Tartuntatautimallit interventioiden väestövaikutusten arvioinnin apuna Kari Auranen Rokoteosasto Kansanterveyslaitos Matematiikan ja tilastotieteen laitos Helsingin yliopisto 16.5.26
Aiheita (1) Miksi tartuntatatautimallinnusta tarvitaan? Yksinkertainen deterministinen malli: alttiit ja tartuttajat: mass action principle epideeminen kynnys ja laumaimmuniteetti uusiutumisluku R
Aiheita (2) Susceptible - Infected - Removed -malli infektiopaine (force of infection) uusiutumisluvun estimointi rokotuksten vaikutukset Sovelluksia laajojen vesirokkorokotusten vaikutukset? SARSin tapaisten epidemioiden seuranta pneumokokkibakteerin epidemiologia
Tartuntatautien matemaattiset mallit Laajoilla rokotuksilla useimmiten myös epäsuoria (heijastus)vaikutuksia, esim. väestöimmuniteetin muutokset sairastuneiden ikäjakauman siirtymät Väestötason kokeet mahdottomia! Tartuntatautimalleja tarvitaan epäsuorien vaikutusten ennustamiseen taudin epidemiologian tiivistämiseen puuttuvan informaation paikantamiseen Epäsuorat vaikutukset syntyvät tartunnoista!
Mitä voidaan arvioida? taudin leviämispotentiaali (R ja R eff ) esim. tuhkarokko Suomessa nyt ja 2 v päästä riittävä rokotuskattavuus laumaimmuniteetti ylipäätään epäsuoria vaikutuksia tyypillisen sairastumisiän muutokset serotyyppien korvautuminen ja taudin uudeelleen juurtuminen (esim. pneumokokkibakteeri) vesirokkorokotukset ja vyöruusu
A simple epidemic model (Hamer, 196) Consider an infection that involves three states/compartments of infection: Susceptible Case Immune proceeds in discrete generations of infection is transmitted in a homogeneously mixing population of size N
Model equations Dependence of generation t+1 on gen. t: C = R x C x S / N t + 1 t t S = S - C + B t+1 t t+1 t S t = number of susceptibles at time t C t = number of cases (infectious individuals) ) at time t B = number of new susceptibles (by birth) t
Dynamics of transmission Dynamics (Ro = 1; N = 1,; B = 3) numbers of individuals 14 12 1 8 6 4 2 1 5 9 13 17 21 time period 25 29 susceptibles cases epidemic threshold Epidemic threshold : S e/n = 1/R
Epidemic threshold S e S - S = - C + B t+1 t t+1 t number of susceptibles increases when C < B decreases when C > B number of susceptibles cycles around the epidemic threshold S = N / R e t+1 t t+1 t R x S / N = 1 R x S / N = 1 e
Epidemic threshold C / C = R x S / N = S / S t+1 t t t e the number of cases increases when S t > S decreases when S t < S e 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 = 1 1/R implies a critical vaccination coverage: what proportion of the population needs to be effectively vaccinated to eliminate infection
Herd immunity threshold and R herd immunity threshold H,9,8,7,6,5,4,3,2,1 1 2 3 4 5 6 7 8 9 1 Ro H = 1-1/R (Assumes homogeneous mixing
Basic reproduction number (R ) the average number of secondary cases that an infected individual produces in a totally susceptible population during his/her infectious period a threshold parameter for a branching process
Basic reproduction number R = 3 C C C C C C C C C C C C C
Endemic equilibrium R x S / N = 1 e C C C
Estimates of R * * * * Anderson and May: Infectious Diseases of Humans, 1991
Effect of vaccination Ro = 1; N = 1,; B = 3 numbers of individuals 2 15 1 5 1 6 11 16 21 26 31 36 41 Hamer model under vaccinatio susc. cases epidemic threshold S = S - C + B (1- VCxV t+1 t t+1 Vaccine efficacy (VE) x Vaccine coverage (VC) = 8% time period Epidemic threshold sustained: : S = N / R e
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 the analogy to modelling the rate of chemical reactions is responsible for indirect effects of vaccination assumes homogenous mixing in the whole population or in appropriate subpopulations/strata
Reproduction number of SARS Aika
Reproduction number of SARS Aika
Reproduction number of SARS Aika Wallinga & Teunis 25
