In this talk, we study the linear and nonlinear speed selection of traveling waves for a two-species Lotka-Volterra competitive system with nonlocal dispersal. By using the method of upper-lower solutions, we obtain some new conditions for linear and nonlinear minimal speed selection. It is shown that the strong competition between two populations will lead to the realization of nonlinear selection. Two examples are presented to illustrate the realization of linear a nd nonlinear selection.
In this talk, we propose a nonlocal reaction-diffusion model of bluetongue disease with seasonality, spatial heterogeneous structure, and periodic extrinsic incubation period (EIP). We introduce the basic reproduction ratio R0 for this model and show that the disease-free periodic solution is globally attractive if R0 <1, while the disease is uniformly persistent if R0>1. Further, we obtain the global attractivity of the positive steady state in the case where all the coefficients are constants. Numerically, we study the bluetongue transmission in Corsica Island, France, and investigate the impact of some model parameters on R0. We find that the disease risk will be underestimated if the spatial heterogeneity is ignored.
In this paper, we formulate a multi-group SIR epidemic model with the consideration of proportionate mixing patterns between groups and group-specific fractional-dose vaccination to evaluate the effects of fractionated dosing strategies on disease control and prevention in a heterogeneously mixing population. The model allows for different dose fractionation usages for different subpopulations and incorporates some important fractional-dose related parameters to reflect the effectiveness of fractionated dose vaccines at the individual level. Furthermore, the basic reproduction number R0, the final size of the epidemic, and the infection attack rate are used as three measures of population-level implications of fractionated dosing programs. Theoretically, we identify the basic reproduction number, R0, establish the existence and uniqueness of the final size and the final size relation with R0, and obtain explicit calculation expressions of the infection attack rate for each group and the whole population. Numerical simulations are provided to present a case study regarding the im- pact of fractionated dosing campaign on control of the pandemic Asian influenza A (H2N2), in 1957–1958. The simulation results suggest that dose fractiona- tion policies take positive effects in lowering the R0, decreasing the final size and reducing the infection attack rate only when the fractional-dose influenza vaccine efficacy is high enough rather than just similar to standard-dose. We find evidences that fractional-dose vaccination in response to influenza vaccine shortages take negative community-level effects. Our results indicate that the role of fractional dose vaccines should not be overestimated even though fractional dosing strategies could extend the vaccine coverage and policy makers should cautiously propose the dose-sparing vaccination policy to optimize the utilization of limited vaccine supplies.
In this talk, I will report our recent reserach on a reaction-diffusion competition model with seasonal succession in the whole space. The time periodicity accounts for the effect of two different seasons. Under the weak competition assumption, by using the method of upper and lower solutions and Schauder's fixed point theorem, we obtain the existence of traveling wave solution when the wave speed c is greater than or equals c*. Moreover, the asymptotic spreading speed of solutions is also given.
A SEIRS reaction-diffusion epidemic model with general incidence is proposed due to the multiple factors such as individual differences, spatial environment and so on. The existence of global solutions, the ultimate boundedness of the solutions and the basic reproduction number $R_0$ are obtained. Next, the threshold dynamics (extinction and persistence) of the disease are obtained in two special cases. Moreover, the global dynamics of the homogeneous space and heterogeneous diffusion model are obtained in terms of $R_0$: the disease-free steady state is globally asymptotically stable in two special cases when $R_0\leq 1$; the endemic steady state is globally asymptotically stable in permanent acquired immunity case when $R_0>1$. Our numerical simulations show the spatial heterogeneity may promote the infection of the disease.
In order to investigate the roles that environmental virus played in the dynamics of hospital infections in China, we proposed a cross infection model with diffusive virus in the environment. Firstly, we prove the global existence, the boundedness and the positivity of solution as well as a global attractor of the model. Secondly, we investigated the threshold dynamics of the limiting system and then established the threshold dynamics of the proposed system by using the theory of chain transitive sets. In particular, if $R_0\leq1$, then the disease-free equilibrium is globally asymptotically stable; if $R_0>1$, then the system has a unique positive equilibrium and it is globally attractive. Finally, we use the numerical method to explore the influence of different diffusion coefficient on $R_0$. The results show that in a less polluted environment, $R_0$ is a decreasing function with respect to the diffusion rate, which implies that the diffusion of virus is beneficial for patients. However, in a more polluted environment, $R_0$ is an increasing function with respect to the diffusion rate, which means increasing diffusion of virus is harmful to the elimination of disease.
In this work, we consider a nonlocal differential equation with multiple delays, which describes the population growth subject to development diapause. We use unimodal birth functions to illustrate the psychological effect on the reproduction of populations. In a bounded domain, we define the basic reproduction number $R_0$ and determine a threshold result on the extinction or persistence of populations. In an unbounded domain, if the equation admits a monostable structure, we consider the existence of spreading speed and monostable traveling waves. Moreover, if the equation admits a bistable structure, the existence of bistable traveling waves and the sign of the wave speed are further determined. At the end of this work, we present some simulation results by choosing parameter values for tick populations.
Natural rivers connect to each other to form river networks. The geometric structure of a river network can significantly influence spatial dynamics of populations in the system. We consider a process-oriented model to describe population dynamics in river networks of trees, establish the fundamental theories of the corresponding parabolic problems and elliptic problems, derive the persistence threshold by using the principal eigenvalue of the corresponding eigenvalue problem, and define the net reproductive rate to describe population persistence or extinction. By virtue of numerical simulations, we investigate the effects of hydraulic, physical, and biological factors, especially the structure of the river network, on population persistence.