随着科技的快速发展以及人民生活水平的不断提高，传染病的预防和治疗受到越来越多关注，尤其是媒介传染病。万物相依而存，媒介传染病是通过媒介生物作为载体传播的传染病，这种传播特点更加符合实际，从而引起了很多研究者的关注。在研究传染病的过程中，传染病动力学是对传染病模型进行理论性定量研究的一种重要的数学方法，根据疾病发生、发展及环境变化等情况，建立能反映其变化规律的数学模型，通过模型动力学性态的研究来显示疾病的发展过程，预测其流行规律和发展趋势，分析疾病流行的原因和关键因素，寻求对其进行预防和控制的最优策略，为人们防制决策提供更加有效、准确、全面、理论基础和数量依据。随着研究的深入，模型逐渐向实际靠拢，加入了更多的实际因素，例如模型考虑在多个群体中的传播和交叉感染等，这类受媒介影响的传染病研究依然有空白：群体内部对群体传染病传播的影响，又或者外部因素对群体传染病传播的影响，都需要进行深入研究。

第一部分为绪论部分，这部分的工作主要对传染病与媒介传染病的背景、研究意义、以及研究现状和本文研究所需的相关储备知识等进行叙述，为后面的研究提供一定的依据。

第二部分的工作是基于考虑种群之间的内部因素，研究了一类具有年龄结构的媒介传染病动力学模型。首先将模型重新定义为感染人群随年龄增长的积分方程，利用特征线积分，常微分方程变量分离法，不动点定理等，定义了基本再生数，证明了当时地方病平衡点的存在唯一性。接着导出了地方病平衡点稳定性分析的特征方程，以为分歧参数，得到了分歧的充分条件。最后，通过建立相应的数值模拟，进一步为本文的结果提供一定的理论依据。

第三部分的工作结合实际，考虑外部影响的因素，结合医院内媒介交叉感染存在的实际问题，建立了医院内以医护人员为传染媒介引起的抗生素耐药性交叉感染模型。近年来，随着医院内就诊人数增加、抗生素滥用等原因，造成医院内引起交叉感染急速增多，因此医院内感染已成为全球关注的热点，这部分所研究的内容符合传染病研究中的热点问题。在模型中研究了以医护人员作为媒介而引起的传染，同时涉及了是否有耐药菌的情况，对模型进行动力学分析，计算了疾病的基本再生数，分别证明了平衡点的局部稳定性与全局稳定性，得到了如果医院携带耐药菌的医护人员和病人都消失，即为零时，不存在交叉感染的结论。最后，根据以上的理论证明，通过确定不同参数，用数值模拟的方法进一步验证。这些结果能够更好有效地防止传染病进一步传播。

With the rapid development of science and technology and the continuous improvement of people's living standard, the prevention and treatment of infectious diseases has aroused the extensive attention of more and more researchers in recent years.Especially medium infectious disease,because of its transmission characteristics,it has aroused the extensive attention of researchers,infectious disease dynamics is an important mathematical method for theoretical and quantitative study of infectious disease models. Through the study of the dynamics of the model, the development process of the disease is displayed, the epidemic law and development trend are predicted, the causes and key factors of the epidemic are analyzed, and the optimal strategies for prevention and control are sought to provide more effective, accurate, comprehensive, theoretical basis and quantitative basis for prevention and control decisions of people.With research,the model is closer to reality.For example, the model considers transmission and cross infection in multiple groups,there is a gap in the research of such medium affected infectious diseases.For example, considering the impact of the age structure of the internal group,or considering the influence of external factors,these we need to continue to research.

The first part is the introduction,this part introduces the research background and significance, as well as the research status and content.

It mainly studies the dynamic behavior of a kind of infectious diseases with age structure and medium in the second part. Firstly, the model was redefined as the integral equation of age - dependent growth of infected population, and the basic reproductive numberwas defined. Then the existence and uniqueness of the endemic equilibrium point are proved. Then the characteristic equation of stability analysis of endemic disease equilibrium is derived as bifurcation parameter, at the same time, a sufficient condition for divergence of delay differential systems is obtained based on a method of delay differential systems in literature. Finally, through the establishment of the corresponding numerical simulation, a certain theoretical basis is proved further.

The cross infection model of hospital vectors is studied in the third part, the research content accords with the hot issues in infectious disease research, infections caused by health care workers acting as vectors are studied in the model, it also involves the presence or absence of drug-resistant bacteria, the dynamic analysis of the model is carried out to calculate the basic reproduction number of the disease, and the local stability and global stability of the equilibrium point are proved respectively. It is concluded that there is no cross-infection if all the medical staff and patients carrying drug-resistant bacteria in the hospital disappear, namely zero. Finally, according to the above theoretical proof, it is further verified by numerical simulation after determining different parameters. These results can prevent the further spread of infectious diseases better.

