Epidemiologinen malli

Tämä sivu on tietokide alatyypiltään arviointi. Sivutunniste: Op_fi4414
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Kysymys

Miten arvioidaan vakavan pneumokokkitaudin tautitaakka tilanteessa, jossa kansalliseeen rokotusohjelmaan on valittu tietyt rokoteserotyypit sisältävä konjugaattirokote?

Rajaus

• Vakava pneumokokkitauti tarkoittaa invasiivista pneumokokkitautia (ks. valintaperusteet).
• Arvioidaan tautitapausten vuosittainen määrä kaikisssa ikäryhmissä.
• Rokotuskattavuus, rokotteen teho ja aika rokotusten aloittamisesta ovat riittävät, jotta rokoteserotyyppien kantajuus ja tauti häviävät.
• Oletetaan, että rokoteserotyyppien kantajuus korvautuu täysin muilla serotyypeillä, joiden taudinaiheuttamiskyky säilyy ennallaan.

Vastaus

Mallin antama ennuste tautitapausten vuosittaisesta määrästä ikäluokittain lasketaan ennustemallin avulla. + LINKKI MALLIIN.

Perustelut

Pneumokokkitaudin rokotusten jälkeinen tautitaakka arvioidaan kantajuuden laskennallisen korvautumismallin avulla. Mallissa rokoteserotyyppien kantajuus häviää vähitellen rokotusohjelman kohdeväestössä sekä rokotetun että rokottamattoman väestönosan keskuudessa. Samalla rokotetyyppien kantajuus korvautuu ei-rokotetyyppien kantajuudella. Tämän korvautumisen seuraukset voidaan laskea ennen rokotuksia vallinneiden seotyppikohtaisten kantajuusosuuksien ja tautimäärien perusteella.

Linkki korvautumissivulle

Laskenta

R-program code for the replcement model as in File S1 of Nurhonen M, Auranen K: Optimal serotype compositions for pneumococcal conjugate vaccination under serotype replacement [[1]

Esimerkkejä

• Example model run

----#: . Yritin selvittää kuinka alla oleva koodi toimii. Huomasin että vaihtaessa arvoja ajoon, niin vaikuttaisi siltä että ajon tulokset eivät kuitenkaan muutu? --Jaakko (keskustelu) 5. kesäkuuta 2014 kello 14.13 (UTC) (type: truth; paradigms: science: comment) ----#: . Taulukot vastaavat näköjään aina lähtömallia, joka on PCv13, mutta pylväskuviossa muutettu malli on mukana. --Markku N. (keskustelu) 9. kesäkuuta 2014 kello 09.18 (UTC) (type: truth; paradigms: science: comment)

