Analysis of Epidemiology Models by Non Standard Finite Difference Scheme and Laplace Adomian Decomposition Method

Abstract

Mathematical modeling has proven to be an essential tool for the development of control strategies and in distinguishing driving factors in disease dynamics. A key determinant of a given model’s potential to aid in such measures is the availability of data to parameterize the model. For developing countries in particular, data are often sparse and difficult to collect. It is therefore important to understand the types of data that are necessary for a modeling project to be successful. Infectious diseases are a persistent problem throughout the world, potentially threatening everyone who comes in contact with them. This thesis attempts to improve our understanding of infectious diseases by developing mathematical models of the cellular dynamics of human infectious diseases. This has been carried out through the investigation of the interaction between infectious agents and cells of the humoral and cell-mediated immune response. Additionally, dynamics of the infectious agent in an infected cell are described through the development of descriptive mathematical models. In this thesis, we consider the models for Smoking, SIR and SEIR measles. Sensitivity analysis of these models are provided by threshold or reproductive number as well as analysed qualitatively also check the stability analysis of these models. A non-linear mathematical model is employed to study and assess the dynamics of smoking and its impact on public health in a community. The analysis of two different states disease free and endemic which means the disease dies out or persist in a population has been used. We developed an unconditionally convergent nonstandard finite difference scheme by applying Mickens approach φ(h) = h + O(h2) instead of h to control the spread of bad impact in society , which is dynamically consistent, easy to implement and show a good agreement to control the bad impact of smoking for long period of time and to eradicate a death killer factor in the world spread by smoking. Finally numerical simulations are also established to investigate the influence of the system parameters on the spread of the disease. We proposed the fractional order model for smoking, SIR and SEIR measles. A nonlinear time fractional model is used in order to understand the outbreaks of this epidemic disease. Verify the non-negative unique solution of the developed fractional order models. The Caputo fractional derivative operator of order α ∈ (0,1] or φ ∈ (0,1] is employed to obtain the system of fractional differential equations. Laplace adomian decomposition method has been employed to solve these fractonal order models. Finally, some numerical results are presented which show the effect of fractional parameter α and φ on our obtained solutions. The caparison of different fractional values can be shown in tables and graphs. The Laplace Adomian Decomposition Method is applied to give an approximate solution of nonlinear system of fractional ordinary differential equation of models at different fractional values.