流感傳播和控制的數(shù)學(xué)模型研究ppt課件

1、流感傳播和控制的數(shù)學(xué)模型研究流感傳播和控制的數(shù)學(xué)模型研究1Introduction to myself研究方向：研究方向： 生物數(shù)學(xué)、傳染病動力學(xué)生物數(shù)學(xué)、傳染病動力學(xué)（1 1） 宏觀艾滋病病毒傳播與微觀宏觀艾滋病病毒傳播與微觀HIVHIV的進展研究的進展研究（2 2）流感（）流感（A/H1N1, H7N9A/H1N1, H7N9）的模型研究）的模型研究（3 3）數(shù)學(xué)工具：）數(shù)學(xué)工具：ODEODE、 DDEDDE、 IDEIDE、Non-smooth ODENon-smooth ODE23綱綱 要要 流感病毒簡介流感病毒簡介 流感的預(yù)防與控制問題的提出流感的預(yù)防與控制問題的提出 傳染病動力學(xué)簡

2、介傳染病動力學(xué)簡介 流感的數(shù)學(xué)模型研究流感的數(shù)學(xué)模型研究3什么是禽流感？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致病性禽流感病毒。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

3、死亡。至今由禽鳥傳人的禽流感有三種：甲型H5N1、甲型H7N7及甲型H9N2。 45什么是什么是H7N9H7N9禽流感病毒？禽流感病毒？流感病毒可分為甲（）、乙（）、丙（）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、日本及美國等地曾發(fā)生過禽間暴發(fā)疫情，但未發(fā)現(xiàn)過人的感染情況。6H7N9H7N9禽流感感染病例的主要

4、臨床表現(xiàn)？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。全球人感染甲型全球人感染甲型H7H7流感病毒情況？流感病毒情況？1996年2009年間，荷蘭、意大利、加拿大、美國和英國曾報道人感染甲型H7流感病例，病毒亞型分別為H7N2、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，我國從未發(fā)現(xiàn)過H7亞型流感病毒感染病例。7H7N9H7N9病毒與既往病毒與既往H1N1H1N1、H5N1H5N1和其他季

5、節(jié)性流感病例相比其毒力和其他季節(jié)性流感病例相比其毒力和傳染性如何？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，對該病毒及其所致疾病的研究資料十分有限，尚無法對該病毒的毒力和人際傳播的能力做出準(zhǔn)確判斷。本次人感染本次人感染H7N9H7N9禽流感的病例的感染來源？禽流感的病例的感染來源？此次人感染的H7N9流感病毒從病毒生物學(xué)上屬于禽源流感病毒，既往國際上所發(fā)現(xiàn)的人感染H7亞型的流感病毒也多來自于禽類，但截至目前，3例確診病例的具體感染來源尚不清楚。H7N9H7N9禽流感病毒是否能夠在人與人之間傳播？禽流感病毒是否能夠在人與人之間傳播？目前尚未證實該病毒具有人傳人的能力。8如

6、何預(yù)防如何預(yù)防H7N9H7N9流感？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，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直接接觸病死禽、畜。目前尚無針對H7N9禽流感病毒的疫苗。目前的治療方法？目前的治療方法？基因序列分析顯示，該病毒對神經(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，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。9 香港在1997年發(fā)生禽流感，有

7、6人死亡，當(dāng)局捕殺上百萬只家禽，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)，往往會造成家禽的大量死亡。潛伏期一般為3-5天。一般病程12天，癥狀變化很大。 病雞可能見有呼吸道癥狀，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，開始侵襲人類社會。10禽流感的潛在威脅可能遠大于禽流感的潛在威脅可能遠大于SARSSARS 首先是這種禽流感一旦變異后可能會成為普種禽流感一旦

