流感传播和控制的数学模型研究(1)

1、1 流感传播和控制的数学模型研究流感传播和控制的数学模型研究 2 Introduction to myself 研究方向：研究方向： 生物数学、传染病动力学生物数学、传染病动力学 （1 1） 宏观艾滋病病毒传播与微观宏观艾滋病病毒传播与微观HIVHIV的进展研究的进展研究 （2 2）流感（）流感（A/H1N1, H7N9A/H1N1, H7N9）的模型研究）的模型研究 （3 3）数学工具：）数学工具：ODEODE、 DDEDDE、 IDEIDE、Non-smooth ODENon-smooth ODE 33 纲纲 要要 流感病毒简介流感病毒简介 流感的预防与控制问题的提出流感的预防与控制问题的

2、提出 传染病动力学简介传染病动力学简介 流感的数学模型研究流感的数学模型研究 4 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。 禽流感病毒可分为高致病性禽流感病毒、低致病性禽流感病毒 和无致病性禽流感病毒。高致病性禽流感病毒目前只发现H5和 H7两种亚型。由于种属屏障，禽流感病毒只在偶然的情况可以 感染人，既往确认感染人的禽流感病毒有H5N1、H9N2、 H7N2、 H7N3、 H7N7、H5N2、H10N7，症状表现各不相同，可以表现为 呼吸道症状、结膜炎、甚至死亡。人感染高致病性H5N1禽流感 病毒后常表现为高热等呼吸道症状，往往很快发展成肺炎，甚 至

3、急性呼吸窘迫综合症和全身器官衰竭，甚至死亡。 至今由禽鸟传人的禽流感有三种：甲型H5N1、甲型H7N7及甲型H9N2。 5 6 什么是什么是H7N9H7N9禽流感病毒？禽流感病毒？ 流感病毒可分为甲（）、乙（）、丙（）三型。其中， 甲型流感依据流感病毒血凝素蛋白（HA）的不同可分为1-16 种亚型，根据病毒神经氨酸酶蛋白（NA）的不同可分为1-9种 亚型，HA不同亚型可以与NA的不同亚型相互组合形成多达144 种不同的流感病毒。而禽类特别是水禽是所有这些流感病毒 的自然宿主，H7N9禽流感病毒是其中的一种。H7N9亚型流感 病毒既往仅在禽间发现，在荷兰、日本及美国等地曾发生过 禽间暴发疫情，但

4、未发现过人的感染情况。 7 H7N9H7N9禽流感感染病例的主要临床表现？禽流感感染病例的主要临床表现？ 目前3例确诊病例主要表现为典型的病毒性肺炎，起病 急，病程早期均有高热（38以上），咳嗽等呼吸道感 染症状。起病5-7天出现呼吸困难，重症肺炎并进行性 加重，部分病例可迅速发展为急性呼吸窘迫综合症并死 亡。 全球人感染甲型全球人感染甲型H7H7流感病毒情况？流感病毒情况？ 1996年2009年间，荷兰、意大利、加拿大、美国和英 国曾报道人感染甲型H7流感病例，病毒亚型分别为H7N2、 H7N3和H7N7，临床表现主要为结膜炎与轻型的上呼吸道 感染。此前，我国从未发现过H7亚型流感病毒感染病

5、例。 8 H7N9H7N9病毒与既往病毒与既往H1N1H1N1、H5N1H5N1和其他季节性流感病例相比其毒力和其他季节性流感病例相比其毒力 和传染性如何？和传染性如何？ 由于目前只发现3例人感染H7N9禽流感病毒确诊病例，对该病 毒及其所致疾病的研究资料十分有限，尚无法对该病毒的毒力 和人际传播的能力做出准确判断。 本次人感染本次人感染H7N9H7N9禽流感的病例的感染来源？禽流感的病例的感染来源？ 此次人感染的H7N9流感病毒从病毒生物学上属于禽源流感病毒， 既往国际上所发现的人感染H7亚型的流感病毒也多来自于禽类， 但截至目前，3例确诊病例的具体感染来源尚不清楚。 H7N9H7N9禽流感

