传染性疾病传播机制与控制的系统研究
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摘要
本文根据媒介传染病、疱疹和禽流感等传染病的发生与传播规律,建立相应的系统动力学模型.并分别对其进行系统研究,揭示相应传染病的传播机制与规律,为控制相应传染病的传播提供理论依据.
全文由三个部分构成：第一部分(含第二章,第三章)主要研究媒介传染病和疱疹传染病；第二部分(含第四章,第五章)主要研究具有不同治疗机制的禽流感传染病；第三部分(含第六章,第七章)主要建立和研究隔离和潜伏期具有传染性的年龄结构的传染病模型.全文共分七章,每章主要结果及创新点如下：
第一章绪论部分,简单介绍本文研究问题的背景、理论和实际意义、国内外现状以及本文主要内容和方法.
第二章建立并研究了具有非线性传染率的媒介传染病系统动力学模型,该模型可描述疟疾,登革热和西尼罗病毒等多种依靠媒介传播的疾病.得到了系统存在后向分支的条件,分析了地方病平衡点的稳定性.结果表明非线性传染率使传染病模型的动力学性态更加复杂,基本再生数小于1不足以使疾病灭绝.
第三章建立并研究了具有非线性传染率的疱疹系统,得到了决定疾病流行与否的基本再生数表达式,证明了当基本再生数小于1时,无病平衡点全局渐近稳定,此时疾病消亡.当基本再生数大于1时,地方病平衡点全局渐近稳定,此时疾病发展为地方病.
第四章建立并研究了带有饱和治疗的禽流感动系统力学模型.得到了系统基本再生数的表达式和无病平衡态稳定性的条件.并运用Bendixson-Dulac定理,证明了系统地方病平衡态的全局渐近稳定性.
第五章建立并研究了药物资源或治疗能力有限情况下的禽流感传播模型,假设当患者数量在治疗能力范围内时,治疗率与染病者数量成正比；当患者数量超过治疗能力范围时,为常数治疗.分析了不同情况下平衡点的存在性和稳定性.
第六章建立并研究了具有隔离的年龄结构传染病模型,给出了基本再生数的表达式,证明了无病平衡点的全局稳定性.得到地方病平衡点的局部稳定的条件.
第七章建立并研究了潜伏期具有传染性的年龄结构传染病模型.运用微分方程和积分方程中的理论和方法,得到了基本再生数的表达式,证明了当基本再生数小于1时,无病平衡点是局部和全局渐近稳定的,此时疾病消亡.当基本再生数大于1时,无病平衡点不稳定,此时系统至少存在一个地方病平衡点,并且在一定条件下证明了地方病平衡点的局部渐近稳定性.
In this thesis, several system dynamics models are formulated based on the occurance and transmission patterns of some infectious diseases such as vector-borne diseases, herpes, avian influenza,etc, and are systemically studied. The transmission mechanisms for these diseases are revealed. The results obtained in this thesis can be applied to guide the control of these diseases.
This thesis consists of three parts:the first part (contains Chapter 2 and Chapter 3) mainly concerns the vector-borne diseases and herpes; the second part (contains Chapter 4 and Chapter 5) mainly focus on avian influenza with different treatment rules; the third part (contains Chapter 6 and Chapter 7) formulate and study the age-structured epidemic models with isolation or infectiousness in latent period.
This thesis contains seven chapters. The main results and innovations in this disser-tation can be summarized as the following:
In chapter 1, the background, significance of the theory and practices, research on-goings at Home and Abroad, main contents and methods in this dissertation are briefly introduced.
In chapter 2, a vector-borne systematic epidemic model with nonlinear incidence is formulated and studied. The model can describe the dynamic of many diseases spreading by vectors such as malaria, dengue, West Nile Fever and so on. We get the conditions ensuring that the system exhibits backward bifurcation and analyze the stability of the endemic equilibrium. Our results imply that a nonlinear incidence make the models more complex dynamic behavior. It is not enough to make the disease die out that the basic reproductive number is smaller than 1.
In Chapter 3, an herpes systematic epidemic model with nonlinear incidence rate is formulated and investigated. We get the expression of the basic reproductive number which decide the spread of the diseases. If the basic reproductive number is smaller than 1, the disease-free equilibrium is globally asymptotically stable and the disease dies out. If the basic reproductive number is bigger than 1, the endemic equilibrium is globally asymptotically stable and the disease persists.
In Chapter 4, an avian systematic epidemic model with saturation treatment is es-tablished and studied. We obtain the basic reproductive number of the model and the stability conditions of the infection-free equilibrium. By using the Bendixson-Dulac the-orem, the global asymptotic stability of the endemic equilibrium is proved.
In Chapter 5, we develop a mathematical model for the spread of the avian influenza when drug resources and treatment capacity are limited. Moreover we assume the treat-ment rate is proportional to the number of infections when the capacity of treatment is not reach and otherwise, adopt a constant treatment. We analyze the existence and the stability of equilibria under different conditions.
In Chapter 6, an age-structured epidemic model with isolation is proposed and an-alyzed. We get the expression of the basic reproductive number and proved globally stability results for the disease-free equilibrium. At last we get the locally stability con-ditions of the endemic equilibrium.
Chapter 7, an age-structured HIV/AIDS epidemic model with infectivity in incuba-tive period is formulated and studied. By using the theory and methods of differential and integral equation, the explicit expression of the basic reproductive number was obtained, it is showed that the disease-free equilibrium is locally and globally asymptotically stable if the basic reproductive number is smaller than 1, at least one endemic equilibrium ex-ists if the basic reproductive number is bigger than 1, the stability conditions of endemic equilibrium are also given.
