Weinian Zhang

Generalization of a retarded Gronwall-like inequality and its applications

Abstract This paper generalizes a Lipovan’s result of Gronwall-like inequalities [J. Math. Anal. Appl. 252 (2000) 389] to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. From our result Bihari’s result and Pinto’s result can be deduced as some special cases. Our result can be applied to give the global existence and estimate solutions. We also apply it to improve an estimation for almost periodicity of an invariant manifold.

Modeling the transmission dynamics and control of hepatitis B virus in China

Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV) and is a major global health problem. HBV is the most common serious viral infection and a leading cause of death in mainland China. Around 130 million people in China are carriers of HBV, almost a third of the people infected with HBV worldwide and about 10% of the general population in the country; among them 30 million are chronically infected. Every year, 300,000 people die from HBV-related diseases in China, accounting for 40-50% of HBV-related deaths worldwide. Despite an effective vaccination program for newborn babies since the 1990s, which has reduced chronic HBV infection in children, the incidence of hepatitis B is still increasing in China. We propose a mathematical model to understand the transmission dynamics and prevalence of HBV infection in China. Based on the data reported by the Ministry of Health of China, the model provides an approximate estimate of the basic reproduction number R(0)=2.406. This indicates that hepatitis B is endemic in China and is approaching its equilibrium with the current immunization program and control measures. Although China made a great progress in increasing coverage among infants with hepatitis B vaccine, it has a long and hard battle to fight in order to significantly reduce the incidence and eventually eradicate the virus.

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135–163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov–Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, one, or two limit cycles. In this paper, we investigate the coexistence of a limit cycle and a homoclinic loop in this model. One of the difficulties is to determine the multiplicity of the weak focus. We first prove that the maximal multiplicity of the weak focus is 2. Then feasible conditions are given for the uniqueness of limit cycles. The coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate Hopf bifurcations and degenerate Bogdanov–Takens bifurcations of limit cycles and homoclinic loops of order 2.

An Age-Structured Model for the Transmission Dynamics of Hepatitis B

Hepatitis B virus (HBV) infection is endemic in many parts of the world. One of the characteristics of HBV transmission is the age structure of the host population. In this paper, we propose an age-structured model for the transmission dynamics of HBV. The host population is stratified by age and is divided into six subclasses: susceptible, latently infected, acutely infectious, carrier, recovered, and vaccinated individuals. By determining the basic reproduction number, we study the existence and stability of the disease-free and endemic steady state solutions of the model. Numerical simulations are performed to find optimal strategies for controlling the transmission of HBV.

Mathematical modelling and control of Schistosomiasis in Hubei Province, China

Hubei Province, along with four other provinces in the central and eastern China where schistosomiasis is endemic (Anhui, Hunan, Jiangsu, and Jiangxi), is located in the lake and marshland regions along the Yangtze River. High population density, large numbers of farm cattle, and huge areas of snail habitat are the main characteristics that maintain the persistence of the disease and the transmission of the parasite Schistosoma japonicum in these regions. Based on the schistosomiasis infection data from Hubei province, we propose a mathematical model for the human-cattle-snail transmission of schistosomiasis. The model is a system consisting of six ordinary differential equations that describe susceptible and infected human, cattle and snail subpopulations. After analyzing the existence of the disease-free equilibrium of the model, we determine the basic reproduction number and use the model to simulate the schistosomiasis infection data from Hubei Province. By carrying out sensitivity analyses of the basic reproduction number on various parameters, we find that the transmission of S. japonicum between cattle and snails plays a more important role than that between humans and snails in the endemicity of schistosomiasis in these regions. This strongly suggests that, to control and eventually eradicate schistosomiasis in the lake and marshland regions in China, a more comprehensive approach needs to include environmental factors in order to break the cattle-snail transmission cycle.

Analytic Solutions of an Iterative Functional Differential Equation which may Violate the Diophantine Condition

Existence of analytic solutions of an iterative functional differential equation is studied by reducing locally the equation to another functional differential equation without iteration of the unknown function. As done previously we first discuss the case that the constant f (0) is not on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Furthermore, we investigate the case that the constant is a unit root, which violates the Diophantine condition.

Decomposition of algebraic sets and applications to weak centers of cubic systems

There are many methods such as Grobner basis, characteristic set and resultant, in computing an algebraic set of a system of multivariate polynomials. The common difficulties come from the complexity of computation, singularity of the corresponding matrices and some unnecessary factors in successive computation. In this paper, we decompose algebraic sets, stratum by stratum, into a union of constructible sets with Sylvester resultants, so as to simplify the procedure of elimination. Applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center, we determine the order of weak centers and discuss the bifurcation of critical periods.

Linearizability and local bifurcation of critical periods in a cubic Kolmogorov system

Since Chicone and Jacobs investigated local bifurcation of critical periods for quadratic systems and Newtonian systems in 1989, great attention has been paid to some particular forms of cubic systems having special practical significance but less difficulties in computation. This paper is devoted to the linearizability and local bifurcation of critical periods for a cubic Kolmogorov system. We use the Darboux method to give explicit linearizing transformations for isochronous centers. Investigating the finite generation for the ideal of all period constants, which are of the polynomial form in six parameters, we prove that at most two critical periods can be bifurcated from the interior equilibrium if it is an isochronous center. Moreover, we prove that the maximum number of critical periods is reachable.

Conditions of infinity to be an isochronous centre for a rational differential system

In this article we find conditions under which infinity is a centre or an isochronous centre for a rational differential system. By a transformation, infinity is taken to the origin and therefore properties at infinity can be studied by using the methods developed for finite critical points. Using the computer algebra system-Mathematica we compute the singular point values and the period constants at the origin for the transformed system and then get the necessary conditions of the problems. Finally, we prove that the conditions are sufficient.

Also set-valued functions do not like iterative roots

In [1] Babbage studied (1.1) for f being the identity mapping. After him a lot of results concerning the general case of (1.1) in various settings have been proved. Many of them can be found in the monographs [12] and [13] by M. Kuczma and M. Kuczma, B. Choczewski, R. Ger, respectively, as well as in the book [19] by Gy. Targonski. Some recent results have been presented in the survey papers [3] and [2].