We propose an impulsive integro-differential model to describe an invading species with a birth pulse in the reproductive stage and a nonlocal dispersal stage. We first establish a threshold-type result on the global dynamics for the model system in a bounded domain, and present an application to insect pests outbreak in terms of the critical domain size. For the spatial spread in an unbounded domain, we then prove the existence of the invasion speed and its coincidence with the minimal speed for monotone traveling waves. We find that the spread rate depends on the strength of the birth pulse at low densities, and if the strength is greater than certain value, it may accelerate the invasion speed, otherwise, it may slow down. Numerical simulations are also carried out to illustrate our analytical results.
West Nile virus (WNv) is a mosquito-borne disease caused by flavivirus. To investigate the combined effects of vertical transmission, temperature-dependent incubation periods and seasonality on the transmission of WNv, we develop a delay differential system with stage-structure and time-varying delays. We then derive the mosquito reproduction number $\mathcal{R}_0^V$ and basic reproduction number $\mathcal{R}_0$, and show that these two numbers serve as threshold parameters that determine whether WNv will spread. As an application, we conduct a case study for the WNv transmission in Los Angeles County, California. We also carry out numerical simulations to identify the situations that require time-periodic delays. Moreover, we find that rising temperatures may potentially increase the risk of disease outbreaks.
Seasonal change has played a critical role in the spatiotemporal dynamics of West Nile virus transmission. In this talk, we discuss 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 for mosquitoes. We first introduce the basic reproduction ratio R0 for this model, and then show that the disease is uniformly persistent if R0 > 1. It is also shown that the disease-free periodic solution is attractive if R0 < 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 stability in terms of R0. Numerically, we study the West Nile virus transmission in Orange County, California, United States.
In this talk, we will introduce two HIV/AIDS epidemic models. By the epidemic characteristics of HIV/AIDS in Yunnan Province of China, the population is divided into two groups: injecting drug users(IDUs) and sexual immoral populations(SIP). The conditions and thresholds for the existence of four equilibria are established. In the simulations, parameters are chosen to fit as much as possible prevalence data publicly available for Yunnan. Our results show that controlling the size of high-risk people is a very effective control measure. For better understanding the HIV/AIDS transmission trend in Yunnan province, we improve the above HIV/AIDS model to an HIV/AIDS epidemic model with 12 compartments. The total population is divided into four subgroups: injecting drug users(IDUs), female sex workers (FSWs), clients of FSWs (C) and men who have sex with men (MSM). Due to behavioral change, susceptible people will move into the other susceptible groups. The simulations indicate that Yunnan will have about 140,000 HIV positives, 18,000 AIDS cases unless there are any stronger or more effective control measures by the end of 2015. Sexual transmission is the main mode from 2006-2015. The HIV prevalence rate among men who have sex with men continues to increase more quickly.
In this talk, we will discuss the propagation dynamics of a nonlocal dispersal population model in a time-periodic shifting habitat. We first show that this model admits a periodic forced wave with the speed at which the habitat is shifting by using the monotone iteration method together with a pair of generalized super- and sub-solutions. Then we establish the nonexistence and uniqueness of periodic forced waves by applying the sliding technique and some analytical skills. Furthermore, with the help of the dynamical systems approach, the global exponential stability of the periodic forced waves is obtained. Finally, the spreading properties of solutions are considered.
The theory of the basic reproduction ratio R0 is established for a large class of almost periodic and time-delayed compartmental population models . We first present some dynamical properties for linear almost periodic functional differential systems. By using the productspace and evolution semigroup approach, we then prove that R0-1 has the same sign as the exponential growth bound of an associated linear system. As an application, we apply the developed theory to an almost periodic SEIR model with an incubation period and obtain a threshold result on its global dynamics in terms of R0. Finally, we present some numerical simulations. Numerical simulations indicate that prolonging the length of incubation period is beneficial for the control of the disease. In addition, a simple model shows that the exponential growth bound or basic reproduction ratio may be underestimated or overestimated if an almost periodic coefficient is approximated by a periodic one.
We consider a two species competition model with different free boundaries and seasonal succession. The condition to determine whether the species spatially spreads to infinity or vanishes at a finite space interval is derived, and when the spreading happens, the asymptotic spreading speed of the species is also given. The obtained results reveal the effect of seasonal succession on the dynamical behavior of the spreading of those two species.
We propose a Chikungunya transmission model with time-varying parameters and extrinsic incubation period (EIP). The model includes the larval and mature stages of mosquitoes. We deduce the vector reproduction number Rvand the basic reproduction number R0, and then show that these two threshold values completely determine the global dynamics of the model system. More precisely, (i) if Rv < 1, then mosquito population will become extinct ultimately; (ii) if Rv >1 and R0 <1, then Chikungunya disease will be eliminated; (iii) if Rv >1 and R0 >1, then the disease will persist, which is manifested as seasonal fluctuation. Numerically, we explore the spread of Chikungunya disease in Delhi, India. The analytic results are in good consistence with our numerical simulations. Further, an interesting finding is that if the time-averaged EIP is used, then R0 may be underestimated, and the number of infectious humans and mosquitoes may be underestimated or overestimated.
In this project, we propose a periodic SIS epidemic model with time delay and transport-related infection in a patchy environment. The basic reproduction number R0 is derived which determines the global dynamics of the model system: the disease-free periodic state is globally attractive if R0< 1, while there exists at least one positive periodic state and the disease persists if R0> 1. Numerical simulations are performed to confirm the analytical results and to explore the dependence of R0 on the transport-related infection parameters and the amplitude of fluctuations.