流感傳播和控制的數(shù)學(xué)模型研究肖燕妮課件

1、流感傳播和控制的數(shù)學(xué)模型研究,Modelling the Transmission Dynamics and Control of Influenza,肖燕妮,2014、夏,,,,Introduction to myself,http://gr.xjtu.edu.cn/web/yxiao,辦公室： 理科樓318yxiao@mail.xjtu.edu.cn,研究方向： 生物數(shù)學(xué)、傳染病動(dòng)力學(xué),（1） 宏觀艾滋病病毒傳播與微觀H

2、IV的進(jìn)展研究（2）流感（A/H1N1, H7N9）的模型研究（3）數(shù)學(xué)工具：ODE、 DDE、 IDE、Non-smooth ODE,3,綱 要,流感病毒簡(jiǎn)介 流感的預(yù)防與控制問(wèn)題的提出 傳染病動(dòng)力學(xué)簡(jiǎn)介 流感的數(shù)學(xué)模型研究,什么是禽流感？    禽流感主要是指禽中流行的由流感病毒引起的感染性疾病。禽流感病毒可分為高致病性禽流感病毒、低致病性禽流感病毒和無(wú)致病性禽流感病毒

3、。高致病性禽流感病毒目前只發(fā)現(xiàn)H5和H7兩種亞型。由于種屬屏障，禽流感病毒只在偶然的情況可以感染人，既往確認(rèn)感染人的禽流感病毒有H5N1、H9N2、 H7N2、 H7N3、 H7N7、H5N2、H10N7，癥狀表現(xiàn)各不相同，可以表現(xiàn)為呼吸道癥狀、結(jié)膜炎、甚至死亡。人感染高致病性H5N1禽流感病毒后常表現(xiàn)為高熱等呼吸道癥狀，往往很快發(fā)展成肺炎，甚至急性呼吸窘迫綜合癥和全身器官衰竭，甚至死亡。,至今由禽鳥傳人的禽流感有三種：甲型H5N1、甲

4、型H7N7及甲型H9N2。,什么是H7N9禽流感病毒？流感病毒可分為甲（Ａ）、乙（Ｂ）、丙（Ｃ）三型。其中，甲型流感依據(jù)流感病毒血凝素蛋白（HA）的不同可分為1-16種亞型，根據(jù)病毒神經(jīng)氨酸酶蛋白（NA）的不同可分為1-9種亞型，HA不同亞型可以與NA的不同亞型相互組合形成多達(dá)144種不同的流感病毒。而禽類特別是水禽是所有這些流感病毒的自然宿主，H7N9禽流感病毒是其中的一種。H7N9亞型流感病毒既往僅在禽間發(fā)現(xiàn)，在荷蘭、日本及美國(guó)

5、等地曾發(fā)生過(guò)禽間暴發(fā)疫情，但未發(fā)現(xiàn)過(guò)人的感染情況。,H7N9禽流感感染病例的主要臨床表現(xiàn)？目前3例確診病例主要表現(xiàn)為典型的病毒性肺炎，起病急，病程早期均有高熱（38℃以上），咳嗽等呼吸道感染癥狀。起病5-7天出現(xiàn)呼吸困難，重癥肺炎并進(jìn)行性加重，部分病例可迅速發(fā)展為急性呼吸窘迫綜合癥并死亡。,全球人感染甲型H7流感病毒情況？1996年～2009年間，荷蘭、意大利、加拿大、美國(guó)和英國(guó)曾報(bào)道人感染甲型H7流感病例，病毒亞型分別為H7N

6、2、H7N3和H7N7，臨床表現(xiàn)主要為結(jié)膜炎與輕型的上呼吸道感染。此前，我國(guó)從未發(fā)現(xiàn)過(guò)H7亞型流感病毒感染病例。,H7N9病毒與既往H1N1、H5N1和其他季節(jié)性流感病例相比其毒力和傳染性如何？由于目前只發(fā)現(xiàn)3例人感染H7N9禽流感病毒確診病例，對(duì)該病毒及其所致疾病的研究資料十分有限，尚無(wú)法對(duì)該病毒的毒力和人際傳播的能力做出準(zhǔn)確判斷。本次人感染H7N9禽流感的病例的感染來(lái)源？此次人感染的H7N9流感病毒從病毒生物學(xué)上屬于禽源

