Vol. 12, No. 2 (2023): Regular Issue

This paper aims to tackle an important problem - the optimization of measures applied to fight disease in one of the most important economically plant cultures: coffee. This is done by formulating a mathematical model of coffee berry disease (Colletotrichum kahawae) with optimal controls which consist of coffee berry and vector population with the interaction of pathogen, temperature variability, and rainfall pattern. First, we investigate the positivity and boundedness of solutions of the proposed model with the absence of controls. Then, optimal control strategies are studied to minimize the burden of coffee berry disease (CBD) and the cost of interventions. The characterization of optimal trajectories is also analytically derived using Pontryagin's Minimum Principle. Finally, we performed numerical simulations to investigate the impact of control strategies in combating CBD. Furthermore, we studied the cost-effectiveness of our control strategies to determine the best approach to minimize the disease burden. The finding of this study shows that any combination of genetic resistance variety, chemical and cultural control strategies are very effective to combat the disease.

The current paper is concerned with dynamic buckling response of axially loaded graphene-enhanced composite (GEC) laminated cylindrical shells under the influence of thermal conditions. Graphene layers are arranged in a functional graded (FG) along the direction of the shell thickness. A modified Halpin-Tsai approach is utilized to estimate the material properties of GECs and these properties are considered to be temperature dependent. Theoretical Framework is conducted based on the shear deformation theory (SDT) in conjunction with the von-Kármán relation and imperfection geometric effect. By utilizing Galerkin manner in concurrent with the Airy's stress function, the derived nonlinear partial differential equations are solved numerically using the fourth-order Runge--Kutta manner. Budiansky--Roth standard is used to predict the dynamic buckling loads. Besides, a specific study was executed to detect the effect of graphene sheets distribution type, temperature, loading rate, imperfection geometric parameter and the geometric parameter on the GECs laminated cylindrical shells. The proposed method was validated via comparing the results obtained with those from other published ones.

In this work, we studied the peristaltic flow of steady non-Newtonian (Bingham model) nanofluid through a non-uniform inclined pipe. Bingham nanofluid flows through a non-Darcy porous medium under the effects of thermal radiation, heat generation, Ohmic dissipation, chemical reaction, mixed convection and thermal diffusion. The mathematical equations which describe the velocity, temperature and nanoparticles concentration are simplified under the assumptions of long wavelength and low Reynolds number. We obtained a semi-analytical solution for the non-dimensional governing equations by using homotopy perturbation method (HPM). The obtained solutions are functions of the entering physical parameters and the effects of these parameters are explained and discussed through a set of figures. It is clear that the physical parameters in our problem play an effective role and control the obtained solutions. On the other hand, the interference between axial velocity distribution and temperature distribution is due to the presence of mixed convection and Ohmic dissipation, it is found that the effect of local temperature Grashof number $G_r$ on the axial velocity $u$ can be controlled by the adjusting the values of Radiation parameter $R$.

A model of an anomalous boundary layer over a flat plate was presented through fractional calculus. It is a generalization of Blasius's system, including a fractional derivative order in the heat equation. A nonlinear differential system with contour condition is usually difficult to solve, in this case, the system has a fractional differential term, therefore a semi-analytical approximation method was used combined with the initial value problem approximation by sequential parameter optimization method. Besides, it was shown that the derivative order has a significant influence on the boundary layer thermal shape and causes a bigger change in the temperature gradient curve.

In this paper, we study fishing model with time varying variable delays and harvesting term on time scales, $$ \mathtt{z}^\triangle(\mathtt{s})=-\upalpha(\mathtt{s})\mathtt{z}(\mathtt{s})+ \sum_{\mathtt{j}=1}^{\mathtt{N}}\frac{\upbeta_\mathtt{j}(\mathtt{s})}{1+ (\frac{\mathtt{z}(\mathtt{s}-\uptau_\mathtt{j}(\mathtt{s}))} {\mathtt{G}(\mathtt{s})})^{\ell_\mathtt{j}}}-\mathtt{h}(\mathtt{s}),~~\mathtt{s}\in\mathbb{T}. $$ We first derive sufficient conditions for the existence of unique positive almost periodic solution for the model by applying contraction principle. In addition, with the help of Gronwall's inequality and functional analysis, we study global exponential stability and then by means of Lyapunov function, we establish asymptotical stability of the addressed model. Finally, numerical simulations are employed to illustrate the obtained results.

