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Applied Mathematics Colloquium: Dr Chuntian Wang

University of Alabama

Friday, April 22, 2022
2:00 PM – 3:00 PM
Title: Recent advances in population and epidemic models of
Markov pure jump and piecewise-deterministic processes (PJMP and PDMP)

Abstract: Markov pure jump processes and Markov piecewise-deterministic processes (PJMP and PDMP) have been introduced and studied
intensively as two of the most important topics in the field of theory of
stochastic processes. The applications of these two classes of Markov
processes in social and human behavioral models have a relative short
history. In particular, PDMP has only been used very recently to
population models, like population migration, species competition etc. In
this talk, I will introduce two recent works on social-dynamics
modeling, simulations, and analysis, based on theory and analysis of
PJMP and PDMP. The first one is the classical stochastic compartmental
susceptible-infected-recovered (SIR) model. The focus here is the finite-
size-effects analysis. Deterministic SIR models give the mean behaviour
of stochastic agent-based models. However, with finite size populations,
chance variations may lead to significant departures from the mean.
With a martingale approach, we provide a theoretical explanation of
finite size effects. Case studies of COVID-19 transmission in smaller
populations provides an illustration of our theory. The second model is a
multiscale human-environment interaction model of residential burglary.
Particularly, we introduce separated spatiotemporal scales for agent
actions and environment parameter reactions. This hybrid scheme
significantly reduces the computational cost. We manage to circumvent
the “dimension curse” with increasing population scale but also avoid
negligence of random fluctuation in finite size effects. The separation of
scales brings the burglary model into the theoretical framework of
PDMP. A martingale approach is again applicable.
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