Infectious Disease Models. Angie Dobbs. A Thesis. Presented to. The University of Guelph. In partial fulfilment of requirements.

Transcription

1 Issues of Computational Efficiency and Model Approximation for Individual-Level Infectious Disease Models by Angie Dobbs A Thesis Presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Mathematics and Statistics Guelph, Ontario, Canada c Angie Dobbs, December, 2011

2 ABSTRACT Issues of Computational Efficiency and Model Approximation for Individual-Level Infectious Disease Models Angie Dobbs University of Guelph, 2011 Advisor: Professor R. Deardon Individual-level models (ILMs) are models that can account for the spatial-temporal nature of disease data to capture the dynamics of such infectious diseases. Parameter estimation is usually done via Markov chain Monte Carlo (MCMC) methods, but correlation between the model parameters negatively affects MCMC mixing. Introducing a normalization constant to alleviate the correlation results in MCMC convergence over fewer iterations. This occurs, however, at the expense of greater computation time per MCMC iteration. It is important that fitting these models is done as efficiently as possible. An uppertruncated distance kernel is introduced to speed up the computation of the likelihood, but in doing so there is a loss of information and thus goodness-of-fit. Both the normalization constant and upper-truncated distance kernel are evaluated as components in various ILMs via a simulation study. The normalization constant is seen not to prove to be worthwhile as the negative effect of increased computation time is not outweighed by the positive effect of reduced correlation. The upper-truncated distance kernel behaves as expected, reducing computation time yet worsening model fit as the truncation distance decreases.

3 Table of Contents 1 Introduction 1 2 Issues of Computational Efficiency and Model Approximation for Spatial Individual-Level Infectious Disease Models Abstract Introduction Individual-Level Models Study Simulation Study Results Study Methods Simulation Study Results Discussion and Future Work iii

4 List of Tables 2.1 Average correlations between parameters (standard errors in parentheses) averaged over the results from the ten epidemics of Set 1 and Set 2, respectively Average effective sample sizes for α and β (standard errors in parentheses) averaged over the results from the ten epidemics of Set 1 and Set 2, respectively Average computation time in seconds and the run-time ratio for α and β averaged over the results from the ten epidemics of Set 1 and Set 2, respectively (standard errors are in parentheses) Computation time in seconds (standard errors are in parentheses) averaged over the results from the ten epidemics of Set 1 and Set 2, respectively. *These results are for only 4 out of 20 epidemics that could be fitted under these models. MCMC was run for 125,000, rather than 20,000, iterations here iv

5 2.5 Run-time ratios for α and β averaged over the results from the ten epidemics of Set 1 and Set 2, respectively (standard errors are shown in parentheses). *These results are for only 4 out of 20 epidemics that could be fitted under these models. MCMC was run for 125,000, rather than 20,000, iterations here Average DIC and AIC values averaged over the results from the ten epidemics of Set 1 and Set 2, respectively. *These results are for only 4 out of 20 epidemics that could be fitted under these models. MCMC was run for 125,000, rather than 20,000, iterations here v

6 List of Figures 2.1 Posterior predictive plots for model P it (ø) (top left) and P it (N,δ,) with δ = 6 (top right), δ = 4 (bottom left) and δ = 2 (bottom right) vi

7 Chapter 1 Introduction The risks associated with infectious disease spread go beyond human health. Although a major incentive behind the study of modelling infectious diseases is in understanding those which cause human mortality, threats to both vegetative and livestock populations are also of great interest. Diseases which affect plants and animals, such as the tomato spotted wilt virus and foot and mouth disease, limit the populations food resources and can cost millions to eradicate. Potentially foodborne infectious disease, for example, amount to estimated costs of $88.8 million a year in New Zealand (Scott et al., 2000). Advancements in antibiotics, vaccinations and other treatments of disease have been successful in increasing human lifespan. However, a multitude of diseases still manage to persist and evolve, and the importance in understanding the dynamics of such diseases is becoming increasingly more apparent. Greater and more accessible computational power, in addition to this increase in interest in public health policy, are factors which contribute to a growing tide of in-depth research on models of disease transmission (Cook et al., 2007). Deardon et al. (2010) provide an example of statistical models which can be used to 1

8 model infectious disease through space and time at an individual level. These individuallevel models (ILMs) aim to predict the spread of an infection based on an individual s geographic location. Spatio-temporal models such as these are generally applied in a Bayesian framework where parameter estimation is done using Markov chain Monte Carlo (MCMC) methodology. This framework is desirable due to its ability to handle complex models and infer unobserved or missing data. Chis-Ster and Ferguson (2007), Jewell et al. (2009) and Deardon et al. (2010), for example, demonstrate this ability by applying an MCMC-based methodology to the 2001 UK Foot and Mouth epidemic. The parameters of the spatio-temporal ILMs of Deardon et al. (2010) have a naturally occurring, underlying correlation which negatively affects the mixing of the MCMC. A normalization constant can be introduced to these models as a means of reducing this correlation, thus resulting in MCMC convergence in fewer iterations. However, it is generally a computationally intensive task to compute this normalization constant and, thus, the computation time required per MCMC iteration tends to increase. The primary ILM used in Deardon et al. (2010) utilizes a geometric distance kernel involving the Euclidean distance between individuals in the population. A truncated distance-kernel can be used, imposing a maximum distance over which disease can transmit between two individuals. With careful programming, such a truncated distance kernel can be used to reduce computation time. This is desirable since calculation for ILMs can be highly computationally-burdensome, especially the likelihood for large populations. The model fit worsens, however, as the upper-truncation distance decreases, and so caution must be taken in selecting an appropriate truncation point. This thesis is designed to evaluate and compare the inclusion of the normalization constant and upper-truncated distance kernel in the form of two simulation studies. 2

