Narrow Results

This paper presents a comprehensive study of disease spreading dynamics through the application of a nonlinear fractional order epidemic SEIRS model. By incorporating the Crowley–Martin type functional response and a saturated treatment function, the model effectively captures the intricacies of real-world epidemics. Our research establishes the existence, uniqueness, non-negativity and boundedness of the solution, while also investigating the model’s fundamental reproduction number. Additionally, we conduct a thorough analysis of the specific conditions governing the local and global stability of the model’s equilibriums. A notable observation is the variation of the reproduction number with the fractional-order $\alpha$, which represents a memory effect on individuals’ dynamic behavior and reveals the influence of interactions between compartments. To validate these theoretical findings, we employ numerical simulations using Matlab, demonstrating that inhibition measures for susceptibles and the saturated treatment parameters play a pivotal role in determining the disease state. Specifically, we observe that as these parameter values increase, the transition from endemic equilibrium to disease-free equilibrium occurs.

In the Robe’s restricted three-body problem, we have considered the motion of the test particle which is moving inside the outermost layer of the heterogeneous body. This heterogeneous body has N layers with different densities and is filled with viscous fluid. The test particle which is taken as the third (or infinitesimal) body is moving under the influence of the heterogeneous body (primary) and point mass (secondary). We are motivated from Ansari,¹ where the linear stability and other important properties of this specific model have been discussed. In this paper, we have extended their work and discussed here the nonlinear stability of the non-collinear stationary points $L_4$ and $L_5$. Therefore, using the Arnold–Moser theorem (Kolmogorov–Arnold–Moser theory), we have done our analysis on the nonlinear stability and obtained significant results.

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.

The Newton–Raphson basins of attraction, associated with the libration points (attractors), are revealed in the generalized Hill problem. The parametric variation of the position and the linear stability of the equilibrium points is determined, when the value of the perturbation parameter $𝜖$ varies. The multivariate Newton–Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the perturbation parameter on the geometry as well as of the basin entropy of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.

In this paper, we extend the basic equilateral four-body problem by introducing the effect of radiation pressure, Poynting–Robertson drag, and solar wind drag. In our setup, three primaries lie at the vertices of an equilateral triangle and move in circular orbits around their common center of mass. Here, one of the primaries is a radiating body and the fourth body (whose mass is negligible) does not affect the motion of the primaries. We show that the existence and the number of equilibrium points of the problem depend on the mass parameters and radiation factor. Consequently, the allowed regions of motion, the regions of the basins of convergence for the equilibrium points, and the basin entropy will also depend on these parameters. The present dynamical model is analyzed for three combinations of mass for the primaries: equal masses, two equal masses, different masses. As the main results, we find that in all cases the libration points are unstable if the radiation factor is larger than 0.01 and hence able to destroy the stability of the libration points in the restricted four-body problem composed by the Sun, Jupiter, Trojan asteroid and a test (dust) particle. Also, we conclude that the number of fixed points decreases with the increase of the radiation factor.

In this paper, we have developed a five-compartmental epidemic model with Chlamydia infection. We have divided the total population into five classes, namely susceptible, exposed, infective in asymptomatic phase, infective in symptomatic phase and recovered class. The basic reproduction number (R₀) is calculated using the next-generation matrix method. The stability analysis of the model shows that the system is locally asymptotically stable at the disease-free equilibrium (DFE) E₀ when R₀ < 1. When R₀ > 1, an endemic equilibrium E₁ exists and the system becomes locally asymptotically stable at E₁ under some conditions. We have also discussed the Chlamydia epidemic model with two treatment controls. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of treatment. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB, which show the reliability of our model from the practical point of view. Epidemiological implications of our analytical findings are addressed critically.

In this paper we introduce two simple versions of the hat game for three players and a generalization for n players. We compute the extensive version of the game and all the equilibrium and friendly equilibrium points as a simple example of the structure function.

In this paper we study and compute E-points in an explicit way for a special kind of diagonal games with 3k + 1, 4k + 3 and sk + 1 players where 1 ≤ s ≤ k.