7、流感病毒，既往國(guó)際上所發(fā)現(xiàn)的人感染H7亞型的流感病毒也多來(lái)自于禽類，但截至目前，3例確診病例的具體感染來(lái)源尚不清楚。H7N9禽流感病毒是否能夠在人與人之間傳播？目前尚未證實(shí)該病毒具有人傳人的能力。,如何預(yù)防H7N9流感？流感是一種急性呼吸道感染性疾病。勤洗手、室內(nèi)勤通風(fēng)換氣、注意營(yíng)養(yǎng)、保持良好體質(zhì)有利于預(yù)防流感等呼吸道傳染病。出現(xiàn)打噴嚏、咳嗽等呼吸道感染癥狀時(shí)，要用紙巾、手帕掩蓋口鼻，預(yù)防感染他人。此外，還要特別注意盡量避免

8、直接接觸病死禽、畜。目前尚無(wú)針對(duì)H7N9禽流感病毒的疫苗。目前的治療方法？基因序列分析顯示，該病毒對(duì)神經(jīng)氨酸酶抑制劑類抗流感病毒藥物敏感。根據(jù)其他型別流感抗病毒治療的經(jīng)驗(yàn)，發(fā)病后早期使用神經(jīng)氨酸酶抑制劑類抗流感病毒藥物可能是有效的，但對(duì)人類新發(fā)現(xiàn)的H7N9禽流感病毒感染的特異性治療手段仍需觀察研究。,香港在1997年發(fā)生禽流感，有6人死亡，當(dāng)局捕殺上百萬(wàn)只家禽，避免疫情惡化。 禽流感是一種主要流行于雞群中的烈性傳染病，一旦爆發(fā)

9、，往往會(huì)造成家禽的大量死亡。潛伏期一般為3-5天。一般病程1～2天，癥狀變化很大。 病雞可能見(jiàn)有呼吸道癥狀，如打噴嚏、竇炎和結(jié)膜炎。病雞頭部常出現(xiàn)水腫，可能同時(shí)出現(xiàn)或不出現(xiàn)腹瀉；病雞體溫升高，羽毛蓬松，雞冠發(fā)紺。有的腿變紅，鼻分泌物增多，呼吸極度困難，甩頭，嚴(yán)重地可窒息死亡。產(chǎn)蛋率明顯下降。 然而近年來(lái)情況卻悄悄地發(fā)生了變化，禽流感的傳播已經(jīng)跨越了原先的范圍，開(kāi)始侵襲人類社會(huì)。,禽流感的潛在威脅可能遠(yuǎn)大于SARS,首先是這種禽流感

10、一旦變異后可能會(huì)成為普通人類流感病毒，而人體對(duì)于新的流感病毒幾乎沒(méi)有任何免疫力。 其次，人類的流感病毒遠(yuǎn)遠(yuǎn)比非典病毒更具傳染性。因?yàn)榱鞲胁《究梢栽诳諝庵醒杆賯鞑?，而非典病毒則通常在近距離接觸后才會(huì)被傳染。,亞洲的自然環(huán)境為病毒滋生和傳播提供便利 亞洲是多山地區(qū)，特別是東南亞地區(qū)遍布雨林，中部亞洲則高原山脈連綿不斷，而這些地帶又恰恰位于全球鳥類遷徙的路線上。 候鳥是禽流感病毒的重要傳播者。2005年，科學(xué)家在青藏高

11、原沿青海湖、扎林湖等候鳥遷徙路線對(duì)禽流感疫情做了跟蹤調(diào)查，發(fā)現(xiàn)每年候鳥因繁殖、越冬而遷徙前后恰好是高致病性禽流感疫情發(fā)生季節(jié)，同時(shí)候鳥遷徙路線與發(fā)生疫情的地點(diǎn)是重疊的。研究組還研究了不同種類、分布在不同地區(qū)，可能傳播高致病性禽流感病毒的候鳥，發(fā)現(xiàn)候鳥遷徙路徑中的沼澤和湖泊是高致病性禽流感存在和傳播的重要地帶。帶有病毒的野鳥在遷徙的路途中很容易和散養(yǎng)放養(yǎng)的家禽接觸，將病毒傳給亞洲那些位于鳥類遷徙路線上的家禽。 另外，亞洲大部分農(nóng)業(yè)

12、國(guó)家分布在亞洲南部，其中包括中國(guó)南方。這些地區(qū)氣溫高、濕度大、病毒生存機(jī)會(huì)高，再加上一些農(nóng)村生活條件和衛(wèi)生環(huán)境惡劣，容易成為各種病毒滋生的溫床。,,問(wèn)題的提出,傳染病能否在某個(gè)地方傳播開(kāi)來(lái)？能否形成地方??？ 傳染病高潮的什么時(shí)候來(lái)臨？傳染病的規(guī)模有多大？ 什么預(yù)防與控制措施最為有效？ 能否給公共衛(wèi)生部門提供定量的建議？,基于流感的傳播機(jī)理建立數(shù)學(xué)模型，利用數(shù)據(jù)估計(jì)系統(tǒng)的參數(shù)，對(duì)疫情的發(fā)展趨勢(shì)給予預(yù)測(cè)，分析人為的干

