Transcript Mathematical modelling - School of Mathematical Sciences

Mathematical Modelling
of Healthcare Associated Infections
Theo Kypraios
Division of Statistics, School of Mathematical Sciences
[email protected]
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
Day 1
Daily transmission
probability p = 0.5
Mathematical Modelling:
Simple Example
Healthy person
Colonised person
Day 2
Daily transmission
probability p = 0.5
Mathematical Modelling:
Simple Example
Healthy person
Colonised person
Day 3
Daily transmission
probability p = 0.5
Mathematical Modelling:
Simple Example
Healthy person
Colonised person
Day 4
Daily transmission
probability p = 0.5
Mathematical Modelling:
Simple Example
Healthy person
Colonised person
Day 5
Daily transmission
probability p = 0.5
Mathematical Modelling:
Simple Example
Healthy person
Colonised person
Day 6
Daily transmission
probability p = 0.5
Mathematical Modelling:
Simple Example
3
2
Cases
1
Plot of new cases
per day
0
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
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.
•
–
The actual process is rarely fully observed .
•
–
•
i.e. more realistic models, more parameters.
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?