TY - JOUR

T1 - COMPARTMENTAL LIMIT OF DISCRETE BASS MODELS ON NETWORKS

AU - Fibich, Gadi

AU - Golan, Amit

AU - Schochet, Steve

N1 - Publisher Copyright:
© 2023 American Institute of Mathematical Sciences. All rights reserved.

PY - 2023/5

Y1 - 2023/5

N2 - We introduce a new method for proving convergence and rate of convergence of discrete Bass models on various networks to their respective compartmental Bass models, as the population size M becomes infinite. In this method, the master equations are reduced to a smaller system of equations, which is closed and exact. The reduced system is embedded into an infinite system, whose convergence to the infinite limit system is proved using standard ODE estimates. Finally, an exact ansatz reduces the infinite limit system to the compartmental model. Using this method, we show that when the network is complete and homogeneous, the discrete Bass model converges to the original 1969 compartmental Bass model, at the rate of 1/M. When the network is circular, the compartmental limit is different, and the convergence rate is exponential in M. For a heterogeneous network that consists of K homogeneous groups, the limit is given by a heterogeneous compartmental Bass model, and the convergence rate is 1/M. Using this compartmental model, we show that when the heterogeneity in the external and internal influence parameters among the K groups is positively monotonically related, heterogeneity slows down the diffusion.

AB - We introduce a new method for proving convergence and rate of convergence of discrete Bass models on various networks to their respective compartmental Bass models, as the population size M becomes infinite. In this method, the master equations are reduced to a smaller system of equations, which is closed and exact. The reduced system is embedded into an infinite system, whose convergence to the infinite limit system is proved using standard ODE estimates. Finally, an exact ansatz reduces the infinite limit system to the compartmental model. Using this method, we show that when the network is complete and homogeneous, the discrete Bass model converges to the original 1969 compartmental Bass model, at the rate of 1/M. When the network is circular, the compartmental limit is different, and the convergence rate is exponential in M. For a heterogeneous network that consists of K homogeneous groups, the limit is given by a heterogeneous compartmental Bass model, and the convergence rate is 1/M. Using this compartmental model, we show that when the heterogeneity in the external and internal influence parameters among the K groups is positively monotonically related, heterogeneity slows down the diffusion.

KW - Stochastic models

KW - compartmental models

KW - convergence

KW - diffusion in networks

KW - heterogeneity

KW - master equations

KW - ordinary differential equations

KW - rate of convergence

UR - http://www.scopus.com/inward/record.url?scp=85147942197&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2022203

DO - 10.3934/dcdsb.2022203

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AN - SCOPUS:85147942197

SN - 1531-3492

VL - 28

SP - 3052

EP - 3078

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 5

ER -