Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Random graphs: a useful tool in epidemic modelling Tom Britton June, 2008 Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Infectious disease spread Infectious diseases: worlds most common death cause Historical: plague, cholera, small-pox, ... Major present: malaria, TB, HIV, ... New: SARS, Ebola, foot & mouth, avian influenza Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Infectious disease spread Infectious diseases: worlds most common death cause Historical: plague, cholera, small-pox, ... Major present: malaria, TB, HIV, ... New: SARS, Ebola, foot & mouth, avian influenza =⇒ Need for understanding spreading dynamics Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Infectious disease spread Infectious diseases: worlds most common death cause Historical: plague, cholera, small-pox, ... Major present: malaria, TB, HIV, ... New: SARS, Ebola, foot & mouth, avian influenza =⇒ Need for understanding spreading dynamics Two features have equal importance: disease agent (transmissability) and social structure Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Infectious disease spread New disease: could either take off or die out Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Infectious disease spread New disease: could either take off or die out “Take off” possible if: R0 > 1, where R0 = average number of new infections caused by an infected during initial phase Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Infectious disease spread New disease: could either take off or die out “Take off” possible if: R0 > 1, where R0 = average number of new infections caused by an infected during initial phase Initial phase of epidemic ≈ branching process If R0 > 1 (super-critical) disease may become endemic Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Intervention – Control Aim: to stop or reduce spreading (either during early stages or when endemic) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Intervention – Control Aim: to stop or reduce spreading (either during early stages or when endemic) How: Reduce R below 1 =⇒ epidemic will not take off, or endemic disease will go extinct Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Intervention – Control Aim: to stop or reduce spreading (either during early stages or when endemic) How: Reduce R below 1 =⇒ epidemic will not take off, or endemic disease will go extinct Means: Vaccination, use of condoms, ...: changes transmittability Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Intervention – Control Aim: to stop or reduce spreading (either during early stages or when endemic) How: Reduce R below 1 =⇒ epidemic will not take off, or endemic disease will go extinct Means: Vaccination, use of condoms, ...: changes transmittability isolation, travelling restrictions, reduce prostitution, sauna clubs, ...: changes social structure Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Stochastic network model Social structure only partly known: modelled using random graph/network with structure Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Stochastic network model Social structure only partly known: modelled using random graph/network with structure Graph/Network: nodes (individuals) and edges (“friendship”) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Stochastic network model Social structure only partly known: modelled using random graph/network with structure Graph/Network: nodes (individuals) and edges (“friendship”) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Stochastic epidemic model Also spreading is uncertain =⇒ stochastic model Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Stochastic epidemic model Also spreading is uncertain =⇒ stochastic model Simplest model: an infected person infects each friend independently with prob p (and then recovers) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Stochastic epidemic model Also spreading is uncertain =⇒ stochastic model Simplest model: an infected person infects each friend independently with prob p (and then recovers) Effect on graph: thinning – each edge is removed with prob 1 − p Interpretation: remaining edges reflect “potential spreading” Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Graph and its thinned version Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Graph and its thinned version The thinned graph is also a random graph! Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Graph and its thinned version The thinned graph is also a random graph! Those connected to index case make upp final outbreak! Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Scientific questions Given social structure (random network) + epidemic model (p): Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Scientific questions Given social structure (random network) + epidemic model (p): Can a big outbreak occur? (R0 > 1?) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Scientific questions Given social structure (random network) + epidemic model (p): Can a big outbreak occur? (R0 > 1?) If so, how many will get infected? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Scientific questions Given social structure (random network) + epidemic model (p): Can a big outbreak occur? (R0 > 1?) If so, how many will get infected? How about when control measures are put into place? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Motivation Stochastic models Scientific questions Given social structure (random network) + epidemic model (p): Can a big outbreak occur? (R0 > 1?) If so, how many will get infected? How about when control measures are put into place? In what follows we will study these questions for some examples Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Heavy tail degree distribution: Motivation Example 1: Heavy tailed degree distribution Social structure described by degree distribution Degree distribution {pk ; k ≥ 0}: D ∼ {pk } is the number of friends of randomly selected individual Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Heavy tail degree distribution: Motivation Example 1: Heavy tailed degree distribution Social structure described by degree distribution Degree distribution {pk ; k ≥ 0}: D ∼ {pk } is the number of friends of randomly selected individual Most empirical networks have heavy tail degree distribution Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Random graph & Epidemic model Model Social structure: Individuals have degree distribution D ∼ {pk } and friends are chosen completely at random Epidemic model: each susc. friend is infected with prob p initially one randomly selected infectious (n − 1 susceptibles) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Random graph & Epidemic model Model Social structure: Individuals have degree distribution D ∼ {pk } and friends are chosen completely at random Epidemic model: each susc. friend is infected with prob p initially one randomly selected infectious (n − 1 susceptibles) What is R0 ? