Special Session 84: Mathematical modeling and analysis in spatial ecology and epidemiology

Modeling and Data Analysis of Emerging Infectious Disease Transmission Using Stochastic Difference Equations

Sha He Shaanxi Normal University Peoples Rep of China Co-Author(s):

Abstract: This study develops stochastic difference equation models to characterize COVID-19 transmission dynamics under random effects. First, we establish a discrete-time stochastic epidemic model using binomial distributions, with parameters estimated from reported data. Next, we construct an improved model that incorporates asymptomatic transmission and imported cases to quantify resurgence risk and evaluate containment measures. An analysis of over 100 Chinese outbreaks reveals small-scale clustered transmission patterns under non-pharmaceutical interventions (NPIs). To investigate how stochastic factors influence the containment process, we derive a stochastic difference equation for newly reported cases based on the stochastic SIR framework and introduce the Stochastic Control Reproduction Number (SCRN). We further estimate the SCRN through Bayesian change-point analysis, and demonstrate via first-passage time theory that controlled randomness can accelerate epidemic containment.

Propagation dynamics for a stage-structured population model in a shifting environment

Leyi Jiang School of Mathematics, Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Leyi Jiang, Yu Jin, Xiao-Qiang Zhao

Abstract: In this talk, we consider an integro-difference system in an unbounded domain for a population with a stage structure consisting of juveniles and adults. The environment shifts between a favorable habitat and an unfavorable habitat at two ends of the domain, which makes the growth and propagation dynamics of the population depend on the changing habitat conditions. Under appropriate assumptions, we establish the existence of spreading speeds and forced traveling waves for such a model. Our analysis of the propagation dynamics is based on the corresponding limiting systems at the far ends of the domain and their relation to the model. It turns out that the habitat shifting speed greatly influences the spreading speeds of the population and may even cause the extinction of the population.

Global dynamics of a predator-prey model with prey-dependent search rate

Weihua Jiang School of Mathematics, Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Qiguo Qian

Abstract: In this talk, we consider a predator-prey model with prey-dependent search rate in advective environments. We investigate the effect of prey-dependent search rate on model dynamics and provide a complete and explicit classification of global dynamics. Moreover, we show that when the limit cycle is stable, the predator can adjust its search rate to stabilize the positive equilibrium, thereby preventing the paradox of enrichment. Specifically, we employ a new planar analysis method to prove that local stability of the positive equilibrium implies global stability.

How Should Species Move? Ideal Free Dispersal in Changing Environments

King Yeung Lam The Ohio State University USA Co-Author(s):    R.S. Cantrell (Miami), C. Cosner (Miami) and H. Zhang (Shanghai Jiaotong).

Abstract: Understanding how organisms distribute themselves in heterogeneous environments is a central question in ecology. The classical principle of ideal free distribution (IFD) predicts that individuals move so as to equalize fitness across space, and such strategies are known to be evolutionarily stable in static environments. However, real habitats are often time-periodic, and the notion of optimal dispersal in such settings is far less understood. In this talk, we study dispersal strategies in a time-periodic, patchy environment through a system of reaction-diffusion (or patch) models with multiple competing species. We introduce a natural definition of ideal free distribution based on pathwise fitness, and characterize when such distributions can exist. We then show that dispersal strategies leading to IFD are evolutionarily stable: populations converge to an IFD determined by the environment, and species that realize a (joint) IFD can outcompete all others. Our results extend classical theory to systems with multiple competitors or multiple prey and predator species in time-periodic environments. Our main tool is the construction of Lyapunov functions inspired by generalized relative entropy inequality. This is joint work with R.S. Cantrell (Miami), C. Cosner (Miami) and H. Zhang (Shanghai Jiaotong).

Bifurcation of rotating wave solutions in reaction-diffusion systems on a circle

Junping Shi College of William & Mary USA Co-Author(s):    Qingying Cai, Junping Shi, Ying Su

Abstract: The existence of rotating wave solutions of a general reaction-diffusion system with nonlocal interaction on a circle are proved using a new bifurcation approach with two parameters, and a normal form for the rotating wave bifurcations is also developed to compute the direction of the bifurcation diagram. Theoretical results are applied to two model systems for the existence of rotating waves which are further verified by numerical simulations.