The SIR epidemic model a continuous time model: overlapping generations permanent immunity after infection the system describes the flow of individuals between three epidemiological compartments formally defined through a set of differential equations Susceptiple Infectious Removed
The SIR model equations (t) ds dt = { µ N I( ) S( t ) N t µ S(t) di dt = I( ) S( t ) N t I(t) µ I(t) dr = I(t) µ R(t) dt N = S( t) + I( t) + R( t) µ = birth rate = rate of clearing infection = rate of infectious contacts by one individual = force of infection
Endemic equilibrium (SIR) 14 numbers of individuals 12 1 8 6 4 2 3. 7, 12, 19, 28, 46 susceptibles infectives epidemic threshold N = 1, µ = 3/1 (per R = 1 (per = 1 (per (per time unit) (per time unit) (per time unit = /( + µ ) = 9.7 time
The basic reproduction number (SIR) R is now a ratio of two rates: R = = rate of infectious contacts x +µ mean duration of infection 1/( +µ) R (usually) not directly observable need to derive relations to observable quantities
Force of infection (SIR) the number of infectious 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 - 1) µ
A simple formula to estimate R assume everyone is infected at age A everyone dies at age L Proportion 1 % Susceptibles A Immunes Age (years) L Stationary proportion of susceptibles S e / N = A / L => R = 1/(S e /N) = L / A
Estimation of and R from seroprevalence data proportion with rubella antibodies (%) 1) Assume equilibrium 1 9 8 7 6 5 4 3 2 1 1 5 1 15 2 25 3 age a (years) model prediction 2) Parameterise force of infection (as a function of age) 3) Estimate 4) Calculate R Ex. constant Proportion not yet infected: 1 - exp(- a), estimate =.1 per year gives reasonable fit to the data
Estimation of R Relation between the average age at infection and R (SIR model) basic reproduction number 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 average age at infection A µ L =1 / µ = = 1/75 (per year) = µ ( R 1) / µ =R 1 R = 1+ /µ R = 1 L/ A + 75
Indirect effects of vaccination proportion p of newborns vaccinated a new reproduction number R vacc = (1-p) x R if p > H = 1-1/R, the infection cannot persist if p < H = 1-1/R, new endemic equilibrium: S = N/R, = (R -1) c» proportion of susceptibles remains untouched(!)» force of infection decreases vac µ vacc
Shift in the average age at infection life length L; proportion p vaccinated at birth, complete protection every susceptible infected at age A Proportion 1 Susceptible s with vaccination: without vaccination: S / N = (1-p) A /L A e S / N = A/ L e => A A = A/(1-p) p A Immunes L Age i.e., increase in the average age of infection
Vaccination at age V > (SIR) assume proportion p vaccinated at age V (instead of at birth) every susceptible infected at age A what proportion p should be vaccinated to obtain herd immunity threshold H? Proportion 1 H = 1-1/R = 1 - A/L proportion immunised by vaccination p (L-V)/L p Susceptibles V Immunes A L Age => p = (L-A)/(L-V) i.e., p bigger than when vaccination at birth
The SIS epidemic model Susceptible Immune herd immunity threshold still the same: H = 1-1/R endemic force of infection: = ( µ + )( R the proportions of susceptibles and immunes 1) different from the SIR model
SIS and SIR R and the force of infection no immunity to infection (SIS) lifelong immunity to infection (SIR) 1,6 1,4 1,2 1 Ro,8,6,4,2.1.2.3.4.5.6.7.8.9 1. force of infection (per year) birth rate µ = 1/75 (per year) rate of clearing infection = 2. (per year) Ro 8 7 6 5 4 3 2 1.2.4.6.8 force of infection (per year) 1. birth rate = 1/75(per year)
Extensions of simple models (1) so far all models assumed homogeneous mixing constant force of infection across age (classes) more realistic models incorporate heterogeneous mixing age-dependent contact/transmission rates social structures: families, day care groups, schools, neighbourhoods etc.
