乙肝病毒感染动力学模型及其预防控制策略研究
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摘要
据世界卫生组织报道,乙型肝炎(乙肝)是一种世界性的疾病。全球有超过20亿的人曾感染过乙肝病毒,慢性乙肝表面抗原携带者多达3.7亿。由于对乙型肝炎的研究采取动物实验形式成本很高,因此,对它的发病机理,流行规律,预测预报就更多的需要理论分析,定量分析,模拟仿真来进行,而上述分析都离不开数学模型。乙型肝炎动力学就是针对乙肝的流行规律进行理论性定量研究的一种重要方法。
本文研究了乙型肝炎动力学模型的建立以及控制乙型肝炎策略两大问题。将脉冲微分方程理论应用到乙型肝炎动力学中,根据疾病发生和在人群中传播的规律建立了具有乙肝动力学特性的数学模型,分析了模型的动力学性质。得到的结论可用于预测其变化发展趋势,找到了对其预防和控制的策略,为制定防治决策提供了理论依据。所做的主要工作有：
乙型肝炎的脉冲预防接种问题的研究。建立了考虑脉冲预防接种的SAIR乙肝模型,分别得到了乙肝最终消失或成为地方病的充分条件。根据乙肝病毒的传播方式以及各种状态间的转化模式,各状态转化所需的时间延迟,并对其采取脉冲预防接种等控制措施,建立了一类具有时滞和脉冲预防接种的SI1I2I3R乙肝模型,分别得到了乙肝最终消失或成为地方病的充分条件。
乙肝病毒和药物及免疫因子相互作用问题的研究。考虑了病毒扩散和肝细胞感染病毒所需的时滞,建立了一类在有限区域里扩散的HBV模型,对时滞和扩散情形的影响进行了模拟验证。得到的结论是,时滞和扩散不影响乙型肝炎的发展。在仓室模型中,分布时滞比离散时滞更具有现实意义。将分布时滞应用于乙型肝炎动力学中,得到了无病平衡点的全局渐近稳定(HBV将消失)和不稳定(HBV持续存在)的充分条件。最后,研究了周期性输入免疫因子在治疗HBV感染时的效果。
乙肝病毒和药物同时扩散问题的研究。基于Capasso和Maddalena提出的病毒感染的反应扩散系统,建立了乙肝病毒和药物相互作用的数学模型,得到了该反应扩散方程行波解存在的条件。在此条件下,方程存在一个乙肝病毒从存在到消失的行波解,得到了乙肝病毒在肝脏内持续存在的条件。
药物脉冲作用下的HBV和免疫因子竞争模型研究。将种群竞争的Lotka-Volterra系统应用到解释病毒和免疫因子的相互作用中,提出的控制措施为：周期脉冲注入药物,达到周期抑制病毒复制增殖的速率,适当缩短脉冲周期,可控制乙型肝炎的发展。考虑到肝脏面积是有限的,将logistic函数应用为健康肝细胞的增长率,建立了一类乙肝模型。分析和模拟结果表明,输入免疫因子可以使得该平衡点稳定,并且可以预测HBV感染的各种临床表现,及时有效的CTL反应对控制乙型肝炎感染是非常关键的。目前还没有文献考虑免疫占优和疫苗在防治HBV感染过程中发挥的作用。最后讨论了具有表面抗原变异的HBV感染的疫苗接种效果,考虑了注射疫苗前后病毒特异性CTL反应和HBV表面抗原变异的影响,得到了注射疫苗前后HBV被从体内彻底清除的完全恢复平衡点的稳定条件。不同组参数模拟表明注射疫苗是一种防治HBV感染的有效的方法。
本文所研究的问题是动力学控制理论在乙型肝炎上应用的重要问题,具有很大的研究价值,属于该领域的前沿问题。文中所用的方法和得到的结果对乙肝动力学模型的研究和控制乙型肝炎病毒感染方面有重要的理论和实用价值。
According to WHO's report, hepatitis B is a worldwide disease. Two billion people have been affected by hepatitis B virus and the number of chronic hepatitis B virus car-riers is up to 0.37 billion in the world. Because animal experimental cost for hepatitis B research is very high, theoretical analysis, quantitative analysis and simulation of mathemat-ical model should be applied to predict development of hepatitis B. Hepatitis B dynamics is an important theoretical analysis method to study hepatitis B development.
In this paper, the establishment of HBV dynamical model and the control strategy of HBV are studied. Applying impulsive differential equation theorem to HBV dynamics, HBV model is established based on HBV characteristic and its dynamical characteristic is also analyzed. The research result can be used to predict development tendency of HBV infection. There are five chapters:Two types of hepatitis B models are established and studied in Chapter 2. Three types of hepatitis B model (spreading, time-delay, periodic inputting immune) are studied in Chapter 3.In Chapter four, HBV model involving time delay and diffusion phenomena is discussed. In the fifth chapter, the role of immune in HBV dynamics is discussed. The main work is as follows:
Impulsive vaccination of hepatitis B is studied. The SAIR model with impulsive vac-cination is constructed. The sufficient conditions under which HBV would be eliminated ultimately or become endemic are derived. According to the propagation mode and the transformation mode between HBV infection states and the transformation delays, an HBV infection model with impulsive vaccination is established and analyzed. The sufficient con-ditions that hepatitis B virus will be eliminated eventually or be persistent are derived.