Proportion of VT carriage removed by vaccination: Proportion of VT carriage removed by vaccination that is replaced by NVT: The true amount of more than 5-year-old carriers is n times larger than estimated. What is n?: Serotypes in a vaccine to look at: 19F 23F 6B 14 9V 4 18C 1 7 6A 19A 3 8 9N 10 11 12 15 16 20 22 23 33 35 38 6C Other + Näytä koodi- Piilota koodi library(OpasnetUtils) library(ggplot2) objects.latest("Op_fi4305", code_name = "alusta") # [[Pneumokokkirokote]] objects.latest("Op_en6007", code_name = "answer") # [[OpasnetUtils/Drafts]] openv.setN(100) ## Read the annual IPD and carriage incidence data. ## The 0 entries in IPD and carriage data are replaced by small values. serotypes<-c( "19F", "23F", "6B", "14", "9V", "4", "18C", "1", "7", "6A", "19A", "3", "8", "9N", "10", "11", "12", "15", "16", "20", "22", "23A", "33", "35", "38", "6C", "Oth") car_under5<-c( 156030, 156030, 126990, 41200, 22290, 12830, 10130, 10, 14180, 54940, 24320, 12160, 1350, 20940, 4050, 72270, 10, 33100, 3380, 1350, 12160, 3380, 680, 30400, 4050, 27470, 24320 ) car_over5<-c( 168100, 314800, 256700, 209800, 114100, 62500, 200700, 100, 100, 158800, 54900, 30800, 8800, 8800, 20800, 97700, 100, 100, 191900, 25200, 72500, 22000, 100, 71300, 100, 79400, 330100 ) ipd_under5<-c( 7.78, 7.88, 24.39, 20.76, 2.91, 2.91, 6.64, 0.31, 3.02, 3.94, 9.88, 1.25, 0.10, 0.83, 0.41, 0.42, 0.21, 1.98, 0.21, 0.01, 0.93, 0.10, 0.42, 0.31, 0.42, 0.01, 0.73 ) ipd_over5<-c( 28.51, 53.72, 29.53, 99.43, 43.07, 76.99, 24.39, 6.58, 46.88, 17.42, 20.54, 55.04, 11.21, 25.20, 6.28, 12.76, 13.89, 9.18, 4.73, 3.29, 29.03, 4.40, 5.64, 12.41, 1.43, 5.50, 11.20 ) ## Combine the data into 2 matrices of dimension 27*2: IPD<-cbind(ipd_under5, ipd_over5) Car<-cbind(car_under5, car_over5) ## Row numbers corresponding to the 3 different PCV formulations ## in matrices IPD and Car. Note: there is no serotype 5 in our data. pcv7rows<-seq(7); pcv10rows<-seq(9); pcv13rows<-seq(12) ## Example S1.2A: Calculate the predicted incidence of IPD for the non-vaccine ## types(NVTs) under PCV13. The predictions are calculated separately for the ## two age classes. These are the values reported on the bottom panel in ## Figure 2 (there given as per 100K incidences). postvacc <-Vaccination(IPD,Car,VT_rows=pcv13rows,p=1,q=1) cat("Incidence of invasive pneumococcal disease after vaccination program (p=1, q=1).\n") oprint(cbind(serotypes,postvacc[[1]])) cat("Number of carriers after vaccination program.\n") oprint(cbind(serotypes,postvacc[[2]])) ## Example S1.2B: Decrease in IPD incidence after adding a single new serotype ## to PCV13 separately for the two age categories. next_under5<-NextVT(IPD[,1],Car[,1], VT_rows=pcv13rows,p=1) next_over5 <-NextVT(IPD[,2],Car[,2], VT_rows=pcv13rows,p=1) cat("Next serotype to add to a vaccine (p=1).