8、變異后可能會成為普通人類流感病毒，而人體對于新的流感病毒幾乎通人類流感病毒，而人體對于新的流感病毒幾乎沒有任何免疫力。沒有任何免疫力。 其次，人類的流感病毒遠遠比非典病毒更具人類的流感病毒遠遠比非典病毒更具傳染性。因為流感病毒可以在空氣中迅速傳播傳染性。因為流感病毒可以在空氣中迅速傳播，而非典病毒則通常在近距離接觸后才會被傳染。1112亞洲的自然環(huán)境為病毒滋生和傳播提供便利亞洲的自然環(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)家在青藏高原沿青海湖、扎

9、林湖等候鳥遷徙路線對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)，同時候鳥遷徙路線與發(fā)生疫情的地點是重疊的。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)國家分布在亞洲南部，其中包括中國南方。這些地區(qū)氣溫高、濕度大、病毒生存機會高，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。13問題的提出問題的提出 傳染病能否在

10、某個地方傳播開來？能否形成地方病？傳染病能否在某個地方傳播開來？能否形成地方?。?傳染病高潮的什么時候來臨？傳染病的規(guī)模有多大？傳染病高潮的什么時候來臨？傳染病的規(guī)模有多大？ 什么預(yù)防與控制措施最為有效？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？能否給公共衛(wèi)生部門提供定量的建議？ 基于流感的傳播機理建立數(shù)學(xué)模型，利用數(shù)據(jù)估計系統(tǒng)的參數(shù)，對疫情的發(fā)展趨勢給予預(yù)測，分析人為的干預(yù)措施對疫情發(fā)展的影響，為制定控制方案提供定量的決策依據(jù)14傳染病模型簡介傳染病模型簡介方法：方法： 將人群分類，建立方程來描述每類人群的數(shù)量的變化將人群分類，建立方程來描述每類人群的數(shù)量的變化 S I

11、 RS(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）15ttNittitNstittiN)()()()()(SIR模型模型很?。┩ǔ?00)0(1rrsi無法求出無法求出 的解析解的解析解)(),(tsti在相平面在相平面 上上研究解的性質(zhì)研究解的性質(zhì)is ttitNststtsN)()()()(00)0(,)0(ssiisidtdsisidtdi160011iisdsdiss000ln1)()(sssissi模型模型400)0(,)0(ssiisidtdsisidtdi/消去消去dtSIR模型模型1,0,0),(isisisD相軌線相軌線 的

12、定義的定義域域)(si相軌線相軌線11si0D在在D內(nèi)作相軌線內(nèi)作相軌線 的圖形，進行分析的圖形，進行分析)(si17si101D相軌線相軌線 及其分析及其分析)(si00)0(,)0(ssiisidtdsisidtdi0011iisdsdiss000ln1)()(sssissi0ln1000sssiss滿足miis,/1傳染病蔓延傳染病蔓延傳染病不蔓延傳染病不蔓延s(t)單調(diào)減單調(diào)減相軌線的方相軌線的方向向0, itP1s0/1imsP1: s01/ i(t)先升后降至先升后降至0P2: s01/ i(t)單調(diào)降至單調(diào)降至01/ 閾值閾值P3P4P2S018Course of number

13、of S, I and R animals in a closed population 19預(yù)防傳染病蔓延的手段預(yù)防傳染病蔓延的手段 (日接觸率日接觸率) 衛(wèi)生水平衛(wèi)生水平 (日日治愈率治愈率) 醫(yī)療水平醫(yī)療水平 傳染病不蔓延的條件傳染病不蔓延的條件s01/ 降低降低 s0提高提高 r0 1000ris 提高閾值提高閾值 1/ 降低降低 (= / ) , 群體免疫群體免疫20Transmission between individuals R0 Basic Reproduction ratio （基本再生數(shù)）基本再生數(shù)） Average number of secondary cases c

