This project is devoted to the study of asymptotic behavior of the principal eigenvalue and basic reproduction ratio for periodic patch models in the case of small and large dispersal rates. We first deal with the eigenspace corresponding to the zero eigenvalue of the connectivity matrix. Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem as the dispersal rate goes to zero and infinity, respectively. We further establish the asymptotic behavior of the basic reproduction ratio for small and large dispersal rates. Finally, we apply these results to a multi-patch Ross-Macdonald model.
In this paper, we investigate a time-delayed differential model of the transmission dynamics of schistosomiasis with seasonality. In order to study the influence of water temperature on egg hatching into miracidia and the development from miracidia to cercariae, we incorporate time-dependent delays into the model to describe the maturation period and the extrinsic incubation period. We first introduce the basic reproduction number R0 for this model and establish a threshold-type result on its global dynamics in terms of R0. More precisely, we show that the disease is uniformly persistent when R0 > 1, while the disease-free periodic solution is globally attractive when R0 < 1. Then we choose parameters to fit the schistosomiasis epidemic data in Hubei province of China. Our numerical simulations indicate that the schistosomiasis will continue to prevail in the near future unless more effective control measures are taken. A further sensitive analysis demonstrates that the parameters with a strong impact on the outcome are baseline transmission rate, recovery rate, schistosome egg output rate, contact rate between miracidia and snails, and cercariae output rate.
We investigate a cross-infection model with diffusion and incubation period. Firstly, we establish the global stability of the disease-free steady state and the positive steady state for the spatially homogeneous system, respectively. Secondly, we study the threshold dynamics for the spatially heterogeneous system in terms of the basic reproduction number R0, and prove that the disease-free steady state is globally attractive if R0<1; and the system is uniformly persistent if R0>1. Finally, we numerically explore the influence of different diffusion coefficients, the spatial heterogeneity and the time delay on R0. Our numerical results show that R0 is decreasing with respect to the diffusion coefficients and the time delay, while it is increasing with respect to the spatial heterogeneity.
This project is concerned with the long time behavior of a diffusive vector-borne disease model in the whole space. Due to the lack of monotonicity, the comparison principle cannot be directly applied to this system. We first establish the spreading speed of the infected hosts and infected vectors when the basic reproduction number of the corresponding kinetic system is larger than one. Concretely, the upper bounds of the spreading speed are obtained by constructing auxiliary systems and using comparison argument. For the lower bounds of the spreading speed, we apply the idea of uniform persistence and generalized principal eigenvalue of a weakly coupled cooperative elliptic system. We further show the final convergence to the positive equilibrium by using the theory of monotone dynamical systems to two auxiliary monotone systems. When the basic reproduction number of the corresponding kinetic system is less than one, the solution approaches to the disease-free equilibrium uniformly.
Competition for resources is a fundamental topic in theoretical ecology. Population dynamics are coupled to dynamics of one or more resources by assuming a constant quota of nutrient per individual. In fact, quotas may vary, leading to variable-internal-stores models. When nutrient is taken up, it is stored internally, and population growth is a positive function of stored nutrient. The competitors live in a flowing habitat with both advection and diffusion, where the nutrient is supplied in the upstream flow, and all constituents flow out at the downstream end. The main difficulties in mathematical analysis for such systems are caused by the singularity at the extinction state. In this talk, we first investigate the nonlinear eigenvalue problem in the special positive cones of functions motivated by the ratio dependence. Then the threshold type result on the extinction/persistence of the species can be determined by the principal eigenvalue of our nonlinear eigenvalue problem. When the habitat is infinite, we shall attempt to study the travelling wave and spreading speeds.
References:
S.-B. Hsu, J. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations 248 (2010), no. 10, 2470-2496.
S.-B. Hsu, K.-Y. Lam and F.-B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol. 75 (2017), no. 6-7, 1775-1825.
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math. 60 (2007), pp. 1-40.
Synchronized maturation has been extensively studied in biological science on its evolutionary advantages. The purpose of our current project is to study the spatial dynamics of species growth with annually synchronous emergence of adults by formulating an impulsive reaction-diffusion model. It turns out that the one-year solution map defines a recursion operator and the spatial dynamics can be proved with the aid of the discrete semiflow on infinite dimensional space. By investigating the properties of the resultant discrete semiflow, the existence of the spreading speed and traveling waves is investigated for the model on an unbounded spatial domain. Furthermore, it is shown that the spreading speed equals to the minimal speed of traveling waves, regardless of the monotonicity of the birth rate function. The critical domain size to reserve species persistence is also determined by investigating the model on a bounded domain with a lethal exterior. Numerical simulations are illustrated to confirm the analytical results and to explore the effects of the emergence maturation delay on the spatial dynamics of the population distribution. This is a joint work with Drs. Yijun Lou and Xiaoqiang Zhao.
Based on the control strategies, transmission mechanism and clinical progression of COVID-19, we propose a compartmental model with time delays. Two time delays are introduced into the model to describe the incubation period of the disease and the quarantine period of uninfected individuals who have contacts with infected people. In order to reveal the spread rule for COVID-19, we study the threshold dynamics for the model. The basic reproduction number R0 is obtained. When R0<1, the disease-free equilibrium is locally asymptotically stable, when R0>1, the disease-free equilibrium is unstable and the disease is uniformly persistent. As the model's applications, we study COVID-19 transmission in the United States. The parameters are chosen to fit public data in the US. The numerical results indicate that an outbreak peak time in the US will appear in the middle of March. Sensitive analysis results show that enhancing the control measures, such as keeping social distance, wearing masks, isolation etc., can significantly contribute to the prevention and control of COVID-19 infection. This is a joint work with Dr. Tailei Zhang.
In this talk, we discuss the asymptotic spreading of KPP fronts in heterogeneous shifting habitats, with possibly any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equation and integro-differential equations with a distributed time-delay. We will first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable s = x/t. In terms of the standard Fisher-KPP equation, our results lead to a new class of ``asymptotically homogeneous" environments which share the same spreading speed with the corresponding homogeneous environments. This is a jointly work with Dr. Adrian Lam. See https://arxiv.org/abs/2101.06698
We study the spreading properties of integro-differential equations with periodic delay. We focus on the case where the kinetic dynamics is of monostable type and the dispersal kernels are algebraically decaying. More precisely, we prove the non-existence of traveling wave solutions and show that the level sets of the solutions eventually locate in between two exponential functions of time. In particular, a sharp bound is obtained. To support these conclusions, the fundamental solutions of integro-differential equations with periodic time-delay are established under certain initial data.
The coevolution or coexistence of multiple viruses with multiple hosts has been an important issue in viral ecology. In this work, we study the mathematical properties of the solutions of a chemostat (ODE) model for two host species and two virus species. By virtue of the global dynamics of its submodels and the theories of uniform persistence, we derive sufficient conditions for the coexistence of two hosts with two viruses and coexistence of two hosts with one virus, as well as occurrence of Hopf bifurcation.
In this talk, I will report our recent research on a nonlocal reaction-diffusion model of bluetongue disease with seasonality, spatial heterogeneous structure, and periodic extrinsic incubation period. 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 may be underestimated if the spatial heterogeneity is ignored. This talk is based on a joint work with Dr. Fuxiang Li.