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Healing the swine flu with
modeFRONTIER
One of the hot topics of the winter
2009 that probably will be remembered
is the outbreak of the so-called “swine
flu”. The new virus A-H1N1 captured
the attention of the Italian media,
which literally bombarded the
population with daily reports on the
number of deaths, the severity of this
virus and other alarms based on the
opinion of some “epidemiology
experts”, spreading in this way the
fear within the population.
During the first weeks of autumn some
sentences such as “We will have an
extraordinary peak of flu diffusion between Christmas and
the new year” or “we will be the victim of a new pandemia
with many deaths” were pronounced.
How is it possible to predict such an “apocalyptic” scenario
so many weeks in advance? The truth is that it is extremely
difficult, especially when no previous knowledge on the virus
behavior is available. However, in epidemiology some simple
mathematical models have been developed and used for
many years; they are mainly based on ordinary differential
equations (shortly ODEs).
Probably, the most known model is the so-called SIR model,
where the population, which is supposed to be large and
homogeneous enough, is divided into three groups
(Susceptible, Infected and Recovered), according to their
status (see [4]). Strong simplifications are present in this
model which can be applied as scale level; in some cases it
could lead to poor results. For this reason, there is a variety
of SIR based models which remove some of these
Newsletter EnginSoft Year 6 n°4 - 35
A photo of the A-H1N1 virus (left) and a swine (right). They do not look so dangerous…
Figure 2: The solution of a classical SIR model: the three categories S, I and R are plotted versus time,
expressed in weeks. It is clear that the disease has a peak between the first and the second week and that the
maximum number of ill people is around 150 over 1000. In this case we adopted the following values for the
parameters (β=10, ν=5 and the number of initial infected is 1.98 for 1000 persons).
simplifications in an attempt to be closer to reality. In this
work, we suggest to add a new category to the standard SIR
model in order to consider the fact that unfortunately, some
infected people may die. The resulting model can be
expressed as:
This is a non-linear system of first
order ODEs; the four categories used to
classify the population are S =
Susceptible, I = Infected, R =
Recovered, D = Dead and they are
expressed in percentage terms. For this
reason the sum of all the categories
has to be always equal to one. The
parameters β, ν and δ are constants
which determine the evolution of the disease. The results
strongly depend on the numerical values of these parameters.
Specifically the peak value of the infected and the week of
the year when it will appear, which
are important information to have
in advance, can be really difficult to
capture if there is not a rigorous
estimation of the above mentioned
parameters.
Obviously, it is mandatory to know
the initial conditions before solving
the system: in other words we have
to know the number of susceptible,
infected, recovered and dead
persons at time zero, when we want
to begin our simulation.
The solution of such equations is
always done, excluding trivial cases,
through numerical techniques which
have been expressively defined to
tackle this kind of problem. To