14、aused by 1 infectious individual during its entire infectious period in a fully susceptible population 21Reproduction ratio, R0R0 = 3R0 = 0.522禽流感的數(shù)學(xué)模型禽流感的數(shù)學(xué)模型 S I R A E Sp Ip傳播框圖人群禽類23121122()()()SSYH SBYXcbSdXXYYBXYHHSBdHbm Y X，Y： denote the susceptible birds, the birds infected with the avian inf

15、luenzaS，B ：the susceptible humans and the humans that are infected with the wild avian influenza.H： denotes the humans infected with the mutant avian influenza;以往簡單模型以往簡單模型24H7N9 H7N9 模型建立模型建立Xiao Y., Sun X., Tang S., Wu J., Transmission potential of the novel avian influenza A(H7N9) infection in ma

16、inland China J. Theor. Biol. 352(2014)1525( )( )( )( )( )( )( )()( )( )pppppppppphhphhhphphhphhhhphhhhhhhhdStS IdtNdItS ImIdtNIdStIt SSdtNNIdEtIt SSEdtNNdItEIdtdDtIdtdRtIdt 模型建立模型建立101()01,( ),pk t TpptTtetTWith 26 As of April 26, 2013, the China Ministry of Agriculture reported that 68,060 bird and

17、 environmental specimens have been tested, 46 (0.07%) were confirmed H7N9-positive by culture We thus assume that I /N =0.07ppA fact: on prevalence in poultry ( )( )( )( )( )()hphhhhphhhhhhhdS tt SIdtdE tt SIEdtdI tEIdt 27數(shù)據(jù)數(shù)據(jù)28參數(shù)估計參數(shù)估計 Using the KaplanMeier (KM) method to data available, we obtaine

18、d the estimation for the mean time from the date of illness onset to death as13days,leading to disease-related death rate =1/13. recovery rate =1/ 11 29Effect of various Intervention timing and intensity30Prediction of the next outbreak 31Prediction of the next outbreak The periodic infection of pou

19、ltry may induce the second outbreak in human population.32 We estimate the reproduction number for human-to-human transmission as 0.467 (95% CI 0.387-0.651). Simulation results indicate that approximate twofold of the current human-to-human transmission rate or periodic outbreaks of avian influenza

20、in poultry may induce an outbreak in human. Though the recent limited transmission potential of the novel avian influenza 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 stra

21、tegies in poultry have to be required.Conclusions33000Theorem 1. If r1, then X and Y converge to X and 0,respectively, as t. On the other hand if r1, then Xand Y converge to X and Y respectively. 000,/rXXc bbm0000(,0,0,0),(,0,0,),bbbEXSEXSH00000Theorem 2. (i) If r1 and R1, then E is globally asympto

22、ticallystable (GAS) which means that the orbit converges to theequilibrium as t for arbitrary initial point.(ii) If r1 and R1, then Eb 0 is GAS.(iii) If r1, then E is GAS.34 Data sources and Non-pharmaceutical interventions (Motivations) Conclusions and discussions Model with interactions between th

23、e university community and the general population Model among university community Fengxiao for mitigating the 2009 H1N1 pandemic in Xian city35A/H1N1 outbreak in mainland China Until 3/31/2010, 0.127 (800)million confirmed cases, with 0.126million local cases and 1228 imported cases. The data on la

24、boratory-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 was reported3601020304050607080901001100200400600800100012001400From Sep 3rd to Dec 20thHospital notific

25、ationsDaily number of hospital notifications of Shaanxi Province, China370510152025303540455055020406080100120140From Sep 3rd to Oct 27thNewly reported casesCommunity casesSporadic casesDaily reported community and sporadic cases of Shaanxi Province, China38024681012141618200510152025303540Days afte

26、r 3rd September, 2009Daily reported cases8th hospital of Xian city0246810121416182005101520Days after 3rd September, 2009 (From 3rd-21st)Daily reported casesFour severe outbreak Universities/School XidianWenliKemaoFuzhongNewly reported cases from 8th hospital 3902468101214161820020406080100120140160

