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R0 and other reproduction numbers for households models ... and epidemic models with other social structures Lorenzo Pellis, Frank Ball, Pieter Trapman MRC Centre for Outbreak analysis and modelling, Department of Infectious Disease Epidemiology Imperial College London Edinburgh, 14th September 2011 Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions R0 in simple models R0 in other models INTRODUCTION Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions SIR full dynamics Basic reproduction number R0 Naïve definition: “ Average number of new cases generated by a typical case, throughout the entire infectious period, in a large and otherwise fully susceptible population ” Requirements: 1) New real infections 2) Typical infector 3) Large population 4) Fully susceptible Branching process approximation Follow the epidemic in generations: X n( N ) number of infected cases in generation n (pop. size N ) For every fixed n , lim X n( N ) X n N where X n is the n -th generation of a simple Galton-Watson branching process (BP) Let be the random number of children of an individual in the BP, and let P k k , k 0,1,... be the offspring distribution. Define R0 E k We have “linearised” the early phase of the epidemic Properties of R0 1 Threshold parameter: If R0 1 , only small epidemics If R0 1 , possible large epidemics Probability of a large epidemic 0.8 R = 0.5 * 0.6 R* = 1 0.4 R* = 1.5 R =2 * Final size: 0.2 1 z e R0 z Critical vaccination coverage: pC 1 1 R0 If X 0 1 , then R0 E X1 lim E X n( N ) N 0 0 0.2 0.4 0.6 0.8 1 Properties of R0 1 Threshold parameter: If R0 1 , only small epidemics If R0 1 , possible large epidemics Probability of a large epidemic 0.8 R = 0.5 * 0.6 R* = 1 0.4 R* = 1.5 R =2 * Final size: 0.2 1 z e R0 z Critical vaccination coverage: pC 1 1 R0 If X 0 1 , then R0 E X1 lim E X n( N ) N 0 0 0.2 0.4 0.6 0.8 1 Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions Multitype epidemic model Different types of individuals Define the next generation matrix (NGM): k11 k12 k k22 21 K kn1 k1n k nn where k ij is the average number of type-i cases generated by a type- j case, throughout the entire infectious period, in a fully susceptible population Properties of the NGM: Non-negative elements We assume positive regularity Perron-Frobenius theory Single dominant eigenvalue , which is positive and real “Dominant” eigenvector V has non-negative components For (almost) every starting condition, after a few generations, the proportions of cases of each type in a generation converge to the components of the dominant eigenvector V , with per-generation multiplicative factor Define R0 Interpret “typical” case as a linear combination of cases of each type given by V Formal definition of R0 Start the BP with a j -case: X n ( j; i ) number of i -cases in generation n X n ( j ) X n ( j; i ) total number of cases in generation n i Then: R0 : lim lim n E X n( N ) ( j ) n N Compare with single-type model: R0 : lim E X1( N ) N Basic reproduction number R0 Naïve definition: “ Average number of new cases generated by a typical case, throughout the entire infectious period, in a large and otherwise fully susceptible population ” Requirements: 1) New real infections 2) Typical infector 3) Large population 4) Fully susceptible Network models People connected by a static network of acquaintances Simple case: no short loops, i.e. locally tree-like Repeated contacts First case is special E X1 1 is not a threshold Define: R0 E X 2 | X1 1 Difficult case: short loops, clustering Maybe not even possible to use branching process approximation or define R0 Network models People connected by a static network of acquaintances Simple case: no short loops, i.e. locally tree-like Repeated contacts First case is