9 Chapter 2 presents the details of these studies. It is written as a manuscript to be submitted to the journal, Spatial and Spatio-temporal Epidemiology. 3

10 Chapter 2 Issues of Computational Efficiency and Model Approximation for Spatial Individual-Level Infectious Disease Models 2.1 Abstract Individual-level models (ILMs) are models that can account for the spatial-temporal nature of disease data to capture the dynamics of such infectious diseases. Parameter estimation is usually done via Markov chain Monte Carlo (MCMC) methods, but correlation between the model parameters negatively affects MCMC mixing. Introducing a normalization constant to alleviate the correlation results in MCMC convergence over fewer iterations. This occurs, however, at the expense of greater computation time per MCMC iteration. 4

11 It is important that fitting these models is done as efficiently as possible. An uppertruncated distance kernel is introduced to speed up the computation of the likelihood, but in doing so there is a loss of information and thus goodness-of-fit. Both the normalization constant and upper-truncated distance kernel are evaluated as components in various ILMs via a simulation study. The normalization constant is seen not to prove to be worthwhile as the negative effect of increased computation time is not outweighed by the positive effect of reduced correlation. The upper-truncated distance kernel behaves as expected, reducing computation time yet worsening model fit as the truncation distance decreases. 2.2 Introduction Modelling the spread of infectious diseases is an ever-growing field as more and more diseases arise posing risks to public health and economic growth. Real time epidemic modelling is also being seen as increasingly useful. It can be important to the understanding of how a disease is spreading as an epidemic unfolds, so as to implement appropriate measures to control or eradicate the disease as quickly as possible. Outbreaks of H1N1 and SARS are just two which have triggered a response in the increase in research on disease transmission models (O Neill, 2010). However, these models are important for understanding not only the dynamics of human diseases, but also for animals and plants. For example, the 2001 Foot and Mouth epidemic in the UK, in which 34 million animals were slaughtered, has been analyzed by many researchers including Keeling et al. (2001), Diggle (2006), Chis-Ster and Ferguson (2007), and Deardon et al. (2010). Similar models can also be used to model invasive species; for example, Cook et al. (2007) implement a fully spatio-temporal model for the spread of 5

12 Heracleum mantegazzianum (Giant Hogweed) in Britain. Spatio-temporal models often provide us with a means of estimating the course a disease will take. However, in order to understand how the disease is spreading with as high a degree of certainty as possible, these models must be sufficiently accurate. Accuracy of these models can be assessed in a variety of ways including posterior predictive plots (Gardner et al., 2011; Gelman et al., 2000). Model comparison can also be carried out in a number of ways, including using Akaike s Information Criterion (Akaike, 1974), the Deviance Information Criterion (Spiegelhalter et al., 2001), or reversible jump MCMC (Green, 1995) to find posterior model probabilities and/or Bayes factors. The individual-level models (ILMs) of Deardon et al. (2010) allow for the modelling of spatio-temporal disease dynamics. They are discrete time models that can utilize information on an individual s geographic location to model the spread of an infectious disease. These models are generally applied in a Bayesian framework and the parameter estimates of the models can be obtained via Markov chain Monte Carlo (MCMC) methods. Among the benefits of using a Bayesian MCMC framework is its ability to handle complex models, as well as its ability to infer unobserved or missing data. It can be difficult to obtain complete data on the infectious status of individuals within a population due to misclassification of disease or under-reporting, as is often the case with common, seasonal diseases such as influenza. Another common issue with disease data is the inaccurate recording of infection times. Often the time of onset of symptoms is recorded as the time of infection, however, some diseases may show symptoms well after an individual has actually contracted the disease (Cauchemez and Ferguson, 2011). The primary ILM used in Deardon et al. (2010) utilizes a geometric distance kernel 6

13 involving the Euclidean distance between all individuals in a population. Due to a naturally occurring, underlying correlation between the parameters of such ILMs, MCMC mixing can be negatively affected. A normalization constant can be introduced to such models to reduce this correlation, resulting in MCMC convergence in fewer iterations. The calculation of the normalization constant, however, is generally computationally intensive and results in an increase in required computation time per MCMC iteration. A second computational issue is that the likelihood itself can take a very long time to calculate, especially for large populations. One way of helping to deal with this problem is to truncate the distance kernel that is, impose a maximum distance over which disease can transmit between two individuals under the model. The lower this maximum distance is set, the greater the opportunity for time saving. However, the lower this maximum distance is set, the more likely it is that long distance infections that occurred in reality will be excluded as a possibility from the fitted model, thus compromising model fit. The purpose of this paper is to evaluate and compare the effects of the normalization constant and the effects of an upper-truncated distance kernel. The effect of a sparks term, a parameter which describes sources of infection unexplained by the model, is also considered. Two studies are conducted for these evaluations: Study 1, described in Section 2.4, examines the effect of including the normalization constant and the tradeoff between improved mixing and increased computation time; and Study 2, described in Section 2.5, evaluates the use of the upper-truncated distance kernel and the trade-off between improved computation time and reduced goodness-of-fit. A simulation study is carried out in which all models are fit to the simulated data via MCMC methods. Computation time is recorded to determine efficiency and goodness-of-fit is determined by study-specific methods. 7

