Stability Analysis and Optimal Control of Selected Deterministic Models

Abstract

In this study, we will develop new and extend some exitance epidemic models to study the transmission dynamics of infectious diseases. The epidemic models and its optimal control of selected different epidemic diseases will be developed by using system of first order nonlinear ordinary differential equation. In the research work we determined four epidemic mathematical models which are categorized as: ”A Dynamic Compartmental Mathematical Model Describing the Transmissibility Of Mers-CoV Virus In Public”. Then ”Modeling and Stability Analysis Of Epidemic Expansion Disease Ebola Virus With Implications Prevention In Population”, and ”Prevention strategy for superinfection mathematical model Tuberculosis and HIV associated with AIDS” are discussed with different schemes and situations. Here we considered the stability and optimal control of selected deterministic epidemic models, their infectious classes, various compartmental changes, threshold number R0, biological region of study, the sensitivity indices analysis of threshold number, points of endemic equilibrium, and stability analysis either local and global stability analysis are discussed. Especially to investigate global stability for our model we have used the theory of Lyapunov-function. Numerical simulation and Matlab programming are being the part of our work used for the validity of models. Chapter I: In the first section we have presented a description of the literature of some of the important fundamental diseases. The historical background, their mutual interaction, infectious classes, and their compartmental behavior change are presented in this chapter. In this subpart of the thesis we also formulated new models, research objectives, as well as, outlines of the thesis. Chapter II: This section is all about the literature survey and methodology. Chapter III: In this chapter of the thesis, We have considered our new mathematical model for Middle East Respiratory Syndrome (Mers-Corona) virus and shown its viii spreading effect from infected camel to individual, family members, clinic and care center staff. In this work, we apprized the transmission associated with different infection stages which generated an epidemiological history of the model. We organized the model and split into infectious and non-infectious categories and presented the properties of our proposed model. Then for local stability as well as for global stability we analyzed the concern model and shown that the infection is reduced in the patients body. While the reproduction key value of the model is derived from the next-generation matrix approach (NGM). The biological region of study R0 is discussed and investigated properly. Then the global stabilities of all types of equilibria are proved. Numerical simulations are performed with and without control or vaccination by an RK-4 method which supports the analytical work and shown the existence of the model. Finally, in the end, we put some remarks in the light of our research article published in Punjab University Journal of Mathematics (ISSN 1016-2526) Vol. 51(4)(2019) pp. 57-71 [1]. Chapter IV: In current subsection, we presented a mathematical model of Ebola virus which proposed by SEIR (susceptible exposed infected recovered) model. In our model, the population is affected by animals either domestic animals or wild animals. The Ebola virus is an infectious agent and one of the viruses that can cause hemorrhagic fever, a severe infectious disease characterized by high fever and bleeding, in humans and some monkeys. Here we assessed the transmissibility associated with the infection stages of the Ebola virus which generated an epidemic model. The threshold property is presented for the reproduction number R0. We also discussed it with sensitivity analysis for dominant parameters in our model. The dynamical behavior is completely analyzed in light of basic reproductive numbers. Numerical simulations are carried out with and without vaccination or control for the proposed model: Cogent Biology (2019), 5: 1619219 https://doi.org/10.1080/23312025.2019.1619219 [2]. Chapter V: In this chapter, we have formalized the model in their infection and non infection classes. We analysis if the value of R0 is less than unity, then the disease dies out. Next, we described all the endemic equilibrium points, as well as, local stability analysis, at disease free equilibrium and, at the endemic equilibrium of the ix related model are shown stable. The global stability either, for disease free equilibria, and endemic equilibria are discussed by constructing Lyapunov function theory which shows the validity of the concern model exists. In such epidemic models we introduced the Matlab programming concept, which shown and detects the area graph involved in any population. While numerical interpretation is shown for the model with the help of RK-4 method. Chapter VI: Here in this chapter, we extended the co-infection mathematical model [1] for optimal control purposes. We initially derived threshold number R0, bounded and biological region for study of the proposed model. Here we developed a method through the considered superinfection problem getting abate while neglecting AIDS because it is a non curable disease. In this order we develop control variables in our model as, V1 used for TB treatment, while to control infection in health care centers, we assigned V2, V3 assigned for TB also for HIV or adopt both HIV anti-retroviral and anti TB drug therapy. In the last we used V4 use to leave any close contact from TB patients are defined with some schemes to minimize and control any infection from the community and population. The optimal control policy is formulated and solved as an optimal control problem. Objective functionals are constructed which aims to (i) minimize infected cell quantity; (ii) minimize free virus particle number, and (iii) maximize healthy cell density in the body. In the last section, numerical simulation with Matlab programming has been presented which supports the given model.