Extensions of simple models (2) seasonal patterns in risks of infection latency, maternal immunity etc. different vaccination strategies different models for the vaccine effect stochastic models to model chance phenomena time to eradication apply statistical techniques/inference
Example: structured models Contact structures (WAIFW) structure of the Who Acquires Infection From Whom matrix for varicella, five age groups (e.g. -4, 5-9, 1-14, 15-19, 2-75 years) table entry = rate of transmission between an infective and a susceptible of respective age groups e.g., force of infection in age group -4: a*i1 + a*i2 + c*i3 + d*i4 + e*i5 I1 = equilibrium number of infectives in age group -4, etc. WAIFW matrix non-identifiable from age-specific incidence!
Ajankohtaisia sovelluksia laajojen vesirokokkorokotusten heijastusvaikutukset? pneumokokkibakteerin epidemiologia
Vesirokkorokotusten heijastusvaikukset? vesirokko lastentauti (9% alle 1v:n iässä) n. 2-25 sairaalahoitoa vaativaa tapausta/vuosi virus jää latentiksi, voi aktivoitua vyöruusuna vyöruusu n. 6 sairaalahoitoa vaativaa tapausta/vuosi vesirokkokohtaamiset pitävät yllä vyöruusuimmuniteettia rokote suojaa tyyppilliseltä (vesirokko)taudilta ja vyöruusulta rokottamattomat voivat altistua vyöruusulle, kun luonnontehoste heikentyy tai lakkaa
Seroepidemiologia: kumul. ilmaantuvuus 1..9.8.7 Proportion.6.5.4.3.2.1. 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2-24 25-29 3-34 35-39 4-49 5-59 6+ Age Group (years) ESEN2
Ketkä tartuttavat keitä? Kumulatiiviseta ilmaantuvuudesta arvioidaan iänmukainen infektiopaine (a) infektiopaine ei kerro, kuka tartuttaa kenet ( Who Acquires Infection From Whom ) tarvitaan oletuksia ja lisätietoa => ikäluokkien välinen kontaktimatriisi i C = = i c ij I j
Vesirokon väestömalli ρ R Z (vyöruusu) S (altis) B(a) z λ(a,t) λ(a,t) σ S (altis) I (vesirokko) R (immuuni) c(a) V (rokotettu) ω S (altis) v b λ(a,t) I (vesirokko) v σ R k λ( a,t) Brisson et al, 2
Mahdollisia heijastusvaikutuksia vesirokkoon sairastuneet keskimäärin vanhempia suurempi osuus tarvitsee sairaalahoitoa pitempi sairauden kesto vyöruusun ilmaantuvuus voi kasvaa lyhyellä/keskipitkällä tähtäimellä jos korkea rokotuskattavuus ja pitkä immuniteetin kesto jos rokotetaan pieniä lapsia (vaihtoehto: rokotetaan esim. 11- vuotiaat) vaikutukset riippuvat rokotekattavuudesta, rokotteen tehosta, immuniteetin kestosta, rokotusohjelman aikataulusta, aikajänteestä
Epidemiology of Streptococcus pneumoniae pneumokokkibakteeri aiheuttaa mm. lasten korvatulehduksia ja vanhusten keuhkokuumetta bakteeri leviää oireettoman nenänielukantajuuden välityksellä 9 keskenään kilpailevaa alatyyppiä uusi rokote ehkäisee tautia ja rokotteen sisältämien alatyyppien kantajuutta ekologia muuttuu laumaimmuniteetti toimii korvautuuko tauti?
PneumoCarr Grand Challenges in Global Health Initiative 25-21 pneumokokin nenänielukantajuuden epidemiologia rokotevaikutusten mittaaminen kantajuutta vastaan rokotusten väestövaikutusten arviointi Rokoteosasto/KTL koordinointi data-analyysi & mallinnus
Collaborators of PneumoCarr * JHSPH, Baltimore ICH, London * * Uni of Witwaterstrand, Soweto KTL, Finland BGU, Israel * KEMRI, Kilifi, Kenya * * * Res Inst of Trop Med, Manila * University of Melbourne Fi