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Modelling as a tool for planning vaccination policies Kari Auranen Department of Vaccines National Public Health Institute, Finland Department of Mathematics and Statistics University of Helsinki, Finland Outline • Basic concepts and models – – – – – dynamics of transmission herd immunity threshold basic reproduction number herd immunity and critical coverage of vaccination mass action principle Outline continues • Heterogeneously mixing populations – more complex models and new survey data to learn about routes of transmission • Use of models in decision making – example: varicella vaccinations in Finland • Ude of models in planning contaiment strategies – example: a simulation tool for pandemic influenza Basic concepts and models A simple epidemic model (Hamer, 1906) • Consider an infection that – involves three “compartments” of infection: Susceptible Case Immune – proceeds in discrete generations (of infection) – is transmitted in a homogeneously mixing (“everyone meets everyone”) population of size N Dynamics of transmission • Numbers of cases and susceptibles at generation t+1 Ct + 1 = R 0 * C t * S t / N S t+1 = S t - C t+1 + B t S t = number of susceptibles at time t (i.e. generation t) C t = number of cases (infectious individuals) at time t B t = number of new susceptibles (by birth) Dynamics of transmission 1400 1200 1000 800 600 400 200 0 susceptibles cases time period 29 25 21 17 13 9 epidemic threshold 5 1 numbers of individuals R0 = 10; N = 10,000; B = 300 Epidemic threshold : S = N/R e 0 Epidemic threshold S e St+1 - S t = - C t+1 + B t • the number of susceptibles increases when C t+1 < B t decreases when C t+1 > B t • the number of susceptibles cycles around the epidemic threshold S e = N / R 0 • this pattern is sustained as long as transmission is possible Epidemic threshold C t+1 / C t = R 0 x St / N = St / Se • the number of cases increases when S > Se decreases when S < S e • the number of cases cycles around B t (influx of new susceptibles) Herd immunity threshold • incidence of infection decreases as long as the proportion of immunes exceeds the herd immunity threshold H = 1- S e / N • a complementary concept to the epidemic threshold • implies a critical vaccination coverage Basic reproduction number (R 0 ) • the average number of secondary cases that an infected individual produces in a totally susceptible population during his/her infectious period • in the Hamer model : R 0 = R0 x 1 x N / N = R 0 • herd immunity threshold H = 1 - 1 / R 0 • in the endemic equilibrium: S e = N / R0 , i.e., Re0 x Se / N 0= 1 Basic reproduction number (2) R0 = 3 Basic reproduction number (3) R0 = 3 endemic equilibrium R0 x Se / N = 1 Herd immunity threshold and R 0 0,8 0,7 0,6 H = 1-1/R 0 0,5 herd immunity 0,4 threshold H 0,3 (Assumes homogeneous mixing) 0,2 Ro 5 4 3 2 1 0 0 0,1 Effect of vaccination Hamer model under vaccination 2000 S t+1 = S t - C t+1 + B (1- VCxVE) susc. 1500 cases 1000 epidemic threshold 500 Vaccine effectiveness (VE) x Vaccine coverage (VC) = 80% time period 36 41 31 21 26 16 11 6 0 1 numbers of individuals Ro = 10; N = 10,000; B = 300 Epidemic threshold sustained: S = N / R e 0 Mass action principle • all epidemic/transmission models are variations of the use of the mass action principle which – – – – captures the effect of contacts between individuals uses an analogy to modelling the rate of chemical reactions is responsible for indirect effects of vaccination assumes homogenous mixing • in the whole population • in appropriate subpopulations (defined by usually by age categories) The SIR epidemic model • a continuous time model: overlapping generations • permanent immunity after infection • the system descibes the flow of individuals between the epidemiological compartments • uses a set of differential equations Susceptiple Infectious Removed The SIR model equations dS dt N I (t) S (t) S (t) N dI I (t) S (t) I (t) I (t) dt N dR dt I (t) R(t) N S (t) I (t) R(t) = birth rate = rate of clearing infection = rate of infectious contacts by one individual = force of infection Endemic equilibrium (SIR) 1200 susceptibles 1000 800 infectives 600 epidemic threshold 400 200 N = 10,000 = 300/10000 (per time unit) = 10 (per time unit) = 1 (per time unit) R 0 time 46 7, 00 12 ,0 0 19 ,0 0 28 ,0 0 0 3. 0 0 numbers of individuals 1400 The basic reproduction number • Under the SIR model, Ro given by the ratio of two rates: R0 = = rate of infectious contacts x mean duration of infection ) • R 0 not directly observable • need to derive relations to observable quantities Force of infection • the number of infective contacts in the population per susceptible per time unit: (t) = x I(t) / N • incidence rate of infection: (t) x S(t) • endemic force of infection = x (R 0 - 1) Estimation of R 0 basic reproduction number Relation between the average age at infection and R0 (SIR model) 90 80 70 = 1/75 (per year) L 1/ 75 60 50 40 30 R0 1) / R 1 R 1 / 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 average age at infection A R0 1 L / A A simple alternative formula • Assume everyone is infected at age A everyone dies at age L (rectangular age distribution) Proportion 100 % Susceptibles Immunes Proportion of susceptibles: Se / N = A / L A Age (years) L R0 = N / Se = L / A Estimation of and Ro from seroprevalence data proportion with rubella antibodies 1) Assume