In this paper a HBV infection model with impulsive vaccination is considered. By using fixed point theorem and stroboscopic map we prove the existence of disease-free T-periodic solution. Also by comparative theorem of impulsive differential equation we get the global asymptotic stability of the disease-free periodic solution and permanence of the disease. Numerical simulations show the influence of parameters on the dynamics of HBV, which provided references for seeking optimal measures to control the transmission of HBV.
Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV) and is a major global health problem. According to the data of World Health Organization (WHO), more than 2,000 million people have been infected with HBV and about 350 million remain infected chronically. Every year there are over 4 million acute clinical cases of HBV and about 25% of carriers. Hepatitis B causes about 1 million people to die from chronic active hepatitis, cirrhosis, or primary liver cancer annually [
The transmission of HBV occurs normally on contact with infected blood or body fluids. In high prevalence populations, transmission is largely vertical, that is, through mother to child during delivery or horizontal through household contact as skin breaches, open sores, or scratches in the early years of life. In contrast, HBV transmission in low endemicity populations typically occurs in adults via parenteral exposures and intravenous drug use or through sexual contact [
Vaccination is an effective control measures for HBV infection, an universal vaccination programme promoted in more than 170 countries since 1982 [
Mathematical models have been used frequently to study the transmission dynamics of HBV, and qualitative results on such models can be found. Anderson and May first used a simple mathematical model to illustrate the effects of carriers on the transmission of HBV [
In the above literatures, most models involving HepB vaccine strategy often assumed that the vaccine is completely effective in preventing the infection of vaccinated individuals. In fact, it is well known to all that the HepB vaccine should be taken in three doses at 0, 1, and 6 months. Usually 30–50% of individuals will gain anti-HBs antibody after the first dose, 80–90% will gain after the second dose, and almost all the individuals will have high anti-HBs concentrations one month after the last dose that 99.8% of vaccinees gained anti-HBs antibody [
Motivated by the above consideration, and based on the natural course of HBV infection, we promote a novel model to describe the transition dynamic of HBV. It is more reasonable to consider the impulsive vaccination strategy for the susceptible individuals; there are fewer literatures that researched HBV infection with impulsive vaccination already [
The remaining parts of this paper are organized as follows. In Section
Based on the fact that HepB vaccination is not completely effective, we improve the model of Zou et al. [
Flow diagram of HBV transmission in a population.
The mathematical model of the transmission dynamics and prevalence of HBV is as follows:
Since the equations for the variables
Subsequently, we introduce the following lemma, which is useful for the later proof.
Consider the following impulsive differential equation:
It is easy to obtain the analytical solution of (
In the following we prove the global stability of the period solution, and it suffices to prove the global stability of the fixed point (
Assume that the sequence
Consider the following equation:
Now we will prove the disease-free periodic solution
Denote
Let
Since
When
Theorem
In this section we say the disease is endemic if the infectious population persists above a certain positive level for sufficiently large time. The endemicity of the disease can be well captured and studied through the notion of uniform persistence.
System (
If
Let
Firstly, from the first equation of system (
Since
We claim that for any
We denote
From (
Next we prove that there exists a
Define
First, if
Second
We hope to show that
In this section, we present some numerical simulations to demonstrate the transmission dynamic of HBV. According to the natural history of HBV transmission and prior research [
The effect of vaccine efficacy on the number of (a) susceptible individuals, (b) latently infected, (c) acute infectors, and (d) chronic sufferers. The parameters are
In order to find better control strategies for HBV infection, we would like to see what parameters can affect the change of the acute infectors number. From Figure
The graphs of the number of acute infectors with some parameters: (a)
In Figure
In Figure
Hepatitis B virus is highly prevalent in many countries of the world; we promote a new epidemic model based on the spread characters of the HBV, and consider the fact that the HepB vaccine is incomplete immunization in the vaccination process. Our model is more approach to the realistic problem and different from [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work is supported by the National Natural Science Foundation of China (no. 1124319).