Outline Population heterogeneity, structure, Introduction: Heterogeneity and population structure Models for population structure and mixing Structure example: rabies in space Models for heterogeneity Jamie Lloyd-Smith • individual heterogeneity and superspreaders Center for Infectious Disease Dynamics • group-level heterogeneity Pennsylvania State University Population structure and mixing mechanisms Pair formation and STD transmission Heterogeneity and structure – what’s the difference? Modelling heterogeneity Group-level heterogeneity and multi-group models Tough to define, but roughly… Break population into sub-groups, each Heterogeneity describes differences among individuals or groups in of which is homogeneous. a population. (often assume that all groups mix randomly) Population structure describes deviations from random mixing in a However, epidemiological traits of each host individual are due to population, due to spatial or social factors. a complex blend of host, pathogen, and environmental factors, and often can’t be neatly divided into groups The language gets confusing: (or predicted in advance). - models that include heterogeneity in host age are called “age-structured”. Individual-level heterogeneity - models that include spatial structure where model parameters Allow continuous variation among individuals. differ through space are called “spatially heterogeneous”. Models for population structure Models for population structure Random mixing Multi-group Spatial mixing Random mixing or mean-field • Every individual in population has equal probability of contacting any Network Individual-based model other individual. • Mathematically simple – “mass action” formulations borrowed from chemistry – but often biologically unrealistic. • Sometimes basic β SI form is modified to power law β S a I b as a phenomenological representation of non-random mixing. 1
Models for population structure Models for population structure Multi-group Spatial mixing or metapopulation • Divides population into multiple discrete groupings, based on spatial or • Used when individuals are distributed (roughly) evenly in space. social differences. • Can model many ways: • To model transmission, need contact matrix or Who Acquires Infection From Whom (WAIFW) matrix: • continuous space models (e.g. reaction-diffusion or contact kernel) β ij = transmission rate from infectious individual in group i to • individuals as points on a lattice susceptible in group j • patch models (metapopulation with spatial mixing) • Or use only within-group transmission (so β ij =0 when i ≠ j ), and • Used to study travelling waves, spatial control programs, influences of explicitly model movement among groups. restricted mixing on disease invasion and persistence Models for population structure Models for population structure Social network Individual-based model (IBM) or microsimulation model • Precise representation of contact structure within a population • The most flexible framework. • “Nodes” are individuals and “edges” are contacts • Every individual in the model carries its own attributes (age, sex, • Important decisions: Binary vs weighted? Undirected vs directed? location, contact behaviour, etc etc) Static vs dynamic? • Can represent arbitrarily complex systems ( = realistic?) and ask detailed • Basic network statistics include degree distribution (number of edges per questions, but difficult to estimate parameters and to analyze node) and clustering coefficient (How many of my friends are model output; also difficult for others to replicate the model. friends with each other?) • STDSIM is a famous example, used to study transmission and control of • Powerful tools of discrete mathematics can be applied. sexually transmitted diseases including HIV in East Africa. Rabies in space Major Terrestrial Reservoirs of Rabies in the United States Rabies is an acute viral disease of mammals, that causes cerebral dysfunction, anxiety, confusion, agitation, progressing to delirium, abnormal behavior, hallucinations, and insomnia. Raccoon Raccoon Transmitted by infected saliva, most commonly through biting. Latent period = 3 – 12 weeks (in raccoons) Infectious period = 1 week (ends in death) Pre-exposure vaccination offers effective protection. • Until mid 1970s, raccoon rabies was restricted to FL and GA. Post-exposure vaccination possible during latent period. • Then rabid raccoons were translocated to the WV-VA border, and a major epidemic began in the NE states. 2
