TÍTULO DÉCIMO

Accurate estimates of the risk of severe outcomes are necessary for policy planning in epidemic responses (Van Kerkhove et al., 2010). Measures of the severity of an epidemic have been

introduced in Chapter 1; the proportion of fatal cases given a specified condition has been mentioned as the quantity of interest that goes under the name of Case Fatality Rate, Case Fatality Ratio or Case Fatality Risk (Nishiura et al.,2009; Porta,2008). It is usually expressed as a percentage and, albeit its name might refer to a person-time measure (rate/ratio), it expresses a probability; therefore the notation of Wong et al. (2013a) is adopted in this thesis, naming this quantity the case fatality risk (CFR).

2.2.1 Methods for severity

Multiple indexes of severity exist, according to the aspect that is to be described. This chapter focuses on the hospitalised case fatality risk (hCFR), that is the probability of death given hospital admission. However, this literature review covers also other measures of severity, such as the infection fatality risk (IFR) or the symptomatic case fatality risk (sCFR), which are the fatality within infected and symptomatic cases respectively, since they have been often estimated together with thehCFRwhile describing the entire severity process.

The IFR is one of the most interesting severity measures because it represents the actual mortality in the whole population of infected individuals. The IFR is rarely computed via direct estimation of the number of infections and number of deaths in the population. The UK Department of Health, together with the Health Protection Agency (HPA) (now Public Health England (PHE)) achieved the goal of obtaining estimates of both the numerator and the denominator during the 2009 influenza A/H1N1 pandemic (Donaldson et al., 2009). This was possible due to the combination of data from several surveillance schemes rapidly activated during the pandemic. Other studies have estimated the IFRduring the 2009 H1N1 pandemic. Presanis et al. (2009) used Bayesian evidence synthesis to obtain estimates of theIFRfrom data of two cities of the USA. They adopted a pyramid approach, estimating the IFRby combining estimates of the probabilities of reaching different stages of severity (e.g. probability of having symptoms given infection, probability of hospitalization given symptoms, . . . ). Moreover their model accounted for specific testing and reporting probabilities for the different stages of severity. During the following years the group applied the same methodology to estimate the severity in the UK (Presanis et al., 2011; Presanis et al., 2014). Similar analyses were performed in New Zealand (Baker et al., 2009), in Finland (Shubin et al., 2014) and in the whole southern hemisphere (Baker, Kelly, and Wilson, 2009) where, thanks to a wide range of surveillance schemes already active from 2008, the infected population and the sCFR could be estimated. This was achieved by combining sentinel data on symptomatic cases from General Practitioner (GP) consultations with population data on GPconsultations and with experimental studies.

Another approach to estimate the IFRis to infer the excess mortality due to an infection. A simple model is applied by Murray et al. (2006) to analyse the influenza pandemic of 1918-1920. The authors calculated the average mortality rate in 1915-17 and 1921-23, and subtracted this average from mortality in 1918-20. This simple computation gave an estimate of the mortality due to influenza under the assumption that all excess deaths are truly associated with influenza. A more complex model is formulated in Wong et al. (2013b) to estimate the IFR of the 2009 influenza A/H1N1 pandemic Virus in Hong Kong. They derived a proxy of the total influenza activity (in terms of weekly incidence rates of pH1N1 infections) from weekly influenza-like illness (ILI) data and weekly proportions of specimens that tested positive for influenza. Statistical

models (namely, linear regression, time-series regression and Poisson regression) are used to model mortality from 2003 to 2009. Mortality is regressed on the proxy of 2009 influenza A/H1N1 pandemic activity and other covariates including other seasonal influenza proxies and weather variables such as temperature and humidity.

Serfling models (Serfling, 1963) are a sub-family of excess mortality methods that describe the excess of deaths (in counts) by comparing time series and including trigonometric functions to model seasonality. However all these excess mortality models rely on the assumption that the excess mortality is due to the virus we are analysing. Therefore, in the case of a mildly severe epidemic, or when other causes can increase mortality (other diseases, wars, . . . ) the estimates of theIFRare biased.

This consideration, together with the fact that data on both the numerator and the denom- inator of the IFR are rarely available, motivates the focus on the hCFR, hereby denoted only byCFR, in the following analysis of severity.

