Epidemiological modelling

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Question

How to predict the net effectiveness of a pneumococcal conjugate vaccination with a given set of serotypes when the vaccine is included in the national immunisation programme?

• Focus is on the number of invasive pneumococcal disease (IPD) cases in different age groups.
• 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. a 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 also, due to substancial herd effects, among the unvaccinated members of the population.
• Vaccine type carriage is fully replaced by carriage of the non-vaccine types and the disease causing potential of different serotypes is not altered by vaccination.

Answer

Predicted number of invasive pneumococcal disease (IPD) cases in different age groups are obtained from the serotype replacement model (Nurhonen and Auranen, 2014).

Rationale

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. To predict post vaccination disease only pre vaccination data on serotype specific carriage and disease is used.

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

1. the relative serotype proportions among the non-vaccine types are not affected by vaccination (proportionality assumption);
2. the case-to-carrier ratios (the disease causing potentials) of individual serotypes remain at their pre-vaccination levels.

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. An exception to this is when a possibility of efficacy against disease without any efficacy against carriage is assumed for certain serotypes.

Figure 1. Illustration of the replacement model. The incidence of pneumococcal carriage (x-axis) and case-to-carrier ratios (y-axis) for vaccine serotypes (VT) and non-vaccine serotypes (NVT) before (panel A) and after vaccination (panel B). The incidences of disease (DVT and DNVT) are obtained by multiplication of the two quantities and correspond to the areas of the rectangles. After vaccination, VT carriage is eliminated and replaced by NVT carriage (panel B). The decrease in IPD incidence after vaccination is obtained as the difference between the eliminated VT disease and the replacing NVT disease. This is the area of the blue rectangle in panel B.

Computation

The following program (modified from File S1 in Nurhonen and Auranen, 2014) illustrates the working of the replacement model. In its current implementation the code asks the user to choose a vaccine composition (labelled "New") and then displays the predicted IPD cases in Finland per year corresponding to this vaccine. The results are shown by serotype and by age category (<5 and 5+ year olds) and the corresponding results for PCV10 (labelled "Current") are displayed for comparison.

Instructions for user: Choose the desired vaccine composition from the list below and then press "Run code". The results will be displayed on the right side of the page. The default choice is PCV10 and this should be changed either by adding or removing serotypes as PCv10 is the vaccine to which the new vaccine is compared. For PCV13, add serotypes 6A, 19A and 3.

Please choose the serotypes for the vaccine composition New. Current=PCV10.: 19F 23F 6B 14 9V 4 18C 1 7F 6A 19A 3 8 9N 10 11 12 15 16 20 22 23 33 35 38 6C Other + Show code- Hide code 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) ## 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) # 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) ## 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) ################################### ## 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"), serotypes %in% servac_user )) )) p_user<-q_user<-adultcarriers<-1 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) if (1==0) { cat("servac\n") oprint(summary(servac)) 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 if (1==0) {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 per year") 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 per year")

Initializing the functions

+ Show code- Hide code 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("the functions Vaccination, NextVT, OptimalSequence, OptimalVacc sekä ovariablet VacCar, VacIPD are now saved. \n")

References

Nurhonen M, Auranen K (2014) Optimal Serotype Compositions for Pneumococcal Conjugate Vaccination under Serotype Replacement. PLoS Comput Biol 10(2): e1003477. doi:10.1371/journal.pcbi.1003477