محمود باردل

‎In this thesis‎, ‎an epidemic model is introduced for interaction between susceptible individuals an​d 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 an​d mass action incidence rate‎. ‎The total population size of the model is variable due to natural death an​d birth‎, ‎disease caused death‎, ‎and immigration‎. ‎The vaccine is temporary an​d vaccinated individuals gradually lose their immunity from disease through time an​d become susceptible‎. ‎The vaccine that includes both new members an​d susceptible individuals is completely effective an​d no vaccinated individual becomes infectious‎. ‎First‎, ‎dynamics of the model are saparately obtained for two incidence rates an​d 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 o​r 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 an​d stochastic differential equations form are presented an​d 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 an​d simulations‎.