【摘要】：傳染病一直伴隨著人類社會的發(fā)展.歷史上,傳染病的不斷爆發(fā)和傳播給人類帶來了巨大的災(zāi)難.盡管當(dāng)今社會科學(xué)技術(shù)持續(xù)發(fā)展、醫(yī)療條件也得到了很大的改善,然而世界衛(wèi)生組織(WHO)宣稱傳染病仍是人類健康的最大威脅.因此,人們有必要了解疾病的分布情況、時空傳播規(guī)律并采取適當(dāng)?shù)目刂撇呗?自1927年美國數(shù)學(xué)家Kermack和蘇格蘭醫(yī)學(xué)家、流行病學(xué)家McKendrick構(gòu)造了著名的SIR“倉室”模型以來,數(shù)學(xué)模型成為研究疾病傳播規(guī)律、評估感染風(fēng)險、優(yōu)化控制策略的重要工具.早期,研究人員主要研究與空間無關(guān)的常微分系統(tǒng),僅反應(yīng)隨時間推移的動力學(xué)特征.為了更加真實地描述現(xiàn)實,研究人員發(fā)現(xiàn)空間擴(kuò)散是影響疾病傳播的重要因素.為此許多學(xué)者建立了一系列偏微分方程模型分析傳染病的動力學(xué)性態(tài).近年來,隨著研究的更加深入,研究人員逐漸意識到空間擴(kuò)散和環(huán)境異質(zhì)性在一些傳染病傳播過程中產(chǎn)生了重要影響,例如:流感、瘧疾、西尼羅河病毒等.除此之外,周期性、對流、媒體報道和有限醫(yī)療資源配置等在傳染病傳播中的作用也引起了廣泛關(guān)注.這篇博士論文主要圍繞空間異質(zhì)性、周期性、對流、非線性恢復(fù)率以及非線性發(fā)生率對于SIS傳染病模型蔓延和消退的影響展開的.本文的主要研究工作組織如下.第一章介紹本文研究主題的一些背景知識和已經(jīng)取得的最新進(jìn)展.第二章主要研究異質(zhì)環(huán)境中的一類具自由邊界和對流項影響的傳染病模型.首先利用拋物方程初邊值問題的Lp理論、Zorn引理、壓縮映像原理得到全局解的存在唯一性和正性性.接著給出反應(yīng)擴(kuò)散系統(tǒng)的基本再生數(shù)及其性質(zhì),引進(jìn)自由邊界問題風(fēng)險指標(biāo)R0F(t)的定義和討論了其解析性質(zhì).借助于風(fēng)險指標(biāo)RF(t),通過構(gòu)造精細(xì)上解、下解得到了疾病蔓延和消退的二擇一定理,給出了蔓延和消退的判據(jù).并且用半波方法得到了當(dāng)疾病蔓延時受對流影響的漸近擴(kuò)張速度.數(shù)值模擬給出了對流強度和擴(kuò)張能力對于染病區(qū)域邊沿的影響.這些結(jié)果與固定區(qū)域上傳染病模型的動力學(xué)性質(zhì)完全不同.第三章深入探討了周期異質(zhì)環(huán)境下具自由邊界的傳染病模型.首先引入基本再生數(shù),并且給出了兩種特殊情形下顯式表達(dá)式.再借助譜半徑的方法給出自由邊界問題的風(fēng)險指標(biāo)R0F(τ),該指標(biāo)與相應(yīng)的周期拋物特征問題的主特征值密切相關(guān).利用最大模原理、上下解方法、譜分析以及偏微分方程其它多種技巧證明了疾病蔓延和消退的充分條件.當(dāng)疾病蔓延發(fā)生時,估計了受對流影響的左右不同的漸近擴(kuò)張速度.最后利用數(shù)值模擬給出對流強度、擴(kuò)散率和擴(kuò)張能力對疾病傳播機(jī)理的影響.第四章提出一個異質(zhì)環(huán)境中具媒體報道影響的SIS傳染病反應(yīng)擴(kuò)散模型.在該模型中,我們用媒體報道影響的因子體現(xiàn)疾病的非線性接觸率.首先利用變分法給出異質(zhì)環(huán)境中具媒體報道和擴(kuò)散影響的基本再生數(shù)的定義及其解析性質(zhì).接著給出無病平衡點和染病平衡點的存在性,再利用上下解方法、單調(diào)迭代序列、經(jīng)典的半群理論和強極值原理證明了當(dāng)R0D1時,無病平衡點全局漸近穩(wěn)定;而當(dāng)R0D1時,證明了當(dāng)ds=dI時染病平衡點的全局漸近穩(wěn)定性.數(shù)值模擬表明如果增加媒體報道強度,疾病的感染風(fēng)險會降低,從而傳染病能夠得以快速有效地控制.第五章考慮了空間異質(zhì)環(huán)境中一類受有限醫(yī)療資源配置影響的具非線性恢復(fù)率的SIS傳染病模型.探討了環(huán)境異質(zhì)性、有限醫(yī)療資源配置等對于疾病蔓延和消退的影響.首先利用變分方法給出與最大、最小恢復(fù)率有關(guān)的閾值R0*和R0*及其性質(zhì).借助于這兩個閾值,以及上下解方法、單調(diào)迭代動力學(xué)、乘乘減積技巧等證明了無病平衡點和染病平衡點的存在性、唯一性和穩(wěn)定性.數(shù)值模擬表明適當(dāng)?shù)牟〈矓?shù)的配置對于疾病的控制是非常關(guān)鍵的.我們的理論結(jié)果對于公共衛(wèi)生管理部門優(yōu)化有限醫(yī)療資源的配置提供了理論依據(jù).第六章中,我們總結(jié)了本文的主要工作,并且在此基礎(chǔ)上對今后的研究工作作了進(jìn)一步的規(guī)劃.
