Two outbreaks of the same disease in different communities can lead to completely different outcomes [39]. The main difficulty in making inferences using typical epidemic models arises in the calculation of probability [3, 52].
Thesis Outline
We compare the results of the hierarchical fit against the fit of the model at each outbreak independently. The diagonal elements, qii, follow from the properties of the transition function and the definition of Q,.
Epidemic modelling
The SIR model
In the case where the population size, N, is constant, the number of cured individuals, R(t), can be easily recovered as R(t) =N−S(t)−I(t). For this model, it is easy to see that the DA representation is more informative than the population representation.
Bayesian inference
The likelihood
When the size of the state space is small, we can exploit exact methods to solve this series of equations. Each of these methods becomes more and more inefficient as the size of the state space increases, so instead we can use Monte Carlo methods [3].
Particle filters
- Bootstrap particle filter
- Alive particle filter
- Importance sampling
- Importance sampling in particle filters
In Section 2.4 we introduced the basic idea of the bootstrap particle filter and how it can be used to estimate the likelihood. This leads to a large improvement in the running time of the particle filter as all simulations are consistent with observations.
The SPSA algorithm
The MAP is the vector of parametersθMAP ∈Θ, where Θ is the support of the posterior, such that,. The second reason is to tune the particle filters used to obtain log-likelihood estimates, thus leading to improved efficiency of pmMH methods.
Tuning particle filters
This optimal number of particles is problem dependent and a suitable choice of particles is one that keeps the variance of the log-likelihood within strict bounds. MAP Estimation as a Minimization Problem Identifying the number of particles to use is a one-dimensional optimization problem.
MAP estimation as a minimisation problem
This noise term also depends on the number of particles used, which can also affect the overall performance of the algorithm. Stochastic optimization techniques allow us to solve problems of this form and obtain estimates of the minimum [16, 69].
SPSA for MAP estimation
An important consideration when performing SPSA is the choices of amplification sequences c(k) and a(k). The perturbation gain sequence, c(k), appears only when computing the lead estimate.
Application to the SIR model
- Choosing the key parameters
- Choosing the number of iterations
- Choosing the number of particles
- Sensitivity to the initial point
We can control the search performance by adjusting the number of iterations and the number of particles used. In Figure 3.6 we see that the number of particles does affect the convergence of the search. As the number of particles increases, we see that more points converge reasonably close to the MAP.
The estimated value of the loss function found during the search is shown by the dashed red line.
Application to the SEIAR model
- Model details
- Data
- Search setup
- Results
It may also be a byproduct of research that compares the current minimum estimate with the new estimate. In Table 3.6 we see that there is an agreement between the average estimate of the SPSA algorithm and the estimate obtained via the AKDE approach. In Table 3.7 we run the particle filter independently 10000 times on the AKDE estimation of the MAP and the mean SPSA estimation and average the results.
This is likely as a result of the step size chosen for the first term being slightly too large.
Summary
The 1995 Ebola outbreak in the Democratic Republic of Congo (DRC) was one of the largest outbreaks in the country [ 13 , 41 ]. A total of 316 individuals were found to have been infected during the outbreak, including the index case. 51] adjusted their model for the entire duration of the outbreak, taking January 6 as the starting day.
Due to the large amount of data, the dimension of the enlarged parameter space is on the order of about 1,000 dimensions.
Model
SIR
We initialize the current time t = 0 and let tn, where n denotes next, be the time of the next observed event. The rate of the original process is then calculated, given the current state of the system (note that we omit the time dependence of the state variables). We need the modified repair. rate must be set to 0 if Z2 =`2 or we must prevent the epidemic from fading.
For more complex models, we may need to impose a chain of events to make the system state consistent with the next observed event [8].
SEIR with partial detection of onset and removal events
You can determine the type of next event by looking at the current state of the system. In calculating these rates, we must consider the following forcing event. You can tell this by looking at the event type at the top of the stack; if one 6= 0, the next event is of type 1 or 4.
Inference on a simulated explosion If en = 0, then one of the events observed in the first modified process is next.