special E X1 1 is not a threshold Define: R0 E X 2 | X1 1 Difficult case: short loops, clustering Maybe not even possible to use branching process approximation or define R0 Network models People connected by a static network of acquaintances Simple case: no short loops, i.e. locally tree-like Repeated contacts First case is special E X1 1 is not a threshold Define: R0 E X 2 | X1 1 Difficult case: short loops, clustering Maybe not even possible to use branching process approximation or define R0 Basic reproduction number R0 Naïve definition: “ Average number of new cases generated by a typical case, throughout the entire infectious period, in a large and otherwise fully susceptible population ” Requirements: 1) New real infections 2) Typical infector 3) Large population 4) Fully susceptible Reproduction numbers Definition of R0 Generalisations HOUSEHOLDS MODELS Model description Example: sSIR households model Population of m households with of size nH Upon infection, each case i : remains infectious for a duration I i I , iid i makes infectious contacts with each household member according to a homogeneous Poisson process with rate L makes contacts with each person in the population according to a homogeneous Poisson process with rate G N Contacted individuals, if susceptible, become infected Recovered individuals are immune to further infection Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions Household reproduction number R Consider a within-household epidemic started by one initial case Define: L average household final size, excluding the initial case G average number of global infections an individual makes “Linearise” the epidemic process at the level of households: R : G 1 L Household reproduction number R Consider a within-household epidemic started by one initial case Define: L average household final size, excluding the initial case G average number of global infections an individual makes “Linearise” the epidemic process at the level of households: R : G 1 L Individual reproduction number RI Attribute all further cases in a household to the primary case G MI L G 0 RI is the dominant eigenvalue of M I : G 4 L RI 1 1 2 G More weight to the first case than it should be Individual reproduction number RI Attribute all further cases in a household to the primary case G MI L G 0 RI is the dominant eigenvalue of M I : G 4 L RI 1 1 2 G More weight to the first case than it should be Further improvement: R2 Approximate tertiary cases: 1 average number of cases infected by the primary case Assume that each secondary case infects b further cases Choose b 1 1 L , such that 1 1 b b b ... 2 1 3 1 b L , so that the household epidemic yields the correct final size Then: G M2 1 G b and R2 is the dominant eigenvalue of M 2 Opposite approach: RHI All household cases contribute equally L RHI : G 1 L Less weight on initial cases than what it should be Opposite approach: RHI All household cases contribute equally L RHI : G 1 L Less weight on initial cases than what it should be Vaccine-associated reproduction numbers RV and RVL Perfect vaccine Leaky vaccine Assume R 1 Assume R 1 Define pC as the fraction of the population that needs to be vaccinated to reduce R below 1 Define EC as the critical vaccine efficacy (in reducing susceptibility) required to reduce R below 1 when vaccinating the entire population Then RV : 1 1 pC Then RVL : 1 1 EC Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions Naïve approach: next generation matrix Consider a within-household epidemic started by a single initial case. Type = generation they belong to. Define 0 1, 1 , 2 ,..., nH 1 the expected number of cases in each generation Let G be the average number of global infections from each case The next generation matrix is: G 1 K G G G G 2 1 n H 1 n H 2 0 0 More formal