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Random graph & Epidemic model Model Social structure: Individuals have degree distribution D ∼ {pk } and friends are chosen completely at random Epidemic model: each susc. friend is infected with prob p initially one randomly selected infectious (n − 1 susceptibles) What is R0 ? R0 = pE (D)? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Random graph & Epidemic model Model Social structure: Individuals have degree distribution D ∼ {pk } and friends are chosen completely at random Epidemic model: each susc. friend is infected with prob p initially one randomly selected infectious (n − 1 susceptibles) What is R0 ? R0 = pE (D)?– Wrong! Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Random graph & Epidemic model Model Social structure: Individuals have degree distribution D ∼ {pk } and friends are chosen completely at random Epidemic model: each susc. friend is infected with prob p initially one randomly selected infectious (n − 1 susceptibles) What is R0 ? R0 = pE (D)?– Wrong! R0 = p(E (D) − 1)? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Random graph & Epidemic model Model Social structure: Individuals have degree distribution D ∼ {pk } and friends are chosen completely at random Epidemic model: each susc. friend is infected with prob p initially one randomly selected infectious (n − 1 susceptibles) What is R0 ? R0 = pE (D)?– Wrong! R0 = p(E (D) − 1)?– Wrong! Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering The basic reproduction number What is the degree distribution of infectives (during early stages)? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering The basic reproduction number What is the degree distribution of infectives (during early stages)? Answer: {p̃k ; k ≥ 1}, where p̃k = const · kpk = kpk /E (D) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering The basic reproduction number What is the degree distribution of infectives (during early stages)? Answer: {p̃k ; k ≥ 1}, where p̃k = const · kpk = kpk /E (D) V (D) − E (D) =⇒ R0 = p(E (D̃) − 1) = · · · = p E (D) + E (D) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering The probability of an outbreak The initial phase of epidemic ≈ branching process =⇒ π = π(p, {pk }) := P(big outbreak) can be computed Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Size of outbreak A big class of random graphs: one giant component Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Size of outbreak A big class of random graphs: one giant component π = P(big outbreak) = P(belong to giant) = rel size of giant Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Size of outbreak A big class of random graphs: one giant component π = P(big outbreak) = P(belong to giant) = rel size of giant =⇒ outbreak size can also be derived Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination Suppose a fraction v are vaccinated prior to outbreak Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination Suppose a fraction v are vaccinated prior to outbreak Effect on graph: thinning – vaccinated nodes (and their surrounding edges) are removed Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination Suppose a fraction v are vaccinated prior to outbreak Effect on graph: thinning – vaccinated nodes (and their surrounding edges) are removed Who? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination Suppose a fraction v are vaccinated prior to outbreak Effect on graph: thinning – vaccinated nodes (and their surrounding edges) are removed Who? a) Randomly chosen individuals =⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0 Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination Suppose a fraction v are vaccinated prior to outbreak Effect on graph: thinning – vaccinated nodes (and their surrounding edges) are removed Who? a) Randomly chosen individuals =⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0 =⇒ if v ≥ 1 − 1/R0 then Rv ≤ 1 =⇒ no outbreak! Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination Suppose a fraction v are vaccinated prior to outbreak Effect on graph: thinning – vaccinated nodes (and their surrounding edges) are removed Who? a) Randomly chosen individuals =⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0 =⇒ if v ≥ 1 − 1/R0 then Rv ≤ 1 =⇒ no outbreak! Critical vaccination coverage: vC = 1 − 1/R0 Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination Suppose a fraction v are vaccinated prior to outbreak Effect on graph: thinning – vaccinated nodes (and their surrounding edges) are removed Who? a) Randomly chosen individuals =⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0 =⇒ if v ≥ 1 − 1/R0 then Rv ≤ 1 =⇒ no outbreak! Critical vaccination coverage: vC = 1 − 1/R0 Problem: If R0 large, vC ≈ 1 Tom Britton =⇒ impossible Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination, cont’d Can we do better? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination, cont’d Can we do better? Yes! Vaccinate social people Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination, cont’d Can we do better? Yes! Vaccinate social people But social structure usually unknown ... Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination, cont’d Can we do better? Yes! Vaccinate social people But social structure usually unknown ... b) Acquaintance vaccination strategy Choose individuals at random Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination, cont’d Can we do better? Yes! Vaccinate social people But social structure usually unknown ... b) Acquaintance vaccination strategy Choose individuals at random vaccinate one of their friends Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Vaccination, cont’d Can we do better? Yes! Vaccinate social people But social structure usually unknown ... b) Acquaintance vaccination strategy Choose individuals at random vaccinate one of their friends Vaccinees will have degree distribution {p̃k } rather than {pk } =⇒ much more efficient Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Uniform vaccination: Proportion infected as function of v Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Acquaintance vaccination: Proportion infected as function of v Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Second example: An STI in heterosexual community Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering An application to Sexually Transmitted Infections A model for sexually transmitted infections D = # sex-partners (e.g. during a year) p = P(transmission in a relationship) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering An application to Sexually Transmitted Infections A model for sexually