[Abstract]:Infectious diseases have been accompanied by the development of human society. In history, the continuous outbreak and spread of infectious diseases have brought great disasters to human beings. Although today's social science and technology continue to develop and medical conditions have been greatly improved, the WHO (WHO) claims that infectious diseases are still the greatest threat to human health. It is necessary to understand the distribution of the disease, the law of space-time transmission and the appropriate control strategy. Since the 1927 American mathematician Kermack and the Scotland medical scientist, the epidemiologist McKendrick constructed the famous SIR "warehouse room" model, the mathematical model has become a study of the law of disease transmission, the assessment of the risk of infection, and the optimization of the control strategy. In the early stage, the researchers mainly studied the space independent ordinary differential system, which only responded to the dynamic characteristics of the time lapse. In order to describe the reality more truly, the researchers found that space diffusion was an important factor affecting the spread of disease. In recent years, with the further research, researchers have gradually realized that space diffusion and environmental heterogeneity have played an important role in the transmission of some infectious diseases, such as influenza, malaria, West Nile virus and so on. In addition, periodicity, convection, media coverage, and the allocation of limited medical resources in the transmission of infectious diseases This thesis mainly focuses on the effects of spatial heterogeneity, periodic, convection, nonlinear recovery and nonlinear incidence on the spread and decline of SIS infectious disease model. The main research work of this paper is as follows. In the second chapter, the second chapter mainly studies an infectious disease model with free boundary and convective effects in the heterogeneous environment. First, the existence and uniqueness and the positive nature of the global solution are obtained by using the Lp theory of the initial boundary value problem of the parabolic equation, the Zorn lemma and the compression mapping principle. It introduces the definition and the analytic properties of the risk index R0F (T) of the free boundary problem. By means of the risk index RF (T), by constructing the fine upper solution and the lower solution, the two alternative theorem of the spread and decline of the disease is obtained, and the criterion of the spread and regression is given. Near expansion speed. The numerical simulation gives the effect of convection intensity and expansion ability on the edge of the infected region. These results are completely different from the kinetic properties of the fixed area. The third chapter discusses the infectious disease model with free boundary in the periodic heterogeneous environment. First, the basic regeneration number is introduced, and two is given. An explicit expression under special circumstances. The risk index R0F (tau) of the free boundary problem is given by means of the aid spectrum radius. The index is closely related to the principal eigenvalues of the corresponding periodic parabolic problem. The maximum modulus principle, the upper and lower solutions, the spectral analysis and the other techniques of the partial differential equation prove the spread and decline of the disease. In the fourth chapter, a diffusion model of SIS infectious disease with the influence of media coverage in a heterogeneous environment is proposed. In the model, we use the factor of media coverage to reflect the nonlinear contact rate of the disease. First, we use the variational method to give the definition and the analytic properties of the basic regeneration number with media coverage and diffusion in the heterogeneous environment. Then we give the existence of the disease-free equilibrium point and the equilibrium point of the disease, and then use the upper and lower solutions and the monotone iterative sequence. The classical semigroup theory and the strong extremum principle prove that the disease free equilibrium point is globally asymptotically stable when R0D1, and when R0D1, the global asymptotic stability of the equilibrium point of the disease is proved. The numerical simulation shows that the risk of infection of the disease will be reduced if the media coverage is increased, thus the infectious disease can be controlled quickly and effectively. In the fifth chapter, the SIS infectious disease model, which is affected by the allocation of limited medical resources in the spatial heterogeneity environment, is considered. The effects of environmental heterogeneity and the allocation of limited medical resources on the spread and regression of the disease are discussed. First, the threshold R0* and R0* related to the maximum and minimum recovery rate are given by the variational method. By means of these two thresholds, as well as the method of the upper and lower solutions, the monotone iterative dynamics, the multiplication and multiplication technique, the existence, uniqueness and stability of the disease free equilibrium point and the equilibrium point of the disease are proved. The numerical simulation shows that the allocation of the appropriate number of beds is not essential for the control of the disease. Our theoretical results are for public health. The management department provides a theoretical basis for the optimization of the allocation of limited medical resources. In the sixth chapter, we have summarized the main work of this paper, and on this basis, we made further plans for the future research work.
【學(xué)位授予單位】：揚州大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175

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