Details of the algorithm

The diagram in Figure 3.6 shows the transition between different states. Note that the S = susceptible At each iteration of the MCMC we randomly choose an individual j

S

I

I, R

Figure 3.6: State diagram for the infection times and the removal times to show the states transitions. I, R corresponds to the state of individuals removed in the interval

[tobs, T ], S represents the susceptibles individual and I the infected but not removed.

1. If j is susceptible (S), we either add an infection time sj ∈ [T0, T ] or both

infection and removal times simultaneously where sj ∈ [T0, T ] and rj ∈

[max(sj, tobs), T ]. The new vector s0 is accepted in the first case with proba-

bility α = min  1,2(T − T0) 3 L(β, λ, ν, s0; r) L(β, λ, ν, s; r)  (3.6.16) while in the latter case the acceptance probability is given by Equation (3.6.16) with 2(T −T0)

3 replaced by

(T −max(sj,tobs))(T −T0)

2 .

2. If j is infected but not removed (I), we either propose with equal probability

to delete the infection time sj or move uniformly sj ∈ [T0, T ] or add a removal

time. The new vector s0 of the infection times is accepted with probability given

by Equation (3.6.16) with 2(T −T0)

3 respectively replaced by

3

2(T −T0), omitted and

replaced by 3(T −max(sj,tobs))

4 .

3. If j corresponds to an individual observed to be removed, we can only propose to

move its infection time sj uniformly in [T0, tobs] and the acceptance probability of

the new infection time is then given by Equation (3.6.16) with 2(T −T0)

3 omitted.

4. Finally, if j is removed after the observation time (I, R), with equal probability

we either delete its removal time rj or delete the coupled infection and removal

time (rj, sj), or we move its infection time sj uniformly in [T0, T ] or move its

removal time rj in [max(sj, tobs), T ]. The acceptance probability is obtained

from Equation (3.6.16) with 2(T −T0)

3 respectively replaced by

4

3(T −max(sj,tobs)) and

1

2(T −max(sj,Tobs))(T −T0) in the first two cases respectively; and omitted in the last

two cases.

Instead of using a uniform proposal distribution, we can propose an independence

sampler which makes use of the likelihood. That is q(rj− sj, r0j− sj) ≈ W eibull(λ, ν).

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 3.7: Trace plots of model parameters for the three algorithms. (a), (d) and (g)

tobs = 60 with uniform proposal; (b), (e) and (h) T = 70 with uniform proposal; (c),

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0 200 400 600 β P oster ior density 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.001 0.002 0.003 0.004 0.005 0.006 0.007 tobs=60

T=70 with uniform proposal T=70 with Weibull proposal

(a) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 5 10 15 λ P oster ior density 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 tobs=60

T=70 with uniform proposal T=70 with Weibull proposal

(b) 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 ν P oster ior density 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 tobs=60

T=70 with uniform proposal T=70 with Weibull proposal

(c)

Figure 3.8: Posterior distribution of the contact rate β (a), the shape parameter λ (b) and the scale parameter ν (c) for different models: Streftaris-Gibson model (black), removal prediction using uniform proposal (red) and prediction using Weibull distribution (blue).

must truncate its distribution to this interval. In this case, the above algorithm is maintained with the only change occurring in the uniform proposal for the removal

replaced by the truncated Weibull distribution. This implies that_{(T −max(s}1

j,tobs)) is then

replaced by fW(rj−sj)

FW(T −sj)−FW(max(sj,tobs)−sj) where fW and FW are respectively the density

and the distribution function of the Weibull distribution. We name the algorithm with uniform proposal and the Weibull proposal distribution respectively as uniform- proposal and Weibull-proposal.

To illustrate the efficiency of our algorithms, we consider the simulated data of the

thresholds model described previously and observe the progress till time tobs = 60. We

impute new infections up to T = 70. We run the MCMC routine for 106 steps with

the first 10000 discarded using both uniform and Weibull proposal distributions. We

then compare the results with the one obtained at tobs. The trace plots in Figure 3.7

show that the chains mixed well and do not show any evidence of non-convergence. From Figure 3.8, we can see that the posterior distributions of the model parameters match in the three cases considered for the model parameters (see figure 3.8). Also, the chain obtained from the model with Weibull-proposal seems to mix better than the one with uniform (see Figures 3.7h and 3.7i). This results confirm that the algorithm performs well. Note that we will use this type of prediction later on to compute measures for control.

