Uncertainty in the Tail of the Variant Creutzfeldt-Jakob Disease Epidemic in the UK

Despite low case numbers the variant Creutzfeldt-Jakob disease epidemic poses many challenges for public health planning due to remaining uncertainties in disease biology and transmission routes. We develop a stochastic model for variant CJD transmission, taking into account the known transmission routes (food and red-cell transfusion) to assess the remaining uncertainty in the epidemic. We use Bayesian methods to obtain scenarios consistent with current data. Our results show a potentially long but uncertain tail in the epidemic, with a peak annual incidence of around 11 cases, but the 95% credibility interval between 1 and 65 cases. These cases are predicted to be due to past food-borne transmissions occurring in previously mostly unaffected genotypes and to transmissions via blood transfusion in all genotypes. However, we also show that the latter are unlikely to be identifiable as transfusion-associated cases by case-linking. Regardless of the numbers of future cases, even in the absence of any further control measures, we do not find any self-sustaining epidemics.


Uncertainty in the Tail of the Variant Creutzfeldt-Jakob Disease Epidemic in the UK -Electronic Supplementary Material
Tini Garske, Azra C. Ghani MRC Centre of Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, Imperial College London, UK

Sensitivity Analyses Assumptions
In the main paper we have made some assumptions regarding effectiveness of control measures and test sensitivity of the prevalence test. Here, we investigate the sensitivity of our results to these assumptions. Table S 1 shows the four scenarios considered and the associated parameters that differ between them. All other parameters were sampled from the ranges as given in the main text.

Parameter fitting
The estimated parameters for the different scenarios are shown in Table S 2. Varying the assumptions regarding effectiveness of control measures and test sensitivity has very little impact on these estimates. However, the basic reproduction number R 0 does vary for the different scenarios, indicating the effect of the different control policies. Nevertheless, for none of the scenarios does the upper limit reach anywhere close to the threshold value. Even in the absence of all control measures we would not be in the situation where a self-sustaining epidemic is possible.

Epidemic scale
The overall shape and timing of the incidence curve is very similar under the four different scenarios (data not shown), and the point estimates do not differ substantially (see Table S 3). However, the upper limits of the credibility intervals vary somewhat between the different scenarios, with the largest upper limit in the case of inefficient leukodepletion. For reduced effectiveness of the donor ban, there is very little difference to the baseline scenario. While this might be surprising at first, it is easily explained by the fact that most of the blood transfusion cases are caused by donors that were infected via the primary route and are therefore not subject to the donor ban, whereas those infected via blood transfusions themselves are on average fairly old and do not cause many further transmissions during their remaining lifetime.
Although the test sensitivity is not a control measure, the prevalence in the population between 1995 and 2000 (as measured in the prevalence study) has an impact on the future course of the epidemic, with larger epidemics possible if the prevalence of infection is high, as the prevalent people might transmit the infection to others, whether they will develop clinical symptoms or not. If the sensitivity of the prevalence test is assumed to be only 50%, the true prevalence in the population is expected to be twice as high as the measured prevalence, and therefore the upper limits of the projected future size of the epidemic is considerably higher under this scenario than for the baseline scenario. Numbers are rounded to two significant digits.

Future transmissions
Due to the long incubation period it is conceivable that there might be a substantial number of cases in the future which have already been infected in the past and would therefore not be preventable with any additional control measures that might be implemented in the future. In order to assess the extent to which infections of future cases have already occurred we performed further simulations for 798500 sets of parameter values in which the transmission rate was set to 0 from 2010 onwards. The total number of future cases is considerably lower than in the baseline scenarios (Table S 4). While the expected number of cases from primary transmission is essentially identical indicating that these transmissions already have occurred, the majority of the transfusion associated cases in the baseline scenario have not been transmitted yet. The time series of the epidemic if no further transmissions occur is shown in Figure S 1, and shows a substantially smaller secondary peak than that of the baseline scenario. The most striking difference to the baseline time series ( Figure 6 in the main text) is the sharp cut-off of unidentifiable transfusion associated cases at around 2020 in the growing phase of the epidemic. This means that theoretically, the immediate introduction of effective control measures preventing the transmission via blood transfusions could reduce the number of future cases by around 260 over the next century as a best estimate, or by up to 2600 if the worst case scenarios turn out to be true.
Numbers are rounded to two significant digits.

