# Applied Mathematics Colloquium: Dr Chuntian Wang

## University of Alabama

Friday, April 22, 2022

2:00 PM – 3:00 PM

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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