3.7

Conclusions

This chapter has reviewed the statistical approach mostly used to draw inference from epidemic data especially when the number of infected individual is unknown. The Bayesian techniques presented here, particularly the Reversible-Jump, play a significant role in epidemic modelling given the flexibility it offers when dealing with missing information, which is relevant when designing control strategies.

In chapter 5, we expand these techniques to the spatio-temporal models and mod- els that present structure within the population, and incorporate the Sellke construc- tion in order to provide an efficient control for real time epidemics. We go further in the chapter 7.2 to provide a statistical method in Bayesian framework, treating the transmission network as missing data.

Chapter 4

Coupling non-spatial epidemics

using latent processes

4.1

Introduction

The main question that arises for an epidemiologist at the outbreak of an epidemic

is which control strategy to adopt in order to prevent future outbreaks. In this

chapter, we will consider strategies for controlling the spread of an epidemic using control strategies based on the removal of individuals diagnosed as infected. The question of epidemic control has been crucial over the history of mankind. Several authors raised this question in the context of non-spatial epidemic models and several strategies for their eradication were proposed. For instance, Bootsma and Ferguson (2007); Daley and Gani (1999); Castilho (2006) developed control model based on education campaigns to contain epidemics such as AIDS and Ebola. Anderson (1982); Smith (1964); Becker and Dietz (1995, 1996), on the other hand, consider different vaccination strategies to optimise the control of diseases in human populations. In particular, Ball et al. (1997a) utilised a technique, called an “equalising” strategy, which aims to maintain the number of susceptibles in each group to be similar as possible. Also, Daley and Gani (1999) and WHO (2003) adopted a strategy whereby hosts are screened and put in quarantine if deemed to be at risk.

Here, we assume for simplicity that there is a perfect diagnostic test which, when applied to an individual, can tell us about its current state (e.g. screening test) with 100% accuracy, although this is unrealistic in practice. In other words the test has 100% sensitivity and specificity. Once infected individuals are identified, they can be removed (cured or isolated depending on the disease). Intuitively, the epidemic spread could be retarded due to the fact that susceptibles that would have been infected by those individuals, removed after testing, are “rescued”, at least for a certain period of time. Mathematically, the infection pressure exerted on a susceptible individual is reduced, causing them to approach their Sellke threshold (see Section 2.2.2) at a

slower rate.

The benefits of using the latent processes, in this case the Sellke thresholds, is that epidemics can be coupled via a common set of thresholds (Andersson and Brit- ton, 2000). The coupling method has found many important applications in several fields of probability theory such as Poisson approximation, renewal processes and Markov processes (Lindvall, 1992). More recently, the technique has been applied successfully to compare retrospectively alternative strategies on historical epidemic data (Cook et al., 2008). With the correct choice of coupling mechanism, the random variables constructed (the outcome of the controls) are highly dependent, leading to high correlation between the outcomes of the controls, reducing the variance of the difference. This surely gives a better estimation of the expected difference between control strategies. The sample size needed to estimate differences in strategies is re- duced compared to independent sampling as a consequence of the high correlation between the outcomes of controls. This is analogous to the well-known paired t-test where two population means are compared on the same set of experimental units.

In the remainder of this chapter, we will first describe a general framework for epidemic controls using Sellke thresholds. Conditioned on the individual thresholds and the model parameters, the focus will then be on comparing different control strategies using different epidemic models including the general stochastic epidemic model (SIR), and a non-typical, multi-type SIS model. In our initial investigations we will assume known parameters. When optimising control strategies we will make use of the technique of Simulated Annealing. This concept has been used in the past in epidemiology by Demon et al. (2011) to identify the sample size that maximises the probability of detecting an invasive pathogen. We then show that, the outcomes of controls are highly correlated when we couple realisations using the Sellke thresholds, which reduces the variability in the estimate of the expected differences as suggested above.