[1] Eihab B.M. Bashier and Kailash C. Patidar. Optimal control of an epidemiological model with multiple time delays[J]. Applied Mathematics and Computation, 2017, 292 : 47-56.

[2] Lijuan Chen and Khalid Hattaf and Jitao Sun. Optimal control of a delayed SLBS computer virus model[J]. Physica A: Statistical Mechanics and its Applications, 2015, 427 : 244-250.

[3] Greenhalgh David and Griffiths Martin. Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model.[J]. Journal of mathematical biology, 2009, 59(1) : 1-36.

[4] 北京协和医院新型冠状病毒感染的肺炎诊治专家组, 北京协和医院关于”新型冠状病毒感染的肺炎”诊疗建议方案, 协和医学杂志,2020,E003-E003.

[5] 魏永越,赵杨,陈峰,沈洪兵.传染病动力学模型的理论基础及在疫情防控中的应用价值[J].中华预防医学杂志,2020(06):E032.[6] 陈兰荪, 孟新柱, 焦建军.生物动力学[M]. 北京: 科学出版社, 2009.

[7] Kazuo Yamazaki. Threshold dynamics of reaction–diffusion partial differential equations model of Ebola virus disease[J]. International Journal of Biomathematics, 2018, 11(8) : 30.

[8] Patrick W. Nelson and Alan S. Perelson. Mathematical analysis of delay differential equation models of HIV-1 infection[J]. Mathematical Biosciences, 2002, 179(1) : 73-94.

[9] 张恭庆, 林源渠. 泛函分析讲义[M]. 北京:北京大学出版, 2012.

[10]李乔, 张晓东. 矩阵十讲[M]. 上海: 中国科技大学出版, 2014, 26-32.

[11]Russell A. Smith. Some applications of Hausdorff dimension inequalities for ordinary differential equations[J]. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1986, 104(3-4) : 235-259.

[12]刘晨,窦霁虹,李玉峰,赵婷婷.一类具有标准发生率和双垂直传播的媒介传染病模型分析[J].纯粹数学与应用数学,2021,37(02):198-208.

[13]Michael Y. Li and James S. Muldowney. A Geometric Approach to Global-Stability Problems[J]. SIAM Journal on Mathematical Analysis, 2006, 27(4) : 1070-1083.

[14]Franceschetti Andrea and Pugliese Andrea and Breda Dmitri. Multiple endemic states in age-structured SIR epidemic models.[J]. Mathematical biosciences and engineering : MBE, 2012, 9(3) : 577-99.

[15]A nonlocal and time-delayed reactiondiffusion model of dengue transmission[J]. SIAM Journal Applied Mathematics,2011, 71(1): 147-168.

[16]Shepard Donald S et al. Cost-effectiveness of a pediatric dengue vaccine.[J]. Vaccine, 2004, 22(9-10) : 1275-80.

[17]Delfim Torres and Cristiana Silva. Optimal control strategies for tuberculosis treatment: A case study in Angola[J]. Numerical Algebra Control & Optimization, 2012, 2(3) : 601-617.

[18]Nirav Dalal and David Greenhalgh and Xuerong Mao. A stochastic model of AIDS and

condom use[J]. Journal of Mathematical Analysis and Applications, 2007, 325(1) :36-53.

[19]Z.Feng,D.L.Smith,F.E.Mckenzic,and S.A.Lcvin,Coupling eology and evolution:malaria and S-gene across time scales[J]. Mathematical Biosciences, 2004, 189(1): 1-19.

[20]Hisashi Inaba and Hisashi Sekine. A mathematical model for Chagas disease with infection-age-dependent infectivity[J]. Mathematical Biosciences, 2004, 190(1) : 39-69.

[21]F.Forouzannia and A.B.Gumel,The age-structured model undergoes the phenomenon of backward bifurcation at R_0=1 under certain condition[J]. Mathematical Biosciences, 2004, 247: 80-94.

[22]Discrete and Continuous Dynamical Systems; Researchers at University of the Basque Country Report New Data on Discrete and Continuous Dynamical Systems[J]. Journal of Technology & Science, 2015.

[23]Toshikazu Kuniya. Hopf bifurcation in an age-structured SIR epidemic model[J]. Applied Mathematics Letters, 2019, 92 : 22-28.

[24]Hisashi Inaba. Age-Structured Population Dynamics in Demography and Epidemiology [M]. Springer, Singapore, 2017.

[25]Xinyu Song and Xueyong Zhou and Xiang Zhao. Properties of stability and Hopf bifurcation for a HIV infection model with time delay[J]. Applied Mathematical Modelling, 2009, 34(6) : 1511-1523.

