# American Institute of Mathematical Sciences

October  2021, 14(5): 867-894. doi: 10.3934/krm.2021027

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

 Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Torino, Italy

Received  March 2021 Revised  July 2021 Published  October 2021 Early access  August 2021

Fund Project: The postdoctoral fellowship of the first author is funded by INdAM (Istituto Nazionale di Alta Matematica "F.Severi", Italy).Both authors are members of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM (Istituto Nazionale di Alta Matematica "F.Severi"), Italy

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.

Citation: Nadia Loy, Andrea Tosin. Boltzmann-type equations for multi-agent systems with label switching. Kinetic & Related Models, 2021, 14 (5) : 867-894. doi: 10.3934/krm.2021027
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The problem of Section 5.1 with $\nu_1 = 0$ and $\alpha>\alpha_\dagger$. The continuous lines are the solutions of the hydrodynamic model (40a)–(40d). In particular, the solutions to (40a)-(40b) are given exactly by (41) while the solutions to (40c)-(40d) have been obtained numerically by a fourth order Runge-Kutta scheme
">Figure 2.  The problem of Section 5.1 with $\nu_1 = 0$ and $\alpha<\alpha_\dagger$. The continuous lines are the solutions of the hydrodynamic model (40a)–(40d), which have been computed as in Figure 1
The problem of Section 5.2 with $\nu_1 = \nu_2$ and $\mu\gg\lambda$
The problem of Section 5.2 with $\nu_1 = \nu_2$ and $\mu = \lambda$
The problem of Section 5.2 with $\nu_1 = 0$ and $\mu\gg\lambda$
The problem of Section 5.2 with $\nu_1 = 0$ and $\mu = \lambda$
Parameters kept constant in all numerical tests of Section 6
 Parameter $N$ $\lambda$ $\Delta{t}$ $\nu_2$ $\gamma$ Value $10^6$ $1$ $10^{-3}$ $0.2$ $0.3$
 Parameter $N$ $\lambda$ $\Delta{t}$ $\nu_2$ $\gamma$ Value $10^6$ $1$ $10^{-3}$ $0.2$ $0.3$
Parameters varying from test to test of Section 6
 Parameter Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 $\mu$ $1$ $1$ $10$ $1$ $10$ $1$ $\alpha$ $0.8$ $0.2$ $0.8\left(1-e^{-v}\right)$ $0.8\left(1-e^{-v}\right)$ $0.8\left(1-e^{-v}\right)$ $0.8\left(1-e^{-v}\right)$ $\beta$ $0.4$ $0.4$ $0.4 e^{-v}$ $0.4 e^{-v}$ $0.4 e^{-v}$ $0.4 e^{-v}$ $\nu_1$ $0$ $0$ $0.2$ $0.2$ $0$ $0$ $\alpha_\dagger$ $0.28$ $0.28$ – – – –
 Parameter Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 $\mu$ $1$ $1$ $10$ $1$ $10$ $1$ $\alpha$ $0.8$ $0.2$ $0.8\left(1-e^{-v}\right)$ $0.8\left(1-e^{-v}\right)$ $0.8\left(1-e^{-v}\right)$ $0.8\left(1-e^{-v}\right)$ $\beta$ $0.4$ $0.4$ $0.4 e^{-v}$ $0.4 e^{-v}$ $0.4 e^{-v}$ $0.4 e^{-v}$ $\nu_1$ $0$ $0$ $0.2$ $0.2$ $0$ $0$ $\alpha_\dagger$ $0.28$ $0.28$ – – – –
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