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## Network Models in Infectious Disease Ecology

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**Network Models in Infectious Disease Ecology**Rowland Kao X5770 rowland.kao@glasgow.ac.uk**How do network models differ from spatial models?**Harry Beck - 1933 Spatial models – what is the physical distance? Data required is usually more robust Network models – what is the epidemiological distance? Data required is usually more appropriate Often but not always closely related**What do we mean by ‘network processes’**• Explicit contact structure • Spatial models are a ‘subset’ of network models (sessile populations) Note: Infectious diseases are a ‘model system’ for ecological processes and the principles apply more broadly**(Static) Network Interpretation of Disease Transmission**Index Case Important for contact tracing Social Network of “Potentially Infected” Nodes/Individuals At Risk but not infected Not at risk Infected**Index Case**The network-based perspective Parsimonious (ODE) Models Networks Micro-simulation Increasing complexity • ODEs (dynamics) • Micro-simulation • Networks • (population structure) different paradigms**Topics for this lecture**• Impact of the network perspective on our understanding of person-to-person transmission • Percolation concepts • “Small-world” networks • “Scale-free” networks • Dynamic networks (time permitting)**Point I. Networks are very different from classic models of**disease transmission Interpreting r0 on networks**Compartmental models of disease**• Assume all individuals can be classified as in a distinct ‘disease state’ E.g. • Susceptible – Infectious – Removed (SIR) • Susceptible – Exposed (not infectious) – Infectious (SEI) • SI, SEIR, etc**Susceptible/Infected/Removed(SIR) Compartmental Model**birth death S transmission death I removal death R**Basic Reproduction Number (R0)**• The fundamental quantity of theoretical epidemiology • Definition (in words): The mean number of secondary infections that result from the introduction of a single infected individual into an otherwise completely susceptible homogeneously mixing population in equilibrium.**The Basic Reproduction Ratio - II**• When R0 is below one, disease cannot be sustained • For most reasonable epidemiological circumstances, this is globally stable unstable locally stable “globally” stable (approx)**The Basic Reproduction Ratio**I E E=0, I=0 is a “globally” stable equilibrium**The Basic Reproduction Ratio**I E=0, I=0 is an Unstable equilibrium E locally stable E=40, I=20 (approx.) is a “globally” stable equilibrium**The Basic Reproduction Ratio**I e.g. logistical limitations Two locally stable solutions E**R0 for the SIR model**Rate of new infections when I=1, S=N Average lifespan of an infected individual**R0 for individual-based SIR model**Same form as for frequency dependent transmission Rate of new infections when I=1, Slocal=k=6 Average lifespan of an infected individual NO! Is this reasonable?**Problem 1: Local saturation**• Compare frequency dependent model (fixed number of contacts but always different) to a spatial model (fixed number of contacts, always the same) where there are 6 neighbours, all equally susceptible • Fixed rate ‘t=1/6’ per day of each ‘neighbour’ becoming infected • Assume the first infected location remains infectious for three days**What happens? Stochastic difference model**Frequency dependent model (assume N=100) • Day one – mean of one infected • Day two – one of 100 already infected, therefore mean of 0.99 (or 99/100) becomes infected • Day three mean of 1.99 of 100 already infected, therefore mean of 0.981 becomes infected • r0 = 2.979 (Individual-based) Spatial model • Day one – mean of one infected • Day two – one of six already infected, therefore mean of approx. 0.833 (or 5/6) becomes infected • Day three, mean of 1.833 of 6 already infected, therefore mean of 0.6944 becomes infected • r0 = 2.527**More formally**Consider exponentially distributed infectious periods (as in ODE models), with transmission to each neighbour at rate t and reversion to susceptibility at rate g. The probability that an infected individual (assume infected at t=0) remains infected at time t is H(t) with probability density function h(t) that it is removed at time t: Similarly, the probability that a susceptible neighbour remains susceptible at time t is S(t) with probability density function s(t) where If there are exactly k connections, then the expected number of secondary infections is:**What difference does it make?**For the mean field model The individual-based term can be written as: See Keeling & Grenfell (2000) When the infection rate is twice the recovery rate, r0 is reduced by 65%**Can you correct for this?