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 and Related Models, 2021, 14 (5) : 867-894. doi: 10.3934/krm.2021027
References:
[1]

G. AlbiM. BonginiF. Rossi and F. Solombrino, Leader formation with mean-field birth and death models, Math. Models Methods Appl. Sci., 29 (2019), 633-679.  doi: 10.1142/S0218202519400025.

[2]

F. Bassetti and G. Toscani, Mean field dynamics of interaction processes with duplication, loss and copy, Math. Models Methods Appl. Sci., 25 (2015), 1887-1925.  doi: 10.1142/S0218202515500487.

[3]

A. V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys. Rev. E, 61 (2000), 4576-4586.  doi: 10.1103/PhysRevE.61.4576.

[4]

M. Burger, Network structured kinetic models of social interactions, Vietnam J. Math., 49 (2021), 937-956.  doi: 10.1007/s10013-021-00505-8.

[5]

C. Cercignani, The Boltzmann Equation and its Applications, no. 67 in Applied Mathematical Sciences, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[6]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.

[7]

M. Delitala, Generalized kinetic theory approach to modeling spread and evolution of epidemics, Math. Comput. Modelling, 39 (2004), 1-12.  doi: 10.1016/S0895-7177(04)90501-8.

[8]

R. Della Marca, N. Loy and A. Tosin, An SIR-like kinetic model tracking individuals' viral load, 2021, Preprint. doi: 10.13140/RG. 2.2.32046.02883.

[9]

G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Wealth distribution under the spread of infectious diseases, Phys. Rev. E, 102 (2020), 022303, 14 pp. doi: 10.1103/physreve. 102.022303.

[10]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.

[11]

C. D. Greenman and T. Chou, Kinetic theory of age-structured stochastic birth-death processes, Phys. Rev. E, 93 (2016), 012112.

[12]

M. Groppi and J. Polewczak, On two kinetic models for chemical reactions: Comparisons and existence results, J. Stat. Phys., 117 (2004), 211-241.  doi: 10.1023/B:JOSS.0000044059.59066.a9.

[13]

M. Groppi and G. Spiga, Kinetic approach to chemical reactionsand inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219. 

[14]

N. Loy and L. Preziosi, Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2020), 373-421.  doi: 10.1007/s00285-019-01411-x.

[15]

N. Loy and L. Preziosi, Stability of a non-local kinetic model for cell migration with density dependent orientation bias, Kinet. Relat. Models, 13 (2020), 1007-1027.  doi: 10.3934/krm.2020035.

[16]

N. Loy and A. Tosin, Markov jump processes and collision-like models in the kinetic description of multi-agent systems, Commun. Math. Sci., 18 (2020), 1539-1568.  doi: 10.4310/CMS.2020.v18.n6.a3.

[17]

N. Loy and A. Tosin, A viral load-based model for epidemic spread on spatial networks, Math. Biosci. Eng., 18 (2021), 5635-5663. 

[18]

M. Morandotti and F. Solombrino, Mean-field analysis of multipopulation dynamics with label switching, SIAM J. Math. Anal., 52 (2020), 1427–1462. doi: 10.1137/19M1273426.

[19]

M. Moreau, Formal study of a chemical reaction by Grad expansion of the Boltzmann equation. I, Phys. A, 79 (1975), 18-38. 

[20]

L. Pareschi and G. Russo, An introduction to Monte Carlo method for the Boltzmann equation, ESAIM: Proc., 10 (2001), 35-75.  doi: 10.1051/proc:2001004.

[21] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. 
[22]

L. PareschiG. ToscaniA. Tosin and M. Zanella, Hydrodynamic models of preference formation in multi-agent societies, J. Nonlinear Sci., 29 (2019), 2761-2796.  doi: 10.1007/s00332-019-09558-z.

[23]

B. Piccoli, A. Tosin and M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory, Z. Angew. Math. Phys., 71 (2020), Paper No. 152, 25 pp. doi: 10.1007/s00033-020-01383-9.

[24]

L. Preziosi, G. Toscani and M. Zanella, Control of tumour growth distributions through kinetic methods, J. Theoret. Biol., 514 (2021), Paper No. 110579, 13 pp. doi: 10.1016/j. jtbi. 2021.110579.

[25]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[26]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[27]

G. Toscani, A. Tosin and M. Zanella, Multiple-interaction kinetic modeling of a virtual-item gambling economy, Phys. Rev. E, 100 (2019), 012308, 16 pp. doi: 10.1103/PhysRevE. 100.012308.

[28]

A. Tosin and M. Zanella, Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles, Multiscale Model. Simul., 17 (2019), 716-749.  doi: 10.1137/18M1203766.

[29]

A. Tosin and M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls, Math. Control Relat. Fields, 11 (2021), 681-713. 

[30]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

show all references

References:
[1]

G. AlbiM. BonginiF. Rossi and F. Solombrino, Leader formation with mean-field birth and death models, Math. Models Methods Appl. Sci., 29 (2019), 633-679.  doi: 10.1142/S0218202519400025.

[2]

F. Bassetti and G. Toscani, Mean field dynamics of interaction processes with duplication, loss and copy, Math. Models Methods Appl. Sci., 25 (2015), 1887-1925.  doi: 10.1142/S0218202515500487.

