Farming is the basic economy of any country. Soil fertility and water are key resources for yielding. The fractional order compartmental model is prepared in Caputo sense to improve the density of yield through diagnosed and undiagnosed soil. To measure the growing intensity of yield, the basic reproduction number is formulated using the next generation matrix method for integer order non-linear dynamical system. Local stability analysis is described for both the equilibrium points; undiagnosed soil free and optimum equilibrium point. Global stability is exhaustively computed by generating a Lyapunov function. With reference to the basic reproduction number, bifurcation analysis has been defined which expresses the chaotic nature of soil fertility. Moreover, optimal control theory is applied in the present fractional model to optimize yield. And optimality conditions are calculated with the help of Pontryagin maximum principle. The numerical simulation for different fractional orders is performed concerning validated data to analyze the behavior with respect to the order.
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.
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.
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