6、病毒是否能够在人与人之间传播？禽流感病毒是否能够在人与人之间传播？ 目前尚未证实该病毒具有人传人的能力。 9 如何预防如何预防H7N9H7N9流感？流感？ 流感是一种急性呼吸道感染性疾病。勤洗手、室内勤通风 换气、注意营养、保持良好体质有利于预防流感等呼吸道 传染病。出现打喷嚏、咳嗽等呼吸道感染症状时，要用纸 巾、手帕掩盖口鼻，预防感染他人。此外，还要特别注意 尽量避免直接接触病死禽、畜。目前尚无针对H7N9禽流感 病毒的疫苗。 目前的治疗方法？目前的治疗方法？ 基因序列分析显示，该病毒对神经氨酸酶抑制剂类抗流感 病毒药物敏感。根据其他型别流感抗病毒治疗的经验，发 病后早期使用神经氨酸酶抑制剂

7、类抗流感病毒药物可能是 有效的，但对人类新发现的H7N9禽流感病毒感染的特异性 治疗手段仍需观察研究。 10 香港在1997年发生禽流感，有6人死亡，当局捕杀上 百万只家禽，避免疫情恶化。 禽流感是一种主要流行于鸡群中的烈性传染病，一 旦爆发，往往会造成家禽的大量死亡。潜伏期一般为 3-5天。一般病程12天，症状变化很大。 病鸡可能见有呼吸道症状，如打喷嚏、窦炎和结膜 炎。病鸡头部常出现水肿，可能同时出现或不出现腹 泻；病鸡体温升高，羽毛蓬松，鸡冠发绀。有的腿变 红，鼻分泌物增多，呼吸极度困难，甩头，严重地可 窒息死亡。产蛋率明显下降。 然而近年来情况却悄悄地发生了变化，禽流感的传 播已经跨越

8、了原先的范围，开始侵袭人类社会。 11 禽流感的潜在威胁可能远大于禽流感的潜在威胁可能远大于SARSSARS 首先是这种禽流感一旦变异后可能会成为普种禽流感一旦变异后可能会成为普 通人类流感病毒，而人体对于新的流感病毒几乎通人类流感病毒，而人体对于新的流感病毒几乎 没有任何免疫力。没有任何免疫力。 其次，人类的流感病毒远远比非典病毒更具人类的流感病毒远远比非典病毒更具 传染性。因为流感病毒可以在空气中迅速传播传染性。因为流感病毒可以在空气中迅速传播， 而非典病毒则通常在近距离接触后才会被传染。 12 13 亚洲的自然环境为病毒滋生和传播提供便利亚洲的自然环境为病毒滋生和传播提供便利 亚洲是多山

9、地区，特别是东南亚地区遍布雨林，中部亚洲则 高原山脉连绵不断，而这些地带又恰恰位于全球鸟类迁徙的路线 上。 候鸟是禽流感病毒的重要传播者。2005年，科学家在青藏高 原沿青海湖、扎林湖等候鸟迁徙路线对禽流感疫情做了跟踪调查， 发现每年候鸟因繁殖、越冬而迁徙前后恰好是高致病性禽流感疫 情发生季节，同时候鸟迁徙路线与发生疫情的地点是重叠的。研 究组还研究了不同种类、分布在不同地区，可能传播高致病性禽 流感病毒的候鸟，发现候鸟迁徙路径中的沼泽和湖泊是高致病性 禽流感存在和传播的重要地带。带有病毒的野鸟在迁徙的路途中 很容易和散养放养的家禽接触，将病毒传给亚洲那些位于鸟类迁 徙路线上的家禽。 另外，亚

10、洲大部分农业国家分布在亚洲南部，其中包括中国南方。 这些地区气温高、湿度大、病毒生存机会高，再加上一些农村生活条件 和卫生环境恶劣，容易成为各种病毒滋生的温床。 14 问题的提出问题的提出 传染病能否在某个地方传播开来？能否形成地方病？传染病能否在某个地方传播开来？能否形成地方病？ 传染病高潮的什么时候来临？传染病的规模有多大？传染病高潮的什么时候来临？传染病的规模有多大？ 什么预防与控制措施最为有效？什么预防与控制措施最为有效？ 能否给公共卫生部门提供定量的建议？能否给公共卫生部门提供定量的建议？ 基于流感的传播机理建立数学模型，利用数据估计系统的 参数，对疫情的发展趋势给予预测，分析人为的

