A mathematical model of dengue diseases transmission will be discussed in this paper. Various interventions, such as vaccination of adults and newborns, the use of insecticides or fumigation, and also the enforcement of mechanical controls, will be considered when analyzing the best intervention for controlling the spread of dengue. From model analysis, we find three types of equilibrium points which will be built upon the dengue model. In this paper, these points are the mosquito-free equilibrium, disease-free equilibrium (with and without vaccinated compartment), and endemic equilibrium. Basic reproduction number as an endemic indicator has been found analytically. Based on analytical and numerical analysis, insecticide treatment, adult vaccine, and enforcement of mechanical control are the most significant interventions in reducing the spread of dengue disease infection caused by mosquitoes rather than larvicide treatment and vaccination of newborns. From short- and long-term simulation, we find that insecticide treatment is the best strategy to control dengue. We also find that, with periodic intervention, the result is not much significantly different with constant intervention based on reduced number of the infected human population. Therefore, with budget limitations, periodic intervention of insecticide strategy is a good alternative to reduce the spread of dengue.

This paper aims to connect the bridge between analytical results and the use of the computer for numerical simulations in economics. We address the analytical properties of a simple dynamic aggregate demand and aggregate supply (AD-AS) model and solve it numerically. The model undergoes a bifurcation as its steady state smoothly interchanges stability depending on the relationship between the impact of real interest rate on demand for liquidity and how fast agents revise their expectations on inflation. Using code embedded into a unique function in Matlab, we plot the numerical solutions of the model and simulate different dynamic adjustments using different parameter values. The same function also accommodates the analysis of the impacts of fiscal and monetary policy and supply side shocks on the steady state and the transition dynamics of the model.

Malaria is one of the three most dangerous infectious diseases worldwide (along with HIV/AIDS and tuberculosis). In this paper we compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in malaria transmission dynamics. Relationships between the basic reproduction number for malaria transmission dynamics between humans and mosquitoes and the extinction thresholds of corresponding continuous-time Markov chain models are derived under certain assumptions. The stochastic model is formulated using the continuous-time discrete state Galton-Watson branching process (CTDSGWbp). The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction from stochastic models contribute crucial knowledge on disease control and elimination and mitigation of infectious diseases. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that malaria outbreak is more likely if the disease is introduced by infected mosquitoes as opposed to infected humans. These insights demonstrate the importance of a policy or intervention focusing on controlling the infected mosquito population if the control of malaria is to be realized.

We consider an insurance company whose reserves dynamics follow a diffusion-perturbed risk model. To reduce its risk, the company chooses to reinsure using proportional or excess-of-loss reinsurance. Using the Hamilton-Jacobi-Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation (VIDE) which we transform into a linear Volterra integral equation (VIE) of the second kind. We then proceed to solve this linear VIE numerically using the block-by-block method for the optimal reinsurance policy that minimizes the ultimate ruin probability for the chosen parameters. Numerical examples with both light- and heavy-tailed distributions are given. The results show that proportional reinsurance increases the survival of the company in both light- and heavy-tailed distributions for the Cramér-Lundberg and diffusion-perturbed models.

Streptococcus pneumoniae is one of the leading causes of serious morbidity and mortality worldwide, especially in young children and the elderly. In this study, a model of the spread and control of bacterial pneumonia under public health interventions that involve treatment and vaccination is formulated. It is found out that the model exhibits the disease-free and endemic equilibria. The disease-free equilibrium is stable if and only if the basic reproduction number and the disease will be wiped out of the population. For ${\mathcal{R}}_{\mathrm{0}}\ge \mathrm{1},$ the endemic equilibrium is globally stable and the disease persists. We infer the effect of these interventions on the dynamics of the pneumonia through sensitivity analysis on the effective reproduction number ${\mathcal{R}}_e,$ from which it is revealed that treatment and vaccination interventions combined can eradicate pneumonia infection. Numerical simulation to illustrate the analytical results and establish the long term behavior of the disease is done. The impact of pneumonia infection control strategies is investigated. It is revealed that, with treatment and vaccination interventions combined, pneumonia can be wiped out. However, with treatment intervention alone, pneumonia persists in the population.

