Simulation study

We use the methods outlined in Section 5.3.1 to illustrate the effects of estimat-ing epidemics governed by the Cauchemez model with the simpler and more widely used basic model. In particular, we wish to assess the accuracy of using the basic model to estimate critical vaccination coverage for the three vaccina-tion strategies outlined in Secvaccina-tion 5.1.3 for epidemics with dynamics governed by the Cauchemez model. To achieve this, epidemic simulations are performed to generate emerging epidemic data and final size data.

Let θ = (λ_G, λ_L, η, γ)be a vector denoting the unknown parameters of a given epidemic under the Cauchemez model. The data from the simulated epidemics that take off are used to determine one of four possible estimators for θ. These are denoted by ˆθ⁽_EME^Cauch), ˆθ⁽_{FI N}^Cauch), ˆθ_EME^(basic) and ˆθ⁽_{FI N}^basic). The subscripts EME and

FIN refer to estimators made from emerging and final size data respectively (and so no estimate of γ is made for the latter case) and the superscripts(Cauch) and(basic)refer to the Cauchemez and basic models respectively. For the basic model estimators, ˆη is fixed at 0. An estimator for the CVC may be calculated from each of these four estimators using the methods of Section 5.1.3.

The parameter choices for the epidemic used in the simulation study are as follows. As outlined in Section 5.3.1, the infectious period, T_I, is set to have an exponential distribution with rate γ =1 and hence E[T_I] = 1. Two population distributions are considered to reflect different household structures in different parts of the world. The first, α_UK = [0.36, 0.30, 0.15, 0.10, 0.05, 0.04], represents a typical city in Western Europe and is based on data taken from the 2011 UK census for the urban area of Nottingham (Office for National Statstics [2011]).

Due to the lack of households of size greater than 6, all households meeting this criterion have been truncated to be of size 6 for the sake of convenience.

The second structure, α_Ghana = [0.20, 0.15, 0.15, 0.14, 0.12, 0.09, 0.05, 0.10], repre-sents a typical city in West Africa and is taken from the combined urban data of the 2010 Ghanaian census (Ghana Statistical Service [2012]). All households of size 8 or above are truncated to size 8, again for the sake of convenience.

The parameters governing disease transmission are set to be λ_G = 0.8, λ_L = 2 and η = 1. These values are chosen such that approximately 50% of an unvac-cinated population becomes infected if the epidemic takes off under the α_UK population structure (see Ball et al. [2010a], Ferguson et al. [2005]). For compar-ison, the pre-vaccination threshold parameter R_∗ = 1.57 under the α_UK popu-lation structure and R_∗ = 2.20 under α_Ghana for epidemics with parameters as outlined above. The value of η = 1 follows the suggestion of Cauchemez et al.

[2004] and Chapter 3.

Consider the following vaccines with efficacy ǫ = 0.84: an all-or-nothing vac-cine, a leaky vaccine (A = 0.16) and a non-random response vaccine with A = B = 0.4. For the epidemic outlined above, Table 5.3 gives the true CVC for the epidemic outlined above under each vaccine action model, each of our three vaccination strategies and both the UK and Ghanaian population struc-tures. Note from Table 5.3 that the CVC values are generally larger under the Ghanaian population structure which contains larger households and thus has potentially larger local outbreaks. The CVC is also smallest under the optimal

Table 5.3: Critical vaccination coverage (CVC) for the epidemic outlined in this simulation study under different population structures, vaccine ac-tion models (all with efficacy ǫ=0.84) and both the UK and Ghana-ian population structures, as outlined in the main text

Population

structure Vaccine action model c⁽_v^ind) c⁽_v^house) c⁽_v^opt)

UK

All-or-nothing, ǫ =0.84 0.2988 0.3976 0.2113 Leaky, A =0.16 0.3100 0.3998 0.2225 Non-random, A =B=0.4 0.3211 0.3998 0.2323

Ghana

All-or-nothing, ǫ =0.84 0.4269 0.5861 0.3492 Leaky, A =0.16 0.4430 0.5889 0.3643 Non-random, A =B=0.4 0.4684 0.5889 0.3903

strategy (as one would hope), followed by the random individuals strategy and finally the random households strategy is least effective. We also see that the all-or-nothing vaccine performs better than the leaky vaccine which in turn gen-erally outperforms the non-random response A = B vaccine, even though all three vaccines have the same efficacy. These observations are reassuring given previous results in the literature (see, for example, Ball et al. [2004a] and Ball and Lyne [2006]) and those seen in Section 5.2.

However, the two non-random response vaccines (Leaky and A = B) perform equally well under the random households strategy. In a household of size n in which every individual has been vaccinated with a non-random response vaccine which has effect (A, B), infectious contacts between a susceptible and infective occur at rate ABλ⁽_Lⁿ⁾. Thus, any two non-random response vaccines with the same efficacy will perform equally well under a random households strategy, since we have already established in Section 5.2 that only the vaccine efficacy ǫ affects the gain of vaccinating a given individual in terms of curtailing global infectious contacts.

