Abstract
As the complexity of the world in which we live increases, system thinking is becoming
a major factor in success and even in survival. This is why robust tools of complex
dynamic systems can give answers to several problems and can be applied to many
dierent areas, such as business, society and ecosystems, as well as in ordinary life such
as compulsive shopping, drug abuse, tobacco addiction, obesity, etc. When experiments
to test the real world cannot be carried out, simulation becomes the best way to learn
about the dynamic of these systems. For this reason I am pleased to present this Ph.D.
Dissertation, in which theory and practice of the dynamic systems are combined. It also
embraces epidemiologic models of some parasitic diseases with transmission vector.
The Toxoplasmosis and the bovine Babesiosis are parasitic diseases (zoonoses), which
are spread through a transmission vector and aect both human beings and livestock.
As a public health problem, Toxoplasmosis causes high health care costs when treating
unborn and newborn babies. It also causes a great amount of sick leaves. In addition to
this, livestock economic sector in tropical countries, such as Colombia, must bear an extra
cost of millions of dollars due to the high mortality rates and to the low productivity
levels in by-products of farming.
Mathematical models try to describe and represent reality using mathematical techniques.
The importance of mathematical modeling when studying the way some diseases
can spread lies in forecasting the behaviour of these biological phenomena and their effects
wherever they may occur. Thus mathematical models supply a valuable tool for
doctors to use for containment methods, estimation and safety, as well as many other
dierent decisions aimed to reduce economic costs.
Three mathematical models, which describe the behaviour of two parasitic diseases with
transmission vector, are presented in this dissertation. Two of these models are dedicated
to Toxoplasmosis and they explore the dynamic of the disease in relation to human
population and pet cats. In this model, cats play the role of infectious agents and carrier
of the protozoan Toxoplasma Gondii. The qualitative dynamic of the model is established
by the basic reproduction threshold R0. If the parameter R0 < 1, then the solution
converges to the equilibrium point disease free. However, if R0 > 1, convergence leads to
the equilibrium point endemic. Numerical simulations of the models illustrate dierent
dynamics according to the threshold parameter R0 and show the importance of this
parameter. Finally, bovine babesiosis is modeled starting from a mathematical model,
which is composed of ve ordinary dierential equations that explain the in
uence of the
epidemiological parameters over the evolution of the disease. The stationary states of
the system and the basic reproduction number R0 are determined. The existence of the
endemic point and the disease free point are calculated and they depend on the threshold
parameter R0, which determines the local and global stability of the equilibrium points.