全文由三个部分构成：第一部分(含第二章,第三章)主要研究媒介传染病和疱疹传染病；第二部分(含第四章,第五章)主要研究具有不同治疗机制的禽流感传染病；第三部分(含第六章,第七章)主要建立和研究隔离和潜伏期具有传染性的年龄结构的传染病模型.全文共分七章,每章主要结果及创新点如下：
第一章绪论部分,简单介绍本文研究问题的背景、理论和实际意义、国内外现状以及本文主要内容和方法.
第二章建立并研究了具有非线性传染率的媒介传染病系统动力学模型,该模型可描述疟疾,登革热和西尼罗病毒等多种依靠媒介传播的疾病.得到了系统存在后向分支的条件,分析了地方病平衡点的稳定性.结果表明非线性传染率使传染病模型的动力学性态更加复杂,基本再生数小于1不足以使疾病灭绝.
第三章建立并研究了具有非线性传染率的疱疹系统,得到了决定疾病流行与否的基本再生数表达式,证明了当基本再生数小于1时,无病平衡点全局渐近稳定,此时疾病消亡.当基本再生数大于1时,地方病平衡点全局渐近稳定,此时疾病发展为地方病.
第四章建立并研究了带有饱和治疗的禽流感动系统力学模型.得到了系统基本再生数的表达式和无病平衡态稳定性的条件.并运用Bendixson-Dulac定理,证明了系统地方病平衡态的全局渐近稳定性.
第五章建立并研究了药物资源或治疗能力有限情况下的禽流感传播模型,假设当患者数量在治疗能力范围内时,治疗率与染病者数量成正比；当患者数量超过治疗能力范围时,为常数治疗.分析了不同情况下平衡点的存在性和稳定性.
第六章建立并研究了具有隔离的年龄结构传染病模型,给出了基本再生数的表达式,证明了无病平衡点的全局稳定性.得到地方病平衡点的局部稳定的条件.
第七章建立并研究了潜伏期具有传染性的年龄结构传染病模型.运用微分方程和积分方程中的理论和方法,得到了基本再生数的表达式,证明了当基本再生数小于1时,无病平衡点是局部和全局渐近稳定的,此时疾病消亡.当基本再生数大于1时,无病平衡点不稳定,此时系统至少存在一个地方病平衡点,并且在一定条件下证明了地方病平衡点的局部渐近稳定性.
In this thesis, several system dynamics models are formulated based on the occurance and transmission patterns of some infectious diseases such as vector-borne diseases, herpes, avian influenza,etc, and are systemically studied. The transmission mechanisms for these diseases are revealed. The results obtained in this thesis can be applied to guide the control of these diseases.
This thesis consists of three parts:the first part (contains Chapter 2 and Chapter 3) mainly concerns the vector-borne diseases and herpes; the second part (contains Chapter 4 and Chapter 5) mainly focus on avian influenza with different treatment rules; the third part (contains Chapter 6 and Chapter 7) formulate and study the age-structured epidemic models with isolation or infectiousness in latent period.
This thesis contains seven chapters. The main results and innovations in this disser-tation can be summarized as the following:
In chapter 1, the background, significance of the theory and practices, research on-goings at Home and Abroad, main contents and methods in this dissertation are briefly introduced.
In chapter 2, a vector-borne systematic epidemic model with nonlinear incidence is formulated and studied. The model can describe the dynamic of many diseases spreading by vectors such as malaria, dengue, West Nile Fever and so on. We get the conditions ensuring that the system exhibits backward bifurcation and analyze the stability of the endemic equilibrium. Our results imply that a nonlinear incidence make the models more complex dynamic behavior. It is not enough to make the disease die out that the basic reproductive number is smaller than 1.
In Chapter 3, an herpes systematic epidemic model with nonlinear incidence rate is formulated and investigated. We get the expression of the basic reproductive number which decide the spread of the diseases. If the basic reproductive number is smaller than 1, the disease-free equilibrium is globally asymptotically stable and the disease dies out. If the basic reproductive number is bigger than 1, the endemic equilibrium is globally asymptotically stable and the disease persists.
In Chapter 4, an avian systematic epidemic model with saturation treatment is es-tablished and studied. We obtain the basic reproductive number of the model and the stability conditions of the infection-free equilibrium. By using the Bendixson-Dulac the-orem, the global asymptotic stability of the endemic equilibrium is proved.
In Chapter 5, we develop a mathematical model for the spread of the avian influenza when drug resources and treatment capacity are limited. Moreover we assume the treat-ment rate is proportional to the number of infections when the capacity of treatment is not reach and otherwise, adopt a constant treatment. We analyze the existence and the stability of equilibria under different conditions.
In Chapter 6, an age-structured epidemic model with isolation is proposed and an-alyzed. We get the expression of the basic reproductive number and proved globally stability results for the disease-free equilibrium. At last we get the locally stability con-ditions of the endemic equilibrium.
Chapter 7, an age-structured HIV/AIDS epidemic model with infectivity in incuba-tive period is formulated and studied. By using the theory and methods of differential and integral equation, the explicit expression of the basic reproductive number was obtained, it is showed that the disease-free equilibrium is locally and globally asymptotically stable if the basic reproductive number is smaller than 1, at least one endemic equilibrium ex-ists if the basic reproductive number is bigger than 1, the stability conditions of endemic equilibrium are also given.
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