13、預(yù)措施對(duì)疫情發(fā)展的影響，為制定控制方案提供定量的決策依據(jù),傳染病模型簡(jiǎn)介,方法： 將人群分類，建立方程來(lái)描述每類人群的數(shù)量的變化,,S,,I,,R,S(t)：易感者的數(shù)量（或比例）I(t)： 感染者的數(shù)量（或比例）R(t)：移除者的數(shù)量（或比例）,SIR模型,,模型4,SIR模型,相軌線 的定義域,在D內(nèi)作相軌線 的圖形，進(jìn)行分析,相軌線 及其分析,s(t)單調(diào)減?相軌線的

14、方向,P1: s0>1/ ? i(t)先升后降至0,P2: s0<1/ ? i(t)單調(diào)降至0,1/ 閾值,Course of number of S, I and R animals in a closed population,預(yù)防傳染病蔓延的手段,? (日接觸率)? ? 衛(wèi)生水平?,?(日治愈率)? ? 醫(yī)療水平?,傳染病不蔓延的條件——s0<1/?,降低 s0,提高 r0,,提高閾值 1/?,,Trans

15、mission between individuals,R0 Basic Reproduction ratio （基本再生數(shù)）,Average number of secondary cases caused by 1 infectious individual during its entire infectious period in a fully susceptible population,,Reproduction rat

16、io, R0,R0 = 3,R0 = 0.5,,,,,,,,,,,,,,,,,,禽流感的數(shù)學(xué)模型,,S,,I,,R,,A,,E,,Sp,,Ip,傳播框圖,人群,禽類,X，Y： denote the susceptible birds, the birds infected with the avian influenzaS，B ：the susceptible humans and the humans that are infe

17、cted with the wild avian influenza.,H： denotes the humans infected with the mutant avian influenza;,以往簡(jiǎn)單模型,H7N9 模型建立,Xiao Y., Sun X., Tang S., Wu J., Transmission potential of the novel avian influenza A(H7N9) infection

18、 in mainland China J. Theor. Biol. 352(2014)1–5,模型建立,With,As of April 26, 2013, the China Ministry of Agriculture reported that 68,060 bird and environmental specimens have been tested, 46 (0.07\%) were confirmed H7N9-

19、positive by culture We thus assume that,A fact: on prevalence in poultry,數(shù)據(jù),參數(shù)估計(jì),Using the Kaplan–Meier (KM) method to data available, we obtained the estimation for the mean time from the date of illness onset to dea

20、th as13days,leading to disease-related death rate α =1/13. recovery rate γ =1/ 11,Effect of various Intervention timing and intensity,Prediction of the next outbreak,Prediction of the next outbreak,The periodic infec

21、tion of poultry may induce the second outbreak in human population.,We estimate the reproduction number for human-to-human transmission as 0.467 (95\% CI 0.387-0.651). Simulation results indicate that approximate twof

22、old of the current human-to-human transmission rate or periodic outbreaks of avian influenza in poultry may induce an outbreak in human. Though the recent limited transmission potential of the novel avian influenza

23、A(H7N9) virus, a new outbreak may be possible due to virus mutation and adaption or periodic outbreaks in poultry, and hence careful surveillance and persistent intervention strategies in poultry have to be required.,Con

24、clusions,Data sources and Non-pharmaceutical interventions (Motivations),Conclusions and discussions,Model with interactions between the university community and the general population,Model among university c

25、ommunity,Fengxiao for mitigating the 2009 H1N1 pandemic in Xi’an city,A/H1N1 outbreak in mainland China,Until 3/31/2010, 0.127 (800)million confirmed cases, with 0.126million local cases and 1228 imported cases.,Th

26、e data on laboratory-confirmed cases of pandemic A/H1N1 influenza from beginning to the end of November were quite accurate. H1N1cases were under-reported since December.,Note:,Before Oct 2009, no death cases

27、 was reported,Daily number of hospital notifications of Shaanxi Province, China,Daily reported community and sporadic cases of Shaanxi Province, China,Newly reported cases from 8th hospital,Daily number of 8th hospital

28、notifications of Xi’an city,The country's experience with the 2003 SARS outbreak enabled the central government to quickly take a set of very strict nonpharmaceutical interventions (NPIs),Intensive contact tracing fo

29、llowed :,,Non-pharmaceutical interventions,by quarantine of suspected individuals who have the high risk of having been exposed to the virus,by isolation of symptomatic individuals,School closure and Fengxiao,Precauti

30、on: travel or hygiene precaution,Fengxiao (封校),Model among university community,Ref : Tang S. et al. (2010). Community based measures for mitigate the 2009 H1N1 pandemic in China. PLoS ONE,Flow diag