The mutual effects of HBV, medicine and immune factors are studied. The HBV in-fection model considers diffusion within a finite domain and the time delay of cell infection and effect of medication. The effects of time delay and diffusion are validated by computer simulations. The research result is that the time delay and diffusion cannot affect HBV de-velopment. Usually only two time delays were considered in literature:disperse delays and continuous distributed delays. In the SIR model, continuous distributed delay is more inter-esting than disperse delay. Distributed delay is applied to HBV dynamics. The reciprocity model considering the mutual effects between uninfected cell, HBV, infected cell and im-munological factors with continuous distributed delays is established. The conditions for the disappearance or persistence of HBV, i.e., the global asymptotic stability or non-stability of the equilibrium points without disease, are derived. The effect of periodic input immune factor in HBV infection is also studied.
The concurrent diffusion of HBV and medicine diffusion is studied. Based on the reaction diffusion system provided by Capasso and Maddalena, a delayed reaction-diffusion model is established to describe HBV infection and control. The sufficient conditions for the existence of traveling wave solutions of reaction-diffusion systems with delay are derived. The travelling wave front, derived here, corresponds to medicine injection that drives HBV to extinction. The parameter identification of interaction system between HBV and medicine with reaction-diffusion phenomenon is investigated.
Under the condition of impulsive input medicine, the competition model between HBV and immune factor is studied. Population competitive Lotka-Volterra system is applied to explain the mutual effect of virus and immune factor. The prevention and control strategy derived is that periodic input medicine, shortening impulsive period can control HBV de-velopment. With limited liver area, logistic function is applied to formulate growth rate of healthy liver cell. Simulation shows that the input immune factor makes it stable. Differ-ent sets of parameters are used in the simulation, and the results show that the model can predict various clinical appearance of HBV infection. The simulation results suggest that a timely and vigorous CTL response is required in the treatment of HBV infection. No one considered the possible role of immune-dominance and vaccine in preventing and treating HBV infection in existing documents. In this part, we focus on aspects of the virus-specific cellular immune response, immune-dominance and influence of hepatitis B surface variant infection in vivo after vaccination. The stability conditions of the complete recovery equi-librium points at which HBV will be eliminated entirely from the body before and after vaccination are discussed. A different set of parameters is used in the simulation, and the results show that vaccination is an efficient way in preventing and treating HBV infection.
As an application of dynamical control theorem to hepatitis B, the problem discussed here is of great importance. The results derived and methods applied provide important theoretical and practical value both in the study of HBV dynamical model and in the control of hepatitis B virus infection.
本文研究了乙型肝炎动力学模型的建立以及控制乙型肝炎策略两大问题。将脉冲微分方程理论应用到乙型肝炎动力学中,根据疾病发生和在人群中传播的规律建立了具有乙肝动力学特性的数学模型,分析了模型的动力学性质。得到的结论可用于预测其变化发展趋势,找到了对其预防和控制的策略,为制定防治决策提供了理论依据。所做的主要工作有：
乙型肝炎的脉冲预防接种问题的研究。建立了考虑脉冲预防接种的SAIR乙肝模型,分别得到了乙肝最终消失或成为地方病的充分条件。根据乙肝病毒的传播方式以及各种状态间的转化模式,各状态转化所需的时间延迟,并对其采取脉冲预防接种等控制措施,建立了一类具有时滞和脉冲预防接种的SI1I2I3R乙肝模型,分别得到了乙肝最终消失或成为地方病的充分条件。
乙肝病毒和药物及免疫因子相互作用问题的研究。考虑了病毒扩散和肝细胞感染病毒所需的时滞,建立了一类在有限区域里扩散的HBV模型,对时滞和扩散情形的影响进行了模拟验证。得到的结论是,时滞和扩散不影响乙型肝炎的发展。在仓室模型中,分布时滞比离散时滞更具有现实意义。将分布时滞应用于乙型肝炎动力学中,得到了无病平衡点的全局渐近稳定(HBV将消失)和不稳定(HBV持续存在)的充分条件。最后,研究了周期性输入免疫因子在治疗HBV感染时的效果。
乙肝病毒和药物同时扩散问题的研究。基于Capasso和Maddalena提出的病毒感染的反应扩散系统,建立了乙肝病毒和药物相互作用的数学模型,得到了该反应扩散方程行波解存在的条件。在此条件下,方程存在一个乙肝病毒从存在到消失的行波解,得到了乙肝病毒在肝脏内持续存在的条件。
药物脉冲作用下的HBV和免疫因子竞争模型研究。将种群竞争的Lotka-Volterra系统应用到解释病毒和免疫因子的相互作用中,提出的控制措施为：周期脉冲注入药物,达到周期抑制病毒复制增殖的速率,适当缩短脉冲周期,可控制乙型肝炎的发展。考虑到肝脏面积是有限的,将logistic函数应用为健康肝细胞的增长率,建立了一类乙肝模型。分析和模拟结果表明,输入免疫因子可以使得该平衡点稳定,并且可以预测HBV感染的各种临床表现,及时有效的CTL反应对控制乙型肝炎感染是非常关键的。目前还没有文献考虑免疫占优和疫苗在防治HBV感染过程中发挥的作用。最后讨论了具有表面抗原变异的HBV感染的疫苗接种效果,考虑了注射疫苗前后病毒特异性CTL反应和HBV表面抗原变异的影响,得到了注射疫苗前后HBV被从体内彻底清除的完全恢复平衡点的稳定条件。不同组参数模拟表明注射疫苗是一种防治HBV感染的有效的方法。
本文所研究的问题是动力学控制理论在乙型肝炎上应用的重要问题,具有很大的研究价值,属于该领域的前沿问题。文中所用的方法和得到的结果对乙肝动力学模型的研究和控制乙型肝炎病毒感染方面有重要的理论和实用价值。
According to WHO's report, hepatitis B is a worldwide disease. Two billion people have been affected by hepatitis B virus and the number of chronic hepatitis B virus car-riers is up to 0.37 billion in the world. Because animal experimental cost for hepatitis B research is very high, theoretical analysis, quantitative analysis and simulation of mathemat-ical model should be applied to predict development of hepatitis B. Hepatitis B dynamics is an important theoretical analysis method to study hepatitis B development.