The disproportionality in the problem parameters of the convection-diffusion-reaction equation may lead to the formation of layer structures in some parts of the problem domain which are difficult to resolve by the standard numerical algorithms. Therefore the use of a stabilized numerical method is inevitable. In this work, we employ and compare three classical stabilized finite element formulations, namely, the Streamline-Upwind Petrov-Galerkin (SUPG), Galerkin/Least-Squares (GLS), and Subgrid Scale (SGS) methods, and a recent Link-Cutting Bubble (LCB) strategy proposed by Brezzi and his coworkers for the numerical solution of the convection-diffusion-reaction equation, especially in the case of small diffusion. On the other hand, we also consider the pseudo residual-free bubble (PRFB) method as another alternative that is based on enlarging the finite element space by a set of appropriate enriching functions. We compare the performances of these stabilized methods on several benchmark problems. Numerical experiments show that the proposed methods are comparable and display good performance, especially in the convection-dominated regime. However, as the problem turns into reaction-dominated case, the PRFB method is slightly better than the other well-known and extensively used stabilized finite element formulations as they start to exhibit oscillations.

The Equivalent Linearization Method (ELM) with a weighted averaging is applied to analyze five undamped oscillator systems with nonlinearities. The results obtained via this method are compared with the ones achieved by Parameterized Perturbation Method (PPM), Min–Max Approach (MMA), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Energy Balance Method (EBM), Harmonic Balance Method (HBM), 4th-Order Runge-Kutta Method, and the exact ones. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully applied to a lot of practical engineering and physical problems.

Every minute counts in an event of fire evacuation where evacuees need to make immediate routing decisions in a condition of low visibility, low environmental familiarity, and high anxiety. However, the existing fire evacuation routing models using various algorithm such as ant colony optimization or particle swarm optimization can neither properly interpret the delay caused by congestion during evacuation nor determine the best layout of emergency exit guidance signs; thus bee colony optimization is expected to solve the problem. This study aims to develop a fire evacuation routing model “Bee-Fire” using artificial bee colony optimization (BCO) and to test the routing model through a simulation run. Bee-Fire is able to find the optimal fire evacuation routing solutions; thus not only the clearance time but also the total evacuation time can be reduced. Simulation shows that Bee-Fire could save 10.12% clearance time and 15.41% total evacuation time; thus the congestion during the evacuation process could be effectively avoided and thus the evacuation becomes more systematic and efficient.

We employ the method of stochastic optimal control to derive the optimal investment strategy for maximizing an expected exponential utility of a commercial bank’s capital at some future date $T>\mathrm{0}$. In addition, we derive a multiperiod deposit insurance (DI) pricing model that incorporates the explicit solution of the optimal control problem and an asset value reset rule comparable to the typical practice of insolvency resolution by insuring agencies. By way of numerical simulations, we study the effects of changes in the DI coverage horizon, the risk associated with the asset portfolio of the bank, and the bank’s initial leverage level (deposit-to-asset ratio) on the DI premium while the optimal investment strategy is followed.

Given a graph, its $2$-core is the maximal subgraph of $G$ without vertices of degree $\mathrm{1}$. A $\mathrm{2}$-path in a connected graph is a simple path in its $\mathrm{2}$-core such that all vertices in the path have degree $\mathrm{2}$, except the endpoints which have degree $⩾\mathrm{3}$. Consider the Erdős-Rényi random graph $\mathbb{G}(n,M)$ built with $n$ vertices and $M$ edges uniformly randomly chosen from the set of $\left(\begin{array}{c}n\\ 2\end{array}\right)$ edges. Let ${\xi }_{n,M}$ be the maximum $\mathrm{2}$-path length of $\mathbb{G}(n,M)$. In this paper, we determine that there exists a constant $c(\lambda )$ such that $\mathbb{E}\left({\xi }_{n,\left(n/\mathrm{2}\right)\left(\mathrm{1}+\lambda n^{-\mathrm{1}/\mathrm{3}}\right)}\right)~c(\lambda )n^{\mathrm{1}/\mathrm{3}},\hspace{0.17em}\hspace{0.17em}\mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{y}\hspace{0.17em}\hspace{0.17em}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}\hspace{0.17em}\hspace{0.17em}\lambda .$ This parameter is studied through the use of generating functions and complex analysis.