The basic reproduction number
The basic reproduction number R 0 is defined as the mean number of onward infections a typical infected individual will cause, and in the limit of an infinite population this has the threshold property that for any values of R 0 < 1, the epidemic will certainly die out, whereas for R 0 ≥ 1, the epidemic will grow exponentially until saturation is reached unless it dies out through stochastic effects in the early stages. Values of R 0 ≥ 1 therefore indicate the potential for a self-sustaining epidemic. In our model the first generation of cases stems from primary infection through the consumption of BSE-contaminated food. Individuals infected via this route are assumed to show different characteristics to those infected in subsequent epidemic generations through human-to-human transmission via red cell blood transfusions, for instance different incubation periods. The concept of the basic reproduction number is valid in the early stages of an epidemic (before any saturation effects become important), but after the stochastic fluctuations of the very first few generations have passed. For calculating R 0 , we therefore only need to consider those infected via red cell transfusions.
Formulating the infection process as a multitype branching process, where the type i of any infected individual is determined by their genotype and age at infection, the basic reproduction number R 0 is given as the leading eigenvalue of the next generation matrix M = (m ij ), where m ij is the expected number of infections of type j caused by a type-i individual [1]. The infection process can be subdivided into two stages, (i) blood donation and (ii) infection following the transfusion.
Neglecting the effects of control measures such as leuko-depletion and the donor ban, individuals donate infectious blood units according to a time-inhomogeneous Poisson process, where the donation rate depends on age and the time since infection, as this determines whether infectivity in the blood has developed yet. The generating function of this Poisson process is given by where n b (a, g) is the expected number of infectious red cell units donated by a person of genotype g infected at age a during their lifetime, which we shall calculate later. The probability that any infectious red cell unit is transfused into a patient of age a is given by where Ψ b (a) is the age distribution of red cell transfusion recipients and b(a) is the mean number of units given in a transfusion to a person of age a. If the unit is transfused to a patient of genotype g, it will cause a new infection with probability βξ g , such that the infection process can be described by a multi-type Bernoulli process with generating function where s = (s a,g ) a∈{0,...,A},g∈{M M,M V,V V } . The total number of further infections of age a and genotype g caused by a person of genotype g who was infected at age a is given as the sum of X iid random variables, where X is the number of infectious blood units donated. The generating function of the two-step process is therefore The expected number of further transmissions of age a and genotype g caused by a person of genotype g infected at age a is given by the partial derivative of f a,g (s) with respect to s a ,g , evaluated at s = 1, that is the vector with all entries 1: m a,g:a ,g = ∂ ∂s a ,g f a,g (1) = βn b (a, g)π g ξ g P b (a ) .
As the next generation matrix M = (m a,g:a ,g ) factorises, in the absence of control measures the basic reproduction number R 0 as its leading eigenvalue can easily be derived as If the donor ban has effectiveness e b , it works by excluding a proportion e b of those who have ever received a blood transfusion from donating blood, and therefore excluding a proportion e b of those who have been infected by the blood-borne transmission route from donating. This means that the expected number of units donated by an average infective is reduced to (1 − e b )n b (a, g), therefore linearly reducing the value of R 0 by this factor. Similarly, the effect of leukodepletion is to reduce the transmission probability β to (1 − e l )β, where e l is the effectiveness of leuko-depletion, reducing R 0 by the factor (1 − e l ), such that with leuko-depletion and the donor ban in place, the basic reproduction number is given by The expected number of infectious red cell units donated For simplicity, the population is structured into age cohorts, with the whole age cohort born at the start of the calendar year, rather than throughout the year. This means that the total population size fluctuates slightly over the year. We have N (t) = N exp(−M φ(t)), where φ(t) = t − t is the phase of the time.
At the end of the calendar year, the population has fallen by the number of births into each cohort N 0 , and therefore M = ln N N −N0 . In the British population, the overall annual death rate is M ≈ 1.3%pa. In a population that has a constant birth rate and constant (albeit age-dependent) death rates, the age-composition of the population is stationary over time. At the start of the calendar year, the number of people of age a is proportional to the survival probability to that age, N a = N S(a) s , where s = A a=0 S(a). At any time t, the number of people aged a is N a (t) = N a exp [−µ a φ(t)], where µ a is the death rate at age a, which is given by µ a = − ln S(a + 1|a) = − ln S(a+1) S(a) . We assume that all age-dependent rates (death, donation etc.) are constant per year, so that they don't depend on the exact age a c , but only on the count of years a = a c . However, as infection can occur at any exact age, not just at the birthday, for evaluating the number of secondary infections we need to take into account the actual age at primary infection, as this determines when the rates change.
The expected number of infectious blood units donated by a person of genotype g who was infected at age a c = a + t 0 , where a is an integer and t 0 ∈ [0; 1[, throughout their lifetime is given as where α(t|a c g) is the rate of donation of infectious blood units by a person aged a c + t, given infection at age a c , and P (t|a c g) is the probability that the person is alive at t 0 + t, given they were infected at t 0 when aged a c .
Assuming that blood donations match the demand for transfusions, where G min g (t) = G(t|κ min , η g ) is the cumulative distribution function of the γ-distributed time to becoming infectious, giving the probability that the person has already entered the infectious stage at time t after infection. It is given as and we assume here that only the scale parameter η depends on the genotype. The mean of this distribution is given by ∆ = κ/η. As the time-dependence of the rate of donations (for constant age) over the year is an artifact of the separation of the population into birth cohorts, we average the exponential factor over the year and get The probability of being alive at time t after infection at age a c is given by where ω 2 is the probability of sub-clinical infection, and G g (t) is the cumulative distribution function of the incubation period (10), which is γ-distributed with the same shape parameter η g , but a different scale parameter κ ≥ κ min than the latent period. The first factor takes into account potential death from disease for pre-clinically infected individuals, whereas the remaining term accounts for death from other causes. Here, we have introduced the shorthand Z g (t) = 1 − G g (t). The expected number of infectious red cell units donated throughout their lifetime by a person infected when age a c = a + t 0 is given by whereas the expected number of infectious red cell units donated by a person infected at some point while they were aged between a and a + 1 is Here, we set Φ b (a > A) = 0 (note a is an integer, and therefore this relates to a c ≥ A + 1) in order to ensure that no transmission can occur after reaching the largest possible age. In order to split up the integral over t 0 , we write u = t , with We therefore have Inserting this back we get [H b (a + u)I − (ag, u) + H b (a + u + 1)I + (ag, u)] , where and the integrals are I − (ag, u) and I + (ag, u). Using the definition of Z g (t) and G g (t), I ± (ag, u) can be rewritten as the sum of integrals of the form