transmitted infections D = # sex-partners (e.g. during a year) p = P(transmission in a relationship) Heterosexual community: Df , Dm , pf , pm Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering An application to Sexually Transmitted Infections A model for sexually transmitted infections D = # sex-partners (e.g. during a year) p = P(transmission in a relationship) Heterosexual community: Df , Dm , pf , pm =⇒ bipartite graph Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering An application to Sexually Transmitted Infections A model for sexually transmitted infections D = # sex-partners (e.g. during a year) p = P(transmission in a relationship) Heterosexual community: Df , Dm , pf , pm =⇒ bipartite graph Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering It can be shown that R0 r (Df ) = pf E (Df ) + V (DEf )−E (Df ) r )−E (Dm ) × pm E (Dm ) + V (DEm (D m) Similar to before Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis However: P(transmission) depends on # sex-acts in relationship Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis However: P(transmission) depends on # sex-acts in relationship Promisquous individuals tend to have fewer sex-acts per partner Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis However: P(transmission) depends on # sex-acts in relationship Promisquous individuals tend to have fewer sex-acts per partner This should reduce R0 ! Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis: continued Data: (Anonymous) study of sexual habits in Gotland ≈ 1000 people (17-28 yrs) Among other things: How many sex-partners during last year and how many sex-acts in each relationship Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis: continued Data: (Anonymous) study of sexual habits in Gotland ≈ 1000 people (17-28 yrs) Among other things: How many sex-partners during last year and how many sex-acts in each relationship =⇒ Include in model: short and long term relationships Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis: continued Data: (Anonymous) study of sexual habits in Gotland ≈ 1000 people (17-28 yrs) Among other things: How many sex-partners during last year and how many sex-acts in each relationship =⇒ Include in model: short and long term relationships =⇒ two types of edges in graph (diff transm prob) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis, cont’d P(transmission in relationship) estimated from P(transmission) = 1 − (1 − p)# sex-acts Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Improved analysis, cont’d P(transmission in relationship) estimated from P(transmission) = 1 − (1 − p)# sex-acts R0 fitted to data and computed as a function of p Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering R0 as function of p (fitted to Gotland data) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Conclusions: 1. Acknowledgeing short and longterm relationships reduces R0 2. Endemicity not possible (for realistic p’s) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Conclusions: 1. Acknowledgeing short and longterm relationships reduces R0 2. Endemicity not possible (for realistic p’s)but maybe in subgroups ... Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Graphs with clustering Third example: Social structures with clustering In many social networks (perhaps not sexual networks!) friends of an individual are quite often friends themselves Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Graphs with clustering Third example: Social structures with clustering In many social networks (perhaps not sexual networks!) friends of an individual are quite often friends themselves c := P(two friends of an individual are friends) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Graphs with clustering Third example: Social structures with clustering In many social networks (perhaps not sexual networks!) friends of an individual are quite often friends themselves c := P(two friends of an individual are friends) How construct a random network with predefined clustering c? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering One solution: bipartite graphs One solution: bipartite graphs Specific construction: 1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering One solution: bipartite graphs One solution: bipartite graphs Specific construction: 1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn) 2. An individual is attached to a group, independently, with prob γ/n Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering One solution: bipartite graphs One solution: bipartite graphs Specific construction: 1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn) 2. An individual is attached to a group, independently, with prob γ/n 3. Project the graph on ”true individuals”: individuals that share a common group are connected Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering One solution: bipartite graphs One solution: bipartite graphs Specific construction: 1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn) 2. An individual is attached to a group, independently, with prob γ/n 3. Project the graph on ”true individuals”: individuals that share a common group are connected 4. An infected individual infects each not yet infected ”friend” with prob p and then recovers. Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Illustration of bipartite graph Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Resulting graph: Conclusions from analysis Positive clustering: c = 1 1+βγ E (D) = βγ 2 Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Resulting graph: Conclusions from analysis Positive clustering: c = 1 1+βγ E (D) = βγ 2 How is the epidemic affected by c? Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Resulting graph: Conclusions from analysis Positive clustering: c = 1 1+βγ E (D) = βγ 2 How is the epidemic affected by c? Next slide: R0 and P(major outbreak) and studied as functions of c, (and E (D) and p) Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering Summary Random graphs are useful Social structure is very important in spreading of diseases Tom Britton Random graphs: a useful tool in epidemic modelling Introduction Heavy tail degree distribution An STI in heterosexual community Social structures with clustering References Britton, T., Janson, S., Martin-Löf A. (2007): Graphs with specified degree distributions, simple epidemics and local vacination strategies. Adv. Appl. Prob., 39, 922-948. Britton T., Nordvik, M.K., and Liljeros, F. (2007): Modelling sexually transmitted infections: the effect of partnership activity and number of partners on R0 . Theor Pop Biol., 72, 389-399. Britton T., Deijfen, M., Lindholm, M. and Nordvall Lagerås, A.: Epidemics on random graphs with tunable clustering. To appear in J. App. Prob. Tom Britton Random graphs: a useful tool in epidemic modelling