Equivariant Turing-Turing Bifurcations and Pattern Formation on a Square Domain

Hongbin Wang School of Mathematics, Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Chen Chen

Abstract: In this talk, I would like to report one of our recent works on equivariant Turing-Turing bifurcations and pattern formation in square spatial domains. For a relatively general reaction-diffusion systems with self-diffusion and cross-diffusion terms, we derive the local center manifolds near equivariant Turing-Turing singularities. We give approximate expressions for superposition-type steady states and their stability conditions. As an application, we analyze a plant-water interaction model to explore the formation of self-organized vegetation patterns in semi-arid regions.

A nonlocal reaction-diffusion pest model with impulsive larvicidal treatments

Ruiwen Wu Jinan University Peoples Rep of China Co-Author(s):

Abstract: Larvicides are pesticides that target immature stages of pests by disrupting their development, thereby reducing survival, causing mortality, and potentially leading to population suppression or extinction. This study constructs a nonlocal periodic delayed reaction-diffusion model to describe the effects of impulsive larvicide applications on pest populations in a spatially and temporally heterogeneous environment. Furthermore, the threshold dynamics of the model are established through dynamical systems analysis. The numerical results indicate that larvicides can play a significant role in pest management; however, their application strategies need to be carefully designed. In addition, ignoring spatial heterogeneity in model formulation may lead to a misestimation of the risk of pest outbreaks.

Modelling heterogeneity and its impact on disease transmission dynamics

Yanni Xiao Xi`an Jiaotong University Peoples Rep of China Co-Author(s):    Yanni Xiao, Biao Tang, Pengfei Song

Abstract: Human heterogeneity is a critical issue in infectious disease transmission dynamics modelling, and it has recently received much attention in COVID-19 studies. In this talk, a novel model with heterogeneity in susceptible individuals is proposed to accurately estimate the final size. Then, a general human heterogeneous disease model with mutation is proposed to comprehensively study the effects of human heterogeneity on basic reproduction number, final epidemic size and herd immunity. We show that human heterogeneity may increase or decrease herd immunity level, strongly depending on some convexity of the heterogeneity function.

Global Dynamics of Nonlocal Dispersal Systems with Time-Varying Domains

Hailong Ye Shenzhen University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will report our recent research on the asymptotic behaviour of a class of nonlocal dispersal systems with time-varying domains. We first establish the comparison principle for generalized sub- and supersolutions of nonautonomous nonlocal dispersal systems. Based on the spectral bound, we rigorously characterize the threshold dynamics of nonautonomous nonlocal dispersal systems in the time-periodic case, which provide a comprehensive framework to examine the threshold dynamics of the original system on asymptotically fixed and time-periodic domains. In the asymptotically unbounded case, we introduce an auxiliary function to overcome the challenges arising from the asymptotically vanishing viscosity in the system and the time-dependent coupling structure in kernel. This talk is based on joint work with Prof. Xiao-Qiang Zhao and Dr. Xiandong Lin, which has been published by JDE (2026).

Convergence to forced waves of integro-difference equations in a shifting environment

Xiao Yu South China Normal University Peoples Rep of China Co-Author(s):

Abstract: We study integro-difference equations in shifting habitats. We establish the existence of forced waves and prove convergence to these waves in a moving coordinate system, provided that both limiting environments admit KPP-type structures and the rightward spreading speed coincides with the habitat speed.

Well-posedness and propagation dynamics of reaction-diffusion equations with Borel-measure coefficients

Ziqi Zhen School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China Peoples Rep of China Co-Author(s):    Xing Liang

Abstract: We study the equation $ u_t = u_{xx} + \beta(x; L) g(u) - a u$, $x \in \mathbb{R}$, where the Borel-measure coefficient $\beta(x;L)$ represents the local density of calcium release sites with $L$ being the separation between release sites and $g(u)$ satisfies a weak Allee effect condition. The model describes intracellular waves in a continuum excitable media with discrete release sites. In this paper, we first establish the well-posedness of the Cauchy problem for general Borel-measure $\beta(x;L)$ within a rigorous mathematical framework, using an approximation procedure. Next, we investigate the propagation dynamics of the model where $\beta(x;L)$ is an $L^1$ perturbation of the periodic Dirac sources $\sum_{n=-\infty}^{+\infty} \delta(x - nL)$ and $g(u) = u^p (1 - u)$ with the integer $p \ge 2$. We focus on the bistable regime of the model, which is determined by the decay strength parameter $a$ and the separation distance $L$ between release sites, in which the model admits exactly three $L$-periodic steady states. We establish the existence of pulsating waves that spatially connect stable steady states by a dynamical systems approach. Furthermore, we prove that any such traveling wave with nonzero speed is both unique and globally exponentially stable via the squeezing method.