On the control of density-dependent stochastic population processes with time-varying behavior

The study of density-dependent stochastic population processes is important from a historical perspective as well as from the perspective of a number of existing and emerging applications today. In more recent applications of these processes, it can be especially important to include time-varying parameters for the rates that impact the density-dependent population structures and behaviors. Under a mean-field scaling, we show that such density-dependent stochastic population processes with time-varying behavior converge to a corresponding dynamical system. We analogously establish that the optimal control of such density-dependent stochastic population processes converges to the optimal control of the limiting dynamical system. An analysis of both the dynamical system and its optimal control renders various important mathematical properties of interest.

[1]  Dirk Aeyels,et al.  Exponential Stability of Slowly Time-Varying Nonlinear Systems , 2002, Math. Control. Signals Syst..

[2]  A. M. Lyapunov.  The general problem of the stability of motion , 1992 .

[3]  Benjamin Armbruster,et al.  An elementary proof of convergence to the mean-field equations for an epidemic model , 2015, 1501.03250.

[4]  Mark S. Squillante,et al.  On the Control of Epidemic-Like Stochastic Processes with Time-Varying Behavior? , 2015, PERV.

[5]  B. Armbruster.  A simple and general proof for the convergence of Markov processes to their mean-field limits , 2016, 1602.05224.

[6]  Donald F. Towsley,et al.  The effect of network topology on the spread of epidemics , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[7]  Ward Whitt,et al.  An Introduction to Stochastic-Process Limits and their Application to Queues , 2002 .

[8]  P. Hartman.  A lemma in the theory of structural stability of differential equations , 1960 .

[9]  K. Dietz,et al.  Daniel Bernoulli's epidemiological model revisited. , 2002, Mathematical biosciences.

[10]  Hamidou Tembine,et al.  Opinion Dynamics in Social Networks through Mean-Field Games , 2016, SIAM J. Control. Optim..

[11]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[12]  P. Hartman.  On the local linearization of differential equations , 1963 .

[13]  Amin Saberi,et al.  How to distribute antidote to control epidemics , 2010 .

[14]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[15]  M. Hirsch.  Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere , 1985 .

[16]  Bruno Gaujal,et al.  Mean Field for Markov Decision Processes: From Discrete to Continuous Optimization , 2010, IEEE Transactions on Automatic Control.

[17]  E. Todeva.  Networks , 2007 .

[18]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[19]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[20]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.