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

Resumenes 112
Travelling Waves in a Model describing “mal de las caderas”
in capybaras (Hydrochaeris hydrochaeris)
including the vector presence.
The aim of this work is to describe the capybara (Hydrochaeris
hydrochaeris) dynamic. The model, proposed by Silvio Pregnolatto in [5],
considers the presence of the ‘‘Mutuca’’ (Diptera: Tabanidae), vector which
transmit the disease called ‘‘mal de las caderas’’ in Estero del Ibera,
Argentina. The study of this problem have ecological and economical
importance: for example de meat and the hide of the capybara can be an
exploitable resource. An analysis of a possible exploitation has been made
in [2].
The model is a system of partial differential equation of reactiondiffusion
and transport. Four classes are studied: susceptible capybaras,
infected capybaras, infected insects and non-infected insects. The dynamic
is as follows: a susceptible capybara moves into the infected class by
contact with an infected insect. A non-infected insect moves into the
infected class by contact with an infected capybara. This is how the disease
spreads.
The mortality of the susceptible and infected capybaras are
considered, and include external factors, such as predation, exploitation, or
disease-related mortality. The mortality considered in the infected and noninfected
insects is assume to be the same. The growth dynamic of both
species is given by a logistic function.
The diffusion is considered in both species, and is greater for the
insects than for capybaras. The diffusion in infected capybaras is not
considered because the disease affects the capybara’s mobility.
At first, a threshold value is determined as a function of the model
parameters, obtaining a critical carrying capacity for the disease
propagation or eradication [10]. As the carrying capacity condition for the
disease existence is satisfied , the existence of travelling wave solution is
studied [3], [8]. Independent velocities are considered for the susceptible
capybaras, the non-infected insect and the disease.
The minimum speed of the disease propagation for this model is
obtained [1], [4], [9], because the travelling wave solution with minimum
speed is stable and represent the dynamic of the system [6], [7]. Control
strategies are discusses to decrease the minimum speed of propagation in
function of the model parameters.
References
[1] Tiemi Takahashi, L. Maidana, N.A., Castro Ferreira Jr., W. Petronio, P. Yang,
H.M. (2005). Mathematical models for the Aedes aegypti dispersal dynamic:
travelling waves by wings and wind. Bulletin of mathematical biology 67.
509-528.
[2] Federico, P. Canziani, G.A. (2005). Modeling the population dynamic of
capybara Hydrochaeris hydrochaeris: a first step towards a management
plan. Ecological Modelling 186. 111-121.
[3] Maidana, N.A. (2004). Algumas aplicaçoes das ondas viajantes a fenômenos
biológicos. Ph. D. Tesis. IMEC-UNICAMP, CAMPINAS, BRASIL .
[4] Abramson, G. Kenkre, V.M. (2003) Traveling waves of infections in the
Hantavirus Epidemic. Bulletin of mathematical biology 65. 519-534.