Vol. 11, No. 3 (2022): Regular Issue

In this study, a nonlinear deterministic mathematical model for the impact of pathogens on plant disease transmission with optimal control is proposed. Mathematically, we analyzed the dynamics of the system properties including boundedness of the solutions, existence of pathogen free and coexistence equilibria, local and global stability of equilibrium points. Furthermore, we computed the basic reproduction number $R_{0}$ and studied its sensitivity with respect to model parameters to identify the most affecting parameters. The local stability of the equilibria is also established via the Jacobian matrix and Routh Hurwitz criteria while the global stability of the equilibria is proved by using an appropriate Lyapunov function. Using center manifold theory, we proved that the model exhibits forward bifurcation. An optimal control problem is proposed which minimizes costs incurred due to the disease and applied controls. The characterization of optimal paths is analytically derived using Pontryagin's Minimum Principle. We then numerically studied the optimal control problem. A comparative study is conducted by considering control strategies: implementation of only quarantine, chemical, execution of both quarantine and chemical. We observe that the inclusive use of quarantine and chemical controls can reduce the infected host population with minimum implementation cost.

Economically motivated crop rotation has been proposed to model coupled agricultural markets. The resulting supply-sided behavior generates a wealth of complex dynamics. Since centuries most Western economies established substantial agricultural subsidies that we show to be both persistent and differentiated by variety. We incorporate differential subsidies and study whether this changes implied dynamics of interacting crop markets. Our findings contrast with single-crop model predictions inasmuch we find subsidy differentials for rotated crops to exert stabilization.

Several types of vibrating systems experiencing some interesting effects of inertial forces generated by vibrations are studied. The behavior of these systems can be easily predicted when we know how and which vibratory forces are affecting them. One example is a pendulum with a vibrating pivot that changes the position of equilibrium, alters the natural frequency, and the vibratory force stabilizes its inverse position. Sometimes, a rotor might not reach the final velocity due to the resonance because of the breaking vibratory moment, or the rotor speed decreases rapidly if the motor is switched off. If the imbalanced rotor has free elements inside it, then they can freely align their position and compensate the initial rotor imbalance. Also, the free elements of the rotor are able to eliminate its vibration if the rotor is placed on a vibrating base. It is observed that it is easier to move an object if it is on a vibrating surface than when the surface is not moving at all, apparently due to a lower friction force. By controlling the vibration components of the plane, the object's trajectory as well as its velocity can be controlled. This model can be used to simulate the locomotion of living systems or active Brownian particles.\newline A novel method of analysing mechanical systems which allows to explain their dynamic behaviour and allows to predict some effects is described, along with some examples of laboratory tests.

Due to the unavailability of proper antiviral therapies and high disease transmission rates, the pandemic COVID-19 is still increasing at a high rate in many countries. In each of the three countries, the USA, Brazil, and India, the COVID-19 positive cases have been crossed one million. With high population density and higher percentages of migrating workers has enabled India to be vulnerable to the disease quite more than other affected countries. In this paper, we have proposed a mathematical model with the help of a system of first-order ordinary differential equations and analyzed the model in the context of the COVID-19 pandemic. We have determined the expression of the basic reproduction number and relates it to establishing the disease-free equilibrium point's asymptotic stability and endemic equilibrium point. As it has been observed that only ten states and union territories are carrying more than $70\%$ infection in India, we have predicted long-term scenarios of the COVID-19 positive cases on those $10$ states and India until the end of the year 2020.

This article investigates the non-parallel distribution compensation issue (N-PDC) of Networked Control Systems (NCSs) under a new event-triggered strategy based on the fuzzy system for a class of nonlinear with appearing the time-varying delays. A unified framework of T-S fuzzy model under the new event-triggered strategy is present, in which \textit{(i)} The observer-based fuzzy controller with the incomplete premise matching is a build-up to obtain the unmeasurable states of the system under study; \textit{(ii)} A designed fuzzy controller has used the same premise such as the observer; and \textit{(iii)} To rationally utilize network resources and elaborately avoid unnecessary continuous monitoring, a new event-triggered mechanism with variable parameter and a sampler considered, respectively. Besides we encounter the nonlinear term with the \textit{Lipschitz condition}. Another derives of this paper introduced the $H_{\infty}$ performances index. Furthermore, in this regard, by taking into account a new \textit{fuzzy Lyapunov-Krasovskii functional} (LKF) in conjunction with free weighting matrices, containing mode-dependent non-integral terms such that the resulting system is stable with the desired performance. Then, the desired observer-based controller is presented in the account of LMI. Finally, an illustrative example is provided to demonstrate the effectiveness and superiority of the proposed method.

In this paper, we approach conceptual mathematical models for COVID-19 eruption and it's abstinence metering; quarantine. In this circumstance, mathematical models are a grandly tool to make use of an impressive strategy in order to fight against this pandemic. We define the positivity, boundedness of solutions and basic reproduction number. Then, we study local and global stability analysis of equilibrium to examine its epidemiological relevance. Our model represent the various transmission route in the infection dynamics, and diligence the role of the environmental reservoir in the transmission and dispersion of this disease. Simulation explication of the model ratify to the analytical results.

This study presents a Susceptible-Infected-Recovered-Susceptible (SIRS) system for the dynamics of epidemics in a non-closed human population. We consider the vertical transmission of the epidemic and incorporate varying periods of the individuals' immunity. The stochastic version of our new epidemic model is obtained by simultaneously introducing the stochastic transmission and the proportional perturbation. For the permanence case, the ergodicity of the perturbed system is proved by employing a new approach different from the Lyapunov method. Under the same condition of the existence of the unique ergodic stationary distribution, the persistence in the mean of the epidemic is shown. When the basic reproduction number $\mathcal{R}_0>1$, the asymptotic behavior of the solution around the endemic equilibrium of the deterministic model is provided. For the extinction case, sufficient conditions for the disappearance of the epidemic are obtained. When $\mathcal{R}_0\leq1$, the analysis of the asymptotic behavior around the disease-free equilibrium of the deterministic model is also investigated. In order to make the readers understand our study better, we present some numerical simulations to illustrate our theoretical results.

