Vol. 12, No. 1 (2023): Regular Issue

In this paper, we develop a COVID-19 mathematical model and divide the entire populations into six classes, namely susceptible, susceptible quarantined, exposed, infected, infected quarantined and recovered. We utilize the concept of "shield immunity" which is a different concept to herd immunity and could play a key role in getting back to normal. We consider that the recovered people are absolutely virus negative, produce antibodies to defend themselves against the virus and are able to interact with susceptible and infected people. We also assume that the recovered people may be infected when they come in contact with the infected people. Moreover, the control parameters are taken as triangular intuitionistic fuzzy numbers to incorporate the uncertainty. The model is converted to intuitionistic fuzzy model and analysed the boundedness, local and global stability, calculated the equilibrium points and basic reproduction number. We also studied optimal control of the model. The MATLAB codes are implemented to solve the system of ordinary nonlinear differential equations and to predict different scenarios for different values of the control parameters involved in the dynamical system. The sensitivities of the control parameters have also been performed to forecast the behaviour of the virus.

In this study, we present a new method for assessing the directional coupling between drive and noisy response chaotic oscillators with significant frequency mismatch. This method is based on the phase-amplitude coupling between the drive oscillator and response oscillator. Specifically, the weak phase-amplitude coupling effect is calculated as the magnitude squared coherence between the intrinsic frequency of the drive oscillator and the oscillation frequency of the dynamics of the bispectral energy of the response oscillator. The bispectral energy of the response oscillator at the selected combination of the frequencies is estimated using the Blackman-Harris sliding window with a sharp peak at the central frequency, and the magnitude squared coherence is calculated based on the moving block-bootstrap (MBB) technique. Application of the proposed approach to master-slave Rössler systems and coupled Van der Pol oscillators with a frequency ratio of $ {\sim}$1:4 show that the new algorithm is well-suited for assessing the presence of coupling with a priori known direction between master and noisy slave oscillators at signal-to-noise ratios (SNR) up to 12 dB, especially in the case of weak and moderate coupling.

While cure is rare, a systemic and proper treatment can prolong the lives of HIV positive individuals and keep them healthy. To reduce toxicity and minimize treatment costs an optimal treatment program is critically important. Here we present and study a mathematical model to find an optimal treatment strategy, target-oriented-treatent (TOT), against HIV infections. We highlight the optimization case when a given CD4$^+$ T cells are required in a treatment period. The model demonstrates the viral dynamics in the presence of an immune boosting nutrition and an antiretroviral drug. It is found that the infected virus particles can be made negligible if the label of CD4$^+$T cells remains non-decreasing via treatment. Unlike other studies, boundary conditions are applied on the state variables to find the optimal solution in these regard. The results are confirmed by maximizing the objective functional via Pontryagin's Maximum Principle.

The problem of illicit drug use with its associated hazardous effects on social behaviour and human population dynamics require an intervention. In this study, a compartmental deterministic model for the dynamics of illicit drug use student population is proposed and analysed. The illicit drug use threshold, $\mathcal R_0$, associated with the model is computed. The Centre Manifold Theorem is used to establish bifurcation phenomenon. By constructing suitable Lyapunov functions, the global asymptotic stability of the illicit drug-free and illicit drug-present equilibria exhibits by the model is established. It is shown that the illicit drug-free equilibrium is globally asymptotically stable if $\mathcal R_0<1$, otherwise the illicit drug-present equilibrium is globally asymptotically stable. Sensitivity analysis is performed to gain insight into how the model parameters contribute to the dynamics of illicit drug use student population.

This paper proposes a computationally technique for simulating solutions of nonlinear nonlocal stochastic dynamical systems driven by variable-order fractional Brownian motion with Hurst index. The value of the Hurst index depends on time $t$ belong to interval $(\frac{1}{2},1)$. The proposed technique is adopted quadratic interpolation for fractional-order derivative. Moreover, it is exploited in the discussion of fractional stochastic financial and pendulum dynamical systems. The proficiency of the presented technique is confirmed by using of investigating statistical indicators for the stochastic approximations for various values of fractional order parameters.

In this paper theoretical analysis of blood flow with heat along with mass transport under the influence of time-dependent magnetism with Casson fluid flow intensity has been elucidated. The unsteady nonlinear partial differential equations (PDEs) of blood flow considers time-dependent stretching velocity, the energy equation also accounts time-dependent temperature of the vessel wall and the concentration equation includes time-dependent blood concentration. The governing nonlinear equation of motion, energy as well as concentration were simplified into ordinary differential equations (ODE) using the approach of similarity transformations. Runge-Kutta Fehlberg method alongside shooting procedure was later used to solve the set of ODE. The effect of physical parameters viz., Casson fluid, permeability, unsteadiness, Prandtl number, Hartmann number, thermal radiation parameter, chemical reaction parameter, and Schmidt number on flow variables viz., velocity flow of blood in the vessel, temperature and concentration of blood has been analyzed and discussed graphically. From the simulation study, the following important results are obtained: velocity of blood flow decreases with both increment of magnetic and unsteadiness parameter. The temperature of the blood decreases in the vessel wall as Prandtl number becomes large. The concentration of the blood degenerates as time-dependent chemical reaction parameter together with the Schmidt number increases.This study is unique because it explores unsteady blood flow extended to a penetrable slim vessel with the impact of time-dependent magnetism. We considered the blood flow in a slim vessel; hence the flow is two dimensional. We have taken the blood vessel to be stretched with a velocity and blood concentration. To the very best of our understanding, the problem of this type has not been elucidated in the past.

