MATHEMATICAL ANALYSIS OF SOME BIOLOGICAL MODELS USING DELAY AND FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

Almost all the man-made and natural processes in physics, chemistry, biology, ecology, economics, engineering, etc. show their effect after the moment of their occurrence. This lag is called time delay. Like it or not, the occurrence of time delays is so often, in almost all real and complex phenomena, that ignoring them would certainly be ignoring reality. Further, in usual epidemic models, the probability of transmitting a disease by all infective individuals is assumed to be the same, i.e., at any time, the system’s state is independent of the previous history of the system. However, during the spread of disease within the human population, the response of any individual should be affected by one’s knowledge and experience of that disease. Therefore, it is not realistic to consider the disease evolution process and its control without any memory effect. So, the motivation of these two ideas led us to study some real dynamical models considering either Delay Differential Equations (DDEs), Fractional Differential Equations (FDEs), or a combination of both. The thesis can be read by dividing it into two major parts. The first half of which deals with the malignant tumor modeling with oncolytic virotherapy where we first study a delayed fractional-order tumor model for its stability and after that, we consider a delayed tumor model with control where we discuss different strategies to control tumor growth and eliminate it. The second half is devoted to the study of an MSEIR chickenpox model for its evolution process and an SEIR epidemic model with media delay and non-instantaneous impulsive vaccination. Chapter 1 is purely introductory in which we put the literature review together with a motivational background behind the problems discussed in the thesis. Further, it contains the biological overview used in the thesis and the organization of the whole thesis chapterwise. This second chapter is completely devoted to mathematical preliminaries including some basic concepts of delay differential equations and fractional calculus that would be required in the subsequent chapters of the thesis. Chapter 3 consists of a delayed fractional model related to oncolytic virotherapy. This problem deals with a nonlinear dynamical system involving virus-infected cells, uninfected cells (though both are cancerous cells), and free virus particles. Detailed stability analysis for non-delay and delayed fractional system is done to show in what cases the virotherapy fails completely and in what cases the virotherapy succeed partially. Further, the effect of the fractional order parameter on tumor stabilization is shown. In the previous chapter, oncolytic virotherapy either failed at all or succeeded partially. So, to enhance the efficiency of virotherapy, this fourth chapter again deals with the same but integer order delayed tumor model with two effective optimal strategies. Shrinking of tumor load using each strategy and complete eradication of the tumor using the combination of both strategies is shown in this chapter. Also, the bifurcation diagrams for delay values and direction and stability of Hopf bifurcation which were absent in the previous chapter are shown here. Numerical simulations using ode45, dde23, and Runge-Kutta 4th order scheme are used to plot the graphs. Chapter 5 discusses an epidemic model which shows the evolution process of the outbreak of chickenpox disease (caused by a virus namely varicella-zoster virus) among the children of schools based in Shenzhen city of China in 2013. An MSEIR model using fractal-fractional order derivative is taken and a comparative study is done to show that the disease evolution process is best described when we use fractal-fractional order derivative in contrast to other fractional order derivatives. In this penultimate chapter, a non-instantaneous vaccination SEIRS model with three delays namely the latent period, immune period, and media delay is studied. The existence of an infection-free periodic solution and the permanence of disease is proven. An example is taken to show that the non-instantaneous impulsive vaccination pacifies the negative effect of increasing media delay on the population. The seventh and last chapter summarizes the whole work along with the future scope of the work.