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In this paper, we propose a time-periodic reaction–diffusion model which incorporates seasonality, spatial heterogeneity and the extrinsic incubation period (EIP) of the parasite. The basic reproduction number R0 is derived, and it is shown that the disease-free periodic solution is globally attractive if R0<1, while there is an endemic periodic solution and the disease is uniformly persistent if R0>1. Numerical simulations indicate that prolonging the EIP may be helpful in the disease control, while spatial heterogeneity of the disease transmission coefficient may increase the disease burden.

The theory of the principal eigenvalue is developed for an elliptic eigenvalue problem associated with a linear parabolic cooperative system with some zero diffusion coefficients. Then the basic reproduction number and its computation formulae are established for reaction-diffusion epidemic models with compartmental structure. These theoretical results are applied to a spatial model of rabies to study the influence of spatial heterogeneity and population mobility on disease transmission. [PUBLICATION ABSTRACT]

This paper is concerned with the spatial dynamics of a nonlocal dispersal population model in a shifting environment where the favorable region is shrinking. It is shown that the species becomes extinct in the habitat if the speed of the shifting habitat edge c > c ∗ ( ∞ ) , while the species persists and spreads along the shifting habitat at an asymptotic speed c ∗ ( ∞ ) if c < c ∗ ( ∞ ) , where c ∗ ( ∞ ) is determined by the nonlocal dispersal kernel, diffusion rate and the maximum linearized growth rate. Moreover, we demonstrate that for any given speed of the shifting habitat edge, the model system admits a nondecreasing traveling wave with the wave speed at which the habitat is shifting, which indicates that the extinction wave phenomenon does happen in such a shifting environment.

A vector-borne disease is caused by a range of pathogens and transmitted to hosts through vectors. To investigate the multiple effects of the spatial heterogeneity, the temperature sensitivity of extrinsic incubation period and intrinsic incubation period, and the seasonality on disease transmission, we propose a nonlocal reaction–diffusion model of vector-borne disease with periodic delays. We introduce the basic reproduction number R 0 for this model and then establish a threshold-type result on its global dynamics in terms of R 0 . In the case where all the coefficients are constants, we also prove the global attractivity of the positive constant steady state when R 0 > 1 . Numerically, we study the malaria transmission in Maputo Province, Mozambique.

There is a growing body of biological investigations to understand impacts of seasonally changing environmental conditions on population dynamics in various research fields such as single population growth and disease transmission. On the other side, understanding the population dynamics subject to seasonally changing weather conditions plays a fundamental role in predicting the trends of population patterns and disease transmission risks under the scenarios of climate change. With the host–macroparasite interaction as a motivating example, we propose a synthesized approach for investigating the population dynamics subject to seasonal environmental variations from theoretical point of view, where the model development, basic reproduction ratio formulation and computation, and rigorous mathematical analysis are involved. The resultant model with periodic delay presents a novel term related to the rate of change of the developmental duration, bringing new challenges to dynamics analysis. By investigating a periodic semiflow on a suitably chosen phase space, the global dynamics of a threshold type is established: all solutions either go to zero when basic reproduction ratio is less than one, or stabilize at a positive periodic state when the reproduction ratio is greater than one. The synthesized approach developed here is applicable to broader contexts of investigating biological systems with seasonal developmental durations.

A nonlocal and time-delayed reaction-diffusion model of dengue fever is first proposed that incorporates the aquatic stage, the winged stage, and the incubation periods of the dengue virus within mosquitos and hosts. Then the basic reproduction number R₀ is established for the model system, and an explicit formula of R₀ is obtained in the case of spatially homogeneous infections. It is shown that this R₀ gives the threshold dynamics in the sense that the disease-free equilibrium is asymptotically stable if R₀ < 1 and the disease is uniformly persistent if R₀ > 1. The influences of diffusion coefficients, time delays, and infection heterogeneity on the spread of the disease are also studied via numerical simulations. It turns out that the infection risk may be underestimated if the spatially averaged parameters are used to compute the basic reproduction number for spatially heterogeneous infections.

Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes becomes very important for predicting and controlling disease transmission risks. In order to investigate the impact of time-dependent delays on disease control, we propose an SEIRS epidemic model with a periodic latent period. We introduce the basic reproduction ratio \\[R_0\\] for this model and establish a threshold type result on its global dynamics in terms of \\[R_0\\]. More precisely, we show that the disease-free periodic solution is globally attractive if \\[R_0<1\\]; while the system admits a positive periodic solution and the disease is uniformly persistent if \\[R_0>1\\]. Numerical simulations are also carried out to illustrate the analytic results. In addition, we find that the use of the temporal average of the periodic delay may underestimate or overestimate the real value of \\[R_0\\].

Much work has focused on the basic reproduction ratio R0 for a variety of compartmental population models, but the theory of R0 remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of R0 for a class of such impulsive models. We first introduce R0 and show that it is a threshold parameter for the stability of the zero solution of an associated linear system. Then we apply this theory to a time-delayed computer virus model with impulse treatment and obtain a threshold result on its global dynamics in terms of R0. Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.

Synchronized maturation has been extensively studied in biological science on its evolutionary advantages. This paper is devoted to the study of the spatial dynamics of species growth with annually synchronous emergence of adults by formulating an impulsive reaction–diffusion model. With the aid of the discrete-time semiflow generated by the 1-year solution map, we establish the existence of the spreading speed and traveling waves for the model on an unbounded spatial domain. It turns out that the spreading speed coincides with the minimal speed of traveling waves, regardless of the monotonicity of the birth rate function. We also investigate the model on a bounded domain with a lethal exterior to determine the critical domain size to reserve species persistence. 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. In particular, the relationship between the spreading speed and the emergence maturation delay is found to be counterintuitively variable.

Seasonal change has played a critical role in the evolution dynamics of West Nile virus transmission. In this paper, we formulate and analyze a novel delay differential equation model, which incorporates seasonality, the vertical transmission of the virus, the temperature-dependent maturation delay and the temperature-dependent extrinsic incubation period in mosquitoes. We first introduce the basic reproduction ratio R 0 for this model and then show that the disease is uniformly persistent if R 0 > 1 . It is also shown that the disease-free periodic solution is attractive if R 0 < 1 , provided that there is only a small invasion. In the case where all coefficients are constants and the disease-induced death rate of birds is zero, we establish a threshold result on the global attractivity in terms of R 0 . Numerically, we study the West Nile virus transmission in Orange County, California, USA.