approach (I) Notation: xn,i average number of cases in generation n and householdgeneration i nH 1 xn i 0 xn,i average number of cases in generation n and any household-generation System dynamics: nH 1 xn,0 G xn1,i i 0 xn,i i xni ,0 Derivation: 1 i nH 1 xn,0 G xn1 xn,i i G xni 1 xn nH 1 x i 0 1 i nH 1 nH 1 n ,i G i xni 1 i 0 More formal approach (II) System dynamics: xn,0 G xn1 xn,i i G xni 1 xn Define x ( n) nH 1 x i 0 nH 1 n ,i xn , xn1 ,..., xnnH 1 System dynamics: x where AnH ( n) G 0 1 1 i nH 1 AnH x G i xni 1 i 0 ( n1) G 1 G 2 G n 0 1 1 0 H 1 More formal approach (III) Let dominant eigenvalue of AnH V (v0 , v1 ,..., vn 1 ) “dominant” eigenvector H Then, for n : x x ( n) (n) ( n) V ( n 1) x x xn xn1 Therefore: R0 Similarity Recall: G 1 K AnH Define: G G G G 0 0 2 1 G 0 1 n H 1 G 1 G 2 n H 2 G n 0 1 1 0 H 1 0 S 1 n H 2 nH 1 Then: K SAnH S 1 So: K An H R 0 Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions Generalisations This approach can be extended to: Variable household size Household-network model Model with households and workplaces ... (probably) any structure that allows an embedded branching process in the early phase of the epidemic ... all signals that this is the “right” approach! Households-workplaces model Model description Assumptions: Each individual belongs to a household and a workplace Rates H , W and G of making infectious contacts in each environment No loops in how households and workplaces are connected, i.e. locally tree-like Construction of R0 W W W W H H H H Define 0 1, 1 , 2 ,..., nH 1 and 0 1, 1 , 2 ,..., nW 1 for the households and workplaces generations Define nT nH nW Then R0 is the dominant eigenvalue of AnH where ck G c0 1 i j k 0 i nH 1 0 j nW 1 c1 1 cnT 2 0 , 0 iH Wj , cnT 3 1 iH Wj i j k 1 1 i nH 1 1 j nW 1 0 k nT 2 Fundamental inequalities Insight COMPARISON BETWEEN REPRODUCTION NUMBERS Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions Comparison between reproduction numbers Goldstein et al (2009) showed that R 1 RVL 1 Rr 1 RV 1 RHI 1 Comparison between reproduction numbers Goldstein et al (2009) showed that R 1 RVL 1 Rr 1 RV 1 RHI 1 In a growing epidemic: R R RVL Rr RV RHI Comparison between reproduction numbers Goldstein et al (2009) showed that R 1 RVL 1 Rr 1 RV 1 RHI 1 In a growing epidemic: R RV RHI Comparison between reproduction numbers Goldstein et al (2009) showed that R 1 RVL 1 Rr 1 RV 1 RHI 1 To which we added RI 1 R0 1 R2 1 In a growing epidemic: RI RV R0HI R2 Comparison between reproduction numbers Goldstein et al (2009) showed that R 1 RVL 1 Rr 1 RV 1 RHI 1 To which we added RI 1 R0 1 R2 1 In a growing epidemic: R RI RV R0 R2 RHI RHI To which we added that, in a declining epidemic: R RI RV R0 R2 Practical implications 1 RV R0 , so vaccinating p 1 is not enough R0 Goldstein et al (2009): R RV RHI Now we have sharper bounds for RV : R RI RV R0 RHI Outline Introduction R0 in simple models R0 in other models Households models Reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions Insight Recall that R0 is the dominant eigenvalue of AnH G 0 1 G 1 G 2 G n 0 1 1 0 H 1 From the characteristic polynomial, we find that R0 is the only positive root of nH 1 g0 ( ) 1 i 0 G i i 1 Discrete Lotka-Euler equation Continuous-time Lotka-Euler equation: r ( ) e d 1 H 0 Discrete-generation Lotka-Euler equation: H ( ) k k ( ) 0 0 k G k 1 for k 0 for e k 0 k 1, 2,..., nH k nH Therefore, R0 e is the solution of r r k k nH 1 G i R i 0 1 0 i 1 1 Fundamental interpretation For each reproduction number RA , define a r.v. X A describing the generation index of a randomly selected infective in a household epidemic iA Distribution of X A is P X A i , 1 L 0 i Non-normalised cumulative distribution of XA 2.8 From Lotka-Euler: st XA XB 2.6 RA 2.4 RB R RI 2.2 R0 2 Therefore: R2 RHI 1.8 R RI R0 R2 RHI 1.6 1.4 