We investigate unidirectionally coupled Lorenz systems in which the drive possesses a chaotic attractor and the response admits two stable equilibrium points in the absence of the driving. It is found that double chaotic attractors coexist in the dynamics. The approach is applicable for chains of coupled Lorenz systems. The existence of four as well as eight chaotic attractors are also demonstrated. Additionally, the time evolutions of the maximum Lyapunov characteristic exponents of the systems under consideration are discussed. This is the first time in the literature that multiple chaotic attractors are obtained for coupled Lorenz systems.

Inspired by various studies on $q$-deformations in physical systems and their wider applications in different fields, we study the $q$-deformed logistic map via a superior fixed-point feedback approach. Different dynamical characteristics such as fixed-points, time-series evolution, period-doubling bifurcation, and Lyapunov exponent of the system are analyzed using a superior approach. The results are carried out analytically as well as experimentally. Under the proposed approach, due to the presence of an additional control parameter $\alpha$, the system exhibits superior dynamical characteristics such as better stability range, suitable lower maximum Lyapunov exponent value, and an improved sensitivity as compared to the traditional approach.

In this paper, we propose a seven compartmental model based on ordinary differential equations (ODEs) to understand the importance of non-pharmaceutical interventions and pharmaceutical protocols. The boundedness and non-negativity of solutions of the model are discussed to ensure the feasibility of solutions of the model. To classify epidemic and endemic cases of the model, we determine the basic reproduction number. Local stability analysis of the non-negative equilibria is performed to gather a dependency of all compartmental populations on time. It is inspected that social awareness parameter controls the symptomatic and asymptomatic populations. It is also found that restrictions on public gathering reduce the transmission of novel coronavirus effectively. Furthermore, the recovery of the COVID-19 infected people is significantly increased when proper medication and adequate clinical support are arranged immediately. Finally, numerical results demonstrate that transmission of novel coronavirus can be prevented and regulated in a densely populated country like Bangladesh when COVID-19 health rules are strictly followed and movement of infected people is minimized as non-pharmaceutical strategies. Apart from non-pharmaceutical interference, medication during quarantine and sufficient clinical support play a pivotal role to minimize the demise of COVID-19 infected people once they are infected.

The flow of ternary nanofluid past a rotating cone has been analysed using the Ternary nanofluid model. The ternary nanofluid is formed by suspending $CuO$, $MgO$ and $TiO_2$ nanoparticles into water. The nanoparticles that are suspended in the base fluid are assumed to be in the shape of a sphere so that there will be minimum friction between the nanoparticles and the surface as a result this will allow the fluid to flow with less frictional force. Such a characteristic flow finds application in automobiles, production industries, metallurgical process, solar appliances etc. Hence, in order to analyse the heat transfer characteristics of ternary nanofluid, a mathematical model is framed with the help of partial differential equations considering thermal radiation and heat source/sink to achieve realistic results. These equations are further transformed to non-linear differential equations that are solved using RKF-45 technique. The results of this study are interpreted graphically for various parameters corresponding to the fluid flow. The outcomes of this study indicated that the increase in convection enhanced the tangential velocity of the flow and the nanofluid temperature. Whereas, the increase in the thermal slip reduced the tangential flow velocity as well as the temperature of the nanofluid.

The Susceptible-Exposed-Infectious-Recovered (SEIR) model is applied in several countries to ascertain the spread of the coronavirus disease 2019 (COVID-19). We consider discrete-time SEIR epidemic model in a closed system which does not account for births or deaths, total population size under consideration is constant. This dynamical system is generated by a non-linear evolution operator depending on four parameters. Under some conditions on parameters we reduce the evolution operator to a quadratic stochastic operator (QSO) which maps 3-dimensional simplex to itself. We show that the QSO has uncountable set of fixed points (all laying on the boundary of the simplex). It is shown that all trajectories of the dynamical system (generated by the QSO) of the SEIR model are convergent (i.e. the QSO is regular). Moreover, we discuss the efficiency of the model for Uzbekistan.

