Resumen de Modeling of parasitic diseases with vector of transmission: toxoplasmosis and babesiosis bovine
Diego Fernando Aranda Lozano
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 di erent 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 a ect 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 di erent 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 di erent 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 di erential 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.
