# Mathematical Modelling of Healthcare Associated Infections Theo Kypraios Division of Statistics, School of Mathematical Sciences

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Mathematical Modelling of Healthcare Associated Infections Theo Kypraios Division of Statistics, School of Mathematical Sciences t.kypraios@nottingham.ac.uk http://www.maths.nott.ac.uk/~tk

Outline 1.Overview. 2.Mathematical modelling. 3.Concluding comments.

Outline 1.Overview. 2.Mathematical modelling. 3.Concluding comments.

Motivation High-profile hospital-acquired infections such as: Methicillin-Resistant Staphylococcus Aureus (MRSA) Vancomycin-Resistant Enterococcal (VRE) have a major impact on healthcare within the UK and elsewhere. Despite enormous research attention, many basic questions concerning the spread of such pathogens remain unanswered.

Can we fill in the gaps? Aim: To address a range of scientific questions via analyses of detailed data sets taken from hospital wards. Methods: Use appropriate state-of-the-art modelling and statistical techniques.

What sort of questions? What value do specific control measures have? isolation, handwashing etc. Is it of material benefit to increase or decrease the frequency of swab tests? What enables some strains to spread more rapidly than others? What effects do different antibiotics play?

What do we mean by datasets ? Information on: Dates of patient admission and discharge. Dates when swab tests are taken and their outcomes. Patient location (e.g. in isolation). Details of antibiotics administered to patients.

Outline 1.Overview. 2.Mathematical modelling. 3.Concluding comments.

Mathematical Modelling – what is it? An attempt to describe the spread of the pathogen between individuals. Includes inherent stochasticity (= randomness). Data enables estimation of model parameters.

Mathematical Modelling: Simple Example Consider population of individuals. Each can be classified healthy or colonised each day. Each colonised individual can transmit pathogen to each healthy individual with probability p per day.

Mathematical Modelling: Simple Example Healthy person Colonised person Daily transmission probability p = 0.5 Day 1

Mathematical Modelling: Simple Example Healthy person Colonised person Daily transmission probability p = 0.5 Day 2

Mathematical Modelling: Simple Example Healthy person Colonised person Daily transmission probability p = 0.5 Day 3

Mathematical Modelling: Simple Example Healthy person Colonised person Daily transmission probability p = 0.5 Day 4

Mathematical Modelling: Simple Example Healthy person Colonised person Daily transmission probability p = 0.5 Day 5

Mathematical Modelling: Simple Example Healthy person Colonised person Daily transmission probability p = 0.5 Day 6

Mathematical Modelling: Simple Example Plot of new cases per day

Mathematical Modelling: Inference Information about p could take various forms: Most likely value of p e.g. p = 0.42 Range of likely values of p e.g. p is 95% likely to be in the range 0.23 – 0.72 In general, how p relates to any other model parameters?

Mathematical Modelling In practice, we deal with more complicated models. – i.e. more realistic models, more parameters. The actual process is rarely fully observed. – difficult to observe colonisation times. Inference becomes much more challenging.

Mathematical Modelling How do we address hypotheses? e.g. Does transmission probability p vary between individuals? Construct two models: one with same p for all, one where each individual has their own p. Can determine which model best fits the data.

Outline 1.Overview. 2.Mathematical modelling. 3.Concluding comments.

Conclusions Models seek to describe process of actual transmission and are biologically meaningful. Scientific hypotheses can be quantitatively assessed. Methods are very flexible but still contain implementation challenges.

Thank you for your attention! Any questions?

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