In this article, we investigate an optimal control problem for acid-mediated cancer invasion model which describes the normal cell density, the tumor cell density, the excess $H^+$ ion concentration, the extracellular matrix, and active metalloproteinases. The main objective of this paper is to minimize the growth of tumor cells by controlling the excess production of $H^+$ ions. First, we establish the existence of weak solutions by using the Faedo-Galerkin approximation method, then we prove the existence of optimal control. Further, we derive the necessary optimality condition for acid-mediated cancer invasion model. Finally, we illustrate the importance of the control term using some numerical simulations.

This study aims to establish exact solutions of a new (3+1) Date-Jimbo-Kashiwara-Miwa equation. The method of the modern group analysis will be implemented to derive exact solutions of the aforementioned equation. In addition, the variational method will be employed to construct conserved vectors of a new (3+1) Date-Jimbo-Kashiwara-Miwa equation. Furthermore, a brief physical interpretation of these conserved vectors will be mentioned.

Modeling of physical phenomena with fractional differential equations is as old as modeling with ordinary differential equations. There are two stages in modeling of a memory process. One of them is short with persistent impact and other is usually governed by fractional mathematical model. It is established that fractional models fit the experimental data for the memory phenomena in better way when compared with the ordinary models, particularly in mechanics, psychology and in biology. Fractional model of viscoelastic nanofluid flow through permeable medium is studied in this communication. Convection parameters in the flow domain are used to account for buoyancy forces. The governing flow equations are computed using a numerical algorithm that combines finite difference and finite element techniques. The governing model's friction coefficient, Sherwood numbers, and Nusselt numbers are calculated. Change in non-integer numbers behave similarly in concentration, temperature, and velocity fields, according to simulations. It is also noted that heat flux, $\delta_{1}$ and mass flux, $\delta_{2}$ numbers have contradictory effects on friction coefficient. Various flow patterns, particularly in the polymer industry and electrospinning for nanofiber manufacture, can be addressed in a similar manner.

This paper presents a method for predicting limit cycles of a multi-degree-of-freedom system possessing a freeplay non-linearity. The approach taken is through casting the problem as an integro-differential equation. The method is a development of that previously reported in the literature for the simpler case of such a system with a cubic hardening non-linearity. The system considered is based on aeroelastic applications where structural non-linearities of this kind are encountered. Limit cycles stability is determined using an implementation of Floquet analysis based on extending the Hill's Determinant approach that may be used in analysing the Mathieu equation. The limit cycle predictions and Floquet multipliers are compared against predictions from numerical integration to show the validity of the method. Fast Fourier transform analysis is used to provide comparisons with the predictions of harmonic components from the analytical results. As the Floquet analysis also produces an approximation to the motion of the system in the neighbourhood of a limit cycle, in the case of an unstable limit cycle, it was possible to approximate the limit cycle stable manifold in situations where the limit cycle amplitude is much greater than the amount of freeplay.

We look at the SQIRP mathematical model for new coronavirus transmission in Bangladesh and India in this study. The basic reproduction number of the SQIRP system is designed using the next cohort matrix process. The SQIRP system has asymptotically stable locally at an infection-free equilibrium point when the basic reproduction number is not more than unity and unsteady when the value is greater than unity. The SQIRP system is found to go through a backward bifurcation, which is a novel perspective for Coronavirus infection transmission. The infection-free equilibrium and endemic equilibrium are shown to be asymptotically stable globally using the Lyapunov function hypothesis and the invariance principle of Lasalle. A SQIRP system with backward bifurcation is explored using stochastic analysis. The ecological stochasticity in the appearance of white noise best describes the system's value. To verify the results, more numerical simulations are run.