1.2 1 0 1 2 3 4 5 6 7 8 9 CONCLUSIONS Why so long to come up with R0? Typical infective: “Suitable” average across all cases during a household epidemic G 1 K G G G G 2 1 n H n H 1 0 0 Types are given by the generation index: not defined a priori appear only in real-time “Fully” susceptible population: the first case is never representative need to wait at least a few full households epidemics Conclusions After more than 15 years, we finally found R0 General approach clarifies relationship between all previously defined reproduction numbers for the households model works whenever a branching process can be imbedded in the early phase of the epidemic, i.e. when we can use Lotka-Euler for a “sub-unit” Allows sharper bounds for RV : R RI RV R0 RHI Acknowledgements Co-authors: Pieter Trapman Frank Ball Useful discussions: Pete Dodd Christophe Fraser Fundings: Medical Research Council Thank you all! SUPPLEMENTARY MATERIAL Outline Introduction R0 in simple models R0 in other models Households models Many reproduction numbers Definition of R0 Generalisations Comparison between reproduction numbers Fundamental inequalities Insight Conclusions R0 in simple models R0 in other models INTRODUCTION Galton-Watson branching processes Threshold property: If R0 1 the BP goes extinct with probability 1 (small epidemic) If R0 1 the BP goes extinct with probability given by the smallest solution s [0,1] of s k s k 1 k 0 Assume X 0 1 . Then: E X n R0 n n E X n R0 E X n1 E X n R0 0.8 R = 0.5 * 0.6 R =1 * 0.4 R = 1.5 * R* = 0.5 0.2 0 0 0.2 0.4 0.6 0.8 1 Galton-Watson branching processes Threshold property: If R0 1 the BP goes extinct with probability 1 (small epidemic) If R0 1 the BP goes extinct with probability given by the smallest solution s [0,1] of s k s k 1 k 0 Assume X 0 1 . Then: E X n R0 n n E X n R0 E X n1 E X n R0 0.8 R = 0.5 * 0.6 R =1 * 0.4 R = 1.5 * R* = 0.5 0.2 0 0 0.2 0.4 0.6 0.8 1 Galton-Watson branching processes Threshold property: If R0 1 the BP goes extinct with probability 1 (small epidemic) If R0 1 the BP goes extinct with probability given by the smallest solution s [0,1] of s k s k 1 k 0 Assume X 0 1 . Then: E X n R0 n n E X n R0 E X n1 E X n R0 0.8 R = 0.5 * 0.6 R =1 * 0.4 R = 1.5 * R* = 0.5 0.2 0 0 0.2 0.4 0.6 0.8 1 Galton-Watson branching processes Threshold property: If R0 1 the BP goes extinct with probability 1 (small epidemic) If R0 1 the BP goes extinct with probability given by the smallest solution s [0,1] of s k s k 1 k 0 Assume X 0 1 . Then: E X n R0 n n E X n R0 E X n1 E X n R0 0.8 R = 0.5 * 0.6 R =1 * 0.4 R = 1.5 * R* = 0.5 0.2 0 0 0.2 0.4 0.6 0.8 1 Properties of R0 1 Threshold parameter: If R0 1 , only small epidemics If R0 1 , possible large epidemics Probability of a large epidemic 0.8 R = 0.5 * 0.6 R* = 1 0.4 R* = 1.5 R =2 * Final size: 0.2 1 z e R0 z Critical vaccination coverage: pC 1 1 R0 If X 0 1 , then R0 E X1 lim E X n( N ) N 0 0 0.2 0.4 0.6 0.8 1 Properties of R0 1 Threshold parameter: If R0 1 , only small epidemics If R0 1 , possible large epidemics Probability of a large epidemic 0.8 R = 0.5 * 0.6 R* = 1 0.4 R* = 1.5 R =2 * Final size: 0.2 1 z e R0 z Critical vaccination coverage: pC 1 1 R0 If X 0 1 , then R0 E X1 lim E X n( N ) N 0 0 0.2 0.4 0.6 0.8 1 HOUSEHOLDS MODELS Within-household epidemic Repeated contacts towards the same individual Only the first one matters Many contacts “wasted” on immune people Number of immunes changes over time -> nonlinearity Overlapping generations Time of events can be important Rank VS true generations sSIR model: draw an arrow from individual to each other individual with probability 1 exp Ii n 1 attach a weight given by the (relative) time of infection Rank-based generations = minimum path length from initial infective Real-time generations = minimum sum of weights 0.8 0.6 1.9