In this talk, I propose to present my recent research notes on lattice and nonlocal dispersal equations. We first characterize the continuity of a map in the space $\mathcal{C}=BC(\mathcal{H},\mathbb{R}^m)$ equipped with the compact open topology. Then we show that linear lattice and nonlocal dispersal equations generate uniformly continuous semigroups in the Banach space $\mathcal{B}=BC(\mathcal{H},\mathbb{R}^m)$ equipped with the supremum norm. Finally, we illustrate how to prove nonlinear lattice and nonlocal dispersal equations generate monotone semiflows with respect to the compact open topology.

Age-structured models with nonlocal diffusion arise naturally in describing the population dynamics of biological species and the transmission dynamics of infectious diseases in which individuals disperse nonlocally and interact and the age structure of individuals matters. In this paper we study the principal spectral theory of age-structured models with nonlocal diffusion. First, we provide two criteria on the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we define the generalized principal eigenvalue and use it to investigate the influence of diffusion rate on the principal eigenvalue. Finally we study the effects of principal eigenvalue on the global dynamics of the model with Fisher-KPP nonlinearity on the birth rate and show that the principal eigenvalue being zero is critical. In addition, we establish the strong maximum principle for age-structured nonlocal diffusion operators.

In this talk, we propose a nonlocal reaction-diffusion model of Chikungunya disease with seasonality (temperature and rainfall), spatial heterogeneous structure, periodic maturation delay, and periodic extrinsic incubation period. We introduce the basic reproduction number $R_m$ for the vector and the basic reproduction ratio $R_0$ for the disease to describe the global dynamics of the model system. We further conduct a case study for the Chikungunya transmission in Cear\'{a}, Brazil. Our numerical simulations are well consistent with the analytic results. The effects of spatial heterogeneity and some control strategies will be also discussed. This talk is based on a joint work with Dr. Xiao-Qiang Zhao.

We consider the spread of an infectious disease in a heterogeneous environment modeled as a network of patches. We focus on the invasibility of the disease, as quantified by the corresponding value of an approximation to the network basic reproduction number, R0, and study how changes in the network structure affect the value of R0. We provide a detailed analysis for two model networks, a star, and a path, and discuss the changes to the corresponding network structure that yield the largest decrease in R0. We develop both combinatorial and matrix analytic techniques, and we illustrate our theoretical results by simulations with the exact R0. https://epubs.siam.org/doi/abs/10.1137/20M1328762

Huang, Wang and Lewis [SIAP, 2017] developed a hybrid continuous/discrete-time model to describe the persistence and invasion dynamics of Zebra mussels in rivers. They used a net reproductive rate R0 to determine population persistence in a bounded domain and estimated spreading speeds by applying the linear determinacy conjecture and using the formula in Neubert and Caswell [Ecology, 2000]. Since the associated solution operator is non-monotonic and non-compact, it is nontrivial to rigorously establish these quantities. In this paper, we analyze the spatial dynamics of this model mathematically. We first solve the parabolic equation and rewrite the model into a fully discrete-time model. In a bounded domain, we show that the spectral radius r of the linearized operator can be used to determine population persistence and that the sign of r-1 is the same as that of R0-1, which confirms that R0 defined in Huang, Wang and Lewis [SIAP, 2017] can be used to determine population persistence. In an unbounded domain, we construct two monotonic operators to control the model operator from above and from below and obtain upper and lower bounds of the spreading speeds of the model. https://doi.org/10.3934/dcdsb.2020362

Industrial bioreactors use microbial organisms as living factories to produce a widerange of commercial products. For most applications, yields eventually become limited by the proliferation of "escape mutants" that acquire a growth advantage by losing the ability to make product. The goal of this work is to use mathematical models to determine whether this problem could be addressed in continuous flow bioreactors that include a "stem cell" population that multiplies rapidly and could be used to compete against the emergence of cheater mutants. In this system, external stimuli can be used to induce stem cell multiplication through symmetric cell division, or to limit stem cell multiplication and induce higher production through an asymmetric cell division that produces one stem cell and one new product-producing "factory cell". Our results show product yields from bioreactors with microbial stem cells can be increased by 18% to 127% over conventional methods, and sensitivity analysis shows that yields could be improved over a broad range of parameter space.

We study a multi-strain reaction-diffusion epidemic model dynamics with spatially heterogeneous infection and recovery rates. The non-existence of coexistence endemic equilibrium (EE) solutions and the final extinction of one or multiple strains are proved under some proper assumptions on the model's parameters. In the case of a two-strains model, under a proper assumption on the parameters, we provide complete information on the large time behavior of classical solutions. Finally, the asymptotic behaviors of the coexistence endemic equilibrium solutions, as the diffusion rate of the populations approaches zero, are also investigated, where the spatial segregation of multiple strains is found and determined.