Inference on a simulated outbreak
We see in Figure 4.7 that the true parameter values (indicated by the gray lines) lie close to the medians of the marginal posterior densities. The values used to simulate the data set appear in the "True Value" column of the table. In Figure 4.8 we see that the average incidence of the simulations is consistent with the observed incidences.
From Figure 4.9 we see that the average of the simulations seems to reasonably capture the undetected events during the missing burst phase.
Inference on the 1995 outbreak
By applying the SPSA algorithm using the parallelized particle filter, we obtained an estimate of the MAP of. A comparison with the results of the three different approaches from the literature is shown in Table 4.3. It is also likely that, as they fit the data after the missing 55-day observation period, the parameter estimates are indicative of the fact that the missing data are only from the secondary phase of the outbreak.
Interestingly, we see that the standard deviations for the incubation period are larger than for the DA-MCMC estimate, which is approximately twice as large as the Chain-Binomial estimate.
Summary
The inference results from the DR Congo outbreak are consistent with those reported by McKinleyet al.[51]. In the next section, we extend the model to a hierarchical context to fit multiple Ebola outbreaks. This chapter builds on the work of Chapter 4 by using a particle filter to infer four Ebola outbreaks using a hierarchical model.
The models we consider for this Ebola outbreak assumed that this date was unknown, and we try to derive as an additional parameter the date of transmission change.
Epidemic model development
We attempted to map a complex transmission parameter directly responsible for hospital closures during the Yambuku and Kikwit outbreaks. The parameterτ is the datum of the midpoint of the change in contact behavior (the midpoint of the curve) and must be derived. After the midpoint, the reduction of the effective transmission parameter slopes off in an exponential manner to a potentially non-zero quantity, which is a more realistic model of the real dynamics.
For the Kikwit eruption, we then divided these probabilities between the first phase of the eruption when no active surveillance was being performed.
Independent inferences
Note that the broad uniform priors on the midpoint parameters τ(i) reflect that the change in transmission is equally likely to occur at any point in the course of the outbreak. We can use slightly fewer particles for the Kikwit model than in the previous chapter, since there is more certainty about the evolution of the eruption over the first 55 days. This is likely due to the strict measures taken in closing the hospitals and how this would have affected individuals' perceptions of Ebola.
The biggest intervention was the closure of the hospital and this happened 123 days after the outbreak.
Hierarchical modelling process
Before identifying the number of particles to use, we can consider using parallelism to reduce the variance of the probability estimates. Let ˆL(·) denote the estimate of the total probability contribution and let ˆLi(·) denote the estimate of the probability of the ith particle filter. For the shared parameters, we assign hyper-prior distributions to the parameters of the prior distributions to capture the variability between outbreaks.
For the hyperpriors on mσ and νσ, we want to allow less variability between outbreaks and focus on deriving a value of the true parameters.
Results
The small change to R0 in the hierarchical model is a result of the assumption of some common admixture behavior in different provinces of the DRC. This is a result of the pooling process, where the information contained in the more informative data sets dominates the shape of the posterior for the shared parameters. We see that the hierarchical model aggregates the estimates to average the independent studies.
Since the Kikwit model dominated that density, it is clear in the boxplot of the marginal posterior densities for 1/σ that the results for the other outbreaks are pulled towards the result of the Kikwit model.
Summary
The computational cost of the search is dominated by the running time of the particle filter. Using multiple feature evaluations requires multiple particle filter runs, and the search execution time grows linearly with the number of particle filter runs. This would help provide a more justifiable choice of the number of iterations and particles to use.
Searching for a larger number of runs of a particle filter in any given iteration will drastically increase the run time of the search.
Particle filtering and Ebola
We used the results of the independent derivations to better inform prior distributions about the shared parameters in the hierarchical model. Essentially, this means that the prior in the hierarchical model reflects the posterior density (for 1/σ) found during the inference on the parameters of the Kikwit model. This may be due to the small number of cases over the course of the outbreak.
An extension of the particle filtering methodology could be to examine a more refined proposal and subsequent selection of the next observed event.