In this paper, the establishment of HBV dynamical model and the control strategy of HBV are studied. Applying impulsive differential equation theorem to HBV dynamics, HBV model is established based on HBV characteristic and its dynamical characteristic is also analyzed. The research result can be used to predict development tendency of HBV infection. There are five chapters:Two types of hepatitis B models are established and studied in Chapter 2. Three types of hepatitis B model (spreading, time-delay, periodic inputting immune) are studied in Chapter 3.In Chapter four, HBV model involving time delay and diffusion phenomena is discussed. In the fifth chapter, the role of immune in HBV dynamics is discussed. The main work is as follows:
Impulsive vaccination of hepatitis B is studied. The SAIR model with impulsive vac-cination is constructed. The sufficient conditions under which HBV would be eliminated ultimately or become endemic are derived. According to the propagation mode and the transformation mode between HBV infection states and the transformation delays, an HBV infection model with impulsive vaccination is established and analyzed. The sufficient con-ditions that hepatitis B virus will be eliminated eventually or be persistent are derived.
The mutual effects of HBV, medicine and immune factors are studied. The HBV in-fection model considers diffusion within a finite domain and the time delay of cell infection and effect of medication. The effects of time delay and diffusion are validated by computer simulations. The research result is that the time delay and diffusion cannot affect HBV de-velopment. Usually only two time delays were considered in literature:disperse delays and continuous distributed delays. In the SIR model, continuous distributed delay is more inter-esting than disperse delay. Distributed delay is applied to HBV dynamics. The reciprocity model considering the mutual effects between uninfected cell, HBV, infected cell and im-munological factors with continuous distributed delays is established. The conditions for the disappearance or persistence of HBV, i.e., the global asymptotic stability or non-stability of the equilibrium points without disease, are derived. The effect of periodic input immune factor in HBV infection is also studied.
The concurrent diffusion of HBV and medicine diffusion is studied. Based on the reaction diffusion system provided by Capasso and Maddalena, a delayed reaction-diffusion model is established to describe HBV infection and control. The sufficient conditions for the existence of traveling wave solutions of reaction-diffusion systems with delay are derived. The travelling wave front, derived here, corresponds to medicine injection that drives HBV to extinction. The parameter identification of interaction system between HBV and medicine with reaction-diffusion phenomenon is investigated.
Under the condition of impulsive input medicine, the competition model between HBV and immune factor is studied. Population competitive Lotka-Volterra system is applied to explain the mutual effect of virus and immune factor. The prevention and control strategy derived is that periodic input medicine, shortening impulsive period can control HBV de-velopment. With limited liver area, logistic function is applied to formulate growth rate of healthy liver cell. Simulation shows that the input immune factor makes it stable. Differ-ent sets of parameters are used in the simulation, and the results show that the model can predict various clinical appearance of HBV infection. The simulation results suggest that a timely and vigorous CTL response is required in the treatment of HBV infection. No one considered the possible role of immune-dominance and vaccine in preventing and treating HBV infection in existing documents. In this part, we focus on aspects of the virus-specific cellular immune response, immune-dominance and influence of hepatitis B surface variant infection in vivo after vaccination. The stability conditions of the complete recovery equi-librium points at which HBV will be eliminated entirely from the body before and after vaccination are discussed. A different set of parameters is used in the simulation, and the results show that vaccination is an efficient way in preventing and treating HBV infection.
As an application of dynamical control theorem to hepatitis B, the problem discussed here is of great importance. The results derived and methods applied provide important theoretical and practical value both in the study of HBV dynamical model and in the control of hepatitis B virus infection.
引文
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