Models of the spatial spread of rabies Spatial invasion of Rabies across the Northeastern U.S. 23 years (1977-1999) Rabies in wildlife in US Simple patch model (+ small long-range transmission term) was able to fit data well. Smith et al (2002) PNAS 99: 3668-3672 Russell et al (2004) Proc Roy Soc B 271: 21-25. The vital few and insignificant many – the 20/80 rule: Individual heterogeneity 100 Macroparasitic diseases: % Transmission potential (R 0 ) Heterogeneity in the worm burdens in individuals are overdispersed, and well- population 20-80 80 described by a negative binomial distribution. Transmission potential 60 Every host is equal STDs and vector-borne diseases: 20-20 Woolhouse et al (PNAS, 1998) analyzed contact rate data and 40 proposed a general 20/80 rule: 20 % of hosts account for 80 % 20% of hosts are responsible for 80% of transmission 20 of pathogen transmission But how to approach other directly-transmitted diseases , 0 for which contacts are hard to define? 0 20 40 60 80 100 Percentage of host population Percentage of host population Slide borrowed from Sarah Perkins Individual reproductive number, ν A model for individual heterogeneity Expected number of cases caused by a particular infectious Basic reproductive number, R 0 individual in a susceptible population. Expected number of cases caused by a typical infectious individual in a susceptible population. Z = actual number of cases caused by a particular infectious individual. Individual reproductive number, ν Expected number of cases caused by a particular infectious Z ~ Poisson( ν ) Stochasticity in transmission � individual in a susceptible population. The offspring distribution defines Pr ( Z = j ) for all j. ν varies continuously among individuals, with population mean R 0 . R 0 ν Z = 0 Z = 1 Z = 2 Z = 3 … ν 3
Branching process : a stochastic model for disease invasion Contact tracing for SARS Observed offspring distribution into a large population. Number of secondary cases, Z For any offspring distribution, it tells you: • Pr(extinction) Estimated distribution of individual reproductive number, ν • Expected time of extinction and number of cases • Growth rate of major outbreak ν Singapore SARS outbreak, 2003 Monkeypox SARS, Beijing Smallpox What about other emerging diseases? Pneumonic plague Hantavirus* Variola minor Dynamic effects: stochastic extinction of disease k = 0.1 Probability of extinction greater heterogeneity k = 1 k →∞ Basic reproductive number, R 0 Read more about individual heterogeneity and superspreading in Lloyd-Smith et al (2005) Nature 438: 355-359. 4
The R -matrix or next-generation matrix Transmission: mechanisms matter Generalized R 0 for a multi-group population Transmission dynamics are the core of epidemic models R ij = E(# cases caused in group j |infected in group i ) Usual approach considers group membership as static. � Take time to think about the mechanisms underlying D i = expected infectious period, spent entirely in group i transmission, and to find the best tradeoff between model β ij = transmission rate from group i to group j simplicity and biological realism. The expected number of cases in group j caused by an individual infected in group i is then: R ij = D i β ij e.g. Between-group transmission in metapopulations But what if the host moves and transmission is strictly local? Frequency-dependent transmission vs pair-formation models D ij = expected time spent in group j by individual infected in group i , while still infectious β j = transmission rate within group j R ij = D ij β j Now Transmission in a metapopulation Analytic approach to R If movement rules are Markovian, so p ij = Pr(move from group i to group j ): x m j = Pr(recover or die while in group j ) The process can be described by an absorbing Markov chain, x x x with overall transition matrix: x ⎡ ⎤ P m × n n x ⎢ ⎥ x ⎣ 0 1 ⎦ The expected residence times D ij are then given by the fundamental matrix: Simulate: = − − 1 D (I P) • range of multi-group population structures • acute and chronic diseases R ij = D ij β j R-matrix: R 0 = 20 1 Acute disease Chronic disease R 0 = 10 Total proportion infected 0.8 R 0 = 5 0.6 0.4 R 0 = 2 0.2 1 group of 1000 25 groups of 40 100 groups of 10 0 −3 −2 −1 0 1 10 10 10 10 10 Acute and chronic diseases movement rate/recovery rate with same R 0 behave very differently when R 0 does not predict invasion for this system! invading a metapopulation. 5
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