Network Models in Infectious Disease Ecology

1. Network Models in Infectious Disease Ecology Rowland Kao X5770 rowland.kao@glasgow.ac.uk

2. 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

3. 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

4. (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

5. Index Case The network-based perspective Parsimonious (ODE) Models Networks Micro-simulation Increasing complexity • ODEs (dynamics) • Micro-simulation • Networks • (population structure) different paradigms

6. 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)

7. Point I. Networks are very different from classic models of disease transmission Interpreting r0 on networks

8. 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

9. Susceptible/Infected/Removed(SIR) Compartmental Model birth death S transmission death I removal death R

10. 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.

11. 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)

12. The Basic Reproduction Ratio I E E=0, I=0 is a “globally” stable equilibrium

13. 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

14. The Basic Reproduction Ratio I e.g. logistical limitations Two locally stable solutions E

15. R0 for the SIR model Rate of new infections when I=1, S=N Average lifespan of an infected individual

16. 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?

17. 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

18. 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

19. 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:

20. 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%

21. 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

22. 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

23. Problem 3: R0 and the contact matrix description of a population

24. One could write down a ‘next generation matrix’ and calculate ‘R0‘

25. 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

26. Percolation • A better way of looking at (disease) population persistence in spatial and network models is as a percolation phenomenom

27. Percolation in spatial models • Coffee percolator • Creation of large scale phenomena from simple structures • Cellular automata • Porous solids

28. 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

29. Percolation Day 1

30. Percolation Day 2

31. Percolation Day 3

32. Percolation Day 4

33. Percolation Day 5 The Largest Component (Patch of Mould) Spans the Popn.

34. Percolation phenomena 34

35. 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

36. 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)

37. 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!

38. Point II. An example of percolation Some livestock are sent farther than others

39. 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

40. 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

41. 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

42. 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)

43. 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?)

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

50. Does this principle apply to the GB livestock movement network?