14 2.3 Individual-Level Models Eight different models are used in this paper that are all examples of the individual-level model (ILM) of Deardon et al. (2010). The first four are emphasized in Study 1, and the remaining four are included for discussion in Study 2. Here we use an SIR framework which assumes discrete time and that each individual is within one of three states at any given time t. If individual i does not currently have the disease but is able to contract it they are susceptible, i S; once they have contracted the disease they are infectious, i I; and once the individual is no longer infectious they are considered to be removed, i R. Removal could represent recovery accompanied by acquired immunity, isolation or death. Two transitions are possible and must occur in the specific order: S I and I R. S(t), I(t) and R(t) are the sets of susceptible, infectious and removed individuals of the population at time t respectively. Deardon et al. (2010) give the probability of a susceptible individual i being infected at discrete time t in the following general ILM: P it = 1 exp η(i) τ( j)κ(i, j) + (i, t), (2.1) j I(t) where, η(i) is a function which describes the potential risk factors related to the risk of susceptible individual i contracting the disease; τ( j) is a function which describes the potential risk factors related to the risk of infectious individual i transmitting the disease to others; κ(i, j) is the infection kernel, and (i, t) is a sparks function which represents the chance of being randomly infected by sources not explained by η(i), τ(t) or κ(i, j). This probability determines the time that the S I transition occurs. In this paper the models considered assume that the main drives of the epidemic are predominantly spatio-temporal, with η(i)τ( j) = α, i, j. α is known as as infectivity 8

15 constant. The parameter α can be thought of as representing the average strength of the epidemic. An increase in α would result in an increase in the probability of infection, P it, which would result in a larger population of infectious individuals and thus a stronger epidemic. Here, (i, t) is set to take one of two values, 0 or, where is itself referred to as the sparks term. Sparks terms describe random behaviour. Here, a model which includes a distance kernel but no sparks term allows the risk of disease transmission to be based solely on the proximity between a susceptible individual and individuals in the infectious population. The sparks term can therefore be thought of as introducing random noise to the system, or, if used in conjunction with a truncated kernel, allowing for infections over longer distance than that allowed by the kernel (see Section 2.5). The infection kernel, κ(i, j), is a function describing the risk of transmission specifically from individual j to individual i. Here, the infection kernel takes one of two forms, as described in Sections 2.4 and 2.5.1, or variants thereof. Both forms are primarily geometric distance kernels based on the Euclidean distance between two individuals i and j. However, one form allows for disease spread to occur across the entire population, and the other, termed the upper-truncated distance kernel, only allows infection between those individuals within a specified distance of one another. For simplicity, the second transition I R is assumed to occur after a constant time lag, γ I, known as the infectious period. Of course, it would be easy, and fairly standard, to extend the model in order to estimate the infectious period. 9

16 Likelihood The likelihood for the general form of the ILM in (2.1) is given as follows: t max π(d θ) = t=1 i S (t+1) 1 P it P it, (2.2) i I(t+1)\I(t) where D is the observed epidemic data, θ is the vector of parameters to be estimated, S (t + 1) is the set of susceptible individuals at time t + 1, and I(t + 1)\I(t) is the set of newly infected individuals at time t Study 1 The purpose of this study is to examine the effect of including, or not including, a normalization constant for the spatial infection kernel in a simple two-parameter spatial ILM. The expected effect of including such a normalization constant is to increase computation time, but reduce the correlation between two parameters, α and β, and thus improve MCMC mixing. The aim here is to determine whether the negative effect of increased computation time is outweighed by the positive effect of improved mixing. Models for Study 1 Each model in this paper is derived from the general ILM given in the previous section with η(i)τ( j) = α, i, j. We allow m to represent the total number of individuals in the population. Additionally we set κ(i, j) = κ 1 (d ij ) = d β ij, a geometric distance kernel which characterizes the risk of infection over distance, where d ij is the Euclidean distance between individuals i and j. β is the geometric rate of decay. It is assumed here 10

17 that (i, t) = 0. This gives our primary model, P it (ø) : P (ø) it = 1 exp α κ 1 (d ij ), α, β > 0. j I(t) The second model has κ(i, j) = κ 1 (d ij ) N (1) where N (1) is a normalization constant. Specifically: where, P (N) it = 1 exp α κ1 (d ij ) N (1), α, β > 0 j I(t) m N (1) = ji m κ 1 (d ij ) i=1 1 and (i, t) = 0. A third model, P () it, has κ(i, j) = κ 1 (d ij ) and (i, t) =. A fourth model, P (N,) it, has κ(i, j) = κ 1 (d ij ) N (1) and (i, t) = to give: P () it = 1 exp α κ1 (d ij ) +, α, β, > 0 j I(t) and, P (N,) it = 1 exp α κ1 (d ij ) N (1) +, α, β, > 0 j I(t) respectively. The likelihood for each of the four models is obtained by substituting P it (ø), P it (N), P it () or P it (N,) for P it into (2.2). 11