Before looking at simulation studies, we should take account of the fact that, in practice, implementing the optimal vaccination strategy relies on having knowledge of λ_L and η. Specifically, the discussion in Section 5.2 shows that an estimate of η that is far enough away from its true value can lead to incorrect guess as to the form of the optimal vaccination strategy. As such, we now

intro-duce another vaccination strategy which will be referred to as the fitted optimal strategy. This scheme uses the estimated parameters of an epidemic rather than the unknown true parameters to determine an “optimal” vaccination strategy.

Thus, the fitted optimal strategy is not necessarily optimal.

Let c⁽_v^opt) denote the CVC for an epidemic under the fitted optimal vaccination strategy and ˆc⁽_v^opt) be its estimator. Also, let ˆc⁽_v^ind) and ˆc⁽_v^house) be estimators of the CVC under the random individuals and random households strategies re-spectively. Note that Table 5.3 does not contain true values for the fitted optimal strategy since it is dependent on observed data and thus is a random variable.

We simulate final size data for a population of m = 500 households for the UK population structure and m = 300 households for the Ghanaian popula-tion structure so that the total populapopula-tion size is similar for both populapopula-tions.

(N = 1150 for the UK structured population whilst N = 1020 for the Ghanian structured population.) In both cases 1000 epidemics were simulated, with the data used to give estimates ˆθ⁽_{FI N}^Cauch) and ˆθ_{FI N}^(basic), which were then used to give estimates of c⁽_v^ind), c⁽_v^house) and c⁽_v^opt) for each of the vaccine action models con-sidered in Table 5.3.

As explained in Chapter 4, estimating epidemic parameters from emerging data is only reliable if the population is large enough such that there is a point in the epidemic at which the proliferation of infected households still resem-bles the branching process outlined in Chapter 2. However, we must also en-sure that there are enough infected households to give a reliable estimate of θ_EME. As such, we consider epidemics in a population of m = 10000 house-holds (N = 23000) for the UK population structure and m = 6000 households (N = 20400) for the Ghanaian population structure, both of which are used to provide emerging epidemic data after 500 recoveries have been observed, (see Section 4.5.3). An estimate ˆr of the real-time growth rate is made as described in Section 5.3.1 by using the polyfit function in MATLAB and ignoring the first 20 recoveries (also see Section 4.5.1). Again 1000 epidemics are simulated and the data used give CVC estimates for each strategy under the basic and Cauchemez models.

For this simulation study, we focus on the value ˆc_v −^cv, i.e. the difference between estimated CVC and its true value. If this value is positive, then the population would be over-vaccinated if the estimated CVC is used, potentially

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

Figure 5.1:Kernel density estimates of ˆcv −^cv for each of our vaccination strategies. These plots are based on 1000 simulations of the epi-demic outlined in this section for the UK population structure and use an all-or-nothing vaccine with efficacy ǫ = 0.84. We consider both final size data and emerging epidemic data and estimate the CVC assuming both a basic and Cauchemez model for the epi-demics. The true CVC values were c⁽v^ind) =0.2988, c⁽v^house)=0.3976 and c⁽v^opt)took values in the range[0.2113, 0.2160]

wasting valuable resources. If it is negative then the population will be under-vaccinated, leaving it vulnerable to a global outbreak. In Figure 5.1 we illustrate how choice of vaccination strategy under our all-or-nothing vaccine affects the potential to under/over-vaccinate a population for our epidemic, using both our simulated final size and emerging epidemic data from the population struc-ture α_UK, by giving kernel density estimates of ˆc_v−^cvfrom our 1000 epidemic simulations.

As one would expect given that we simulated epidemics from the Cauchemez model, the distribution of ˆc_v−^cv is centred around zero for the fitted opti-mal, random individuals and random households strategies if the Cauchemez model is used to estimate the unknown parameters of the epidemic and this is observed from both the final size and emerging epidemic simulations. If a basic model is used for parameter estimation, both our final size and emerging plots suggest that the fitted optimal strategy is more likely to under-vaccinate the population. The distribution of CVC estimates for random individuals and ran-dom households strategies are both still loosely centred around the true CVC value when the basic model is used for parameter estimation. The plots sug-gest that over-vaccination may be more likely under these strategies if CVC estimates are made using the basic model from final size data.