[26]C.Zou, D. Towsley, W. Gong,Email virus propagation modeling and analysis, Technical report TR-CSE-03-04, University of Massachusetts, Amherst, 2003.

[27]C.Jin,J. Liu,and Q.H. Deng,Network virus propagation model based on effects of removingtime and user vigilance,International Journal of Network Security. 2009, 156-163.

[28]Qintao Gan et al. Travelling waves in an infectious disease model with a fixed latent period and a spatio–temporal delay[J]. Mathematical and Computer Modelling, 2011, 53(5-6) : 814-823.

[29]马佳辉,薛亚奎.医院内抗生素耐药性传染的稳定性与Hopf分支[J].中北大学学报(自然科学版),2015,36(03):261-267.

[30]Severe dengue: the need for new case definitions.[J]. The Lancet infectious diseases, 2006, 6(5): 297-302.

[31]吕茵. 具有接种疫苗和再次感染的媒介传染病模型分析[D].信阳师范学院,2011.

[32]白梦. 几类媒介传染病模型的全局稳定性[D].中北大学,2016.

[33]漆一鸣. 我国的重要虫媒传染病[C]//.当代昆虫学研究——中国昆虫学会成立60周年纪念大会暨学术讨论会论文集,2004:642-647.

[34]闪俊华. 一类媒介传染病模型的稳定性与分支[D].信阳师范学院,2010.

[35]Erika M C D'Agata et al. The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria.[J]. PLoS ONE, 2017, 3(12) : e4036.

[36]Keesing Felicia et al. Impacts of biodiversity on the emergence and transmission of infectious diseases.[J]. Nature, 2010, 468(7324) : 647-52.

[37]Keesing F and Holt R D and Ostfeld R S. Effects of species diversity on disease risk.[J]. Ecology letters, 2006, 9(4) : 485-98.

[38]Josef Hofbauer and Joseph W.-H. So. Uniform persistence and repellors for maps[J]. proc, 1989, 107(4) : 1137-1142.

[39]Ma Junling and Ma Zhien. Epidemic threshold conditions for seasonally forced seir models.[J]. Mathematical biosciences and engineering , 2006, 3(1) : 161-72.

[40]Webb Glenn F et al. A model of antibiotic-resistant bacterial epidemics in hospitals[J]. Proceedings of the National Academy of Sciences of the United States of America, 2005, 102(37) : 13343-8.

[41]Véronique Sébille and Sylvie Chevret and Alain-Jacques Valleron. Modeling the Spread of Resistant Nosocomial Pathogens in an Intensive-Care Unit[J]. Infection Control & Hospital Epidemiology, 1997, 18(2) : 84-92.

[42]Sébille V and Valleron A J. A computer simulation model for the spread of nosocomial infections caused by multidrug-resistant pathogens[J]. Computers and biomedical research, an international journal, 1997, 30(4) : 307-22.

[43]Austin D J et al. Vancomycin-resistant enterococci in intensive-care hospital settings: transmission dynamics, persistence, and the impact of infection control programs[J]. Proceedings of the National Academy of Sciences of the United States of America, 1999, 96(12) : 6908-13.

[44]Boldin B and Bonten M J M and Diekmann O. Relative effects of barrier precautions and topical antibiotics on nosocomial bacterial transmission: results of multi-compartment models[J]. Bulletin of mathematical biology, 2007, 69(7) : 2227-48.

[45]Bonten M J and Austin D J and Lipsitch M. Understanding the spread of antibiotic resistant pathogens in hospitals: mathematical models as tools for control[J]. Clinical infectious diseases : an official publication of the Infectious Diseases Society of America, 2001, 33(10) : 1739-46.

[46]Lipsitch M and Bergstrom C T and Levin B R. The epidemiology of antibiotic resistance in hospitals: paradoxes and prescriptions[J]. Proceedings of the National Academy of Sciences of the United States of America, 2000, 97(4) : 1938-43.

[47]Nigar Ali and Gul Zaman and Obaid Algahtani. Stability analysis of HIV-1 model with multiple delays[J]. Advances in Difference Equations, 2016, 2016(1) : 1-12.

[48]Fred Brauer and P. van den Driessche and Lin Wang. Oscillations in a patchy environment disease model[J]. Mathematical Biosciences, 2008, 215(1) : 1-10.

[49]Jennifer L. Kyle and Eva Harris. Global Spread and Persistence of Dengue[J]. Annual Review of Microbiology, 2008, 62(1) : 71-92.

[50]T. Kuniya.Stability analysis of an age-structured SIR epidemic model with a reduction method to ODEs[J].Mathematics 2018,6(9):147.