18 2.4.1 Simulation Study The purpose of this simulation study is to evaluate and compare each model when fitted to data generated from P it (ø) (the true model). Observed Data Two sets of ten epidemics are generated from the primary model, P (ø) it. The first ten epidemics form Set 1, which spread through a population of 225 individuals who are spaced equidistantly apart on a grid for each epidemic. That is, the 225 individuals collectively have coordinates (x, y) for every combination of x, y = 1,...,15. We consider the entire population to be in the set of susceptible individuals S(t) at time t=0 and one individual becomes infectious at time t=1. The individual at the approximately central location situated at (7,8) on the grid is chosen to be infected first in each of the 10 epidemics. Each epidemic is run for ten time steps t=1,...,10. The model parameters are set to be α=1.0, β=3.0, and the infectious period γ I is set to 1 time unit. The second set of ten epidemics, Set 2, are also generated from P (ø) it. They differ from Set 1 in that four individuals are infected at time t = 1 instead of one. The population is increased to a grid of 400 individuals with coordinates (x, y) for every combination of x, y = 1,...,20. We again consider the entire population to be in S(t) at time t=0, and the four aforementioned individuals become infectious at time t=1. The four individuals which begin the epidemics are chosen arbitrarily and are situated near the top two corners, the centre, and the bottom right corner of the grid with coordinates (3,17), (18,19), (10,10) and (19,2) respectively. 12

19 Fitting the Models Each of the four models are fit to the twenty observed epidemics using a random walk Metropolis-Hastings Markov chain Monte Carlo (MH-MCMC) algorithm to obtain samples from the posterior distribution. The posterior distribution is given (up to proportionality) by multiplying the likelihood by the marginal prior densities for each parameter. Here, vague half-normal priors with mode 0 and variance 10 6 are used for each parameter. The MH-MCMC algorithm is as follows: 1. Set i = 0 and choose initial values α (0) and β (0). 2. Generate random values Z α and Z β from uniform proposal densities, U[-a,a] and U[-b,b], respectively, where a,b R Assign candidate values based on the current state of both parameters as follows: α = α (i) + Z α β = β (i) + Z β. 4. Calculate the acceptance probability, ψ, as: π(d α,β ) ψ = min 1,. π(d α (i),β (i) ) 5. Sample u from a Uniform(0,1) distribution and set α (i+1) = α and β (i+1) = β if u <ψ, otherwise set α (i+1) = α (i) and β (i+1) = β (i). 6. Set i = i + 1 and then repeat steps 2 to 6 for the required number of iterations. For all models, a U[-0.1, 0.1] proposal is used for updating β. For updating α, all models which do not incorporate a normalization constant use a proposal of U[-0.1, 13

20 0.1]. For the models which do, a proposal variance of U[-100, 100] is used for the epidemics of Set 1 and U[-200, 200] is used for Set 2. For models which contain a sparks term, a U[-0.01, 0.01] proposal is used for updating. These proposals were arrived at through tuning. A total of 20,000 MCMC iterations are run for each model with a burn-in period of 1,000 iterations. Note that parameters are updated together in one block. MCMC convergence is checked visually. Model Evaluation The Effective Sample Size (ESS) is obtained using the coda package in R from each converged MCMC chain. The ESS is the estimate of the number of independent samples our dependent MCMC sample is equivalent to, and a large ESS is desirable. It is used to determine if the normalization constant is effective in improving the mixing of the MCMC chains. The correlations between all parameters are also computed to determine if the normalization constant reduces the correlation between α and β. Computation time is recorded to determine the effect of the normalization constant on efficiency. Using the ESS and computation time, the run-time ratio (RTR) is calculated for both parameters. RTR is a measure of the average time taken to produce one pseudoindependent sample (McKinley et al., 2009) for both α and β. It is derived by dividing the computation time by the ESS for each parameter Results Table 2.1 shows the estimated posterior correlation of α and β averaged over the 10 replicate epidemics for the four models for both sets of epidemics. As expected, the normalization constant reduces this correlation. Comparing P (ø) it to P (N) it as well as 14

21 P it () to P it (N,), we see the correlation approximately halved for both Sets 1 and 2. Table 2.1 also shows that the correlation between β and is similar under P it () and P it (N,), but the correlation between α and is reduced. This result is observed in all 20 epidemics separately, as well as on average (results not shown). Set 1 Model Corr(α,β) Corr(α,) Corr(,β) (ø) P it (0.0075) NA NA (N) P it (0.0096) NA NA () P it (0.0065) ( ) (0.0204) (N,) P it (0.0164) (0.0073) (0.0233) Set 2 Model Corr(α,β) Corr(α,) Corr(,β) (ø) P it (0.0062) NA NA (N) P it (0.0093) NA NA () P it (0.0068) (0.0129) (0.0217) (N,) P it (0.0182) (0.0091) (0.0151) Table 2.1: Average correlations between parameters (standard errors in parentheses) averaged over the results from the ten epidemics of Set 1 and Set 2, respectively. Table 2.2 displays the average ESS for α and β for both sets of epidemics. Including a normalization constant appears to improve mixing for both parameters. Comparing P (ø) it to P (N) it, the ESS of both α and β are seen to increase showing that mixing is improved under P (N) it. Similarly, the ESS of both parameters increases from P () it to P (N,) it showing that mixing is improved in P (N,) it. The sparks term has a negative effect on mixing. This is demonstrated by the decrease in ESS for both α and β from P (ø) it to P () it, as well as from P (N) it to P (N,) it. Table 2.3 summarizes the computation times as well as the run-time ratio (RTR) for both α and β. Calculating the normalization constant requires a summation of a geometric transformation of the Euclidean distance between each individual and every other individual in the population. As expected, this calculation causes a great increase 15