**• This can be rewritten as a correction factor • Creates ‘correct’r0, however: } corrected k= 5, 10, 20 mean field } uncorrected k= 5, 10, 20 From Green, Kiss, Kao 2006**Problem 2: Overlapping generations**• If generations of infection overlap, this can also reduce R0 • Assume an infected premises is infectious for two days (discrete model) • Any premises infected on day 1 reduces R0. Day 2 Day 2 Day 1**Problem 3: R0 and the contact matrix description of a**population**One could write down a ‘next generation matrix’ and**calculate ‘R0‘**Problems with NGM interpretation of the Contact Matrix**• Matrix entries > 1 (what does it mean?) • Bottlenecks (no exponential growth phase) • Need to have meaningfully large subpopulations to make NGM work Initial infection 2 Dead end 2**Percolation**• A better way of looking at (disease) population persistence in spatial and network models is as a percolation phenomenom**Percolation in spatial models**• Coffee percolator • Creation of large scale phenomena from simple structures • Cellular automata • Porous solids**What do we mean by a large epidemic?**• Percolation: formation of large scale structures from small elements • Networks are about transmission of “information” • Gossip • the Internet • biomass • Epidemics**Percolation**Day 1**Percolation**Day 2**Percolation**Day 3**Percolation**Day 4**Percolation**Day 5 The Largest Component (Patch of Mould) Spans the Popn.**Percolation and transmission network interpretation**• Identify weighted probability of transmission between premises and “thin” network • Strength of link (e.g. number of livestock moved) • Vulnerability to and potential for transmission (e.g. size of farm, species mix) • Reduces complicated systems to a simple analytically tractable unweighted, directed network • Stochastically generated (requires multiple realisations) • BUT hides transmission dynamics**Giant Weakly Connected Component**Other Giant Strongly Connected Component Giant Connected Components Transmission (NOT social) network Upper/lower estimates of final epidemic size (if disease enters the GSSC)**Percolation Interpretation of R0**• Below percolation threshold, GSCC size (NGSCC) fixed w.r.t. total population size (Npop), i.e. • Above percolation threshold Percolation Threshold Large Epidemic R0 =1 Sometimes!**Point II. An example of percolation**Some livestock are sent farther than others**1000000**100000 10000 Number of Actors 1000 100 10 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Links to Actor Kevin Bacon Sean Connery “obscure actor” Kevin Bacon Game Ivan Stalenin Average link distance: 2.9 824,270 actors 2.3% not linked (IMDB) Scandal? (1929) Julia Eisenstein Battleship Potemkin (1925) Andrei Fajt Silnye dukhom (1967) Viktoriya Fyodorova Target (1985) Matt Dillon Loverboy (2005) Kevin Bacon**Kevin Bacon Game**Distance via Kevin Bacon is 3 Kevin Bacon Without Kevin Bacon, distance is 6! A few actors are at the centre of the Hollywood Universe**Small world Networks**• Relatively few “long distance” connections can connect the world Avg. Path length Avg. Clustering coefficient Epidemics move farther and faster than expected Watts & Strogatz, 1998**A simple percolation example**• Consider a one dimensional lattice (i.e. a chain of sites) • Let p be the probability that an individual is infectious (site percolation). • The pattern of disease transmission in a simulated epidemic is equivalent to retaining all links with probability p (otherwise discarding)**Cluster probability**• If the probability of an infectious link is “p”, then the probability that a link is not infectious is (1-p). • the expected number of clusters of length C per site (the cluster number) is: (What happens when p=1?)**Calculating the effect of jumps**Recall the one dimensional transmission chain model, where the percolation threshold occurred at p=1 (all infected) now allow for random jumps await from any infected location (in addition to the nearest neighbour links) with some probability prand. 44**Calculating the mean cluster size**The expression gives the expected number of clusters of size C per infected site (known as the cluster number) The probability that any given site belongs to a cluster of size C is Therefore the probability that a site is infected AND in a cluster of size C (i.e. ignoring uninfected sites) 45**Calculating the mean cluster size**If we consider all possible cluster sizes, then the mean cluster size is: For an additional probability pLR that a given infected site will have a long distance jump causing infection at a randomly chosen site (i.e. a ‘long way away’). Percolation occurs when 46**Small world percolation threshold**Percolation on the simple chain model with jumps and percolation on the (infinitely large) small world network are the same: percolation is the result of localised clusters with jumps to form new clusters**i**i i Percolation on Small world transmission networks • In the absence of random connections, consider each localised cluster to be a “super node” (SN) • What is the distribution of cluster sizes? Probability cluster of size “C” probability a link is infectious 1-D case Approach related to Moore & Newman, 2000, Ball et al, 1997**plink < pperc**plink > pperc Percolation Interpretation of R0 on Small Worlds Transmission networks • Giant strongly connected component (NGSCC) – largest group of fully connected nodes in a (directed) network plink =0.05 plink =0.25 plink =1.00 plink =0.75 Percolation Threshold R0 > 1 “Large Epidemic” (in some cases) See also Schwartz et al. 2002, Kao et al. 2006**Does this principle apply to the GB livestock movement**network?