equilibrium 2) Parameterise force of infection 100 90 80 70 60 50 40 30 20 10 0 3) Estimate 4) Calculate Ro observed [8] model prediction 1 5 10 15 20 25 30 age a (years) Ex. constant Proportion not yet infected: 1 - exp(- a) , estimate = 0.1 per year gives reasonable fit to the data Estimates of R 0 Infection Location R0 Measles Rubella Poliomyelitis Hib England and Wales (1950-68) England and Wales (1960-70) USA (1955) Finland, 70’s and 80’s 16-18* 6-7* 5-6* 1.05 *Anderson and May: Infectious Diseases of Humans, 1991 Critical vaccination coverage to obtain herd immunity • Immunise a proportion p of newborns with a vaccine that offers complete protection against infection • R vacc = (1-p) x R 0 • If the proportion of vaccinated exceeds the herd immunity threshold, i.e., if p > H = 1-1/R 0 , infection cannot persist in the population (herd immunity) Critical vaccination coverage as a function of R0 1 Critical vaccination coverage 0.9 0.8 0.7 p = 1 – 1/R0 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 Basic reproduction number 12 13 14 15 Indirect effects of vaccination • If p < H = 1-1/R0 , in the new endemic equilibrium: S e = N/R 0 , vacc= (Rvacc -1) e » proportion of susceptibles remains untouched » force of infection decreases Effect of vaccination on average age A’ at infection (SIR) • Life length L; proportion p vaccinated at birth, complete protection • every susceptible infected at age A Susceptibles S e / N = (1-p) A’/L Proportion 1 S e / N = A/ L => A’ = A/(1-p) Immunes p A’ L Age (years) i.e., increase in the average age of infection Vaccination at age V > 0 (SIR) • Assume proportion p vaccinated at age V • Every susceptible infected at age A • How big should p be to obtain herd immunity threshold H Proportion H = 1 - 1/R = 1 - A/L 1 H = p (L-V)/L => p = (L-A)/(L-V) Susceptibles p i.e., p bigger than when vaccination at birth Immunes V A L Age (years) Modelling transmission in a heterogeneously mixing population More complex mixing patterns • So far we have assumes (so called) homogeneous mixing – “everyone meets everyone” • More realistic models incorporate some form of heterogeneity in mixing (“who meets whom”) – e.g. individuals of the same age meet more often each other than individual from other age classes (assortative mixing) Example: WAIFW matrix • structure of the Who Acquires Infection From Whom matrix for varicella , five age groups (0-4, 5-9, 10-14, 15-19, 20-75 years) table entry = rate of transmission between an infective and a susceptible of respective age groups e.g., force of infection in age group 0-4: a a c d e a b c d e a*I1 + a*I2 + c*I3 + d*I4 + e*I5 I1 = equilibrium number of infectives in age group 0-4, etc. c c c d e d d d d e e e e e e POLYMOD contact survey • Records the number of daily conversations in study participants in 7 European countries • Use the number of contacts between individuals from different age categories as a proxy for chances of transmission • Is currently being used to aid in modelling the impact of varicella vaccination in Finland POLYMOD contact survey: the mean number of daily contacts Country DE FI IT LU NL PL GB Number of daily contacts 7.95 11.06 19.77 17.46 13.85 16.31 11.74 Relative (95% CI) 1 1.34 (1.26-1.42) 2.33 (2.19-2.48) 2.02 (1.90-2.14) 1.78 (1.63 -1.95) 1.90 (1.79 – 2.01 1.40 (1.31 – 1.48) POLYMOD contact survey: numbers of daily contacts Finland 70+ 45-49 40-44 35-39 30-34 25-29 Age of contact 65-69 60-64 55-59 50-54 00-04 50-54 55-59 60-64 65-69 70+ 45-49 25-29 30-34 35-39 40-44 15-19 20-24 00-04 05-09 10-14 20-24 15-19 10-14 05-09 Age of participant 0.00-0.31 0.31-0.63 0.63-0.94 1.56-1.88 1.88-2.19 2.19-2.50 0.94-1.25 1.25-1.56 POLYMOD contact survey: where and for how long Duration 100% 100% 90% 90% 80% 80% 70% 70% 60% 60% 50% non-physical physical 40% 30% non-physical physical 50% 40% 30% 20% 20% 10% 10% 0% 0% <5 min 5-15 min 15-60 min 1-4 h 4+ h 100% home school leisure work transport other multiple 100% 90% 90% non-physical physical 80% 70% 80% 60% 60% 50% 50% 40% 4+ h 70% 1-4 h 15-60 min 5-15 min 40% 30% <5 min 30% 20% 20% 10% 10% 0% daily weekly monthly less often first time 0% daily weekly monthly less often first time Use of models in policy making • Large-scale vaccinations usually bring along indirect effects – the mean age at disease increases – population immunity changes • Population-level experiments are impossible Need for mathematicl modelling – to predict indirect effects of vaccination – to summarise the epidemiology of the infection – to identify missing data or knowledge about the natural history of the infection References 1 Fine P.E.M, "Herd immunity: History, Theory, Practice", Epidemiologic Reviews, 2 3 4 5 6 15, 265-302,1993 Fine P.E.M., "The contribution of modelling to vaccination policy, Vaccination and World Health, Eds. F.T. Cutts and P.G. Smith, Wiley and Sons, 1994. Nokes D.J., Anderson R.M., "The use of mathematical models in the epidemiological study of infectious diseases and in the desing of mass immunization programmes", Epidemiology and Infection, 101, 1-20, 1988 Anderson R.M. and May R.M., ”Infectious Diseases of Humans”; Oxford University Press, 1992. Mossong J et al, Social contacts and mixing patterns relevant to the spread of infectious diseases: a multi-country population-based survey, Plos Medicine, in press Duerr et al, Influenza pandemic intervention planning using InfluSim, BMC Infect Dis, 2007