22 Set 1 Set 2 Model ESS (α) ESS (β) ESS (α) ESS (β) (ø) P it (12.174) (4.723) (13.057) (10.583) (N) P it (35.978) (7.981) (35.804) (20.581) () P it (9.767) (9.492) (21.671) (14.603) (N,) P it (25.506) (12.068) (49.135) (15.748) Table 2.2: Average effective sample sizes for α and β (standard errors in parentheses) averaged over the results from the ten epidemics of Set 1 and Set 2, respectively. Set 1 Model Time (s) RTR (α) RTR (β) (ø) P it (1.15) (0.010) (0.004) (N) P it (0.93) (0.031) (0.011) () P it (1.45) (0.022) (0.018) (N,) P it (4.14) (0.038) (0.034) Set 2 Model Time (s) RTR (α) RTR (β) (ø) P it (2.61) (0.010) (0.010) (N) P it (2.75) (0.023) (0.036) () P it (3.88) (0.033) (0.040) (N,) P it (10.94) (0.063) (0.044) Table 2.3: Average computation time in seconds and the run-time ratio for α and β averaged over the results from the ten epidemics of Set 1 and Set 2, respectively (standard errors are in parentheses). in computation time as the normalization constant must be re-calculated at each MCMC iteration for each new candidate β value. For Set 1, the ratio of computation time for P (ø) it to P (N) it is 1:2.31, and for Set 2 it is 1:2.50. Similarly, the ratio of computation time for P () it to P (N,) it is 1:2.27 for Set 1 and 1:2.48 for Set 2. This implies that the negative impact the normalization constant has on computation time worsens even further with population size. The RTR indicates that for both parameters the reduction in correlation is not worth the additional computation time required to calculate the normalization constant. The 16

23 ratios of RTR for P (ø) it to P (N) it for Set 1 are 1:1.380 for α and 1:1.787 for β. Also from Set 1, the ratios of RTR for P () it to P (N,) it are 1:1.406 and 1:2.032 for α and β, respectively. This shows that it takes more time to obtain one pseudo-independent sample for both parameters in models which include a normalization constant. The same is shown for Set 2, where the ratios of RTR for P (ø) it to P (N) it are 1:1.319 for α and 1:1.939 for β, and the ratios of RTR for P () it to P (N,) it for α and β are 1:1.422 and 1:1.917, respectively. Again, there appears to be an increasing negative effect with population size, however this is only seen for β when comparing P (ø) it to P (N) it and for α when comparing P () it to P (N,) it. The computation times in Table 2.3 indicate that including a sparks term increases efficiency, which is contradictory to what might be expected since the inclusion of a third parameter,, might be expected to make MCMC mixing worse. However, further analysis shows that this is an artifact of the way the prior is coded and the mixing properties of the MCMC chain. To save time, if a proposed value of any parameter, θ (.) < 0 at any MCMC iteration, then the proposed value is rejected before the Metropolis Hastings acceptance probability is calculated. It turns out that the tuned proposals used for models containing a sparks term result in a relatively high rejection rate due to proposed values of < 0. Thus, although the inclusion of the parameter leads to worse mixing, since the likelihood is not calculated for < 0 there is a reduction in computation time. This would not stand if the marginal posterior of had most of its mass further from zero, of course. Analyzing the RTR values for both α and β in Table 2.3 shows that the effect of including a sparks term is an improvement for larger population sizes. The ratios of RTR for P (ø) it to P () it for α and β are 1:1.323 and 1:1.320 respectively for Set 1, and 1:1.208 and 1:1.317 for Set 2. Similarly, the ratios of RTR for P (N) it to P (N,) it for α and 17

24 β are 1:1.351 and 1:1.501 respectively for Set 1, and are 1:1.302 and 1:1.302 for Set 2. Therefore, the decrease in computation time reduces the relative negative effect that the sparks term has on mixing for large populations. 2.5 Study 2 The purpose of this study is to examine the effect of using an upper-truncated distance kernel that is, a distance kernel for which κ(d ij ) = 0 for all d ij >δwhere δ is some given constant. It is possible to utilize such a kernel to speed up the computation of the likelihood, with this reduction in computation time becoming greater for decreasing δ. Of course, although the computation time is reduced as δ decreases, it is likely that model fit will also deteriorate if the true underlying kernel is not truncated. The purpose of this study is therefore to examine this trade-off between reduced computation time and reduced goodness-of-fit Methods Here we introduce the remaining four models which are considered in addition to those described in Section 2.4 to form the full set of models for Study 2. Models for Study 2 Setting κ(i, j) = κ 2 (d ij ) yields the model: P (δ) it = 1 exp α κ 2 (d ij ), α, β > 0, where j I(t) 18

25 d β ij 0 < d ij δ κ 2 (d ij ) = 0 otherwise Here, (i, t) is again assumed to be 0 and δ is the upper-truncation distance. We use the notation P it (δ k ) to indicate that the distance δ = k in that particular instance of P it (δ). Note that P it (δ) P it (ø) as δ. P it (N,δ) also contains κ 2 (d ij ) and also incorporates its appropriate normalization constant, N (2), to give: P (N,δ) it = 1 exp α κ2 (d ij ) N (2), α, β > 0, j I(t) where and, once again, (i, t) = 0. m N (2) = ji m κ 2 (d ij ) i=1 1 P (δ,) it also contains the kernel κ(i, j) = κ 2 (d ij ), as well as the sparks term (i, t) =. The final model, P (N,δ,) it, has κ(i, j) = κ 2 (d ij ) N (2) and uses the non-zero (i, t): P (δ,) it = 1 exp α κ2 (d ij ) +, α, β, > 0 j I(t) and, P (N,δ,) it = 1 exp α κ2 (d ij ) N (2) +, α, β, > 0. j I(t) For models which utilize an upper-truncated distance kernel but do not contain a sparks term, it is potentially the case that the likelihood will equal zero and thus, the model cannot be fitted to the observed data. This is because an infected individual s probability of being infected under the fitted model might be zero (P it = 0) if there are 19