31、ram for the pandemic H1N1 (Model 1),When quarantine is implemented, a proportion of E1 is quarantined. These individuals move to the compartment QE1. Those in the QE1 class then progress to the QE2 and will be hospitaliz

32、ed once they develop symptoms.,When effective precautionary measures are taken, a proportion of the individuals exposed to the virus is protected from the infection,The Model 1,Control reproduction number,Meta-p

33、opulation Model (or patch model),,Spatially stratified compartment model,Extend our baseline model to a meta-population model, where coupling among patches is through dispersal on a dispersal network,Use this m

34、odel framework : The spread among a network of universities/colleges within in a city (Xi’an),To evaluate the effectiveness of NPIs and interactions of different spatially relevant interventions: Fengxiao,

35、 quarantine, precaution and mobility control.,,,Dispersal networks,The random network or small-world network introduced by Watts (Watts and Strogatz, 1998) is employed to generate matrix G with an average number of

36、 connections per vertex (degree) of four,Numerical integration for meta-population model,,Numerical integrations for the network models were carried out using the Runge-Kutta method in Matlab 7.0.,Dispersal rates: all

37、 simulations were initiated with pseudorandomly generated dispersal rates independently and identically distributed among all patches on the interval,Implementation of Fengxiao: we randomly generated the dispersal rates

38、 among communities from the interval to represent the Fengxiao, from the interval ( ) to describe weak (strong) dispersal.,Implementation of Fengxiao,Implementing Fengxia

39、o as well as strengthening local interventions in any university/college in terms of hospital notifications,How to trigger or suspend Fengxiao strategy?,Define the upper threshold of the hospital notifications so as to s

40、witch on Fengxiao strategy,Define the low threshold of the hospital notifications so as to switch off Fengxiao strategy,,h=4, no local control measures,h=14, with or without local control measures,,500 independent simula

41、tions are carried out, and mean sum of I class and H class are plotted,The effectiveness of Fengxiao and Local measures,(A-B ) Fengxiao alone. Magenta curve (without Fengxiao) ; Green curve (25,10); Blue curve (10,5),The

42、 effectiveness of Fengxiao and Local measures,(C-F) Magenta curve (local control only) ; Green /blue curve (local and Fengxiao),The effectiveness of Fengxiao and Local measures,(G-H) Relatively strong local control me

43、asures implemented when Fengxiao is suspended,The effectiveness of Fengxiao and local measures,,Early Fengxiao can delay the epidemic peak significantly,Late implementation of Fengxiao has little effect on the outbreak,T

44、he magnitudes of the outbreaks become weaker and weaker as Fengxiao and strengthening control measures are switched on and off, and the sooner the local control measures the less severe the outbreaks,Local control str

45、ategies affects the peak magnitudes while Fengxiao influences the peak timing and prevents the disease spread to the general population,Model with interactions between the university community and the general population

46、 Hospital notifications of 8th hospital of Xi’an city is employed in this part,Ref: Tang, S, Xiao, Y., Yuan, L., Cheke, R. A, Wu, J., Campus quarantine (Fengxiao) for curbing emergent infectious diseases: Lessons

47、 from mitigating A/H1N1 in Xi'an, China, J. Theor. Biol. 295(2012), 47-58.,,The model for the University population,Using the next generation matrix method,,Model Equations,,Parameter determination,Recovery rate for

48、 the symptomatic class,Recovery rate for the hospitalized class,Estimation of the reproduction number,,Model-based estimate of,The adaptive Metropolis-Hasting algorithm is employed to carry out the MCMC procedure, and af

49、ter a burn-in period of 500000 iterations the next 500000 samplers gives estimates (Haario 2006, Stat Comput),Parameter estimates based on data from 8th hospital and sensitivity to the duration,Range of reproduction num

50、ber and data fitting,The mean R0 for the period under consideration to lie in the range 1.273-1.784,,Model involving general population,,,,,,,University population,,,,,,,,,,,,,General population,Model involving gen

51、eral population,The effects of duration of Fengxiao and off-campus rate on H1N1 outbreak,LHS/PRCC uncertainty and sensitivity analysis,Conclusions and discussions,Effectiveness of NPIs (quarantine, isolation, hygiene pre

52、caution and Fengxiao ),Using model-based and likelihood-based methods and on the basis of Laboratory-confirmed data sets for Shaanxi province and 8th hospital of Xi’an city to estimate the control reproduction numbers,S

53、ensitivity analyses: effect of parameter changes on reproduction number and two wave patterns,Our findings suggest that prompt implementation of multiple NPIs is required to reduce a potential risk of new outbreak and