27、180200Days after 3rd September,2009Hospital notifications from 8th hospital of Xian cityDaily number of 8th hospital notifications of Xian city40The countrys experience with the 2003 SARS outbreak enabled the central government to quickly take a set of very strict nonpharmaceutical interventions (NP

28、Is)Intensive contact tracing followed :Non-pharmaceutical interventions by quarantine of suspected individuals who have the high risk of having been exposed to the virus by isolation of symptomatic individualsSchool closure and FengxiaoPrecaution: travel or hygiene precaution41Fengxiao (封校)There has

29、 been one A/H1N1 confirmed case in our University, university students, faculty, and staff members do not allow to leave their campuses, and disallow on-campus visits42Model among university community Ref : Tang S. et al. (2010). Community based measures for mitigate the 2009 H1N1 pandemic in China.

30、 PLoS ONE 43IsolatedQuarantinedSusceptible( )SExposed, notyet infectious1()EInfectiouswith symptoms( )IInfectiouswith symptoms()HRecovered()RInfectious, not yet symptomatic2()EInfectious, notyet symptomatic2()EQExposed, not yet infectious1()EQFlow diagram for the pandemic H1N1 (Model 1) When quarant

31、ine 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 hospitalized once they develop symptoms. When effective precautionary measures are taken, a proportion of the individuals exposed to the v

32、irus is protected from the infection 44111222221112112222311121223212(1)(),(1)()(),(),() ,epeEEpEEEEIE SSNIE SEq ENEEqEIEIQq EQQq EQQHQIHRIH The Model 1 451121231(1)(),()()()cepRFVqqControl reproduction number46121122121111111222212231111(1)(),(1)()(),(),(),ijijijijiiiniiiiijSijjiniiiieiijEijijinipi

33、ijEijijniiijIijjeiiEEEIESSS d DNIESEq EE dDNEEqEE dDIEII d DQq EQQq122212232121,iiijpiiEiEiEiiniiijRijjEQQHQIHRIHR d DMeta-population Model (or patch model)47Spatially stratified compartment model Extend our baseline model to a meta-population model, where coupling among patches is through dispersal

34、 on a dispersal network Use this model framework : The spread among a network of universities/colleges within in a city (Xian) To evaluate the effectiveness of NPIs and interactions of different spatially relevant interventions: Fengxiao, quarantine, precaution and mobility control.48Dispersal netwo

35、rks 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 connections per vertex (degree) of four49Numerical integration for meta-population model Numerical integrations for the network models were carried

36、out using the Runge-Kutta method in Matlab 7.0. Dispersal rates: all simulations were initiated with pseudorandomly generated dispersal rates independently and identically distributed among all patches on the interval (0, 2)h Implementation of Fengxiao: we randomly generated the dispersal rates amon

37、g communities from the interval to represent the Fengxiao, from the interval ( ) to describe weak (strong) dispersal.14(0, 2)9(0, 2)4(0, 2)50Implementation of FengxiaoImplementing Fengxiao as well as strengthening local interventions in any university/college in terms of hospital notificationsHow to

38、 trigger or suspend Fengxiao strategy? Define the upper threshold of the hospital notifications so as to switch on Fengxiao strategy Define the low threshold of the hospital notifications so as to switch off Fengxiao strategy51h=4, no local control measuresh=14, with or without local control measure

39、smaxHStrengtheningthe local controlmeasures minHRelaxing the local controlmeasures 520501001500100200300(E) (10,5)05010015005001000150020002500Time (days)(F)0501001500100200300(G) (10,5)05010015005001000150020002500Time (days)(H)0501001500200400600(C) (25,10)0501001500200040006000Time (days)(D)05010

40、015001234x 104Mean sum of Ii(t)(A)050100150024681012x 104Time (days)Mean sum of Hi(t)(B)500 independent simulations are carried out, and mean sum of I class and H class are plotted The effectiveness of Fengxiao and Local measures530501001500100200300(E) (10,5)05010015005001000150020002500Time (days)

