چكيده به لاتين
In this thesis, an epidemic model is introduced for interaction between susceptible individuals and infectious individuals in a population in which a vaccination program is in effect. The susceptible individuals are infected to the disease with two rates, standard incidence rate and mass action incidence rate. The total population size of the model is variable due to natural death and birth, disease caused death, and immigration. The vaccine is temporary and vaccinated individuals gradually lose their immunity from disease through time and become susceptible. The vaccine that includes both new members and susceptible individuals is completely effective and no vaccinated individual becomes infectious. First, dynamics of the model are saparately obtained for two incidence rates and their asymptotic stability is analyzed. Next, a discrete-time version of the model with mass action incidence is studied in the cases that the total population size is variable or is constant. In addition to the stability of the equilibria, the occurrence of transcritical, period-doubling, and the Neimark-Saker bifurcations are considered. Two stochastic models in Markov chains form and stochastic differential equations form are presented and their asymptotic behaviors are studied. The probabilities of extinction are separately obtained for two incidence rates in Markov chain models. An equivalent model is also introduced after expression the procedure in which the stochastic differential equations model is constructed. Also, the theoretical results are confirmed by using several numerical examples and simulations.
