| This dissertation mainly consists of two parts.;Part one concentrates on some important mathematical modelling problems in mathematical biology, ecology and epidemiology. The author (A) Puts forward two general principles in mathematical modeling; (B) Based on the two principles, gives answers to some important questions that were left unasked, and thus unanswered, regarding the Volterra integral equation models which have occurred in the literature in the past decade. It turns out that those models are, in some sense, improper; (C) Shows that the problems which have so far been, or new problems which have a chance of being, modeled as Volterra integral equations should be modeled as autonomous functional differential equations; (D) Based on the two principles, shows how to formulate autonomous functional differential equation models; (E) Gives two important additional conditions the probability function P(t), which has been widely used in the past decade, must satisfy. The two new conditions imply the improperness of some widely used choices of the probability function P(t) in existing literature.;Part two of this dissertation studies the model of functional differential equations formulated in Part one for fatal infectious diseases (such as AIDS). The model is studied both theoretically and numerically. The studies show that a fatal infectious disease can exhibit the following three totally different behaviors: (1) The disease is almost periodic. It can be so serious that almost the whole population will be infected and die at some point; (2) The disease will persist. The number of disease victims and the number of healthy individuals will, respectively, approach fixed levels as time goes by; (3) The disease will gradually die out.;The key factors that determine which one of the above three behaviors a fatal disease may exhibit have been explicitly identified and studied in depth. |
