title: Applications of the stochastic Galerkin method to epidemic models with uncertainty in their parameters
Creators: name: Harman, David
Publisher: name: Griffith University
Year: 2018
Subjects: subject: Medical Infection Agents (incl. Prions)
subject: Medical Virology
subject: Infectious Diseases
subject: FOS: Health sciences
subject: 110802
subject: 110804
subject: 110309
subject: Stochastic Galerkin method
subject: Epidemic models
subject: Compartmental models
subject: Orthogonal polynomials
subject: Infectious diseases

description: Infectious diseases are a serious problem throughout the world and are responsible for a large number of deaths annually. It is estimated that tuberculosis was respon- sible for 1.3 million deaths in 2016 worldwide. Malaria, a vector-transmitted dis- ease (transmitted to humans by mosquitoes), is responsible for almost half a million deaths annually. There are also many sexually transmitted diseases, such as chlamy- dia, genital herpes and gonorrhea, as well as the much more serious HIV/AIDS. In order to help prevent the spread of an infectious disease, we rst need to understand how the disease is spreading through the population, as well as how fast it is spreading. To do this, we need to build mathematical models for the disease. These are referred to as epidemic models. These models can also help predict the e ectiveness of interventions, such as treatment and vaccines. One of the most widely used methods for constructing an epidemic model is the use of compartmental models. Each person within the population is assigned to a speci c compartment, based upon their current status with regard to the disease. As their status changes, for example, if they contract the disease, they are moved to the appropriate compartment. Using the compartment model, a system of ordinary di erential equations can be derived that models the disease. As the system of ordinary di erential equations is usually non-linear, numerical solvers often need to be used as an analytic solution is rarely obtainable. While it is usually a relatively easy process to derive a model for a particular dis- ease, the parameters within these compartment models are rarely known and usually have to be estimated. Even for well-studied or seasonal diseases, these parameter values are usually not known with certainty and are instead given as probability distributions or simply as a plausible range of values. Since the parameter values are not known with certainty, it is important for this uncertainty to be included in the model. Not including the uncertainty in the model could lead to inaccurate and misleading predictions. This thesis looks at using the stochastic Galerkin method to incorporate uncer- tainty in epidemic modelling. While there is extensive literature on the stochastic Galerkin method, there has been very little research into its applications to epidemic modelling. The stochastic Galerkin method employs a spectral approach, using orthogo-nal polynomials as basis functions. Substituting the spectral expansion into the epidemic model, and performing a Galerkin projection, results in a deterministic system of di erential equations. While this system of di erential equations is usu- ally much larger than the original system of di erential equations, it is deterministic and hence needs only to be solved once. This results in a signi cant speed increase over sampling techniques. From the spectral expansion, the mean and variance can quickly be calculated without the need for sampling. To demonstrate how to apply the stochastic Galerkin method to epidemic models, as well as to analyse its accuracy, in this thesis the stochastic Galerkin method is applied to the SIR epidemic model using di erent combinations of uncertain parameters and initial conditions. Uniform, gamma and normal distributions are considered for the uncertain parameters. The stochastic Galerkin method is shown to produce accurate results when compared to those obtained using Monte Carlo sampling, while also obtaining the results much more quickly. While applying the stochastic Galerkin method to various epidemic models, it was found that the stochastic Galerkin method could not always be calculated and could instead `blow-out'. While this has been noted as a possibility by some re- searchers when using unrealistic parameter values, it is shown here that these blow- outs can occur even when care is taken to ensure that the parameter values are sensible and realistic. These blow-outs are a signi cant draw-back to the stochastic Galerkin method and have not previously been investigated. By using low order stochastic Galerkin expansions, this work has found that the stochastic Galerkin solution cannot be cal- culated if the stable attractor is not present in the resulting system of deterministic di erential equations. It is shown that for low order expansions, it is possible to analytically predict a range of parameter values for which the stochastic Galerkin solution can be calculated, but unfortunately this does not help predict where blow- outs would occur for higher order expansions. A data set from an epidemic that went through a small boarding school was then investigated. Rather than simply considering the parameter values that produced the `best' t to the data, a range of parameter values that resulted in a reasonably accurate t to the data was instead considered. It was found that this range of plausible values formed a simple closed shape on a 2D plot. It was then shown that by simply nding the border of this shape, probability distributions of the uncertain parameters could be calculated. This eliminated the need to test many of the parameter values. As these probability distributions were non-standard, this study next extended the stochastic Galerkin method to work with probability distributions other than uniform, gamma, normal and beta distributions. The orthogonal polynomials as- sociated with the non-standard probability distributions could be quickly calcu-lated using the Gram-Schmidt orthonormalisation method. Extending the stochas- tic Galerkin method to work with non-standard probability distributions allowed for much greater exibility in representing the uncertainty in the parameters. Next, using the stochastic Galerkin method, predictions were calculated about what might have happened if the disease had spread outside of the boarding school. The investigation of the boarding school epidemic was then extended by assuming that the uncertain parameters were no longer independent. While it can be argued that the parameters in an epidemic model are independent, based upon the real world conditions that they are attempting to model, it was clear from the ranges of plausible values that the parameters were dependent, and should be treated that way. Dependent distributions for the parameters were determined and the associated orthogonal polynomials derived. The stochastic Galerkin method was applied and predictions obtained. While the predictions were not particularly accurate, it is hoped that this approach will be helpful and could be studied further to increase the accuracy of its predictions. Finally, predictions were made on each day of the boarding school epidemic using only the data that would have been available on that day. These predictions were then compared to known data points. In order to obtain the predictions, probability distributions for the uncertain parameters needed to be determined on each day of the epidemic and the associated orthogonal polynomials derived before the stochastic Galerkin method could be applied. It was found that predictions made before the peak of the epidemic had very large variances, but predictions made after the peak of the epidemic were relatively accurate and had much smaller variances.