Mathematical Modeling With Delay Differential Equations

Ordinary differential equations (ODEs), partial differential equations (PDEs) and delay differential equations (DDEs) are extensively used in the field of scientific modeling by mathematicians to obtain solutions for problems which arise from tech-nology, biology, medicine, economics and social sciences. Although theories related to differential equations are upgraded considerably, still there are some limitations to obtain solutions for DDEs. Therefore, qualitative properties of the solutions such as local stability, global stability, permanence and bifurcation analysis are interest-ed by many scientists. Mathematical tools mainly Method of Lyapunov functional, Lyapunov-LaSalle invariance principle and Hopf bifurcation theories have exten-sively been applied to obtain vital dynamic properties of the solutions specially in the field of biomathematics.Various kinds of mathematical models by using DDEs have been introduced in biology to investigate epidemiological properties, to study virus dynamics behaviour and to describe some ecological problems in the recent past. However, some bio-logical issues in epidemiology, virology and ecology are not exceptionally addressed to determine dynamical behaviour of the solutions in a reliable and precise man-ner. Therefore, in this dissertation, we focus our attention on developing several biologically reasonable models and we further analyse them by using theorems from functional analysis.As it is well known that analysis of real world problems is interesting, firstly, we focus our attention on new eco-epidemiological deterministic delay differential equation model considering biological controlling approach on mosquitoes for en-demic dengue disease with variable host (human) and variable vector (Aedes ae-gypti) populations, and stage structure for mosquitoes is proposed. In this model, predator-prey interaction is considered by using larvae as prey and mosquito-fish as predator. We have given complete classification of equilibria of the model. Further, a delayed model for vector borne dengue out break with non-linear incidence rates and latent delays of both host and vector was also studied in details. The global stability properties of disease the free equilibrium as well as the endemic equilibrium are obtained for the both models by means of appropriate Lyapunov functionals and LaSalle’s invariance principle for delay differential equations. The global dynamics of equilibria of the model are totally concluded by the basic reproductive number. We proved that if the basic reproductive number less than or equal unity, the disease free equilibrium globally asymptotically stable and the basic reproductive number greater than unity, the endemic equilibrium is globally asymptotically stable.Further, in this context, stability properties of a class of HIV virus infection model with Beddington-DeAngelis functional response and absorption effect are investigated. As a development to this model, we extended our model by using non-linear separable functional response. And stability results for both models were critically analyzed and noticeable results were obtained. Our mathematical analysis shows that stability properties are completely determined by the basic reproduction number of the model. Utilizing characteristic equation of the model, we established that the infection free equilibrium and the chronic infection equilibrium is locally asymptotically stable if the basic reproduction number of the model is less than or equal to unity and the basic reproduction number of the model greater than unity, respectively. Moreover, by means of Lyapunov functionals and LaSalle invariance principle, it is derived that, if the basic reproduction number of the model is less than or equal, the infection free equilibrium is globally asymptotically stable.For aforementioned models, numerical simulations were also carried out to show the validity of theoretical analysis.