In the paper, we present a method based on eigenvalue criterion to test the asymptotic stability of fractional linear Langevin systems represented by the fractional differential equation in the sense of Caputo fractional derivative. Also, this method is extended to nonlinear equations and finally some sufficient conditions ensuring asymptotical stability of fractional-order nonlinear Langevin systems are proposed. Some numerical examples are provided to illustrate the effectiveness of the proposed method.

A speculative investigation has been presented on explore the salient features of multiple slip effects on MHD Casson nanofluid flow over a porous stretching sheet in the presence of Soret effect, thermal radiation and chemical reaction are numerically examined. We consider an applied magnetic field and stretching sheet time-dependent, which moves with non-uniform velocity. For the transformation of governing partial differential equations into a system of coupled nonlinear ordinary differential equations, suitable similarity variables are used. The transformed equations are then solved numerically by applying Runge-Kutta Fehlberg method with shooting technique. The influences of the various physical parameters on the velocity, temperature, and concentration profiles as well as on the skin friction coefficient, Nusselt and Sherwood numbers are discussed by the aid of graphs and tables. The imposed magnetic field produces a Lorentz force which drags the velocity of an electrically conducting Casson fluid. Due to the magnetic field strength (B$_{0}$), a higher value of Casson parameter decelerates the velocity field. Also, increase in thermal radiation parameter enhances the temperature distribution when the plate is hot.

This work presents an investigation on the effect of an initial geometric imperfection; wavelength, amplitude, and degree of localization on the in-plane nonlinear resonance responses of an imperfect beam. The beam was modeled as an Euler Bernoulli beam resting on an elastic foundation, hinged at one end and supported by a torsional spring at the other end. The governing model accounts for the effect of the axial force induced by mid-plane stretching. The imperfection was introduced as a rise with different shapes, different amplitudes, and wavelengths. A quantitative and qualitative analysis of the steady-state responses and their stability is obtained by computing force-frequency response curves using the Harmonic Balance method $HB$ and the method of Multiple Scales $MMS$. Results have shown that the behavior of the steady-state responses may change from hardening to softening types depending on the geometrical and physical parameters of the beam under consideration. Results are presented in a dimensionless form for the steady-state forced vibration in the aid of time histories, phase planes, frequency spectrums, and Poincare maps for selected values of physical parameters. It is shown that the beam response may exhibit complicated dynamic behaviors including period multiplying and period de multiplying bifurcations, period three and period six motions, jump phenomenon, and chaos.

In this paper, We construct the numerical solution of a particular type of different equations, called 'Fuzzy Neutral Delay Differential Equations' by using fifth order of Runge-Kutta method. To validate the findings, relevant examples and algorithms have been discussed in a crystal clear way. The discussion has been pursued on the statement of the analyzation between the approximate and the exact solutions and the error analysis also is examined to a bring better solution.

A Cooperative Supportive network system consisting of the main system supported by a subsystem whose dynamics are described in two separate neuronal fields is considered. Time delays are introduced in intra-network propagation (communication) of information in the main system. The qualitative behavior of the solutions of the system is analyzed. The existence and uniqueness of equilibria and their stability is investigated. Several independent criteria on the parameters and the functional relations of the systems are obtained, to establish the global stability of the equilibrium. Estimates on time delay parameter for which the system remains stable are also obtained, using Lyapunov functional. Several numerical examples are provided to illustrate the results. Levenberg-Marquardt Algorithm is utilized for training the network and obtaining simulation results for the examples provided.

The diffusive Lotka-Volterra model of a prey and predator interaction is analyzed. The model is based on nonlinear differential equation with reaction diffusion term. The approximate analytical expression for diffusive Lotka-Volterra differential equation model has been derived. We employ the Variational Iteration Method to solve this nonlinear boundary value problem. Numerical simulation is obtained through MATLAB software. Both the analytical results and numerical simulation are compared and there is a satisfactory agreement between them.

This paper presents the steady flow of viscous incompressible fluid through a permeable channel with the effect of an exponential reabsorption at the walls. The fluid reabsorption across the channel walls is assumed by taking the flux as a function of axial length. The approximate solutions for velocity, mean pressure drop, wall shear stress and streamlines are solved by regular perturbation technique and finite difference method. The effect of reabsorption on the velocity profiles at different locations of axis, wall shear stress, mean pressure drop and streamlines are discussed through graphs. It is observed that the pressure drop and wall shear stress decrease as reabsorption increases. Further the streamlines show that the reabsorption significantly influences the flow pattern.

This work explores the steady boundary layer flow and heat transfer of a couple-stress fluid flowing over a bidirectional movable surface. The Lie group of symmetry transformations are employed for determining the possible invariant solutions of the governing equations for fluid flow and heat transfer. The self-similar equations are solved numerically and plotted graphically. It is observed that fluid velocities in both $x$ and $y$ directions increase due to the increase of the couple-stress parameter. The stretching ratio parameter suppresses the flow boundary layer in the $x$-direction, but it extends in the $y$-direction. The thermal boundary layer thickness reduces for increasing values of couple stress parameter, stretching ratio parameter, Prandtl number, and power-law index parameter on wall temperature, but opposite behavior occurs for radiation parameter.