This paper is devoted to the study of asymptotic behavior of the basic reproduction ratio for periodic reaction-diffusion systems in the case of small and large diffusion coefficients. We first establish the continuity of the basic reproduction ratio with respect to parameters by developing the theory of resolvent positive operators. Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem for large diffusion coefficients. We further obtain the asymptotic behavior of the basic reproduction ratio as the diffusion coefficients go to zero and infinity, respectively. We also investigate the limiting behavior of positive periodic solution for periodic and cooperative reaction-diffusion systems with the Neumann boundary condition when the diffusion coefficients are large enough. Finally, we apply these results to areaction-diffusion model of Zika virus transmission.
In this paper, we propose a periodic reaction-diffusion model of Zika virus with seasonal and spatial heterogeneous structure in host and vector populations. We introduce the basic reproduction ratio R0 for this model and show that the disease-free periodic solution is globally asymptotically stable if R0<1 or R0=1, while the system admits a globally asymptotically stable positive periodic solution if R0 >1. Numerically, we study the Zika transmission in Rio de Janeiro Municipality, Brazil, and investigate the effects of some model parameters on R0. We find that the neglect of seasonality underestimates the value of R0 and the maximum carrying capacity affects the spread of Zika virus.
In this paper, we study the invasion dynamics of a diffusive pioneer-climax model in monotone and non-monotone cases. For parameter ranges in which the system admits monotone properties, we establish the existence of spreading speeds and their coincidence with the minimum wave speeds by monotone dynamical system theories. The linear determinacy of the minimum wave speeds is also studied by constructing suitable upper solutions. For parameter ranges in which the system is non-monotone, we further determine the existence of spreading speeds and traveling waves by the sandwich technique and upper-lower solution method. Our results generalize the existing results established under monotone assumptions to more general cases.
The secondary spread of an invasive species after initial establishment is a major factor in determining its distribution and impacts. In this study we constructed an individual-based model for the spread of the invasive green alga Codium fragile ssp. fragile along a straight coastline, in order to understand the factors governing spreading speed. Codium can spread locally through non-buoyant propagules, while long-distance dispersal depends on the wind-driven dispersal of buoyant fragments. Since fragment buoyancy is determined by light conditions, we first modelled the buoyancy of fragments, yielding a dispersal time dependent on light conditions. We then used this dispersal time, along with empirical wind speeds and directions to calculate a dispersal kernel for fragments. Finally, we incorporated this dispersal kernel into a population growth model including survival rate and fragmentation rate, to calculate a population spreading speed. We obtained the spreading speeds under current environmental conditions along the east coast of Canada and also conducted a sensitivity analysis to investigate the potential influence of environmental shifts associated with climate change on the spread of Codium.
In this talk, we will devleop the basic theory for age-structured population dynamics with nonlocal diffusion. In particular, we study the semigroup of linear operators associated to an age-structured model with nonlocal diffusion and use the spectral properties of its infinitesimal generator to determine the stability of the zero steady state. An application will be discussed. This is a joint work with Dr. Hao Kang and Dr. Shigui Ruan.
In this talk, I shall present a recent work devoted to the investigation of dengue spread via a time-space periodic reaction-advection-diffusion model. The existence of the spreading speeds and its coincidence with the minimal speed of almost pulsating waves will be established. We also illustrate the analytic results by numerical simulation, for the temporal periodic case and the temporal and spatial periodic case, respectively. This talk is based on the work joint with Drs. Jian Fang (Harbin Institute of Technology, China), Xiulan Lai (Renmin University of China).
In this talk, I will report our recent research on basic reproduction ratio R0 for time-delayed compartmental population models in a periodic environment. It is proved that R0 serves as a threshold value for the stability of the zero solution of the associated periodic linear systems. An extension of such a theory to spatial population models and a general algorithm for the numerical computation of R0 will be discussed. As an application, we propose a Malaria transmission model with temperature-dependent incubation period, and then establish a threshold type result on its global dynamics in terms of R0. We also do a case study of the Malaria transmission in Maputo Province, Mozambique.
The basic reproduction number (or ratio) R0 is an important concept in population biology. As a threshold quantity for population dynamics, it is unquestionably one of the most valuable mathematical ideas brought to theoretical ecology and epidemiology. In this talk, I will briefly review the development of the R0 theory for population models with compartmental structure. Then I will focus on the definition, stability equivalence, characterization and numerical algorithm of R0 for reaction-diffusion systems. I will further address the asymptotic behavior of R0 for small and large diffusion coefficients. As an illustrative example, I will discuss a periodic reaction-diffusion model of Zika virus transmission for the global stability in terms of R0 and the limiting profile of R0.
The dynamics of linear positive systems map the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. This raises the following question: what linear systems map the set of vectors with k sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called k-positive linear systems, that reduces to positive systems for k=1. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case k=2 establish the Poincare-Bendixson property for any bounded trajectory. This talk is based on Eyal Weiss and Michael Margaliot's recent paper.
In this talk, I will report our recent research on monotone evolution systems without spatial translation invariance. Under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of such systems. Then we apply the developed theory to study traveling waves and spatio-temporal propagation patterns for a reaction-diffusion equation in a cylinder under shifting environment and an asymptotically homogeneous KPP-type equation.
A reaction-diffusion model which is called the field-road model was introduced by [Berestycki, Roquejoffre and Rossi, JMB, 2013] to describe biological invasion with fast diffusion on a line. In this paper, we investigate this model in a heterogeneous landscape and establish the existence of the spreading speeds $c^*_{\pm}$ as well as its coincidence with the minimal speeds of pulsating fronts. We start with a truncated problem with an imposed Dirichlet boundary condition. We prove the existence of spreading speeds $c^*_{R,\pm}$ which coincide with the minimal speeds of pulsating fronts in the strip. In particular, $c^*_{R,\pm}$ turn out to be the same due to the variational formulas in terms of the principal eigenvalues of certain elliptic problems. The arguments combine the dynamical system method and PDE's method. Finally, we turn back to the original problem in the half-plane via generalized principal eigenvalue approach as well as asymptotic method.
In this talk, I will report our recent research on the uniqueness and stability of bistable traveling waves for monotone semiflows in an abstract setting. Under appropriate assumptions, we establish the uniqueness and stability of bistable waves for discrete and continuous-time semiflows in a continuous habitat by appealing to a global convergence theorem for monotone semiflows. We also extend such a result to time-periodic semiflows, and apply the general theory to a class of reaction-diffusion-advection systems in a cylinder.
Cholera is a water- and food-borne infectious disease caused by V. cholerea. To investigate multiple effects of human behavior change, seasonality and spatial heterogeneity on cholera spread, we propose a reaction-diffusion-advection model that incorporates human hosts and aquatic reservoir of V. cholerea. We derive the basic reproduction number R0 for this system and then establish a threshold type result on whether the cholera spreads in terms of R0. Further, we show that the relative rate of bacterial loss at the downstream end of the river due to water flux can reduce the disease risk. This is a joint work with Drs. Xueying Wang and Xiaoqiang Zhao.