nottheaverageactuary

Actuarial news and views from Cape Town and beyond

Models of infectious diseases

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This post is intended to provide some interesting information about one of the less common types of model we’re likely to find in actuarial work. I will attempt to describe why infectious disease models are relevant to the reader and highlight some innovations in their design and usage.

What is epidemiology and why is it relevant to actuaries?

“Epidemiology is the study of the distribution and determinants of disease in human populations” writes Mark Woodward in his book on the subject. “The essential aim of epidemiology is to inform health professionals and the public at large in order for improvements in general health status to be made.” [3]

Medical scheme administrators, as purchasers of healthcare, are interested in the cost-effectiveness of treatments and their efficacy on individual patients. They are also interested in the relation between individual treatments and population-level disease dynamics because this affects their future claims burden. Lastly they are interested in public health policy to the extent that it affects regulation and influences long-term disease trends. Life offices also have an interest in significant trends in the burden of disease, especially HIV in South Africa.

Example:

Garnett [2] points out that the health outcomes of a particular treatment should not be naively interpreted. “Many health economic models make linear assumptions – that is, treating one more individual reduces the number of cases by one. However, there are knock on effects which depend upon the epidemiological context.” He uses an example of a (hypothetical) low efficacy HIV vaccine and shows that a vaccine which protects v% of the population from all infectious challenges has a greater impact on prevalence than one which protects all the population from v% of infectious challenges.

Garnett HIV Vaccine predictions

Why are models important in infectious disease epidemiology?

The primary tools of research in epidemiology are observational and experimental studies.

Observational studies typically use data collected from purpose-built surveys or surveillance data such as HIV data from ante-natal clinics. From these studies it is very difficult to infer causal links or gain understanding of the complex transmission dynamics of an infectious disease.

Experimental studies, usually randomised controlled trials, produce stronger causal conclusions but they are difficult to design, expensive to run, and are often limited by ethical constraints. “It is rarely ethically acceptable to force people to be exposed (or unexposed) to a risk factor” writes Woodward. [3]

Mathematical models are most useful in conjunction with observational and experimental data, to which they are fitted or ‘calibrated’. Models have the following advantages:

  • They are quicker and cheaper to implement than other studies.
  • They may test hypotheses which are not morally permissible in experiments.
  • They can describe complex dynamic systems, particularly relating to infectious transmission.
  • Models can generate counter-intuitive results and inform the design of future studies. [1]
  • They can validate an intervention study by isolating the effect of the intervention from other trends present. [2]

Examples:

While it may sometimes be easy to know the efficacy of a treatment on an individual on a single occasion (for example the efficacy of a Malaria prophylactic drug, or the effectiveness of condom usage in a single sexual contact) the unique transmission mechanisms of the disease will have a large impact on its macroscopic health outcomes.

Models have been designed which account for the following unusual features of some infectious diseases:

  • Contact times of varying durations with an infected host, including allowance for concurrency.
  • Sources of heterogeneous mixing, for example ‘assortative mixing’ where contact is more likely between individuals of the same risk group.
  • Demographic, biological or behavioural heterogeneity in host population.
  • Pair models capture key features of STIs by explicitly simulating partnerships in the model. [2]
  • Network models explicitly represent the contact structure of individuals and use analytic techniques from physics, fluid dynamics and population biology. [2]

What interesting features do these models have?

“One of the paradoxes in modelling infectious diseases is that, despite their quantitative nature, the best that we can often expect is qualitative insights” – Mishra et al. [1]

The models have an unusual purpose in that they aren’t always meant for giving quantitative predictions. They are often very useful in discovering the direction of effects, or the relative strengths of interventions under different scenarios, especially when these results are counter-intuitive. Purposes include filling in gaps in data and understanding left by empirical research, validating past research and informing further research. [1,2]

Models of infectious diseases are most naturally built on an individual basis rather than by use of representative ‘points’ which are later scaled up. Numbers of new infections are a dynamic function of the prevalence of the disease at the time. Any complexity in the transmission mechanisms requires the model to work on the level of individual agents. [1,2]

Stochastic and deterministic models are both used. Stochastic models are particularly advantageous because of the importance of random fluctuations. For example, where a deterministic model may predict the eradication of a disease (the differential equations indicate a stable state of zero infections) there may remain tiny numbers of infected individuals which can cause an outbreak later. [1]

In the modelling discussions, a distinction is made between sensitivity analysis (testing large random deviations in parameter inputs to determine, for example, whether the model can generalise to a different population), uncertainty analysis (testing small random deviations in parameter inputs which may arise from measurement error) and scenario analysis (the exploration of illustrative examples with chosen sets of parameters). In other modelling discussions this distinction is often not made. [1,2]

The models are part of an interdisciplinary research framework so they must be communicated with a diverse audience in mind.