\n") oprint(rbind(serotypes, next_under5, next_over5)) # Nämä taulukot kannattaisi transposata niin näyttäisivät siistimmiltä. ## Example S1.3A: The optimal sequence for under 5 year olds when replacement is 100%. ## The output shows the decreases in IPD incidence for each step, ## corresponding to Figure 5(C). The last serotype (row 27, the category "Other") ## is excluded from any vaccine composition but is taken into account as a ## replacing serotype at each stage. opt<-OptimalSequence(IPD[,1],Car[,1],VT_rows=0,Excluded_rows=27,p=1.0,HowmanyAdded=20) cat("Optimal sequence of serotypes to add to a vaccine (p=1, HowmanyAdded=20).\n") oprint(rbind(serotypes[opt[1,]],opt[2,])) ## Example S1.3B: The optimal sequence for the whole population when ## replacement is 50% and the current composition includes the PCV7 serotypes. opt<-OptimalSequence(IPD,Car, VT_rows=pcv7rows,Excluded_rows=length(serotypes), p=0.5,HowmanyAdded=17) cat("Optimal sequence of serotypes to add to a vaccine (p=0.5, HownamyAdded=17).\n") oprint(rbind(serotypes[opt[1,]],opt[2,])) ################################### ## Read the annual IPD and carriage incidence data. ## The 0 entries in IPD and carriage data are replaced by small values. IPD <- Ovariable("IPD", ddata = "Op_fi4305.pneumokokki_vaestossa") IPD@data <- IPD@data[IPD@data$Observation == "Incidence" , colnames(IPD@data) != "Observation"] Car <- Ovariable("Car", ddata = "Op_fi4305.pneumokokki_vaestossa") Car@data <- Car@data[Car@data$Observation == "Carrier" , colnames(Car@data) != "Observation"] servac <- Ovariable("servac", data = data.frame( Vaccine = rep(c("Current", "New"), each = length(serotypes)), Serotype = serotypes, Result = as.numeric(c( serotypes %in% c("19F", "23F", "6B", "14", "9V", "4", "18C", "1", "7","6A", "19A", "3"), serotypes %in% servac_user )) )) p <- Ovariable("p", data = data.frame(Result = p_user)) q <- EvalOutput(Ovariable("q", data = data.frame(Result = q_user))) # EvalOutput must be used because q is mentioned twice in the code and there will otherwise be a merge mismatch. # The true number of adult carriers may actually be larger than estimated. This adjusts for that. Car <- Car * Ovariable("adjust", data = data.frame(Age = c("Under 5", "Over 5"), Result = c(1, adultcarriers))) VacCar <- EvalOutput(VacCar) VacIPD <- EvalOutput(VacIPD) cat("Number of carriers\n") oprint(summary(VacCar)) cat("Incidence of invasive pneumococcal disease.\n") oprint(summary(VacIPD)) if("Iter" %in% colnames(VacCar@output)) N <- max(VacCar@output$Iter) else N <- 1 ggplot(VacCar@output, aes(x = Serotype, weight = result(VacCar) / N, fill = Vaccine)) + geom_bar(position = "dodge") + theme_gray(base_size = 24) + labs(title = "Carriers", y = "Number of carriers in Finland") ggplot(VacIPD@output, aes(x = Serotype, weight = result(VacIPD) / N, fill = Vaccine)) + geom_bar(position = "dodge") + theme_gray(base_size = 24) + labs(title = "Incidence of invasive pneumococcal disease", y = "Number of cases / 100000 py") ggplot(VacIPD@output, aes(x = Vaccine, weight = result(VacIPD) / N, fill = Age)) + geom_bar(position = "stack") + theme_gray(base_size = 24) + labs(title = "Incidence of invasive pneumococcal disease", y = "Number of cases / 100000 py")