26 no infected individuals within a distance δ. As δ decreases, this possibility becomes an increasingly likely eventuality. Similarly, there is a high chance of avoiding this issue in practice for large δ. The inclusion of the sparks term,, allows for transmission of the disease to be possible beyond the upper-truncation distance, which makes the probability of infection for each individual non-zero, thus resulting in a non-zero likelihood. A sparks term is therefore often a practically necessary component of the model (depending on the specific data set being analyzed), as well as an intuitively and theoretically desirable one. Here, the likelihood is computed in the same way as is described in Section 2.3, but P it is now replaced in (2.2) by whichever of the eight appropriate models P (ø) (N,δ,) it,...,p it is being fit to the observed data. The Upper-Truncated Distance Kernel The upper-truncated distance kernel is introduced primarily to reduce computation time. Its ability to do so is as a result of storing information on the distances between individuals so as to reduce the amount of computation required when computing the likelihood. This is done in the code before the MCMC analysis is commenced. This works as follows. First, a series of m vectors, A i, of length n i, i=1,...,m, where n i < m (and m is the total number of individuals in the population), are created. The Euclidean distance between an individual i and another individual j, d i, j, is calculated. If d i, j <δ, j is recorded as an element in A i, where A i is the vector corresponding to individual i. This indicates that individual j is within the specified distance δ of individual i, and that transmission of disease from individual j to i is possible. The distance between i and 20

27 the next individual j + 1 is then computed. If d i, j+1 <δthe value j + 1 is appended to A i, and if not the next individual, j + 2, is considered. This is repeated until every individual that is within δ of individual i is identified in A i = [ j : d i, j <δ]. This is then done for each i = 1,...,m. These A i can be used in the calculation of P it. As δ decreases, so does n i, the length of the A i vectors. This results in a smaller set of infected individuals, I(t), at each time t who are considered as potential transmitters of the disease to individual i, allowing for a quicker calculation of P it and therefore the likelihood. Note that if a naive approach is taken and the search for all j: d i, j <δfor each i is done in the likelihood calculation itself, the computational advantage is, of course, lost Simulation Study The purpose of this simulation study is again to evaluate and compare each of the models from Study 2. Fitting the models Each model is fit to the same twenty epidemics of Study 1, from Set 1 and Set 2. The models which contain κ 2 (d ij ) are fit three times for each of δ = 2, 4 and 6. All models of Study 2 are fit in the same way as is described for Study 1, with the exception that (N,δ 125,000 MCMC iterations are required for model P 2,) it. The small δ value makes it seemingly difficult to find proposal variances to enable the chain to reach convergence in fewer iterations for this model. Additionally, the issue concerning the use of an upper-truncated distance kernel without incorporating a sparks term, as described in Section 2.5.1, poses a problem when fitting two of the models to the simulated data. Models P (δ) it and P (N,δ) it cannot be 21

28 fitted to any of the simulated epidemics for δ<6. For δ = 6, they are able to be fitted to only 3 of the 10 epidemics from Set 1, and only 1 epidemic from Set 2. As a result of this, models P it (δ) and P it (N,δ) with δ = 2 and 4 are excluded from further analysis. Model Evaluation Two methods of evaluation are implemented for Study 2 in addition to the computation time and run-time-ratio (RTR) for α and β that are used in Study 1. First, after fitting the models to the observed data, the posterior predictive distribution (Gelman et al., 2000) of the epidemic curves is simulated to assess goodness-of-fit. Observations from this distribution are obtained by generating an epidemic with a randomly sampled θ from the MCMC-estimated posterior and recording the number of newly infected individuals at each time point. Here, θ = (α, β) or θ = (α, β, ), depending on the model being fitted. This is repeated 100 times for newly sampled θ, to give an estimate of the posterior predictive distribution of the epidemic curves. Second, the Deviance Information Criterion (DIC) (Spiegelhalter et al., 2001) and Akaike Information Criterion (AIC) (Akaike, 1974) are both computed as a means of comparing the model fit. The AIC is calculated using the number of model parameters (k) and the maximized likelihood function value for each model which is denoted below as L: AIC = 2k 2ln(L). In order to calculate the DIC, the deviance, D(θ), the expectation, D, and the effective number of parameters, p D, must be defined: D(θ) = 2log(π(D θ)) + C, 22

29 D = E[D(θ)] and p D = D D( θ), where π(d θ) is the likelihood function for given θ, as before, D is the observed epidemic data and θ is defined above in Section 2.5.2; C is a constant which cancels out in the calculations done to compare each model, and θ denotes the expectation of θ. From this information we can calculate the DIC: DIC = p D + D. Favoured models will be those which have a relatively low computation time, precise posterior predictive epidemic curve densities in line with the data, and low DIC and AIC values Results Table 2.4 summarizes the computation times for fitting the various models to the epidemics of both Set 1 and Set 2. The upper-truncated distance kernel does appear to (δ have a positive effect on computation time, as expected. Fitting P 6,) it reduces the average time to seconds from the seconds it takes to fit P () it to the epidemics (δ of Set 1. The ratio of computation time for P 6,) it to P () (N,δ it is 1:2.00. Fitting P 6,) it to these epidemics reduces the average time to seconds from the seconds it takes to fit P (N,) (N,δ it. The computation time ratio between P 6,) it and P (N,) it is (δ 1:1.17. For Set 2 these ratios increase. The computation time ratio for P 6,) () it to P it (N,δ becomes 1:3.12, and for P 6,) it and P (N,) it it becomes 1:1.27. This not only shows that the upper-truncated distance kernel improves computation time, but also that it has 23