2.2.2 Methods for estimating the Case Fatality Risk

The World Health Organization (WHO) proposed the following estimator of the CFR in the case of an epidemic (WHO,2015):

[

CFR(who)(s) = cumulative number of deaths(s)

cumulative number of (hospitalised) cases(s) (2.5) with s ∈ (0, S) being the time of analysis. Estimator 2.5 assumes constant CFR and it is well known to be biased until the end of the epidemic, here denoted by S (Lipsitch et al.,

2015). This bias is due to right censoring that happens when the analysis is carried out at time s < S, when some patients at risk have not experienced any event such as death or recovery yet. To understand the effect of right censoring on the estimator 2.5 a simulated dataset has been plotted in Figure 2.4. This dataset contains the time from hospitalization to death and recovery generated using a parametric survival model for death and recovery from simulated hospitalization counts mirroring the counts of cases during 2012/13 epidemic. When analysed early, for example on the 100th day from the beginning of the epidemic, the number of hospitalizations is increasing according to the epidemic dynamics, and many individuals have not experienced the final event yet.

Several papers have addressed this problem and most of them have used survival analysis approaches, both parametric and non-parametric.

The problem of estimation of the CFR from survival data has been addressed under two perspectives. The first one assumes that the data-generating process is a mixture model for survival data (Farewell, 1982): the individuals belong to the group of people that die with probabilityCFRand to the group of people that survive with probability 1-CFR. Their time-to- event is then defined conditionally on the group to which the individuals belong. This approach has been adopted by Donnelly et al. (2003) within a parametric-survival context. The other approach takes a prospective perspective and assumes that the data-generating parameter is a competing risk process (which is a special case of MSMs introduced in Section 2.3.1). Ghani et al. (2005) and Jewell et al. (2007) proposed an estimator for theCFRin this context. Garske et al. (2009) briefly reviewed the underestimation error of theCFRestimates and the solutions proposed by Donnelly et al. (2003) Ghani et al. (2005) and Jewell et al. (2007).

● ● ●●●● ●● ●●●●● ●● ●● ●●●●● ●● ●●● ●●●●●●●●●●●●● ●● ●●●●● ●●●●● ●● ●● ●●●●● ●●●●●●●●● ●● ●● ●●●●● ●●●●●●● ●● ●● ●●●●● ●●●● ●●●●● ●● ●● ●●●●●●● ●● ●●●●●●● ●●● ●●●●●●●●● ●●●●● 0 20 40 60 80 100 120 140

Time from hospitalization to Death or Recovery

time (in days)

individuals ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ●●●●●●●● ● ●●●● ● ●● ●● ● ●●●●●●●●●● ●● ● ● ● ●● ● ● ● ● ●●● ●● ●●● ●●● ● ● ●● ● ●● ●●● ●●●● ●● ● ●●●●●●●●●●●●●●●● ●● ● ●● ● ●● ●● ●●●●●● ● ● ● ● ●●●● ●● ● ● ● ● Hospitalization Recovery Death 40 60 80 100 120 140 160 180 200 220

(a) Simulated survival data during an epidemic. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 10 20 30 40 50

60 Time from hospitalization to Death or Recovery

time (in days)

individuals ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Hospitalization Recovery Death 60 80 100 120

(b) Same data observed until day 100.

0.00 0.05 0.10 0.15

0.20 WHO estimation of the CFR

time (in days)

CFR

True CFR CFR(WHO)

40 60 80 100 120 140 160 180 200 220

(c) EstimatedCFR at several time

points during the epidemic. Figure 2.4: A simulated dataset with daily flu-related hospitalizations, deaths and recovery. In Panel (a) and (b) the x axis is the calendar time from the beginning of the epidemic and the y axis is the ordered (by hospital admission date) number of individuals. Panel (b) is the bottom left corner of Panel (a), when data are only observed until day 100. Panel (c) reports

in green the estimator 2.5 calculated at times s = 60, 80, . . . ,220 from the beginning of the

epidemic and in red the true value used to generate the data.

Estimators of theCFRstarting from count data have been proposed by Yip et al. (2005b) and Yip et al. (2005a), using a counting process approach and relaxing the assumption of constant

CFR on which survival data estimators are based. In the same context, Lam et al. (2008) proposed a test for constantCFRin the case of an emerging epidemic.

The papers listed above differ mainly in the data they analyse (individual time-to event data vs population count data) but also in their assumptions on the CFR (constant versus time- varying). However, a point in common is that they all attempt to overcome the problem of the biased estimators due to right censoring.