Macrochaetes (large bristles) are sensor organs of the Drosophila peripheral nervous system with a function of mechanoreceptors. An adult mechanoreceptor comprises four specialized cells: shaft (trichogen), socket (tormogen), neuron, and glial cell (thecogen). All these cells originate from a single cell, the so-called sensor organ precursor (SOP) cell. Separation of the SOP cell from the encompassing cells of the imaginal disc initiates a multistage process of sensory organ development. A characteristic feature of the SOP cell is the highest amount of the proneural proteins AS-C as compared with the encompassing ectodermal cells. The accumulation of proneural proteins and maintenance of their amount in the SOP cell at a necessary level is provided by the gene network with the achaete–scute gene complex (AS-C) as its key component. The activity of this complex is controlled by the central regulatory circuit (CRC). The CRC comprises the genes hairy, senseless (sens), charlatan (chn), scratch (scrt), daughterless (da), extramacrochaete (emc), and groucho (gro), coding for the transcription factors involved in the system of direct links and feedbacks and implementation of activation–repression relationships between the CRC components. The gene phyllopod (phyl), involved in degradation of the AS-C proteins, is also associated with the CRC functioning. In this paper, we propose a mathematical model for the CRC functioning as a regulator of the amount of proneural AS-C proteins in the SOP cell taking into account their degradation. The modeling has demonstrated that a change in the amount of proneural proteins in the SOP cell is stepwise rather than strictly monotonic. This prediction can be tested experimentally.

We analyze the existence of equilibrium points for a particle or dust grain in the framework of unperturbed and perturbed Robe’s motion. This particle is moving in a spherical nebula consisting of a homogeneous incompressible fluid, which is considered as the primary body. The second primary body creates the modified Newtonian potential. The perturbed mean motion and equations of motion are found. The equilibrium points (i.e. collinear, noncollinear and out–of–plane points), along with the required conditions of their existence are also analyzed. We emphasize that this analysis can be used to study the oscillations of the Earth’s core under the attraction of the Moon and it is also applicable to study the motion of underwater vehicles.

In this paper, a fuzzy mathematical model has been developed by considering forest biomass, human population and technological effort for the conservation of forest biomass as separate compartments. We have assumed that the forest biomass and human population grows logistically. We have considered that forest biomass decreases due to industrialization, food, shelter, etc., for humans. For the conservation of forest biomass, some modern technological efforts have been used in this model. Also, time delay of use of modern technological effort for the conservation of forest biomass has been considered on forest biomass. According to the assumptions, a fuzzy mathematical model on forest biomass is formulated. Next we have determined different possible equilibrium points. Also, the stability of our proposed system around these equilibrium points has been discussed. Finally, some numerical simulation results have been presented for better understanding of our proposed mathematical model.

This work presents the nonlinear dynamical system of continuous fermentation from glycerol to 1,3-propanediol. The impulsive control scheme of continuous culture is introduced. By employing impulsive control, Lyapunov's method and comparison technique, sufficient conditions are established for the asymptotical stability and synchronization of the dynamical system of continuous fermentation. The upper bound of the impulse interval is also estimated. An example will illustrate the effectiveness of the results in Sec. 4.

In this paper, we have introduced a six-compartmental epidemic model with hand, foot and mouth disease (HFMD) infection. The total population is divided into six subclasses, namely susceptible, exposed, infective in asymptomatic phase, infective in symptomatic phase, quarantined and recovered class. Some basic properties such as boundedness and non-negativity of solutions are discussed. The basic reproduction number ($R_0$) of the system is obtained using next generation matrix method. Then the deterministic dynamical behaviors of the system are studied. Our study includes the existence and stability analysis of equilibrium points of the system. The sensitivity analysis of our system helps us to find out the parameters of greater interest. Next, we deal with the epidemic model with three controls (two treatment controls with quarantine control). We show that there exists an optimal control, which is effective in controlling the disease outbreak in a cost effective way. Numerical simulation is presented with the help of MATLAB, which shows the reliability of our model from the practical point of view.