11、干预措施对疫 情发展的影响，为制定控制方案提供定量的决策依据 15 传染病模型简介传染病模型简介 方法：方法： 将人群分类，建立方程来描述每类人群的数量的变化将人群分类，建立方程来描述每类人群的数量的变化 S I R S(t)：易感者的数量（或比例） I(t)： 感染者的数量（或比例） R(t)：移除者的数量（或比例） 16 ttNittitNstittiN)()()()()( SIR模型模型 很小）通常 000 )0(1rrsi 无法求出无法求出 的解析解的解析解 )(),(tsti 在相平面在相平面 上上 研究解的性质研究解的性质 is ttitNststtsN)()()()( 00 )0

12、(,)0(ssii si dt ds isi dt di 17 0 0 1 1 ii sds di ss 0 00 ln 1 )()( s s sissi 模型模型4 00 )0(,)0(ssii si dt ds isi dt di / 消去消去dt SIR模型模型 1,0,0),(isisisD 相轨线相轨线 的定义的定义 域域 )(si 相轨线相轨线 1 1 s i 0 D 在在D内作相轨线内作相轨线 的图形，进行分析的图形，进行分析 )(si 18 s i 1 01 D 相轨线相轨线 及其分析及其分析)(si 00 )0(,)0(ssii si dt ds isi dt di 0 0

13、1 1 ii sds di ss 0 00 ln 1 )()( s s sissi 0ln 1 0 00 s s siss 满足 m iis,/1 传染病蔓延传染病蔓延 传染病不蔓延传染病不蔓延 s(t)单调减单调减相轨线的方相轨线的方 向向 0, it P1 s0 /1 im s P1: s01/ i(t)先升后降至先升后降至 0 P2: s01/ i(t)单调降至单调降至0 1/ 阈值阈值 P3 P4 P2 S0 19 Course of number of S, I and R animals in a closed population 20 预防传染病蔓延的手段预防传染病蔓延的手段

14、(日接触率日接触率) 卫生水平卫生水平 (日日治愈率治愈率) 医疗水平医疗水平 传染病不蔓延的条件传染病不蔓延的条件s01/ 降低降低 s0 提高提高 r0 1 000 ris 提高阈值提高阈值 1/ 降低降低 (= / ) , 群体免疫群体免疫 21 Transmission between individuals R0 Basic Reproduction ratio （基本再生数）基本再生数） Average number of secondary cases caused by 1 infectious individual during its entire infectious p

15、eriod in a fully susceptible population 22 Reproduction ratio, R0 R0 = 3 R0 = 0.5 23 禽流感的数学模型禽流感的数学模型 S I R A E Sp Ip 传播框图 人群 禽类 24 12 11 22 () ( ( ) ( ) ) SSYH S BY Xcb Sd XXY Y B XY HHSBdH bm Y X，Y： denote the susceptible birds, the birds infected with the avian influenza S，B ：the susceptible huma

16、ns and the humans that are infected with the wild avian influenza. H： denotes the humans infected with the mutant avian influenza; 以往简单模型以往简单模型 25 H7N9 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 mainland China J. Theor. Bio

17、l. 352(2014)15 26 ( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ppp p ppp p p p hh phhh ph p hh phhhh ph h hh h h h h dStS I dtN dItS I mI dtN I dStI t SS dtNN I dEtI t SSE dtNN dIt EI dt dDt I dt dRt I dt 模型建立模型建立 1 01 () 01 , ( ) , p k t Tp p tT t etT With 27 As of April 26, 2013, the China Ministry of A

18、griculture reported that 68,060 bird and environmental specimens have been tested, 46 (0.07%) were confirmed H7N9- positive by culture We thus assume that I /N =0.07 pp A fact: on prevalence in poultry ( ) ( ) ( ) ( ) ( ) () h phhh h phhhh h hh dS t t SI dt dE t t SIE dt dI t EI dt 28 数据数据 29 参数估计参数

19、估计 Using the KaplanMeier (KM) method to data available, we obtained 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 30 Effect of various Intervention timing and intensity 31 Prediction of the next out

20、break 32 Prediction of the next outbreak The periodic infection of poultry may induce the second outbreak in human population. 33 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 hu

21、man-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 A(H7N9) virus, a new outbreak may be possible due to virus mutation and adaption or periodic outbreaks in

22、poultry, and hence careful surveillance and persistent intervention strategies in poultry have to be required. Conclusions 34 00 0 Theorem 1. If r1, then X and Y converge to X and 0, respectively, as t. On the other hand if r1, then X and Y converge to X and Y respectively. 000 ,/rXXc b bm 0000 (,0,

