Zhien Ma

Four predator prey models with infectious diseases

Four modifications of a predator prey model to include an SIS or SIR parasitic infection are developed and analyzed. Thresholds are identified and global stability results are proved. When the disease persists in the prey population and the predators have a sufficient feeding efficiency to survive, the disease also persists in the predator population.

Global dynamics of an SEIR epidemic model with saturating contact rate.

Heesterbeek and Metz [J. Math. Biol. 31 (1993) 529] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number R0 is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If R0 < or =1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium is also analyzed.

A discrete epidemic model for SARS transmission and control in China

Abstract Severe acute respiratory syndrome (SARS) is a rapidly spreading infectious disease which was transmitted in late 2002 and early 2003 to more than 28 countries through the medium of international travel. The evolution and spread of SARS has resulted in an international effort coordinated by the World Health Organization (WHO). We have formulated a discrete mathematical model to investigate the transmission of SARS and determined the basic reproductive number for this model to use as a threshold to determine the asymptotic behavior of the model. The dependence of the basic reproductive number on epidemic parameters has been studied. The parameters of the model have been estimated on the basis of statistical data and numerical simulations have been carried out to describe the transmission process for SARS in China. The simulation results matches the statistical data well and indicate that early quarantine and a high quarantine rate are crucial to the control of SARS.

Periodic solutions for delay differential equations model of plankton allelopathy

Abstract In this paper, we propose a modified delay differential equation model of the growth of two species of plankton having competitive and allelopathic effects on each other. By using the continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are obtained for the existence of positive periodic solutions for this model.

Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity

In this paper, a delayed SIRS epidemic model with saturation incidence and temporary immunity is investigated. The immunity gained by experiencing a disease is temporary, whenever infected the diseased individuals will return to the susceptible class after a fixed period. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.

A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China

Abstract We propose a compartmental model BloComp(2,7) that mimics the SARS control strategies implemented by the Chinese government after the middle of April 2003: the division of the whole population into two parallel blocks corresponding to the so-called free environment and the isolated environment and the partition of these blocks further into the compartments of susceptible, exposed, infective, possible, diagnosed, removed and the health care workers. We introduce a novel approach to calculate the transfer rate from the free environment to the isolated environment, and we incorporate into the model the fact that many individuals were misdiagnosed as SARS suspected and hence were mistakenly put in the isolated environment due to lack of fast and effective SARS diagnostic tests. We develop some methods for the parameter identification using the daily reported data from the Ministry of Health of China. Simulations based on these parameters agree with the accural data well, thus provide additional validation of the model. We then vary some parameters to assess the effectiveness of different control measures: these new parameters correspond to the situation when the quarantine measures in the free-environment were prematurely relaxed (we thus observe the second outbreak with the maximal number of daily SARS patients much higher than the first outbreak) or when the quarantine time of SARS patients is postponed (we observe delayed peak time but with much higher number of SARS patients at the peak). We also calculate the basic reproductive number and the basic adequate contact rate.

GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA

An SIS epidemic model with a simple vaccination is investigated in this article. The efficiency of vaccine, the disease-related death rate and population dynamics are also considered in this model. The authors find two threshold Ro and Rc (Rc may not exist). There is a unique endemic equilibrium for Ro > 1 or Rc = Ro; there are two endemic equilibria for Rc < Ro < 1; and there is no endemic equilibrium for Ro < Rc < 1. When Rc exists, there is a backward bifurcation from the disease-free equilibrium for Ro = 1. They analyze the stability of equilibria and obtain the globally dynamic behaviors of the model. The results acquired in this article show that an accurate estimation of the efficiency of vaccine is necessary to prevent and controll the spread of disease.

The effect of stage-structure on the permanence of a predator–prey system with time delay

Abstract A delayed predator–prey system with stage-structure for the prey and time delay due to the gestation of the predator is investigated. By using an iteration technique and comparison arguments respectively, sufficient conditions are derived for the global stability of the positive equilibrium and the two boundary equilibria of the proposed model. Numerical simulations are presented to illustrate the main results.

GLOBAL DYNAMICS OF AN SEIR EPIDEMIC MODEL WITH IMMIGRATION OF DIFFERENT COMPARTMENTS

Abstract The SEIR epidemic model studied here includes constant inflows of new susceptibles, exposeds, infectives, and recovereds. This model also incorporates a population size dependent contact rate and a disease-related death. As the infected fraction cannot be eliminated from the population, this kind of model has only the unique endemic equilibrium that is globally asymptotically stable. Under the special case where the new members of immigration are all susceptible, the model considered here shows a threshold phenomenon and a sharp threshold has been obtained. In order to prove the global asymptotical stability of the endemic equilibrium, the authors introduce the change of variable, which can reduce our four-dimensional system to a three-dimensional asymptotical autonomous system with limit equation.

Global stability of a reaction-diffusion predator-prey model with a nonlocal delay

In this paper, a reaction-diffusion predator-prey system with nonlocal delay due to the gestation of the predator and homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each boundary steady state is established. The existence of Hopf bifurcations at the positive steady state is also discussed. Sufficient conditions are derived for the global stability of the positive steady state and the semi-trivial steady state of the proposed problem by using the method of upper-lower solutions and its associated monotone iteration scheme, respectively. Numerical simulations are carried out to illustrate the main results.