We also observe that the variation of the estimators ˆθ⁽_{FI N}^basic), ˆθ⁽_{FI N}^Cauch), ˆθ_EME^(basic) and ˆθ⁽_EME^Cauch) can all lead to rather large errors in CVC estimators. These errors are greater under the random individuals and random households strategies al-though this may be attributed to the fact that these strategies generally require more individuals to be vaccinated than the fitted optimal strategy so this is not entirely unexpected. (Generally speaking, variance of ˆc_v is reduced when c is closer to 0 or 1.) It is also worth noting that the variance of the estimators of θ reduces as the number of households m increases and so this is a less problem-atic issue if large enough data are available (cf. Chapters 3 and 4). We discuss this further in Section 5.3.3.

Figure 5.1 only shows kernel density estimate of the probability density func-tion of the random variable ˆc_v−^cv under the all-or-nothing vaccine. We now turn our attention to comparing our three vaccine action models, as given in Table 5.3. Kernel density estimates of the distribution of ˆc⁽_v^opt)−^c(v^opt) for our epidemic under the α_UK population structure are given in Figure 5.2 for each

−0.20 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 2

4 6 8 10 12

ˆ c^{(f it)}_v − c^{(f it)}v

All−or−nothing (Basic) All−or−nothing (Cauchemez) Leaky (Basic)

Leaky (Cauchemez) Non−random, A=B (Basic) Non−random, A=B (Cauchemez)

Figure 5.2:Kernel density estimates ˆc⁽v^opt)−^c(v^opt) for the 1000 final size data simulations of the epidemic outlined in this section under the UK population structure. Plots are shown for estimates based on as-sumption of a basic model and a Cauchemez model for local epi-demic dynamics and three vaccine action models are considered, each with efficacy ǫ = 0.84. These are the all-or-nothing vac-cine, a leaky vaccine and a non-random response vaccine with A = B = 0.4. The true CVC under the fitted optimal strategy, c⁽v^opt), took values in the range[0.2113, 0.2160]for the all-or-nothing vaccine, [0.2225, 0.2275] for the leaky vaccine and [0.2323, 0.2379] under the non-random response vaccine with A=B

of our vaccine action models, using final size data.

Despite the differences between the three vaccine action models that we have observed in Section 5.2 and Table 5.3, we observe that the three vaccine models exhibit very similar behaviour with respect to the distribution of ˆc⁽_v^opt)−^c(v^opt)

under each vaccine model. Figure 5.2 shows this to be the case whether the correct Cauchemez model or incorrect basic model are used to estimate θ. Sim-ilar plots for emerging epidemic data and alternative vaccination strategies are omitted but show similar results. Thus we conclude that whilst the vaccine action model affects the value of the CVC under any given vaccination strat-egy/population structure/data type, it does not appear to have much bearing on the probability of one under/over-estimating the CVC, if the methods out-lined in this chapter are used.

Finally, we use our simulation study to consider the effects of population struc-ture when estimating the CVC for an epidemic. Kernel density estimates of the distribution of ˆc⁽_v^opt)−^c(v^opt) for our epidemic under the all-or-nothing vaccine are given in Figure 5.3 for both the α_UK and α_Ghana population structures. In Section 5.3.3 we discuss how population structure affects whether one would expect to over-estimate or under-estimate the CVC if an incorrect model choice for local contact rates is assumed. However, from this figure, we see that the variance of ˆc⁽_v^opt) −^c(v^opt) appears to be greater under the complex Ghanaian population structure than the simpler UK structure, especially when final size data are used. Similar plots under other vaccination models and strategies re-veal a similar trend and thus are omitted.

The illustrations from this simulation study show that highly inaccurate CVC estimates are plausible for realistic sizes of data, even if the correct model is chosen. This is particularly likely if there is significant variety in the household sizes within a population and if the more practical random households or ran-dom individuals vaccination strategies are selected, rather than attempting to find an optimal vaccination strategy. In the following section, we investigate ex-actly when parameterising a Cauchemez model epidemic using the basic model would be expected to lead to the most severe cases of under/over-estimation of the CVC and also illustrate the effects of under-vaccination in terms of the expected final outcome if an epidemic in an under-vaccinated population takes off.

−0.250 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 2

4 6 8 10 12

ˆ c^{(f it)}_v − c^{(f it)}v

Final size

UK (Basic) UK (Cauchemez) Ghana (Cauchemez) Ghana (Basic)

−0.250 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

2 4 6 8 10 12

ˆ c^{(f it)}v − c^{(f it)}v

Emerging

Ghana (Basic) UK (Basic) Ghana (Cauchemez) UK (Cauchemez)

Figure 5.3:Kernel density estimates of ˆc⁽v^opt)−^c(v^opt) based on 1000 simula-tions of the outlined epidemic under the αUKand α_Ghanapopulation structures, using both the basic and Cauchemez models to estimate the epidemic parameters. Estimates are based on an all-or-nothing vaccine with efficacy, ǫ = 0.84. The true CVC under the fitted op-timal strategy, c⁽v^opt), took values in the range[0.2113, 0.2160]under αUKand[0.3492, 0.3669]under α_Ghana