23、0,0),(,0,0,), bbb EXSEXSH 000 00 Theorem 2. (i) If r1 and R1, then E is globally asymptotically stable (GAS) which means that the orbit converges to the equilibrium as t for arbitrary initial point. (ii) If r1 and R1, then E b 0 is GAS. (iii) If r1, then E is GAS. 35 Data sources and Non-pharmaceuti

24、cal interventions (Motivations) Conclusions and discussions Model with interactions between the university community and the general population Model among university community Fengxiao for mitigating the 2009 H1N1 pandemic in Xian city 36 A/H1N1 outbreak in mainland China Until 3/31/2010, 0.127 (80

25、0)million confirmed cases, with 0.126million local cases and 1228 imported cases. The 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 was rep

26、orted 370102030405060708090100110 0 200 400 600 800 1000 1200 1400 From Sep 3rd to Dec 20th Hospital notifications Daily number of hospital notifications of Shaanxi Province, China 38 0510152025303540455055 0 20 40 60 80 100 120 140 From Sep 3rd to Oct 27th Newly reported cases Community cases Spora

27、dic cases Daily reported community and sporadic cases of Shaanxi Province, China 39 02468101214161820 0 5 10 15 20 25 30 35 40 Days after 3rd September, 2009 Daily reported cases 8th hospital of Xian city 02468101214161820 0 5 10 15 20 Days after 3rd September, 2009 (From 3rd-21st) Daily reported ca

28、ses Four severe outbreak Universities/School XidianWenliKemaoFuzhong Newly reported cases from 8th hospital 40 02468101214161820 0 20 40 60 80 100 120 140 160 180 200 Days after 3rd September,2009 Hospital notifications from 8th hospital of Xian city Daily number of 8th hospital notifications of Xia

29、n city 41 The countrys experience with the 2003 SARS outbreak enabled the central government to quickly take a set of very strict nonpharmaceutical interventions (NPIs) Intensive contact tracing followed : Non-pharmaceutical interventions by quarantine of suspected individuals who have the high risk

30、 of having been exposed to the virus by isolation of symptomatic individuals School closure and Fengxiao Precaution: travel or hygiene precaution 42 Fengxiao (封校) There has been one A/H1N1 confirmed case in our University, university students, faculty, and staff members do not allow to leave their c

31、ampuses, and disallow on-campus visits 43 Model among university community Ref : Tang S. et al. (2010). Community based measures for mitigate the 2009 H1N1 pandemic in China. PLoS ONE 44 IsolatedQuarantined Susceptible ( )S Exposed, not yet infectious 1 ()E Infectious with symptoms ( )I Infectious w

32、ith symptoms ()H Recovered ()R Infectious, not yet symptomatic 2 ()E Infectious, not yet symptomatic 2 () E Q Exposed, not yet infectious 1 () E Q Flow diagram for the pandemic H1N1 (Model 1) When quarantine is implemented, a proportion of E1 is quarantined. These individuals move to the compartment

33、 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 virus is protected from the infection 45 1 1 12 2 2 2 2 111 21122 2231 11 212 232 12 (1)() , (1)(

34、) (), (), () , , , , , e p eE E pEE E E IE S S N IE S Eq E N EEqE IEI Qq EQ Qq EQQ HQIH RIH The Model 1 46 1 12 1231 (1) (), ()() () c ep RFV qq Control reproduction number 47 1 2 1 1 2 2 1 2 111 1 1 11222 2 1 2231 1 11 (1)() , (1)() (), (), (), , ij ij ij ij i i i n iiii ijSij j i n iiii eiijEij i

35、j i n ipiijEij i j n iiijIij j eiiE E E IES SS d D N IES Eq EE dD N EEqEE dD IEII d D Qq EQ Qq 12 2 212 232 12 1 , , , i i ij piiEiE iEii n iiijRij j EQQ HQIH RIHR d D Meta-population Model (or patch model) 48 Spatially stratified compartment model Extend our baseline model to a meta-population mode

36、l, where coupling among patches is through dispersal 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, quarantin

37、e, precaution and mobility control. 49 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 connections per vertex (degree) of four 50 Numerical integration for meta-population model Num

38、erical integrations for the network models were carried 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 o

39、f Fengxiao: we randomly generated the dispersal rates among 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) 51 Implementation of Fengxiao Implementing Fengxiao as well as strengthening local interventions

40、 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 switch on Fengxiao strategy Define the low threshold of the hospital notifications so as to switch off Fengxiao strategy 52 h=4

