Why Model Infectious Diseases ?
Thomas House, Leon Danon, Nadia Inglis, Matt Keeling
Mathematics Institute and School of Life Sciences, University of Warwick, Coventry
In the physical sciences, mathematical models are powerful and commonly
used tools – allowing detailed predictions of many processes from the
stability of skyscrapers to the air-flow past aircraft. In the life-sciences
models have begun to be used in the same way. Epidemiology and modelling
the dynamics of infectious diseases has pioneered this development. Here
and in our other two posters, we illustrate the value that modelling can have
for public heath.
3. Optimal Control
In a similar manner to scenario modelling, we can use an accurate model to sweep
through a variety of targeting or control options. This allows us to fine-tune public-health
responses to obtain the “best” results. The idea is that it is too timely and costly to
experiment with changes strategies for real, and therefore we run experiments within a
computer simulation.
Figure 2 shows a surprising result for prophylactic vaccination. We find that for a limited
stock-pile it may be optimal to target the vaccination campaign at single communities,
In general, we can partition the benefits of modelling into four elements:
giving them complete protection while others are unvaccinated. Obviously such a
strategy is ethically unacceptable, but it illustrates the complexities of optimal control.
1. Prediction
Probably the most straight-forward use of mathematical models is in prediction. For
example, from early case reports and a detailed understanding of the pathogen, can
Community Size
we predict the course of a novel epidemic? While such modelling is relatively simple,
Small (100,000)
there are huge problems with estimating the relevant parameters. Figure 1 shows the
Medium(200,000)
impact of uncertainty in the level of susceptibility in the population; while uncertainty in
Large (300,000)
reporting of infection can change the timing of the peak and the estimation of case
fatality – both these problems were experienced during the early H1N1 epidemic.
Despite these difficulties, mathematical models remain our best tool for extrapolating
from the limited data early in an epidemic to predict the most-likely outcomes and the
associated demand on a range of health-care services.
Figure 2. What is the optimal allocation (between communities) of prophylactic
vaccine? Surprisingly mathematical models show that it may be optimal to
concentrate vaccination in a limited number of communities. How do we
compromise between an optimal and an ethically acceptable strategy?
4. Greater Understanding
One surprising use of mathematical models is to improve our understanding of
epidemiological processes. The use of mathematics as a “language” forces us to
rigorously define our assumptions. This has three main implications. It can elucidate
holes in our understanding of the epidemiology; it can highlight processes that are
poorly quantified; or by simulating the mathematical model it can show inconsistencies
between our assumptions and reality. Such as process is a useful way to discriminate
between competing hypotheses, and may even allow us to reject some ideas with
Figure 1. Simple epidemic profiles showing how the same early growth rate, but
uncertainty in population-level immunity can give rise to a range of potential
epidemics.
2. Scenario Modelling
Once we have the ability to predict an epidemic, we can consider what happens as we
change health-care responses. For example, we may wish to compare two vaccination
strategies to assess which is “best” – where best could have lots of different meanings,
but generally refers to increasing QALYs subject to economic constraints. Such
questions must be addressed with mathematical models, as:
i) quantitative information is needed to obtain a meaningful comparison and to feed into
economic models.
ii) it is only through such models that the non-linear complexity of the epidemic process
can be realised.
As an example of the power of such models, JCVI (the government’s Joint Committee
on Vaccination and Immunisation) has recently been examining the use of prepandemic vaccines against H5N1 “bird” flu. The key question was: “Should we
vaccinate at-risk individuals (who are most likely to suffer adverse effects of infection)
or should we vaccinate school-age children (who are most likely to get infected and
transmit the virus)?” Mathematical models were able to answer such questions and
even account for the uncertainty in both the epidemiology of the pandemic and the
uncertainty in the performance of the vaccine. The conclusion was that if 2 doses of
vaccine are required it is safest to vaccinate the at-risk groups (due to the
uncertainties), but if it is possible to give just 1 dose then vaccinating both groups is
preferable despite the expected lower efficacy.
statistical confidence.
Unsolved Problems in Mathematical Modelling
Although mathematical models are highly successful, there are still a range of
problems to be resolved. Better statistical methods would allow for more rapid and
reliable predictions. A better understanding of human movements and interactions
would enable models to be formulated based on the contacts between people. Finally,
we have a relatively poor understanding of how people react to infection and epidemics
– do ill people take to their beds and minimise transmission? Do people panic or alter
their behaviour during a major epidemic?
Simple Messages from Mathematical Modelling
1) Models are only as good as the data used to construct them and our understanding
of the epidemiology. Accurate models need accurate data.
2) Epidemics grow exponentially, therefore, for limited resources, it is usually best to
apply controls early.
3) The most fundamental epidemiological quantity is the basic reproductive ratio, R0,
defined as “the expected number of secondary cases caused by an average
infected individual when the population is totally susceptible”
4) R0 allows us to calculate the final-size of the epidemic and the necessary amount of
immunisation needed to eradicate an infection or stop an epidemic.
5) Targeting of control measures (such as age-specific vaccination) can offer dramatic
improvements to efficiency.