Mathematical Modeling and Qualitative Analysis of Dynamical Systems

Abstract

Many harmful and dangerous incidents have occurred throughout human history, restrict- ing people’s way of life and killing millions of people worldwide. It is helpful for analyzing a variety of natural disasters using mathematical methods and creating strategies for these pandemic. This perspective uses differential equations and specific epidemiological con- straints to develop mathematical models. Mathematical modeling have many extensions as epidemiological models, economic models, climate models, biological models, engineer- ing models etc. An important tool for understanding the dynamics of infectious illnesses is the mathematical modeling of epidemiological systems. These models helps in the eval- uation of control strategies, making projections of disease transmission, and the oversight of public health initiatives. The simplest epidemiological dynamical model introduced by Kermack and McKendrick in 1927 is the SIR model. This compartmental model offers basic insights into the spread of disease and possible points of intervention by describing how individuals go through stages of susceptibility, infection and recovery. The SIR model has multiple modifications such as ebola epidemic SEIR model by Rafiq et al. that is clas- sified into susceptible, exposed, infected and recovered. This model tells us that how the susceptible individuals contact with infected people and becomes infected after exposed of ebola symptoms then moves from infected class to recovery class with a specific recov- ery rate. Another example is corruption dynamics with optimal control strategies (SCJH) model by Duressa and Legesse is classified as S (susceptible), C (corrupted), J (jailed) and H (honest). In this model we deal with corruption as an infectious disease and we use different strategies for overcome corruption dynamics. After analyzing the epidemio- logical dynamical model, we find reproductive number (Ro) which denotes the number of secondary infections caused by an infected individual in a population who is completely susceptible, is crucial to these models by using the next generation matrix method with the help of transmission and disease matric. If Ro > 1, then the disease invade and prolif- erate. If Ro < 1, then disease will die out. Understanding the behavior of the disease at ix both small and large scales requires an understanding of local and global stability analy- ses. Local stability is concerned with how solutions behave close to an equilibrium point whereas global stability takes into account the system’s general behavior under any ini- tial conditions. An endemic or disease-free equilibrium can be established and sustained with the help of these analyses. The simulations that numerical analysis offers can han- dle complex models and real-world data which is a complement to theoretical approaches. Researchers can investigate real world problems that are analytically unsolvable, analyze them and through numerical simulations predict the effects of various solutions. In this study, epidemiological modeling is compiled with an emphasis on numerical simulations, stability evaluations and reproductive number calculation. In infectious disease control by offering a thorough review, this study intends to improve knowledge and implementation of mathematical models. The present study focuses on the dynamical analysis of different dynamical models. Ini- tially, the dynamical analysis of epidemiological models will be discussed by evaluating the reproductive number and disease-free and endemic equilibrium points to check the sta- bility of the system. In addition, modeling in a variety of scientific domains, including chemistry and physics, will be covered. Secondly, we will apply various numerical tech- niques to the models, such as the non-standard finite difference approach, and compare the outcomes with the model’s theoretical predictions. Influenza virus, an infection which be- comes a global threat, causing seasonal outbreaks and pandemic. Current vaccines against influenza A and B viruses are vulnerable to epidemic strains that are poorly matched by the vaccine. A vaccination that is less sensitive to virus antigenic(virus’s surface proteins) evolution would be a significant improvement. The present study introduces a nonlinear differential model by including hospitalized compartment. In this paper we have to prove the non-negativity and boundedness of the model where non-negativity tells us that the parameters in the model cannot take negative values for example population cannot be neg- ative means it represents reality in a logical way. When variables in a model are bounded, it means that they have upper bounds and are not able to increase infinitely. This is a reflection of real-world constraints such as finite resources, physical limitations, or envi- ronmental conditions. Also find the sensitivity analysis which describes how changes in a model’s input parameters affect its output and demonstrates which parameters have the greatest influence on the model’s behavior or output, and which have little to no impact. x We have perform the stability analysis to explore the influential parameters on the growth of influenza virus by calculating the influenza free equilibrium (IFE) and influenza en- demic equilibrium (IEE) points, and computes the fundamental reproduction number of influenza (Rin f .) by applying next generation matrix method that indicates how many new infections each infected individual will typically generate in a population that is fully sus- ceptible. We show that the IFE is locally asymptotically stable (LAS) if Rin f . < 1 implies that disease will not spread widely or will eventually die out, as each infected individual is, on average, causing less than one new infection. When the Rin f . > 1 , the IEE is sta- ble locally in an asymptotic manner means the infection will grow and the population will not return to the disease-free state. We generally proves the global stability analysis with help of Lyapunov function ensures that the system will respond in a predictable manner over time, either moving in the direction of the disease’s eradication or stabilising in a state where it is constantly present but under control. We verify these theoretical results through numerical simulation by applying non-standardized finite difference (NSFD) scheme