass-varying interacting multi-agent systems. Our agents are characterised by a microscopic state, which changes due to their mutual interactions, and by a label, which identifies a group to which they belong. Besides interacting within and across the groups, the agents may change label according to a state-dependent Markov-type jump process. We derive general kinetic equations for the joint interaction/label switch processes in each group. For prototypical birth/death dynamics, we characterise the transient and equilibrium kinetic distributions of the groups via a Fokker-Planck asymptotic analysis. Then we introduce and analyse a simple model for the contagion of infectious diseases, which takes advantage of the joint interaction/label switch processes to describe quarantine measures.</p>

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A Hamilton–Jacobi approach to nonlocal kinetic equations

Loy,

Perthame

2024

Nonlinearity

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Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, kinetic model with BGK type operators implementing a velocity-jump process for cell migration, directed by the nonlocal sensing of either an external signal or the cell population density itself. We describe, in an asymptotic regime, the precise profile of these concentrations which, at the macroscale, are Dirac masses. Because Dirac concentrations look like Gaussian potentials, we use the Hopf–Cole transform to calculate the potential adapted to the problem. This potential, as in other similar situations, is obtained through the viscosity solutions of a Hamilton–Jacobi equation. We begin with the linear case, when the heterogeneous external signal is given, and we show that the concentration profile obtained after the diffusion approximation is not correct and is a simple eikonal approximation of the true H–J equation. Its heterogeneous nature leads us to develop a new analysis of the implicit equation defining the Hamiltonian and a new condition to circumvent the ‘dimensionality problem’. In the nonlinear case, when the signal occurs from the cell density itself, it is shown that the already observed linear instability (pattern formation) occurs when the Hamiltonian is convex-concave, a striking new feature of our approach.

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Stability of a non-local kinetic model for cell migration with density dependent orientation bias

Cited by 8 publications

References 24 publications

An SIR-like kinetic model tracking individuals' viral load

An SIR-like kinetic model tracking individuals' viral load

Boltzmann-type equations for multi-agent systems with label switching

A Hamilton–Jacobi approach to nonlocal kinetic equations

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