Teaching-learning-based optimization (TLBO) is a population-based metaheuristic search algorithm inspired by the teaching and learning process in a classroom. It has been successfully applied to many scientific and engineering applications in the past few years. In the basic TLBO and most of its variants, all the learners have the same probability of getting knowledge from others. However, in the real world, learners are different, and each learner’s learning enthusiasm is not the same, resulting in different probabilities of acquiring knowledge. Motivated by this phenomenon, this study introduces a learning enthusiasm mechanism into the basic TLBO and proposes a learning enthusiasm based TLBO (LebTLBO). In the LebTLBO, learners with good grades have high learning enthusiasm, and they have large probabilities of acquiring knowledge from others; by contrast, learners with bad grades have low learning enthusiasm, and they have relative small probabilities of acquiring knowledge from others. In addition, a poor student tutoring phase is introduced to improve the quality of the poor learners. The proposed method is evaluated on the CEC2014 benchmark functions, and the computational results demonstrate that it offers promising results compared with other efficient TLBO and non-TLBO algorithms. Finally, LebTLBO is applied to solve three optimal control problems in chemical engineering, and the competitive results show its potential for real-world problems.

Some methodological approaches based on generalized polynomial chaos for linear differential equations with random parameters following various types of distribution laws are proposed. Mainly, an internal random coefficients method ‘IRCM’ is elaborated for a large number of random parameters. A procedure to build a new polynomial chaos basis and a connection between the one-dimensional and multidimensional polynomials are developed. This allows handling easily random parameters with various laws. A compact matrix formulation is given and the required matrices and scalar products are explicitly presented. For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator and right-hand sides depending only on one random parameter. This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution. The conditional expectation method is also elaborated for reference solutions as well as the Monte-Carlo procedure. The applicability and effectiveness of the developed methods are demonstrated by some numerical examples.

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.

This paper explores aspects of two-sided similarity modeling using cylindrical geometry for radiating shock waves embedded in a medium with a magnetic field. Two-sided similarity solution techniques may be used to link states influenced by long range near instantaneous fields that continually modify the pre- and postshock zones. Emergent radiation scaling relations are immediately available from consistent homologies. For both small angle and large angle measurements, an approximate analytic technique in the vicinity of luminous fronts together with the high symmetry implications delineated in Lemma provides direct access to the homology parameters. The parameters obtained using this process can augment the constraint relations and contribute to establishing relevant similarity homologies.

Contraction theory regards the convergence between two arbitrary system trajectories. In this article we have introduced partial contraction theory as an extension of contraction theory to analyze coupled identical fractional order systems. It can, also, be applied to study the synchronization phenomenon in networks of various structures and with arbitrary number of systems. We have used partial contraction theory to derive exact and global results on synchronization and antisynchronization of fractional order systems.

Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type (MRKT) methods for solving $y^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\left(x\right)=f\left(x,y,y^{\mathrm{\prime }}\right)$ are derived in this paper. These methods are constructed which exactly integrate initial value problems whose solutions are linear combinations of the set functions $e^{\omega x}$ and $e^{-\omega x}$ for exponentially fitted and $\sin \left(\omega x\right)$ and $\cos \left(\omega x\right)$ for trigonometrically fitted with $\omega \in R$ being the principal frequency of the problem and the frequency will be used to raise the accuracy of the methods. The new four-stage fifth-order exponentially fitted and trigonometrically fitted explicit MRKT methods are called EFMRKT5 and TFMRKT5, respectively, for solving initial value problems whose solutions involve exponential or trigonometric functions. The numerical results indicate that the new exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods are more efficient than existing methods in the literature.