In this paper we propose a new sufficient condition of the bifurcation for nonlinear autonomous differential equations with delay in three dimension, which just depends in coefficient on the characteristic equation, who can verify them analytically at the level of the application to a dynamics system. For the application of these conditions we propose a SEIR epidemic model with delay and nonlinear incidence rate. The resulting model has two possible equilibria. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of a disease-free equilibrium is established. Finally we affirm the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. In the end, the numerical simulations are proposed to illustrate our results.

In this paper, a (1+3)-dimensional Boiti-Leon-Manna-Pempinelli equation is investigated. Exact solutions are acquired using the Lie symmetry method and multiple exp-function method via symbolic computation. In addition to exact solutions, we also present conservation laws and their physical ramifications are also discussed. The obtained results enlarge the known category of solutions of the (1+3)-dimensional Boiti-Leon-Manna-Pempinelli equation.

In this paper, the nonlinear behavior of an eleven-term 4-dimensional hyperchaotic Lorenz-type dynamic system is studied. The dynamic response of a hyperchaotic model is investigated. The phase portrait and Lyapunov exponent of the 4-D system are discussed. The phase portrait of the presented system shows the behavior which is like the Lorenz system's phase portrait. The nonlinear model is proved to be hyperchaotic since it has two positive Lyapunov exponents. The main goal of this research is to depict bifurcation and Poincare maps so as to investigate the occurrence of the periodic behavior, period-doubling, crisis, and chaotic motion. Therefore, bifurcation analysis is shown that by increasing the $\theta _2$, the behavior of the system will change from periodic into chaotic and vice versa. Also, the period-doubling, and crisis happened by changing the bifurcation parameter in a specific range. Finally, the Poincare map indicates that the chaotic motion appears when $\theta _2$ is 0.17.

The problem of illicit drug use with its associated hazardous effects on social behavior and human population dynamics require an intervention. This paper presents the optimal intervention strategy for controlling the menace of illicit drug use at population levels. Two time-dependent intervention measures are incorporated into a nonlinear system of differential equations modelling illicit drug use population dynamics. The first measure aims to prevent the influence of drug users on non drug users while the second intervention measure is concerned with behavioral therapy. The control problem is analyzed using a popular Pontryagin's maximum principle to find the necessary condition for optimum solution. Assessment of single implementation of each the two measures and combination of both intervention measures are conducted to investigate the dynamic behaviour of illicit drug users in the population. Moreover, cost-effectiveness analysis based on incremental cost-effectiveness ratio is conducted to find the most cost effective intervention measure capable of averting good numbers of illicit drug users in the population at lowest cost. The results show that combination of both control variables reduce the spread of drug abuse mostly. However, single implementation of preventive control is the most cost-effective intervention.

In this paper, a delayed differential equation model for HIV infection of CD4$^+$ T cells with full logistic proliferation of healthy CD4$^+$ T cells, cure rate of HIV infected CD4$^+$ T cells, and fusion effect is investigated. At first, we have proved the basic properties of the model like non-negativity and uniform boundedness of solutions. It is found that our model exhibits two equilibria: HIV infection-free equilibrium and HIV-infected equilibrium point, later is obtained whenever basic reproduction number is greater than one. The stability criteria for both equilibria are investigated and basic reproduction number is found to be a threshold parameter. Our study indicates that delay can destabilize HIV infected equilibrium and lead to occurrence of Hopf bifurcation. Also, we have calculated the length of delay to preserve stability by using Nyquist criterion. Moreover, explicit formulae are derived using the normal form theory and center manifold argument in order to determine the direction, stability, and period of periodic bifurcating solutions. Finally, numerical simulations are done to illustrate the analytical results.

The present study explores the impact of heat generation and thermal radiation on the time-dependent free convective flow of a hybrid nanofluid past a vertically oscillating porous plate.The nonlinear partial differential equations, which control the transport phenomena, are transmuted into dimensionless form with appropriate parameters. The resulting equations with appropriate boundary conditions are solved numerically with the aid of the Galerkin finite element method. The impact of sundry parameters such as Thermal Grashof number (Gr), Radiation Parameter $(N)$, Heat generation Parameter $(Q)$, Time ($t$), Nanoparticle Volume fraction of $Al_{2}O_{3}$ ($\delta _{2} $), Suction Parameter ($\lambda $), Nusselt number ($Nu$), Skin friction coefficient ($C_{f} $) that controls the flow velocity and temperature distributions are evaluated extensively with the aid of graphs and tables. Moreover, a comparison between the regular Silicon dioxide water $(SiO_{2} -H_{2} O)$ nanoliquid and Silicon dioxide/Aluminum oxide water $(SiO_{2} -Al_{2} O_{3} -H_{2} O)$ hybrid nanoliquid is provided in skin friction and heat transfer enhancement. The present numerical solution is in good agreement with the analytical solution for the special cases of the problem. The current investigation is of immense apropos in the cooling of nuclear reactors, microelectronics, electromechanical systems, and drug delivery in chemotherapy.