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Statistical Modelling for InfectiousDisease Management - Infection period

Question

One or several persons start to show symptoms of COVID-19. When didthe person become infected?

Generating a data frame with dates and infection probability usingget_infection_density for one person

The functionget_infection_density() can be used tocalculate a data frame containing the infection probability when theperson shows symptoms.

Inputs

The functionget_infection_density() expects thefollowing input arguments:

First, thesymptom_begin_date is needed, which defineswhen the person started to have symptoms.

Then, themax_incubation_days has to be set, whichdefines the interval length of the distribution output.

The remaining inputsmeanlog andsdlog arethe parameters of the log-normal distribution for the infectionprobability.

symptom_begin_date<-as.Date("2021-12-28")max_incubation_days<-18meanlog<-1.69sdlog<-0.55infec_date_df<-get_infection_density(symptom_begin_date,                                       max_incubation_days,                                       meanlog,                                       sdlog)

Methodology

The default values of log-normal distribution are taken from thepaper Xin et al [1]. In this paper the authors made a systematic reviewof the current literature and estimated those parameters based on theirmeta-analysis.

Output

The data frame shows for each hour from the earliest potential startof infection up to the symptom begin date the resulting density of thelog-normal distribution. This density can be used for calculating themost probable period of the infection.

values 100 to 109 of resulting data frame
datesdistribution
1002021-12-14 04:00:000.0122820
1012021-12-14 05:00:000.0124346
1022021-12-14 06:00:000.0125891
1032021-12-14 07:00:000.0127457
1042021-12-14 08:00:000.0129043
1052021-12-14 09:00:000.0130650
1062021-12-14 10:00:000.0132277
1072021-12-14 11:00:000.0133926
1082021-12-14 12:00:000.0135597
1092021-12-14 13:00:000.0137289

Generating a data frame with dates and probability of infectionusingget_misc_infection_density for several persons

The functionget_misc_infection_density() creates a dataframe containing the mixture probability of all considered persons. Itcan be used to give an overview of the infection probability of severalpersons with symptom onset dates, e.g., one person with symptom onset on24.12.2021 and two persons with symptom onset on 28.12.2021.

Inputs

The following arguments are needed for using the functionget_misc_infection_density():

The first parametersymptom_begin_dates contains thedates when the persons got symptoms.

The second parameterpersons contains the number ofpersons having symptoms on each date.

The remaining inputs are the same as inget_infection_density.

symptom_begin_dates<-c(as.Date("2021-12-24"),as.Date("2021-12-28"))persons<-c(1,2)max_incubation_days<-18misc_infec_date_df<-get_misc_infection_density(symptom_begin_dates,                                                 persons,                                                 max_incubation_days)

Methodology

This function uses theget_infection_density functionand generates a mixture distribution [2]. This probability distributionis obtained by a sum of the infection probability distribution for eachsymptom onset day multiplied by the percentage of persons, which havestarted to show symptoms on this day.

Output

The data shows the mixture log-normal distribution and thus gives anoverview of the potential infection time points for all consideredpersons. However, it does not necessarily have to imply that they hadtheir infection on the same time point. In fact, there did not have tobe an event, where the persons met. It shows when the persons gotinfected and it is possible that there is more than one infection date,which can be seen based on several maxima.

values 100 to 109 of resulting data frame
datesdistribution
1002021-12-10 04:00:000.0066933
1012021-12-10 05:00:000.0067743
1022021-12-10 06:00:000.0068564
1032021-12-10 07:00:000.0069395
1042021-12-10 08:00:000.0070237
1052021-12-10 09:00:000.0071090
1062021-12-10 10:00:000.0071953
1072021-12-10 11:00:000.0072828
1082021-12-10 12:00:000.0073713
1092021-12-10 13:00:000.0074610

