MATHEMATICAL MODELS AND COMPUTER SIMULATIONS OF THE DYNAMICS OF INFECTIOUS DISEASES

Abstract

In this thesis, the dynamics of infectious diseases has been examined using mathematical models and computer simulation. Models of interest follow the basic principle that the population can be subdivided into a set of distinct classes in accordance with their feasibility to describe the evolution of a particular communicable disease. The underline model is called SIR model which categorizes a given population into three classes: susceptible, infective, and recovered. The dynamics of the population is described by a set of three coupled differential equations with adjustable parameters. Five classical SIR models have been rewritten such that they can described as a single ordinary differential equation having a bilinear structure. Numerical simulation for the selected bilinear models has been performed by employing the fourth-order Runge-Kutta iterative algorithm after testing its convergence and accuracy relative to other methods. Moreover, the concept of stability analysis has been used to explore the free and endemic equilibrium of the SIR-II models. The leastsquares fitting algorithm has been developed to estimate model parameters and extract prominent features of the epidemiological data such as Influenza and Measles with epidemic equilibriums. For Influenza epidemic the estimated parameters for SIR-I, 𝛽 = 1.9721, and 𝛾 = 0.47443 , SIR-II, 𝛽 = 1.9721, 𝛾 = 0.47443, and δ = 0.00, SIR-III, 𝛽 = 1.9729, 𝛾 = 0.47468, δ = 0.00039, and α = 0.00, SIR-IV, 𝛽 = 1.9669, 𝛾 = 0.47392, δ = 0.00, and C = 0.00002, and SIR-V, 𝛽 = 1.9669, 𝛾 = 0.47137, α = 0.00, b = 0.11159, m = 0.00167, p = 0.14815, and b ′ = 0.09841. Finally, a graphical user interface (GUI) has been designed using GUIDE utility of MATLAB to interactively visualize the numerical solutions and estimate parameters for the bilinear models