Guinea worm disease is one of the neglected tropical diseases that is on the verge of elimination. Currently the disease is endemic in four countries, namely, Ethiopia, Mali, Chad, and South Sudan. Prior studies have demonstrated that climate factors and limited access to safe drinking water have a significant impact on transmission and control of Guinea worm disease. In this paper, we present a new mathematical model to understand the transmission dynamics of Guinea worm disease in South Sudan. The model incorporates seasonal variations, educational campaigns, and spatial heterogeneity. Both qualitative and quantitative analysis of the model have been carried out. Utilizing Guinea worm disease surveillance data of South Sudan (2007-2013) we estimate the model parameters. Meanwhile, we perform an optimal control study to evaluate the implications of vector control on long-term Guinea worm infection dynamics. Our results demonstrate that vector control could play a significant role on Guinea worm disease eradication.

We consider a stochastic SIVR (susceptible-infected-vaccinated-recovered) epidemic model with imperfect vaccine. First, we obtain critical condition under which the disease is persistent in the mean. Second, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Third, we study the extinction of the disease. Finally, numerical simulations are given to support the analytical results.

Quadratic programming with interval variables is developed from quadratic programming with interval coefficients to obtain optimum solution in interval form, both the optimum point and optimum value. In this paper, a two-level programming approach is used to solve quadratic programming with interval variables. Procedure of two-level programming is transforming the quadratic programming model with interval variables into a pair of classical quadratic programming models, namely, the best optimum and worst optimum problems. The procedure to solve the best and worst optimum problems is also constructed to obtain optimum solution in interval form.

We provide a theoretical framework to study how criminal behaviors can be treated as an infectious phenomenon. There are two infectious diseases like models that mimic the role of convicted criminals in contaminating individuals not yet engaged in the criminal career. Equilibrium analyses of each model are studied in detail. The models proposed in this work include the social, economic, personal, and pressure from peers aspects that can, theoretically, determine the probability with which a susceptible individual with criminal propensity engages in a criminal career. These crime-inducing parameters are treated mathematically and their inclusion in the model aims to help policy-makers design crime control strategies. We propose, to the best of our knowledge by the first time in quantitative criminology, the existence of thresholds for the stability of crime-endemic equilibrium which are the equivalent to the “basic reproduction number” widely used in the mathematical epidemiology literature. Both models presented the phenomena of backward bifurcation and breaking-point when the contact rates are chosen as bifurcation parameters. The finding of backward bifurcation in both models implies that there is an endemic equilibrium of criminality even when the threshold parameter for contagion is below unit, which, in turn, implies that control strategies are more difficult to achieve considerable impact on crime control.

This paper deals with a nonlinear differential equation, by using the fixed point theory. The existence of the periodic solution of the nonlinear differential equation is obtained; these results are new.

We investigate an efficient numerical method for solving a class of nonlinear Volterra integro-differential equations, which is a combination of the parametric iteration method and the spectral collocation method. The implementation of the modified method is demonstrated by solving several nonlinear Volterra integro-differential equations. The results reveal that the developed method is easy to implement and avoids the additional computational work. Furthermore, the method is a promising approximate tool to solve this class of nonlinear equations and provides us with a convenient way to control and modify the convergence rate of the solution.

The mathematical model for describing internal waves of the ocean is derived from the assumption of ideal fluid; i.e., the fluid is incompressible and inviscid. These internal waves are generated through the interaction between the tidal currents and the basic topography of the fluid. Basically the mathematical model of the internal wave problem of the ocean is a system of nonlinear partial differential equations (PDEs). In this paper, the analytical approach used to solve nonlinear PDE is the Homotopy Analysis Method (HAM). HAM can be applied to determine the resolution of almost any internal wave problem involving tidal forces. The use of HAM in the solution to basic fluid equations is efficient and simple, since it involves only modest calculations using the common integral.

We show that an order $m$ dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order $m$ dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with $n\ge \mathrm{3}$ and propose some problems for further research.

Researchers using ordinary differential equations to model phenomena face two main challenges among others: implementing the appropriate model and optimizing the parameters of the selected model. The latter often proves difficult or computationally expensive. Here, we implement Particle Swarm Optimization, which draws inspiration from the optimizing behavior of insect swarms in nature, as it is a simple and efficient method for fitting models to data. We demonstrate its efficacy by showing that it outstrips evolutionary computing methods previously used to analyze an epidemic model.

