Analysis of Discrete System Modelling Followed by Spread of Infectious Diseases Problem in Fuzzy Environments

In this chapter, the authors discuss the solution of spread of infectious diseases in terms of SI model in fuzzy environment, which is modelled in a typical discrete system. As the system is discrete in nature, the concept of difference equation has been embarked. In order to understand the underlying uncertainty perspective, they explored the fuzzy difference equations to study the problem.

Dynamic Analysis of the Effect of Quitting Smoking Applications on Smoking Cessation

In this chapter, the authors considered a smoking cessation model formulated with a non-linear system of differential equations and obtained the continuous fractional order model and through discretization its discrete form to study the effectiveness of quitting smoking applications in giving up smoking. The existence of smoking free equilibria and smoking present equilibria are discussed, and the dynamical analysis of these two equilibria is put forward with the assistance of the smoking generation number. The numerical simulations aided by time series, phase portraits, and bifurcation diagrams confirm the results that are obtained analytically.

Controlling Asthma Due to Air Pollution

The health of our respiratory systems is directly affected by the atmosphere. Nowadays, eruption of respiratory disease and malfunctioning of lung due to the presence of harmful particles in the air is one of the most sever challenge. In this chapter, association between air pollution-related respiratory diseases, namely dyspnea, cough, and asthma, is analysed by constructing a mathematical model. Local and global stability of the equilibrium points is proved. Optimal control theory is applied in the model to optimize stability of the model. Applied optimal control theory contains four control variables, among which first control helps to reduce number of individuals who are exposed to air pollutants and the remaining three controls help to reduce the spread and exacerbation of asthma. The positive impact of controls on the model and intensity of asthma under the influence of dyspnea and cough is observed graphically by simulating the model.

Mathematical Model to Analyze Effect of Demonetization

Demonetization is a fundamental regulatory act of stripping in which a currency unit's status as an exchange is professed worthless. Generally, it is done whenever there is a change of national currency, often to be replaced of the old notes or coins with a new one. Sometimes, a country totally replaces the old currency with new currency. For example, in India recently the government demonetized RS. 500 and 1000 notes. So, one has to deposit their cash within limited time in the banks. The demonetization affects individuals mildly or potentially, which in turn affects banking sector. So, SMPB-model is proposed and analyzed for demonetization. The SMP-model is formulated with the system of nonlinear differential equations. The effect of demonetization is studied by calculating threshold using next generation matrix. The local and global stability for demonetization free equilibrium and demonetization equilibrium is worked out. The existence of the equilibrium is investigated. The model is validated with numerical simulation.

Vertical Transmission of Syphilis With Control Treatment

Syphilis is a sexually transmitted disease having different signs and symptoms with four main stages, namely primary, secondary, latent, and tertiary. Congenital (vertical) transmission of syphilis from infected mother to fetus or neonatal is still a cause of high perinatal morbidity and mortality. A model of transmission of syphilis with three different ways of transmission, namely vertical, heterosexual, and homosexual, is formulated as a system of nonlinear ordinary differential equations. Treatment is also incorporated at various stages of infection. Total male and female population is divided in various classes (i.e., were susceptible, exposed, primary and secondary infected, early and late latent, tertiary, infected treated, latent treated, infected child [newborn], and treated infected child [at birth time]). Stability of disease-free equilibrium and endemic equilibrium is established. Control treatment is applied. It is observed that safe sexual habits and controlled treatment in each stage including pregnancy are effective parameters to curb disease spread.

Control of Pest Population by Sterile Insect Technique Considering Logistic Growth With Spatial Spread Invasion and Optimal Production Policies

In this chapter, the authors have proposed a SIT model to eradicate the pest population. It has been assumed that the females after mating with wild males grow logistically. Pest population is being controlled with the release of sterile insects in their habitat. The model is formulated with the system of differential equations, and the authors have discussed the local stability analysis of deterministic logistic growth rate model. Further, they have also obtained a potential function by incorporating one-dimensional insect release with an invasion on patch size L, which has a toxic exterior as its surrounding. It has been obtained that, in the presence of spatial spread over a finite patch size, the sterile release of the insects produces a sudden declination of the pest population. Finally, the authors have obtained the optimal production of sterile male population using Pontryagin's maximum principle. The applicability of the proposed model is finally illustrated through numerical solution.

Analytical Study of Large-Scale Household Yagya Effects on Ambient Air Pollution

We all are living in such a world where the pollution and global warming are threats. Every year in India, at the time of festival seasons of Dussehra and Deewali, the smog and pollution are so much that millions of people suffer from different health issues. Also, the farmers of Punjab and Hariyana burn the Parali of their crops due to less awareness, and it becomes a challenge in the national capital, Delhi, to breathe. The government invests resources and the vehicles are allowed as per their even odd numbers. The authors team, including government officials, educationists, academicians, and students, along with IT experts, performed significant experiments on the ancient Indian Vedic science of Yajna and Mantra, and they found surprising results in the reduction of pollution on respective days. The chapter is an effort to present that scientific study conducted in 2018 and 2019 in random days after doing Yajna, and it was found that the pollution level was drastically decreased.

Fractional-Order Model to Visualize the Effect of Plastic Pollution on Rain

In this era, one of the biggest issues faced by humans is due to plastic pollution as it dwells in environment and depletes the ecosystem. This affects the climate and disturbs the chain of rain, which is the common source of obtaining water body. Also, this resulting pollution causes the toxicity in rain. Accordingly, the mathematical model is framed by considering fractional order derivative. Pollution free and endemic equilibrium points are worked out for integer order system of non-linear differential equations. Local stability of equilibrium points brings attention on dynamical behavior of model with sufficient condition. With the help of basic reproduction number, bifurcation is analyzed, which shows the chaotic nature of this model. Providing Caputo derivative of fractional order, a numerical simulation has been done by taking different values of order for the system.

Stability Analysis of Co-Infection of Malaria-Dengue

Dengue and malaria most commonly occur in tropical and sub-tropical areas. Dengue is a viral infection in a human being caused by a bite of a female aedes mosquito whereas malaria is caused by plasmodium parasite transmitted by a bite of infected mosquito. In this chapter, a mathematical model of co-infection of malaria and dengue is described by deterministic system of non-linear ordinary differential equations. This system considers the force of infection which is applied to dengue susceptible individuals. Moreover, two sub-models, namely malaria-only and dengue-only, are also constructed to study the transmission dynamics. Basic reproduction number is calculated for these models to investigate the existence of the models. The system is proved to be locally and globally stable at its equilibrium points. Stability of these models is also shown through numerical simulation.

Spread of Tuberculosis Among Smokers

Smoking tobacco has some hazardous implications on an individual's physical, physiological, and psychological health; health of the passive smokers near him or her; and on the surrounding environment. From carcinomas to auto-immune disorders, smoking has a role to play. Therefore, there arises a need to frame a systemic pathway to decipher relationship between smoking and a perilous disease such as tuberculosis. This research work focuses on how drugs or medications can affect individuals who are susceptible to tuberculosis because of smoking habits and also on individuals who have already developed symptoms of tuberculosis due to their smoking addiction. The mathematical model is formulated using non-linear ordinary differential equations, and then threshold is calculated for different equilibrium points using next generation matrix method. Stability analysis along with numerical simulations are carried out to validate the data.