Libro de Resúmenes / Book of Abstracts (Español/English)

Resumenes 96
Nuestros objetivos principales son estudiar el efecto espacial sobre la
bifurcación hacia atrás, es decir si al considerar vecindades específicas de
contagio éstas tienen algún tipo de efecto sobre la bifurcación hacia atrás.
Adicionalmente, analizamos el efecto de la implementación de alguna
política de vacunación sobre la bifurcación hacia atrás.
An epidemiological model using cellular automats
When we make epidemiological process models, usually we use
differential equations. However, these models have had interesting results,
there are elements which can be studied using other tools. For example, in
the differential equations models the mass action principle is used to
represent the form that diseases spread in the susceptible population. But,
in many cases diseases do not spread in this way. On the other hand, if the
model has a vaccination process, it is not always in a homogenous form in
the real world. Understanding these facts we are modeling some
epidemiological processes using Cellular Automats which permit us to have
explicit spatial control, so we can prove some vaccination policies and have
a more realistic model. Particularly stochastic effects are introduced in the
vaccination process, birth place, death and transmission of the disease.
When epidemiological processes are studied using differential
equations, one of the most important parameters is called Basic
Reproductive Number (Ro), which is a combination of the parameters
involved in the model. The importance of this parameter is in the fact that it
can be used as a bifurcation parameter and we have that if Ro is less than
one, the trivial equilibrium point, (free of disease equilibrium) is stable and
if it is greater than one, it is unstable. It means basically that if Ro is less
than one the disease will disappear in some time and if Ro is greater than
one there will be an endemic outbreak. However, in some models with
specific values of the parameters, one phenomenon appears, known as
backward bifurcation. Backward bifurcation is by itself an interesting
phenomenon to study. This phenomenon appears when for values of the
parameter Ro less than one there is a non trivial bifurcation point which is
stable. Understanding why the backward bifurcation appears, but yet more,
how can it disappear, is very important for people who design disease
control policies. This phenomenon has been studied widely in some works
(Kribs-Zaleta and Jorge X. Velasco-Hernández, Math Biosci. Book 164, Issue
2, May -June 2000, pp. 183-201; Julien Arino et al the, SIAM Appl. Math.,
Vol. 64, Issue 1, pp. 260-276; J Dushoff, W. Huang and C. Castillo-Chavez,
J. Math. Biol., Book 36, pp. 227-248). However their studies are limited to
models with differential equations.
In this work we study, using cellular automats, a model SIV,
(Susceptible, Infected, Vaccinated) which present backward bifurcation. To
show it we need to answer the following questions: what will be understood
by bifurcation parameter? and what will be an equilibrium point in this
model?
Our principal goals are to understand the spatial effect on the
backward bifurcation. When we consider specific transmission
neighborhoods, does have they some effect on the backward bifurcation?
And, we analyze the effect of the implementation of some vaccination policy
on the backward bifurcation.