The intention of the present paper is to establish an approximation method to the limiting power functions of tests conducted based on Kolmogorov-Smirnov and Cramér-von Mises functionals of set-indexed partial sums of multivariate regression residuals. The limiting powers appear as vectorial boundary crossing probabilities. Their upper and lower bounds are derived by extending some existing results for shifted univariate Gaussian process documented in the literatures. The application of multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions is demonstrated. The rate of decay of the power function to a presigned value $\alpha$ is also studied. Our consideration is mainly for the trend plus signal model including multivariate set-indexed Brownian sheet and pillow. The simulation shows that the approach is useful for analyzing the performance of the test.

Spanning trees have been widely investigated in many aspects of mathematics: theoretical computer science, combinatorics, so on. An important issue is to compute the number of these spanning trees. This number remains a challenge, particularly for large and complex networks. As a model of complex networks, we study two families of generalized small-world networks, namely, the Small-World Exponential and the Koch networks, by changing the size and the dimension of the cyclic subgraphs. We introduce their construction and their structural properties which are built in an iterative way. We propose a decomposition method for counting their number of spanning trees and we obtain the exact formulas, which are then verified by numerical simulations. From this number, we find their spanning tree entropy, which is lower than that of the other networks having the same average degree. This entropy allows quantifying the robustness of the networks and characterizing their structures.

An $L(\mathrm{2,1})$-coloring of a simple connected graph $G$ is an assignment $f$ of nonnegative integers to the vertices of $G$ such that $\left|f\left(u\right)-f\left(v\right)\right|⩾\mathrm{2}$ if $d(u,v)=\mathrm{1}$ and $\left|f\left(u\right)-f\left(v\right)\right|⩾\mathrm{1}$ if $d(u,v)=\mathrm{2}$ for all $u,v\in V(G)$, where $d(u,v)$ denotes the distance between $u$ and $v$ in $G$. The span of $f$ is the maximum color assigned by $f$. The span of a graph $G$, denoted by $\lambda (G)$, is the minimum of span over all $L(\mathrm{2,1})$-colorings on $G$. An $L(\mathrm{2,1})$-coloring of $G$ with span $\lambda (G)$ is called a span coloring of $G$. An $L(\mathrm{2,1})$-coloring $f$ is said to be irreducible if there exists no $L(\mathrm{2,1})$-coloring g such that $g(u)⩽f(u)$ for all $u\in V(G)$ and for some $v\in V(G)$. If $f$ is an $L(\mathrm{2,1})$-coloring with span $k$, then $h\in \left\{\mathrm{0,1},\mathrm{2},\dots ,k\right\}$ is a hole if there is no $v\in V(G)$ such that $f(v)=h$. The maximum number of holes over all irreducible span colorings of $G$ is denoted by $H_{\lambda }(G)$. A tree $T$ with maximum degree $\mathrm{\Delta }$ having span $\mathrm{\Delta }+\mathrm{1}$ is referred to as Type-I tree; otherwise it is Type-II. In this paper, we give a method to construct infinitely many trees with at least one hole from a one-hole tree and infinitely many two-hole trees from a two-hole tree. Also, using the method, we construct infinitely many Type-II trees with maximum number of holes one and two. Further, we give a sufficient condition for a Type-II tree with maximum number of holes zero.

One of the most challenging PDE forms in fluid dynamics namely Burgers equations is solved numerically in this work. Its transient, nonlinear, and coupling structure are carefully treated. The Hermite type of collocation mesh-free method is applied to the spatial terms and the 4^th-order Runge Kutta is adopted to discretize the governing equations in time. The method is applied in conjunction with the Gaussian radial basis function. The effect of viscous force at high Reynolds number up to 1,300 is investigated using the method. For the purpose of validation, a conventional global collocation scheme (also known as “Kansa” method) is applied parallelly. Solutions obtained are validated against the exact solution and also with some other numerical works available in literature when possible.