Funktioiden alustus

+ Näytä koodi- Piilota koodi library(OpasnetUtils) #S1.4. The R-functions ############################################################################### ## ## R code for the core methods introduced in ## Markku Nurhonen and Kari Auranen: ## "Optimal serotype compositions for pneumococcal conjugate ## vaccination under serotype replacement", ## PLoS Computational Biology, 2014. ## ############################################################################### ## List of arguments common to most functions: ## ## IPD = matrix of IPD incidences by age class (columns) and serotype (rows) ## Car = corresponding matrix of carriage incidences ## VT_rows = vector of the row numbers in matrices IPD and Car ## corresponding to vaccine types (VT_rows=0 for no vaccination) ## p = proportion of lost VT carriage which is replaced by NVT carriage ## q = proportion of VT carriage lost either due to elimination or replacement ## ## This code includes 4 functions: ## Vaccination, NextVT, OptimalSequence and OptimalVacc. ## Vaccination<-function(IPD,Car,VT_rows,p,q) { ## ## Result: ## A list of 2 matrices: IPD and carriage incidences ## after vaccination (corresponding to matrices IPD and Car). ## [Markku Nurhonen 2013] ## if (VT_rows[1]>0) { IPD<-as.matrix(IPD); Car<-as.matrix(Car) # Post vaccination carriage incidences Car_Total<-t(matrix(apply(Car,2,sum),dim(Car)[2],dim(Car)[1])) Car2<-Car; Car2[VT_rows,]<-0 Car_NVT<-t(matrix(apply(Car2,2,sum),dim(Car2)[2],dim(Car2)[1])) Car_VT<-Car_Total-Car_NVT CarNew<-q*(1+p*Car_VT/Car_NVT)*Car2+(1-q)*Car # Post vaccination IPD incidences NVT_rows<-seq(dim(IPD)[1])[-1*VT_rows] # CCR=Case-to-carrier ratios CCR<-IPD/Car ; IPDNew<-0*IPD # Apply the equation appearing above # equation (1) in text for each serotype. # First term applies to NVTs. IPDNew[VT_rows,]<-(1-q)*IPD[VT_rows,] # Second term applies to NVTs. IPDNew[NVT_rows,]<-((Car_NVT+p*q*Car_VT)*(Car/Car_NVT)*CCR)[NVT_rows,] } else { IPDNew<-IPD; CarNew<-Car } list(IPDNew,CarNew) } NextVT<-function(IPD,Car,VT_rows,p) { ## ## Result: ## A vector of decreases in IPD due to adding a serotype ## to the vaccine. If VT_rows=0, initially no vaccination. ## For row indexes incuded in VT_rows, the result is 0. ## [Markku Nurhonen 2013] ## IPD<-as.matrix(IPD); Car<-as.matrix(Car) ## VaccMat = IPD and Car matrices after vaccination VaccMat<-Vaccination(IPD,Car,VT_rows,p,1) IPD<-VaccMat[[1]]; Car<-VaccMat[[2]] ## Total_IPD,Total_Car = Matrices corresponding to ## overall IPD and carriage in each age class. Total_IPD<-t(matrix(apply(IPD,2,sum),dim(IPD)[2],dim(IPD)[1])) Total_Car<-t(matrix(apply(Car,2,sum),dim(Car)[2],dim(Car)[1])) ## Effect = decrease in IPD when one serotype is added to the vaccine. ## See equation (3) in text. Effect<-(Total_IPD-IPD)*((IPD/(Total_IPD-IPD))-(p*Car/(Total_Car-Car))) ## Special case when only one NVT remains. IPD_nonzero<-which(apply(IPD,1,sum)!=0) if (length(IPD_nonzero)==1) {Effect[IPD_nonzero,]<-IPD[IPD_nonzero,]} ## Result is obtained after summation over age classes. apply(Effect,1,sum) } OptimalSequence<-function(IPD,Car,VT_rows,Excluded_rows,p,HowmanyAdded) { ## ## Starting from VTs indicated by the vector VT_rows ## (VT_rows=0, for no vaccination) sequentially add new VTs ## to the vaccine composition s.t. at each step the optimal ## serotype (corresponding to largest decrease in IPD) is added. ## ## Excluded_rows = Vector of indexes of the rows in matrices ## IPD and Car corresponding to serotypes that are not to ## be included in a vaccine composition, e.g. a row ## corresponding to a group of serotypes labelled "Other". ## Enter Excluded_rows=0 for no excluded serotypes. ## HowmanyAdded = number of VTs to be added. ## ## Result: ## Matrix of dimension 2*HowmanyAdded with 1st row indicating ## the row numbers of added serotypes in the order they appear ## in the sequence. The 2nd row lists the decreases in IPD ## due to addition of each type. [Markku Nurhonen 2013] ## IPD<-as.matrix(IPD); Car<-as.matrix(Car) ## First check the maximum possible number of added VTs. VT_howmany<-length(VT_rows) if (VT_rows[1]==0) {VT_howmany<-0} Excluded_howmany<-length(Excluded_rows) if (Excluded_rows[1]==0) {Excluded_howmany<-0} HowmanyAdded<-min(HowmanyAdded,dim(IPD)[1]-(VT_howmany+Excluded_howmany)) BestVTs<-BestEffects<-rep(0,HowmanyAdded) ## Sequential procedure: at each step find the best additional VT. for (i in 1:HowmanyAdded) { ## Effects = Decrease in IPD after addition of each serotype Effects<-NextVT(IPD,Car,VT_rows,p) ## Set Effects for VTs and excluded types equal to small values ## so that none of these will be selected as the next VT. minvalue<- -2*max(abs(Effects)) if (Excluded_howmany>0) {Effects[Excluded_rows]<-minvalue} if (VT_rows[1]>0) {Effects[VT_rows]<-minvalue} ## BestVTs[i] = Index of serotype with maximum decrease in IPD. BestVTs[i]<-order(-1*Effects)[1] ## BestEffects[i] = Decrese in IPD due to addition of BestVTs[i] ## to the vaccine. BestEffects[i]<-Effects[BestVTs[i]] VT_rows<-c(VT_rows,BestVTs[i]) if (VT_rows[1]==0) {VT_rows<-VT_rows[-1]} VaccMat<-Vaccination(IPD,Car,VT_rows,p,1) IPD<-VaccMat[[1]]; Car<-VaccMat[[2]] } t(matrix(c(BestVTs,BestEffects),HowmanyAdded,2)) } OptimalVacc<-function(IPD,Car,VT_rows,p,q,HowmanyAdded) { ## ## Result: ## A list of 3 elements: (1) Row numbers of serotypes in the optimal ## vaccine composition (2)-(3) IPD and carriage incidences ## by serotype and age class corresponding to the optimal ## vaccine formed using the sequential procedure in the ## function OptimalSequence. [Markku Nurhonen 2013] ## Additional_VTs<-OptimalSequence(IPD,Car,VT_rows,p,HowmanyAdded)[1,] All_VTs<-c(VT_rows,Additional_VTs) if (All_VTs[1]==0) All_VTs<-All_VTs[-1] VaccMat<-Vaccination(IPD,Car,All_VTs,p,q) list(All_VTs,VaccMat[[1]],VaccMat[[2]]) } VacCar <- Ovariable("VacCar", dependencies = data.frame(Name = c( "IPD", # incidence of pneumococcus disease "Car", # number of carriers of pneumococcus "servac", # ovariable of serotypes in vaccine (1 for serotypes in a vaccine, otherwise result is 0) "p", # proportion of eliminated VT carriage that is replaced by NVT carriage "q" # proportion of of VT carriage eliminated by vaccine )), formula = function(...) { ## Result: ## An ovariable of carriage incidences ## after vaccination (corresponding to Car). ## [Markku Nurhonen 2013, Jouni Tuomisto 2014] # Post vaccination carriage incidences # Sum over serotypes and drop extra columns Car_Total<- unkeep(oapply(Car, cols = "Serotype", FUN = sum) * 1, prevresults = TRUE) # Car2 is a temporary ovariable with NVT carriers only Car2 <- unkeep(Car * (1 - servac), prevresults = TRUE) # Take only NVT carriers Car_NVT <- oapply(Car2, cols = "Serotype", FUN = sum) # Carriers of serotypes not in vaccine (NVT) Car_VT <- Car_Total - Car_NVT # Carriers of vaccine serotypes CarNew <- q * (1 + p * Car_VT / Car_NVT) * Car2 + (1 - q) * Car return(CarNew) } ) VacIPD <- Ovariable("VacIPD", dependencies = data.frame(Name = c( "IPD", # incidence of pneumococcus disease "Car", # number of carriers of pneumococcus "servac", # ovariable of serotypes in vaccine (1 for serotypes in a vaccine, otherwise result is 0) "p", # proportion of eliminated VT carriage that is replaced by NVT carriage "q" # proportion of of VT carriage eliminated by vaccine )), formula = function(...) { ## Result: ## An ovariable of IPD incidence ## after vaccination (corresponding to ovariable IPD). ## [Markku Nurhonen 2013, Jouni Tuomisto 2014] # Post vaccination carriage incidences (same code as in VacCar) Car_Total <- unkeep(oapply(Car, cols = "Serotype", FUN = sum) * 1, prevresults = TRUE) # Sums over serotypes Car2 <- unkeep(Car * (1 - servac), prevresults = TRUE) Car_NVT <- oapply(Car2, cols = "Serotype", FUN = sum) # Carriers of serotypes not in vaccine (NVT) Car_VT <- Car_Total - Car_NVT # Carriers of vaccine serotypes CarNew <- q * (1 + p * Car_VT / Car_NVT) * Car2 + (1 - q) * Car # Post vaccination IPD incidences # CCR=Case-to-carrier ratios CCR <- IPD / Car # Apply the equation appearing above # equation (1) in text for each serotype. # First term applies to VTs. IPDNewVT <- (1 - q) * IPD * servac # Second term applies to NVTs. IPDNewNVT <- (Car_NVT + p * q * Car_VT) * (Car / Car_NVT) * CCR * (1 - servac) IPDNew <- IPDNewVT + IPDNewNVT return(IPDNew) } ) objects.store(Vaccination, NextVT, OptimalSequence, OptimalVacc, VacCar, VacIPD) cat("Funktiot Vaccination, NextVT, OptimalSequence, OptimalVacc sekä ovariablet VacCar, VacIPD tallennettu. \n")