30 a stronger effect with larger population sizes. The asterisks in this, and other, tables indicate those models that are able to be fit to only a subset of the epidemics, as described in Section Of the few occurrences where these models can be fit to the data, the upper-truncated distance kernel did reduce the computation time on average. With model P it (δ,), computation time decreases with increasing δ. This is not the case, however for P it (N,δ,). The computation time varies very little when fitting model P it (N,δ 4,) compared to P it (N,δ 6,) across all epidemics for both sets. Six of the 10 epidemics from both Set 1 and Set 2 follow the trend seen with P it (δ,) and decreases with δ. The other 4 epidemics from each set show the reverse. These changes in computation time, regardless of the direction, are small. Model P it (N,δ 2,) has a much higher computation time due to the fact that 125,000 MCMC iterations are run instead of the 20,000 used for the other models. Model Time (s) Set 1 Set 2 (δ P 6 ) it (0.71)* (0)* (ø) P it (1.15) (2.61) (N,δ P 6 ) it (0.70)* (0)* (N) P it (0.93) (2.75) (δ P 2,) it 8.55 (0.13) (0.13) (δ P 4,) it (0.32) (0.30) (δ P 6,) it (1.24) (1.92) () P it (1.45) (3.88) (N,δ P 2,) it (0.77) (0.86) (N,δ P 4,) it (2.19) (0.80) (N,δ P 6,) it (5.32) (13.68) (N,) P it (4.14) (10.94) Table 2.4: Computation time in seconds (standard errors are in parentheses) averaged over the results from the ten epidemics of Set 1 and Set 2, respectively. *These results are for only 4 out of 20 epidemics that could be fitted under these models. MCMC was run for 125,000, rather than 20,000, iterations here. 24

31 Table 2.5 summarizes the run-time ratio (RTR) for parameters α and β for each model averaged over the ten epidemics of each of Set 1 and Set 2. Including the normalization constant worsens the RTR for both parameters, a result which is consistent with Study 1. The upper-truncated distance kernel, however, appears to improve the RTR (α) and RTR (β) as can be seen by, for example, comparing P (N,) it to P (N,δ 6,) it. With the exception of P (δ,) it, the RTR worsens with decreasing δ. This might be expected as a result of the poorer model fit and, thus, increased posterior variance that might be obtained with smaller δ values. Overall, model P (δ,) it performs best in terms of RTR (α) and RTR (β) with values lower than the standard model for all values of δ. Further, this model improves with decreasing δ, which might not be expected since the ESS of both parameters decreases with decreasing δ. Further analysis shows that the reduction in computation time overpowers this reduction in ESS, resulting in improved RTR (α) and RTR (β). The inclusion of the upper-truncated distance kernel appears to improve the RTR for the models which can be fit only to a subset of the epidemics. However, since they are averaged over a non-random sample of epidemics they cannot be compared fairly with the other models in terms of these results. It is also, of course, undesirable to be fitting a model which can only be fit to a subset of possible epidemic realizations. Table 2.6 contains the average DIC and AIC values for each model for both sets of epidemics. A low value of the AIC and DIC is an indication of good fit. The values marked by asterisks again indicate those models which are able to be fitted to only a subset of the simulated epidemics, and although models P it (δ 6 ) and P it (N,δ 6 ) have very small AIC and DIC values they cannot be compared fairly with the other models for reasons described above. Of the remaining models, P it (ø) is ranked first in terms of both AIC and DIC under 25

32 Model Set 1 Set 2 RTR (α) RTR (β) RTR (α) RTR (β) (δ P 6 ) it (0.017)* (0.005)* (0)* (0)* (ø) P it (0.010) (0.004) (0.010) (0.010) (N,δ P 6 ) it (0.161)* (0.055)* (0)* (0)* (N) P it (0.031) (0.011) (0.023) (0.036) (δ P 2,) it (0.004) (0.010) (0.004) (0.004) (δ P 4,) it (0.009) (0.008) (0.009) (0.009) (δ P 6,) it (0.016) (0.015) (0.007) (0.008) () P it (0.022) (0.018) (0.033) (0.040) (N,δ P 2,) it (8.766) (1.919) (44.610) (12.296) (N,δ P 4,) it (0.035) (0.043) (0.112) (0.051) (N,δ P 6,) it (0.034) (0.038) (0.029) (0.027) (N,) P it (0.038) (0.034) (0.063) (0.044) Table 2.5: Run-time ratios for α and β averaged over the results from the ten epidemics of Set 1 and Set 2, respectively (standard errors are shown in parentheses). *These results are for only 4 out of 20 epidemics that could be fitted under these models. MCMC was run for 125,000, rather than 20,000, iterations here. the upper-truncated distance kernel models. This is to be expected since it is the true model from which the epidemics are generated. The DIC and AIC increase with decreasing δ. This is also expected since the fitted model diverges from the true model as the truncation distance becomes smaller. Figure 2.1 shows four posterior predictive densities using an arbitrarily selected (and typical) epidemic from Set 1 for P (ø) it and P (N,δ,) it with δ = 6, 4 and 2. The posterior predictive distribution moves further from the true data as δ decreases, which is reflective of the deviation from the true kernel as δ decreases. This result is consistent among all epidemics from both Set 1 and Set 2 (results not shown). Both the AIC and DIC rank models P (ø) it, P (N) it and P (N,) it as the top three best fitting models. The RTR also suggests that they are reasonable models. (N,δ In fourth place, the DIC results put P 6,) it. The AIC results put P () it in fourth 26