41、(F)0501001500100200300(G) (10,5)05010015005001000150020002500Time (days)(H)0501001500200400600(C) (25,10)0501001500200040006000Time (days)(D)05010015001234x 104Mean sum of Ii(t)(A)050100150024681012x 104Time (days)Mean sum of Hi(t)(B)(A-B ) Fengxiao alone. Magenta curve (without Fengxiao) ; Green cu

42、rve (25,10); Blue curve (10,5)The effectiveness of Fengxiao and Local measures(C-F) Magenta curve (local control only) ; Green /blue curve (local and Fengxiao)540501001500100200300(E) (10,5)05010015005001000150020002500Time (days)(F)0501001500100200300(G) (10,5)05010015005001000150020002500Time (day

43、s)(H)0501001500200400600(C) (25,10)0501001500200040006000Time (days)(D)05010015001234x 104Mean sum of Ii(t)(A)050100150024681012x 104Time (days)Mean sum of Hi(t)(B)The effectiveness of Fengxiao and Local measures(G-H) Relatively strong local control measures implemented when Fengxiao is suspended55T

44、he effectiveness of Fengxiao and local measures Early Fengxiao can delay the epidemic peak significantly Late implementation of Fengxiao has little effect on the outbreak The magnitudes of the outbreaks become weaker and weaker as Fengxiao and strengthening control measures are switched on and off,

45、and the sooner the local control measures the less severe the outbreaks Local control strategies affects the peak magnitudes while Fengxiao influences the peak timing and prevents the disease spread to the general population56Model with interactions between the university community and the general p

46、opulation Hospital notifications of 8th hospital of Xian city is employed in this partRef: Tang, S, Xiao, Y., Yuan, L., Cheke, R. A, Wu, J., Campus quarantine (Fengxiao) for curbing emergent infectious diseases: Lessons from mitigating A/H1N1 in Xian, China, J. Theor. Biol. 295(2012), 47-58. 57The m

47、odel for the University populationUniversity populationuSuEuIuRquSquEquH58Using the next generation matrix method21(1)(1)u uuucqucR11211121212SuU UUEuU UUSuU UUEuU UUP S IquuNPS IuuNuuuQS IqquuNQS IqquNqqquuuuquuuSSEEIEISSEEHIEHRIH Model Equations59Parameter determination 0510150102030405060Duration

48、 of symptoms,daysNumber of hospital notifications(A)051015020406080100120140Duration of hospitalization,daysNumber of hospital notifications(B)Recovery rate for the symptomatic class 11/ 6.56Recovery rate for the hospitalized class 21/ 7.4860Estimation of the reproduction number cRModel-based estima

49、te of cRThe adaptive Metropolis-Hasting algorithm is employed to carry out the MCMC procedure, and after a burn-in period of 500000 iterations the next 500000 samplers gives estimates (Haario 2006, Stat Comput)The likelihood-based method using the following likelihood (White and Pagano (2008), Likel

50、ihood-based estimate of cR1exp()(,|)(1)tNTttcttL RpN61Parameter estimates based on data from 8th hospital and sensitivity to the duration6202468101214161820050100150200250Days after 3rd September 2009Hospital notifications and fitting(D)02468101214161820050100150200250Days after 3rd September 2009Ho

51、spital notifications and fitting(C)02468101214161820050100150200(B)Days after 3rd September 2009Hospital notifications and fitting02468101214161820050100150200(A)Days after 3rd September 2009Hospital notifications and fittingSep 3-Sep 18Sep 3-Sep 19Sep 3-Sep 20Sep 3-Sep 21Range of reproduction number and data fittingThe mean R0 for the period under consideration to lie in the range 1.273-1.78463Model involving general populationUniversity populationuSuEuIuRquSquEquHgSgEgIgRgHGeneral population64Model in