[3]

A. V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys. Rev. E, 61 (2000), 4576-4586.  doi: 10.1103/PhysRevE.61.4576.

[4]

M. Burger, Network structured kinetic models of social interactions, Vietnam J. Math., 49 (2021), 937-956.  doi: 10.1007/s10013-021-00505-8.

[5]

C. Cercignani, The Boltzmann Equation and its Applications, no. 67 in Applied Mathematical Sciences, Springer, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[6]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.

[7]

M. Delitala, Generalized kinetic theory approach to modeling spread and evolution of epidemics, Math. Comput. Modelling, 39 (2004), 1-12.  doi: 10.1016/S0895-7177(04)90501-8.

[8]

R. Della Marca, N. Loy and A. Tosin, An SIR-like kinetic model tracking individuals' viral load, 2021, Preprint. doi: 10.13140/RG. 2.2.32046.02883.

[9]

G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Wealth distribution under the spread of infectious diseases, Phys. Rev. E, 102 (2020), 022303, 14 pp. doi: 10.1103/physreve. 102.022303.

[10]

G. FurioliA. PulvirentiE. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27 (2017), 115-158.  doi: 10.1142/S0218202517400048.

[11]

C. D. Greenman and T. Chou, Kinetic theory of age-structured stochastic birth-death processes, Phys. Rev. E, 93 (2016), 012112.

[12]

M. Groppi and J. Polewczak, On two kinetic models for chemical reactions: Comparisons and existence results, J. Stat. Phys., 117 (2004), 211-241.  doi: 10.1023/B:JOSS.0000044059.59066.a9.

[13]

M. Groppi and G. Spiga, Kinetic approach to chemical reactionsand inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219. 

[14]

N. Loy and L. Preziosi, Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2020), 373-421.  doi: 10.1007/s00285-019-01411-x.

[15]

N. Loy and L. Preziosi, Stability of a non-local kinetic model for cell migration with density dependent orientation bias, Kinet. Relat. Models, 13 (2020), 1007-1027.  doi: 10.3934/krm.2020035.

[16]

N. Loy and A. Tosin, Markov jump processes and collision-like models in the kinetic description of multi-agent systems, Commun. Math. Sci., 18 (2020), 1539-1568.  doi: 10.4310/CMS.2020.v18.n6.a3.

[17]

N. Loy and A. Tosin, A viral load-based model for epidemic spread on spatial networks, Math. Biosci. Eng., 18 (2021), 5635-5663. 

[18]

M. Morandotti and F. Solombrino, Mean-field analysis of multipopulation dynamics with label switching, SIAM J. Math. Anal., 52 (2020), 1427–1462. doi: 10.1137/19M1273426.

[19]

M. Moreau, Formal study of a chemical reaction by Grad expansion of the Boltzmann equation. I, Phys. A, 79 (1975), 18-38. 

[20]

L. Pareschi and G. Russo, An introduction to Monte Carlo method for the Boltzmann equation, ESAIM: Proc., 10 (2001), 35-75.  doi: 10.1051/proc:2001004.

[21] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. 
[22]

L. PareschiG. ToscaniA. Tosin and M. Zanella, Hydrodynamic models of preference formation in multi-agent societies, J. Nonlinear Sci., 29 (2019), 2761-2796.  doi: 10.1007/s00332-019-09558-z.

[23]

B. Piccoli, A. Tosin and M. Zanella, Model-based assessment of the impact of driver-assist vehicles using kinetic theory, Z. Angew. Math. Phys., 71 (2020), Paper No. 152, 25 pp. doi: 10.1007/s00033-020-01383-9.

[24]

L. Preziosi, G. Toscani and M. Zanella, Control of tumour growth distributions through kinetic methods, J. Theoret. Biol., 514 (2021), Paper No. 110579, 13 pp. doi: 10.1016/j. jtbi. 2021.110579.

[25]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Phys. A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[26]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[27]

G. Toscani, A. Tosin and M. Zanella, Multiple-interaction kinetic modeling of a virtual-item gambling economy, Phys. Rev. E, 100 (2019), 012308, 16 pp. doi: 10.1103/PhysRevE. 100.012308.

[28]

A. Tosin and M. Zanella, Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles, Multiscale Model. Simul., 17 (2019), 716-749.  doi: 10.1137/18M1203766.

[29]

A. Tosin and M. Zanella, Uncertainty damping in kinetic traffic models by driver-assist controls, Math. Control Relat. Fields, 11 (2021), 681-713. 

[30]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

Figure 1.  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
Figure 3.  The problem of Section 5.2 with $ \nu_1 = \nu_2 $ and $ \mu\gg\lambda $
Figure 4.  The problem of Section 5.2 with $ \nu_1 = \nu_2 $ and $ \mu = \lambda $
Figure 5.  The problem of Section 5.2 with $ \nu_1 = 0 $ and $ \mu\gg\lambda $
Figure 6.  The problem of Section 5.2 with $ \nu_1 = 0 $ and $ \mu = \lambda $
Table 1.  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 $
Table 2.  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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