33 Model Set 1 Set 2 Average DIC Average AIC Average DIC Average AIC (δ P 6 ) it * * * * (ø) P it (N,δ P 6 ) it * * * * (N) P it (δ P 2,) it (δ P 4,) it (δ P 6,) it () P it (N,δ P 2,) it (N,δ P 4,) it (N,δ P 6,) it (N,) P it Table 2.6: Average DIC and AIC values averaged over the results from the ten epidemics of Set 1 and Set 2, respectively. *These results are for only 4 out of 20 epidemics that could be fitted under these models. MCMC was run for 125,000, rather than 20,000, iterations here. place in Set 1, but P (N,) it in Set 2. However the former model has a much lower RTR for both parameters than the latter suggesting it is to be preferred. The RTR (α) is smaller (N,δ for P 6,) it than P () (N,δ it, but only slightly. The RTR (β) for P 6,) it, however, is nearly twice that of the model P () it. This may suggest that the model P () it is to be favoured (N,δ over P 6,) it. (δ,) (N,δ,) Models P it and P it have nearly equivalent AIC values for each different upper-truncated distance radius δ. However, in analyzing the RTR (α) and RTR (β) results for both models at each value of δ, P (δ,) it consistently out performs P (N,δ,) it. The DIC still ranks P (δ,) it as the poorest fitting model (specifically for δ = 2) despite the fact that it has the most desirable RTR values for both α and β. 27

34 Number of New Cases Fitted Model True Model Number of New Cases Fitted Model True Model Time Time Number of New Cases Fitted Model True Model Number of New Cases Fitted Model True Model Time Time Figure 2.1: Posterior predictive plots for model P it (ø) (top left) and P it (N,δ,) with δ = 6 (top right), δ = 4 (bottom left) and δ = 2 (bottom right). 2.6 Discussion and Future Work The goal of this paper was to evaluate the inclusion of a normalization constant, and/or the use of an upper-truncated distance kernel, in a spatial individual-level infectious disease model. Each model was evaluated in terms of MCMC efficiency and goodness-offit via posterior predictive plots, run-time ratios, AIC and DIC, when fitted to simulated data. The inclusion of the normalization constant, although resulting in improved mixing and reduced correlation between parameters as expected, proves not to be worthwhile as these improvements do not outweigh the increase in computation time. Furthermore, this negative effect worsens with increasing population size suggesting that the inclusion 28

35 of the normalization constant in models for large populations is misguided. Here, block updates that assume independence between parameters are used in a Metropolis-Hastings MCMC algorithm. Adjusting these block updates to account for correlation between parameters may be a better method to be used to improve MCMC efficiency. Of course, although such an approach would forgo the need to include a normalization constant there would be a downside; the need to run at least two MCMC chains sequentially in order to estimate the correlation structure before using the adjusted block updates. To assist in reducing computation time, the upper-truncated distance kernel was considered. We show that the use of the upper-truncated distance kernel, κ 2 (d ij ), leads to a model which behaves as expected with a decrease in required computation time and run-time ratios for both parameters. However, this comes at the cost of increasingly poorer model fit as δ is reduced. Overall, the results suggest that it is not computationally worthwhile to include the normalization constant, but the truncation of the distance kernel is a good method for reducing computation time. However, care must be taken in selecting sensible truncation distances. Additionally, a sparks term is required for models which incorporate the upper-truncated distance kernel in order that the models can be fit to all data that the true model can generate. There is certainly room for more work in this area. All models were fit to simulated data where the infectious and latent periods were known. It would of course be more realistic to estimate the latent and/or infectious period to account for such uncertainty. Another approach would be to implement data augmented MCMC to incorporate uncertainty about infection times and infectious times for each individual, as is done by, for example, Jewell et al. (2009). 29

36 For both simulation studies, the population was set as a square grid where individuals remained stationary, which may be realistic for certain plant populations or some controlled animal experiments. However, considering different population structures, for example clusters of individuals where the cluster size could aid in giving an indication of sensible values of δ, would be well-reasoned. Exploring the use of methods which partition the data and likelihood through linearization of the kernel, as is done in Deardon et al. (2010), have also proven useful in decreasing computation time. However, the nature of these methods means they are unable to be used in a data-augmented setting, whereas the upper-truncated distance kernel can be. Approximate Bayesian methods, for example estimating the likelihood at each MCMC iteration via simulation, are increasingly being considered in infectious disease modelling (McKinley et al., 2009; Toni et al., 2009). Such methods can lead to a drastic reduction in computation time if data can be simulated from the model much more quickly than the likelihood can be calculated. As far as the authors are aware, these methods have not yet been used for individual-based models, making them a potential area for future work. Of course, the downside to such an approach is that final posterior results are with respect to an approximated posterior rather than the true posterior. Further, the degree of accuracy of this approximation is often difficult, if not impossible, to ascertain in practice. Finally, it would be interesting to apply the models to real data, comparing results and conclusions derived under a full model, alongside upper-truncated distance kernel varieties. 30