In the interaction between plants and herbivores that live in the same ecosystem, understanding the conditions in which co-existence equilibrium occurs answers a major question in Ecology. In this interaction, plants serve as food for herbivores on the food chain. Then the livelihood of herbivores highly depends on the availability of food, in this case the availability of plants. Moreover, the abundance of the plant density alone does not guarantee the non-extinction of the herbivore population as they are assumed to reproduce sexually. With this motivation, in this paper a predator–prey mathematical model is reformulated such that the death rate of the herbivore population is dependent on the plant density and their emergence is also governed by the Allee effect. Using the mathematical theory of dynamical system, threshold conditions are obtained for the non-extinction of the herbivore population and a trapping region is obtained to ensure co-existence of the population. Moreover, it has been shown that the dynamics of the population is significantly sensitive to the feeding rate and the harvest rate of the herbivore population.

Towards the end of 2019, the world witnessed the outbreak of Severe Acute Respiratory Syndrome Coronavirus-2 (COVID-19), a new strain of coronavirus that was unidentified in humans previously. In this paper, a new fractional-order Susceptible–Exposed–Infected–Hospitalized–Recovered (SEIHR) model is formulated for COVID-19, where the population is infected due to human transmission. The fractional-order discrete version of the model is obtained by the process of discretization and the basic reproductive number is calculated with the next-generation matrix approach. All equilibrium points related to the disease transmission model are then computed. Further, sufficient conditions to investigate all possible equilibria of the model are established in terms of the basic reproduction number (local stability) and are supported with time series, phase portraits and bifurcation diagrams. Finally, numerical simulations are provided to demonstrate the theoretical findings.

Internet worms have received a lot of attention because of their massive hazards to the internet. As the internet worms propagate quickly, therefore, the automatic control on the internet is required. In this paper, we present a new epidemic model called susceptible-breaking out-quarantine-treatment-recovered (SBQTR) model which integrates dynamic quarantine and treatment approaches to control the spread of internet worms. By using the SBQTR model, we can calculate the basic reproduction number $R_0$, which determines the existence of worms. The model has two equilibrium states: virus-free equilibrium (VFE) and viral equilibrium (VE). The stability analysis of both the equilibrium points is carried out and impact of treatment is examined under various parametric values. The sensitivity analysis of model is carried out to determine the impact of model parameters on virus transmission. Furthermore, we established nonstandard finite difference (NSFD) schemes and Runge–Kutta algorithm of order 4 (RK-4) for SBQTR model. The stated NSFD technique retains all of the significant properties of system (1), whereas RK-4 mechanism gives negative solutions and doesn’t converge to the equilibrium points of system (1). Numerical results together with comparison for various values of step size ($h$) are available. Bifurcation value of infection coefficient is also investigated. It is shown that an appropriate treatment is essential to control the spread of worms in computer network.

Media plays a vital role in disseminating any crucial information to the society. This helps the unaware population of the society to gain awareness regarding a particular information. But many times it is seen that, even though an information is being transmitted through the print or electronic media for a very long period of time, many youngsters although aware about the negative aspects of the information, act as unaware and sometimes face a fatal result as in the case of the Blue Whale Game.

In this paper, we have proposed a mathematical model depicting the effect of transmitting an information by media, particularly “Blue Whale Game”. We have concluded with the help of some simple calculations and figures, that because of the prolonged repeated transmission by the media regarding the blue whale game, curiousness got generated among the young generation and they tried to know more and more about the game, which in turn made them follow the game as designed by the game developer and in many cases turned out to be lethal.

The Lagrangian points for the Sun-Earth system are considered due to their special importance for the scientific community for the design of space missions. The location of the Lagrangian points with the trajectories and stability regions are computed numerically for the initial conditions very close to the points. The influence of belt, effect of radiation pressure due to the Sun and oblateness effect of second primary(finite body Earth) is presented for various values of parameters. The collinear points are asymptotically stable within a specific interval of time t correspond to the values of parameters and initial conditions.

We propose a general model with n parallel food chains through the stage structured maturation time delay, which can cover most of the prey-predator models in the literature. We discuss some basic dynamical properties of the system with single or multiple patches and with general or some particular functional responses, including the existence of equilibrium points and their local and global stabilities. Numerical simulations are given to compliment the theoretical analysis.