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 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-free periodic solution is globally attractive if R0<1; while t he system admits a positive periodic solution and the disease is uniformly persistent if R0>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 R0.
In this talk, we consider a two-species Lotka-Volterra competition system with nonlocal dispersal. For the strong-weak competition case, we provide another invasion way of the stronger species to the weaker one by proving the existence of invasion entire solutions, which behave as two monotone waves with different speeds and coming from both sides of the x-axis. For the strong competition case, we study the existence of bistable traveling waves and their asymptotic behavior at infinity. Furthermore, we prove the monotonicity of the wave profile and the uniqueness of wave speed via the strong comparison principle and the sliding method. If time permits, we will also discuss some spreading properties.
An almost periodic Ross-Macdonald model with age structure for the vector population in a patchy environment is considered. It is proved that the disease persists when the basic reproduction ratio R0 > 1, and the disease will die out when R0 < 1. Numerical simulations imply that the biting rates affect the spread of the disease, and the migration may be helpful to the extinction of the disease or result in the persistence of the disease. Accordingly, we give the condition which determines whether it is needed to control migration and find the threshold of the biting rate. It is also shown that the longer the maturity of the vector is, the more conducive it is to the disease control. The threshold of inducing the outbreak of the disease is obtained by numerical results. Finally, the comparison between the almost periodic and periodic situations implies that the periodic case may overestimate or underestimate the risk of disease.
In this talk, a novel stage-structured single population model with state-dependent maturity delay is formulated and analyzed. The delay is related to the size of population and taken as a non-decreasing differential bounded function. The model is quite different from previous state-dependent delay models in the sense that a correction term, $1-\tau'(z(t))z'(t)$, is included in the maturity rate. Firstly, positivity and boundedness of solutions are proved without additional conditions. Secondly, existence of all equilibria and uniqueness of a positive equilibrium are discussed. Thirdly, local stabilities of the equilibria are obtained. Finally, permanence of the system is analyzed, and explicit bounds for the eventual behaviors of the immature and mature populations are established.
In this talk, we discuss a diffusive mutualistic model with advection and different free boundaries in one dimensional space. These two free boundaries may intersect each other as time evolves and can be used to describe the spreading of invasive and native species directly. We first prove the existence and uniqueness, regularity and uniform estimates of global solutions. Then provide the criteria governing spreading and vanishing. At last we investigate long time behaviors and asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries.
Bluetongue is a midge-borne disease that is transmitted by biting midges of the Culicoides family. In this paper, we propose a bluetongue model with seasonality and temperature-dependent incubation period. We introduce the basic disease reproduction ratio for the whole system and the basic disease reproduction ratio in the absence of sheep, and then establish the threshold type results on the global dynamics in terms of these two ratios. Numerically, we study the bluetongue transmission case in France, and perform some sensitivity analysis of the basic disease reproduction ratio. Our numerical simulations are carried out to illustrate the analytic results.
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 (EIP) and intrinsic incubation period (IIP), 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 R0 for this model and then establish a threshold type result on its global dynamics in terms of R0 In the case where all the coefficients are constants, we also prove the global attractivity of the positive constant steady state when R0>1. Numerically, we study the malaria transmission in Maputo Province, Mozambique.
River ecologists are interested in the drift paradox problem, which asks how stream-dwelling organisms can persist, without being washed out, when they are continuously subject to the unidirectional water flow. Water managers are tasked with meeting water needs for industry, agriculture, and urban consumption while mitigating ecosystem impacts. We develop mathematical models that couple hydraulic flow to population growth and dispersal, and establish theories of persistence metrics, critical domain size, and spreading speeds to describe population persistence and spread in a river environment. We then apply the theories as well as numerical calculations to analyze how biological, physical and hydrological factors affect persistence and spread of single and multiple populations in temporally and/or spatially varying river environments. Our work also provides a mathematical basis for water management.