I hope you’ve enjoyed this discussion and gained some interesting knowledge about ‘not the average actuarial model.’


P.S. After writing this I found this excellent article in The Actuary called Quantifying Pandemic Risk. It describes the way in which a sufficiently well-built epidemiological model can be submitted to stress-testing under the framework of catastrophe modelling. Essentially this is a kind of scenario testing where a ‘stochastic catalog’ of initial parameters is built from all conceivable, plausible scenarios, and the model is tested for all of these outcomes. Ideally, the epidemiological model would be integrated with asset-liability models to stress-test a dynamic array of pandemic-related outcomes (from mortality and morbidity to economic slowdown).

References

[1] Garnett, G.P. An introduction to mathematical models in sexually transmitted disease epidemiology. Sex Transm Inf 2002; 78:7-12

[2] Mishra, S., Fisman, D.N., Boily, M. The ABC of terms used in mathematical models of infectious diseases. J Epidemiol Community Health 2011; 65:87-94

[3] Woodward, M. Epidemiology: Study Design and Data Analysis. Chapman & Hall 1999

Thanks to my project supervisor Dr Leigh Johnson for supplying me with these references.

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3 thoughts on “Models of infectious diseases

  1. This is a very brilliant and interesting piece of work!!! Two of the points that stand out for me are: sensitivity analysis and scenario analysis. Uncertainty analysis is often carried out, in my field, by demographers making extensive use of the Spectrum Suite of Models when national population reports are prepared. I did a bit of sensitivity analysis when I constructed a 3-region population model (adapting the ASSA model to work as a 3-region migration model for South Africa), and quite frankly, the changes in assumptions have a significant bearing on the final estimates delivered by the model. I started off from the premise that migrants moving between South Africa’s regions represent the regional/provincial epidemic (HIV) to which they are moving (this is the initial assumption on which the ASSA model was built. I then changed this by seeking to determine the provincial population age structure under the assumption that migrants move with the HIV profile they represent. The results were very interesting and on average, the model generated a -1.4% decline in mid-year population estimates for Gauteng between 2009 and 2025, for instance.

    So I fully agree with Simon. as actuaries, demographers and statisticians, we should lean towards the right hand of the 4-quadrant diagram without completely discounting the Uncertainty Avoiders and the Intuitive Decision Makers.

    • Thanks for your comments Thabo. It’s invaluable to receive input from an active researcher. I was prompted to read a bit about the Spectrum AIDS Impact Model. It looks like a wonderful model in that, I gather, it’s generalizable to arbitrary populations and well documented so that anyone can use it. The ability for the model to perform well under a wide range of inputs is, I imagine, particularly useful for research in regions without expertise for custom-built models. I also like that it has special functionality for uncertainty analysis and scenario testing. These methods have very different (and important) purposes in demographic and disease modelling.
      Do the results you refer to from the ASSA model imply that the places of origin of migrants to Gauteng on average have lower prevalence than their destination? Or is it that migrants themselves are a high prevalence group, and the decrease in Gauteng estimates is due to their no longer being counted in the Gauteng estimates? Which of the two assumptions do you think is more realistic?

  2. Hi Simon, none of the two points you mentioned necessarily apply. Note that one of thethe assumptions in the ASSA model is that migrants represent the HIV epidemic to which they are moving, which, of course, does not necessarily hold. What I did was figure out a way to get the migrants to represent epidemics of the provinces of origin. This I did by working out the time- dependent weighted average of the provincial. prevalence. For instance, migrants moving to Gauteng each year would represent the weighted average of the HIV prevalence of their provinces of origin. I did this because some of the return migrants would want to die in the home provinces, close to family. The multiregional AIDS model would then rely on a plausible assumption in this regard.

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