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A nonconservative kinetic framework for a closed-market society subject to shock events

Marco Menale, Ana Jacinta Soares, Romina Travaglini

Abstract

Recently, several events have shockingly impacted society, carrying tough consequences. However, not all individuals are similarly affected by shock events. Among other factors, the consequences can vary depending on the income class. In our presented work, the approach typical of kinetic theory is used to analyze the dynamics of a closed-market society exposed to various types of shock events. To achieve this, we introduce non-conservative equations, incorporating proliferative and destructive binary interactions as well as external actions. Specifically, the latter term reproduces the shock events, and to accomplish this, we introduce an appropriate external force field into the kinetic framework, modeled using Gaussian functions. Several numerical simulations are presented to illustrate the behavior of the solution predicted by the model and an application in comparison to real data relative to the Hurricane Katrina catastrophe is carried out.

A nonconservative kinetic framework for a closed-market society subject to shock events

Abstract

Recently, several events have shockingly impacted society, carrying tough consequences. However, not all individuals are similarly affected by shock events. Among other factors, the consequences can vary depending on the income class. In our presented work, the approach typical of kinetic theory is used to analyze the dynamics of a closed-market society exposed to various types of shock events. To achieve this, we introduce non-conservative equations, incorporating proliferative and destructive binary interactions as well as external actions. Specifically, the latter term reproduces the shock events, and to accomplish this, we introduce an appropriate external force field into the kinetic framework, modeled using Gaussian functions. Several numerical simulations are presented to illustrate the behavior of the solution predicted by the model and an application in comparison to real data relative to the Hurricane Katrina catastrophe is carried out.
Paper Structure (12 sections, 1 theorem, 58 equations, 8 figures, 3 tables)

This paper contains 12 sections, 1 theorem, 58 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Kinetic framework
  3. Society under shock events
  4. The framework
  5. The construction of the kinetic system
  6. Numerical Simulations
  7. Baseline scenario
  8. Slow shock scenario
  9. Sudden shock scenario
  10. Two successive sudden shocks scenario
  11. Application to a real scenario: the Hurricane Katrina case
  12. Conclusions and research perspectives

Key Result

Theorem 1

Let us consider the Cauchy problem cauchy. Let the following assumptions hold: Moreover, let the assumption assump hold. Then, there exists a unique bounded and nonnegative function $\mathbf{f}(t)$, local solution of the Cauchy problem cauchy in the time interval $[0,\, t_0]$, for $t_0>0$.

Figures (8)

  • Figure 1: Baseline scenario. Time evolution of the population distribution functions without undergoing any shock. Left column: three-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu1']}-\ref{['ParsAl1']} and initial data \ref{['InData1']}. Right column: five-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu2']}-\ref{['ParsAl2']} and initial data \ref{['InData2']}. Panels (a) - (d): distribution functions; panels (b) - (e): ratio between distribution functions and total density; panels (c) - (f): total density.
  • Figure 2: Slow shock scenario. Time evolution of the population distribution functions when a slow shock occurs at $t=50$ with $\sigma=20$, compared to the reference case (dotted lines). Left column: three-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu1']}-\ref{['ParsAl1']} and initial data \ref{['InData1']}. Right column: five-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu2']}-\ref{['ParsAl2']} and initial data \ref{['InData2']}. Panels (a) - (d): distribution functions; panels (b) - (e): ratio between distribution functions and total density; panels (c) - (f): total density.
  • Figure 3: Sudden shock scenario. Time evolution of the population distribution functions when a sudden shock occurs at $t=50$ with $\sigma=0.2$, compared to the reference case (dotted lines). Left column: three-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu1']}-\ref{['ParsAl1']} and initial data \ref{['InData1']}. Right column: five-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu2']}-\ref{['ParsAl2']} and initial data \ref{['InData2']}. Panels (a) -(d): distribution functions; panels (b) - (e): ratio between distribution functions and total density; panels (c) - (f): total density.
  • Figure 4: Two successive sudden shocks scenario. Time evolution of the population distribution functions when a sudden shock occurs at $t=50$ with $\sigma_1=0.2$, followed by another one at $t=90$ with $\sigma_1=\sigma_2=0.2$, compared to the reference case (dotted lines). Left column: three-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu1']}-\ref{['ParsAl1']} and initial data \ref{['InData1']}. Right column: five-classes population with parameters as in \ref{['parmetri conservativi']}-\ref{['ParsMu2']}-\ref{['ParsAl2']} and initial data \ref{['InData2']}. Panels (a) - (d): distribution functions; panels (b) - (e): ratio between distribution functions and total density; panels (c) - (f): total density.
  • Figure 5: Total income and mean wealth. First row: behavior of total income for the cases of $n=3$ (panel (a)) and $n=5$ (panel (b)) in the four considered scenarios. Second row: behavior of mean wealth for the cases of $n=3$ (panel (c)) and $n=5$ (panel (d)) in the four considered scenarios.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Remark 1