Visualization example of the data frame ofget_infection_density

Source
.calculate_qstart_qend<-function(probability, df) {    hdr_df<-hdr(den =data.frame(x =1:length(df$distribution),y = df$distribution),p = probability*100)$hdr    qstart<- (hdr_df[1:(length(hdr_df)/2)*2]-1)/24    qend<- (hdr_df[1:(length(hdr_df)/2)*2-1]-1)/24return(list("qstart"= qstart,"qend"= qend))}
Source
.shade_curve<-function(df, qstart, qend,fill ="red",alpha =0.4) {  subset_df<- df[floor(qstart*24):ceiling(qend*24), ]geom_area(data = subset_df,aes(x = x,y = y),fill = fill,color =NA,alpha = alpha)}
Source
  symptom_begin_date<-as.Date("2021-12-28")  df<- infec_date_df  period_80<-.calculate_qstart_qend(0.8, df)  period_95<-.calculate_qstart_qend(0.95, df)  symp_date_posixct_start<-as.POSIXct(format(as.POSIXct(symptom_begin_date,tz ="CET"),"%Y-%m-%d"))  symp_date_posixct_end<-as.POSIXct(format(as.POSIXct(symptom_begin_date+1,tz ="CET"),"%Y-%m-%d"))  symp_date_posixct_mid<- symp_date_posixct_start-as.numeric(difftime(symp_date_posixct_start,                           symp_date_posixct_end,units ="hours"))/2*3600
Source
  g<-ggplot()+scale_x_datetime(breaks = scales::date_breaks("1 days"),labels = scales::date_format("%d %b"))+theme(axis.text.x =element_text(angle =90))+# scale_x_continuous(breaks = x_tick,#                    labels = x_label) +# theme(axis.ticks.x = element_line(color = c(rbind(rep("black", length(x_label) / 2), rep(NA, length(x_label) / 2))),        linetype = 2, size = 1))+geom_path(aes(x = df$dates,y = df$distribution,color ="red"))+.shade_curve(df =data.frame(x = df$dates,y = df$distribution),                 period_80$qstart,                 period_80$qend)+.shade_curve(df =data.frame(x = df$dates,y = df$distribution),                 period_95$qstart,                 period_95$qend,alpha =0.2)+geom_rect(data =data.frame(xmin = symp_date_posixct_start,xmax = symp_date_posixct_end,ymin =-Inf,ymax =Inf),aes(xmin = xmin,xmax = xmax,ymin = ymin,ymax = ymax),fill ="brown",alpha =0.3)+geom_label(aes(x = symp_date_posixct_mid,y =0.9*max(df$distribution),label ="symptom\nonset"),colour ="brown",fill ="white",size =5,label.size =NA)+ylab("probability")+xlab("timeline")+labs(color ='Verteilung')+# ggtitle("Visualization of get_infection_density ") +theme(legend.position ="none",text =element_text(size =16*5/5))+theme(axis.text.x =element_text(colour ="black",face ="bold",angle =30,hjust =1))+theme(axis.title.x =element_text(colour ="black",face ="bold"))+theme(axis.text.y =element_text(colour ="gray50"))+theme(axis.title.y =element_text(colour ="gray50"))  g

Visualization example of the data frame ofget_misc_infection_density

Source
  df<- misc_infec_date_df  period_80<-.calculate_qstart_qend(0.8, df)  period_95<-.calculate_qstart_qend(0.95, df)  symp_date_posixct_start<-as.POSIXct(format(as.POSIXct(symptom_begin_date,tz ="CET"),"%Y-%m-%d"))  symp_date_posixct_end<-as.POSIXct(format(as.POSIXct(symptom_begin_date+1,tz ="CET"),"%Y-%m-%d"))  symp_date_posixct_mid<- symp_date_posixct_start-as.numeric(difftime(symp_date_posixct_start,                           symp_date_posixct_end,units ="hours"))/2*3600
Source
  g<-ggplot()+scale_x_datetime(breaks = scales::date_breaks("1 days"),labels = scales::date_format("%d %b"))+theme(axis.text.x =element_text(angle =90))+# scale_x_continuous(breaks = x_tick,#                    labels = x_label) +# theme(axis.ticks.x = element_line(color = c(rbind(rep("black", length(x_label) / 2), rep(NA, length(x_label) / 2))),        linetype = 2, size = 1))+geom_path(aes(x = df$dates,y = df$distribution,color ="red"))+.shade_curve(df =data.frame(x = df$dates,y = df$distribution),                 period_80$qstart,                 period_80$qend)+.shade_curve(df =data.frame(x = df$dates,y = df$distribution),                 period_95$qstart,                 period_95$qend,alpha =0.2)+ylab("probability")+xlab("timeline")+labs(color ='Verteilung')+# ggtitle("Visualization of get_infection_density") +theme(legend.position ="none",text =element_text(size =16*5/5))+theme(axis.text.x =element_text(colour ="black",face ="bold",angle =30,hjust =1))+theme(axis.title.x =element_text(colour ="black",face ="bold"))+theme(axis.text.y =element_text(colour ="gray50"))+theme(axis.title.y =element_text(colour ="gray50"))  g

Literature

[1] Xin H, Wong JY, Murphy C, Yeung A, Taslim Ali S, Wu P, CowlingBJ. The Incubation Period Distribution of Coronavirus Disease 2019: ASystematic Review and Meta-Analysis. Clinical Infectious Diseases, 2021;73(12): 2344-2352.

[2]https://en.wikipedia.org/wiki/Mixture_distribution

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