[51]Aadil Lahrouz et al. Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination[J]. Applied Mathematics and Computation, 2011, 218(11) : 6519-6525.

[52]Tailei Zhang and Junli Liu and Zhidong Teng. Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure[J]. Nonlinear Analysis: Real World Applications, 2010, 11(1) : 293-306.

[53]Edoardo Beretta and Yang Kuang. Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters[J]. SIAM J. Math. Analysis, 2002, 33(5) : 1144-1165.

[54]Shah Nita H. et al. Fractional SIR-Model for Estimating Transmission Dynamics of COVID-19 in India[J]. J, 2021, 4(2) : 86-100.

[55]Urszula Ledzewicz and Heinz Schttler. On optimal singular controls for a general SIR-model with vaccination and treatment[J]. Conference Publications, 2011, 2011 : 981-990.

[56]H.Y.Shu,L.Wang,J.H.Wu.Global dynamics of Nicholson’s blowfies equation revisited:On set and termination of nonlinear oscillations[J]. J. Diff. Equs, 2013, 255: 2565-2586.

[57]Ezer Miller and Amit Huppert. The effects of host diversity on vector-borne disease: the conditions under which diversity will amplify or dilute the disease risk[J]. PLoS ONE, 2017, 8(11) : e80279.

[58]H.L Freedman,Shigui Ruan,Moxun Tang.Uniform Persistence and Flows Near a Closed Positively Invaxiant Set[J]. Journal of Dynamics and Differential Equations, 1994, 6: 583-600.

[59]Mathematics; Kocaeli University Researchers Have Published New Study Findings on Mathematics (Asymptotic Behavior and Stability in Linear Impulsive Delay Differential Equations with Periodic Coefficients)[J]. Journal of Mathematics, 2020, : 537.

[60]Maoan Han et al. Bifurcation Theory And Methods Of Dynamical Systems[M]. SG Singapore : World Scientific Publishing Company, 1997.

[61]C. Connell McCluskey. Global stability for an SIR epidemic model with delay and nonlinear incidence[J]. Nonlinear Analysis: Real World Applications, 2009, 11(4) : 3106-3109.

[62]Castillo-Chavez Carlos and Song Baojun. Dynamical models of tuberculosis and their applications.[J]. Mathematical biosciences and engineering : MBE, 2004, 1(2) :361-404.

[63]Diekmann O and Heesterbeek J A and Metz J A. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations.[J]. Journal of mathematical biology, 1990, 28(4) : 365-82.

[64]P.van den Driessche,J.Watmough.Reproduction numbers and sub-threshod endemic

equilibria for compartmental models of disease transmission[J].Math.Bio.Sci.2002,180:

29-44.

[65]J.R.Rohra,D.J.Civitelloa,P.W.Predation and indirect effects drive Crumrine,Predator diversity,intraguild parasite transmission[J]. PNAS, 2015, 112(10): 3008-3013.

[66]Hutson Vivian and Schmitt Klaus. Permanence and the dynamics of biological systems[J]. Mathematical Biosciences, 1992, 111(1) : 1-71.

[67]E.Beretta,Y.Kuang.Geometric stability switch criteria in delay differential systems with delay dependent parameters[J].Siam.J.Math Anal.2002,33:1144-1165.

[68]白梦,薛亚奎. 具有接种疫苗和再次感染的媒介传染病模型的稳定性分析[J].中北大学学报, 1673-3139（2016）02-0120-06.

[69]钟彦,徐世兰,王妍潼,宗志勇.四川省23所医院医院感染管理现状调查[J].中国循证医学杂志,2014,14(02):174-177.

[70]Abid Ali Lashari,Muhammad Ozair,Gul Zaman,李学志.潜伏期和染病期均具有传染性的媒介传染病模型的全局稳定性分析[J].应用泛函分析报,2012,14(04):321-329.

[71]Li M Y, Graef J R,Wang L,et al. Global dynamics of a SEIR model with varying total population size[J]. Mathematical biosciences,1999,160(2):191-213.

[72]Hou Q, Sun X, Wang Y, et al. Global properties of a general dynamic model for animal diseases: A case study of brucellosis and tuberculosis transmission[J]. Journal of Mathematical Analysis and Applications, 2014,414(1): 424-433.

[73]Elmojtaba I M, Mugisha J Y T, Hashim M H A. Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan[J].Applied Mathematics and Computation,2010,217

(6):2567- 2578.

[74]Abid Ali Lashari,Muhammad Ozair,Gul Zama等. 潜伏期和染病期均具有传染性的媒介传染病模型的全局稳定性分析[J]. 应用泛函分析学报,2012,14(4):321-329.