41、, no local control measures h=14, with or without local control measures max H Strengthening the local control measures min H Relaxing the local control measures 53 050100150 0 100 200 300 (E) (10,5) 050100150 0 500 1000 1500 2000 2500 Time (days) (F) 050100150 0 100 200 300 (G) (10,5) 050100150 0 5

42、00 1000 1500 2000 2500 Time (days) (H) 050100150 0 200 400 600 (C) (25,10) 050100150 0 2000 4000 6000 Time (days) (D) 050100150 0 1 2 3 4 x 104 Mean sum of Ii(t) (A) 050100150 0 2 4 6 8 10 12 x 104 Time (days) Mean sum of Hi(t) (B) 500 independent simulations are carried out, and mean sum of I class

43、 and H class are plotted The effectiveness of Fengxiao and Local measures 54 050100150 0 100 200 300 (E) (10,5) 050100150 0 500 1000 1500 2000 2500 Time (days) (F) 050100150 0 100 200 300 (G) (10,5) 050100150 0 500 1000 1500 2000 2500 Time (days) (H) 050100150 0 200 400 600 (C) (25,10) 050100150 0 2

44、000 4000 6000 Time (days) (D) 050100150 0 1 2 3 4 x 10 4 Mean sum of Ii(t) (A) 050100150 0 2 4 6 8 10 12 x 10 4 Time (days) Mean sum of Hi(t) (B) (A-B ) Fengxiao alone. Magenta curve (without Fengxiao) ; Green curve (25,10); Blue curve (10,5) The effectiveness of Fengxiao and Local measures (C-F) Ma

45、genta curve (local control only) ; Green /blue curve (local and Fengxiao) 55 050100150 0 100 200 300 (E) (10,5) 050100150 0 500 1000 1500 2000 2500 Time (days) (F) 050100150 0 100 200 300 (G) (10,5) 050100150 0 500 1000 1500 2000 2500 Time (days) (H) 050100150 0 200 400 600 (C) (25,10) 050100150 0 2

46、000 4000 6000 Time (days) (D) 050100150 0 1 2 3 4 x 10 4 Mean sum of Ii(t) (A) 050100150 0 2 4 6 8 10 12 x 10 4 Time (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 suspended 56 The effectivenes

47、s 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, and the sooner

48、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 population 57 Model with interactions between the university community and the general population Hos

49、pital notifications of 8th hospital of Xian 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 from mitigating A/H1N1 in Xian, China, J. Theor. Biol. 295(2012), 47-58. 58 The model for th

50、e University population University population u S u E u I u R q u S q u E q u H 59 Using the next generation matrix method 21 (1)(1) u uuu cqu c R 1 121 11 212 12 SuU U U EuU U U SuU U U EuU U U P S Iq uuN PS I uuN uuu QS Iqq uuN QS Iqq uN qqq uuuu q uuu SS EE IEI SS EE HIEH RIH Model Equations 60 P

51、arameter determination 051015 0 10 20 30 40 50 60 Duration of symptoms,days Number of hospital notifications (A) 051015 0 20 40 60 80 100 120 140 Duration of hospitalization,days Number of hospital notifications (B) Recovery rate for the symptomatic class 1 1/ 6.56 Recovery rate for the hospitalized

52、 class 2 1/ 7.48 61 Estimation of the reproduction number c R Model-based estimate of c R The 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 lik

53、elihood-based method using the following likelihood (White and Pagano (2008), Likelihood-based estimate of c R 1 exp() (,|) (1) t N T tt c t t L Rp N 62 Parameter estimates based on data from 8th hospital and sensitivity to the duration 63 02468101214161820 0 50 100 150 200 250 Days after 3rd Septem

54、ber 2009 Hospital notifications and fitting (D) 02468101214161820 0 50 100 150 200 250 Days after 3rd September 2009 Hospital notifications and fitting (C) 02468101214161820 0 50 100 150 200 (B) Days after 3rd September 2009 Hospital notifications and fitting 02468101214161820 0 50 100 150 200 (A) D

55、ays after 3rd September 2009 Hospital notifications and fitting Sep 3-Sep 18Sep 3-Sep 19 Sep 3-Sep 20 Sep 3-Sep 21 Range of reproduction number and data fitting The mean R0 for the period under consideration to lie in the range 1.273-1.784 64 Model involving general population University population u S u E u I u R q u S q u E q u H g S g Eg I g R g H General population 65 Model involving general population 66 0102030405060708090100110 0 200 400 6