Data

Pneumokokin esiintyvyys suomalaisessa väestössä. Carrier: (oireettomien) kantajien lukumäärä, Incidence: invasiivisen pneumokokkitaudin ilmaantuvuus 100000 henkilövuotta kohti.

Näytä yksityiskohdat

Pneumokokki väestössä(#, #/100000py) Obs Serotype Age Carrier Incidence 1 19F Under 5 156030 7.78 2 23F Under 5 156030 7.88 3 6B Under 5 126990 24.39 4 14 Under 5 41200 20.76 5 9V Under 5 22290 2.91 6 4 Under 5 12830 2.91 7 18C Under 5 10130 6.64 8 1 Under 5 10 0.31 9 7 Under 5 14180 3.02 10 6A Under 5 54940 3.94 11 19A Under 5 24320 9.88 12 3 Under 5 12160 1.25 13 8 Under 5 1350 0.1 14 9N Under 5 20940 0.83 15 10 Under 5 4050 0.41 16 11 Under 5 72270 0.42 17 12 Under 5 10 0.21 18 15 Under 5 33100 1.98 19 16 Under 5 3380 0.21 20 20 Under 5 1350 0.01 21 22 Under 5 12160 0.93 22 23A Under 5 3380 0.1 23 33 Under 5 680 0.42 24 35 Under 5 30400 0.31 25 38 Under 5 4050 0.42 26 6C Under 5 27470 0.01 27 Oth Under 5 24320 0.73 28 19F Over 5 168100 28.51 29 23F Over 5 314800 53.72 30 6B Over 5 256700 29.53 31 14 Over 5 209800 99.43 32 9V Over 5 114100 43.07 33 4 Over 5 62500 76.99 34 18C Over 5 200700 24.39 35 1 Over 5 100 6.58 36 7 Over 5 100 46.88 37 6A Over 5 158800 17.42 38 19A Over 5 54900 20.54 39 3 Over 5 30800 55.04 40 8 Over 5 8800 11.21 41 9N Over 5 8800 25.2 42 10 Over 5 20800 6.28 43 11 Over 5 97700 12.76 44 12 Over 5 100 13.89 45 15 Over 5 100 9.18 46 16 Over 5 191900 4.73 47 20 Over 5 25200 3.29 48 22 Over 5 72500 29.03 49 23A Over 5 22000 4.4 50 33 Over 5 100 5.64 51 35 Over 5 71300 12.41 52 38 Over 5 100 1.43 53 6C Over 5 79400 5.5 54 Oth Over 5 330100 11.2

Katso myös

Tämä rokotehankintakeskustelu toimitettiin tiedoksi kansalliselle rokotusasiantuntijaryhmälle 8.9.2014.

Pneumokokki eli Streptococcus pneumoniae -bakteeri on maailmanlaajuisesti merkittävä lapsikuolleisuuden aiheuttaja. Siksi toimiva pneumokokkibakteerin rokotusohjelma on kansanterveystyönä kustannustehokkaimpien joukossa. Imeväisten rokottamisessa on käytetty kolmea eri konjugaattirokotetta, jotka sisältävät joko 7, 10 tai 13 pneumokokkiserotyyppiä. Rokotteet ovat sekä vähentäneet pneumokokin aiheuttamien sairauksien haittoja että muuttaneet pneumokokkiserotyppien keskinäisiä kantajuusosuuksia.

THE EPIDEMIOLOGICAL MODEL

The epidemiological model for pneumococcal carriage and disease is based on the assumption that vaccination completely eliminates the vaccine type carriage in a vaccinated population and this carriage is replaced by non-vaccine type carriage. The implications of this replacement on the decrease or increse in pneumococcal disease then depend on the disease causing potential of the replacing types compared to that of the replaced types.

The model is assumed to be valid in a population in which an infant pneumococcal conjugate vaccination has been in use for several years s.t. the new steady-state after vaccination has been reached. Coverage of vaccination and vaccine efficacy against carriage are assumed to be high enough to justify the assumtion of full elimination of vaccine type carriage among both the vaccinated and the unvaccinated members of the population.

The implications of serotype replacement in the model depend on two key assumptions regarding the new steady-state after vaccination:

(A1) the relative serotype proportions among the non-vaccine types are not affected by vaccination (proportionality assumption);

(A2) the case-to-carrier ratios (the disease causing potentials) of individual serotypes remain at their pre-vaccination levels.

It follows from assumptions (1) and (2) that the case-to-carrier ratios remain at their pre-vaccination values also for the aggregate VT and NVT sets.

The implications of vaccination on disease incidence are assumed to be solely due to the elimination of vaccine type carriage and its replacement by non vaccine type carriage. The only exception to this is when a possibility of efficacy against disease without any efficacy against carriage is assumed for certain serotypes to study sensitivity of results to alternative assumptions.

Let q be the proportion of VT carriage eliminated by vaccination and p the proportion of eliminated VT carriage that is replaced by NVT carriage in the new steady state. Under assumptions (A1) and (A2), the disease incidence after vaccination is (for clarification, see Figure 1)