We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo’s fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts of $T-indicators$ which analyses the economic performance with the presence of memory. The reaction of economic agents due to recurrence identical alteration is minimized by using the modified Caputo’s derivative operator of order $\lambda$ instead of integer order derivative $n$. The two sides of Caputo’s derivative are expressed by a brief time-line. The degree of attenuation is further depressed by involving the modified Caputo’s operator.

An analytical approach on the evaluation of interphase Poisson’s ratio for fibrous composites, consisting of polymeric matrix and unidirectional continuous fibers, is performed. The simulation of the microstructure of the composite was carried out by means of a modified form of Hashin-Rosen cylinder assemblage model. Next, by the use of this three-phase model the authors impose some limitations to the polynomial variation laws which are commonly adopted to approximate the thermomechanical properties of the interphase layer of this type of polymeric composites and then propose an nth-degree polynomial function to approximate the Poisson’s ratio of this layer.

Partial least squares regression (PLS regression) is used as an alternative for ordinary least squares regression in the presence of multicollinearity. This occurrence is common in chemical engineering problems. In addition to the linear form of PLS, there are other versions that are based on a nonlinear approach, such as the quadratic PLS (QPLS2). The difference between QPLS2 and the regular PLS algorithm is the use of quadratic regression instead of OLS regression in the calculations of latent variables. In this paper we propose a robust version of QPLS2 to overcome sensitivity to outliers using the Blocked Adaptive Computationally Efficient Outlier Nominators (BACON) algorithm. Our hybrid method is tested on both real and simulated data.

In this paper, we used Bernstein polynomials to modify the Adomian decomposition method which can be used to solve linear and nonlinear equations. This scheme is tested for four examples from ordinary and partial differential equations; furthermore, the obtained results demonstrate reliability and activity of the proposed technique. This strategy gives a precise and productive system in comparison with other traditional techniques and the arrangements methodology is extremely straightforward and few emphasis prompts high exact solution. The numerical outcomes showed that the acquired estimated solutions were in appropriate concurrence with the correct solution.

The problem of efficiently touring a theme park so as to minimize the amount of time spent in queues is an instance of the Traveling Salesman Problem with Time-Dependent Service Times (TSP-TS). In this paper, we present a mixed-integer linear programming formulation of the TSP-TS and describe a branch-and-cut algorithm based on this model. In addition, we develop a lower bound for the TSP-TS and describe two metaheuristic approaches for obtaining good quality solutions: a genetic algorithm and a tabu search algorithm. Using test instances motivated by actual theme park data, we conduct a computational study to compare the effectiveness of our algorithms.

A modified tanh-coth method with Riccati equation is used to construct several explicit solutions of (3+1)-dimensional Kudryashov-Sinelshchikov equations in bubble gas liquid flow. The solutions include solitons and periodic solutions. The method applied can be used in further works to obtain entirely new solutions to many other nonlinear evolution equations.

This work systematically discusses basic properties and qualitative dynamics of vector-borne disease models, particularly those with vertical transmission in the vector population. Examples of disease include Dengue and Rift Valley fever which are endemic in Sub-Saharan Africa, and understanding of the dynamics underlying their transmission is central for providing critical informative indicators useful for guiding control strategies. Of particular interest is the applicability and derivation of relevant population and epidemic thresholds and their relationships with vertical infection. This study demonstrates how the failure of $R_{\mathrm{0}}$ derived using the next-generation method compounds itself when varying vertical transmission efficiency, and it shows that the host type reproductive number gives the correct $R_{\mathrm{0}}$. Further, novel relationships between the host type reproductive number, vertical infection, and ratio of female mosquitoes to host are established and discussed. Analytical results of the model with vector stages show that the quantities $Q_{\mathrm{0}}$, $Q_{\mathrm{0}}^v$, and $R_{\mathrm{0}}^c$, which represent the vector colonization threshold, the average number of female mosquitoes produced by a single infected mosquito, and effective reproductive number, respectively, provide threshold conditions that determine the establishment of the vector population and invasion of the disease. Numerical simulations are also conducted to confirm and extend the analytical results. The findings imply that while vertical infection increases the size of an epidemic, it reduces its duration, and control efforts aimed at reducing the critical thresholds $Q_{\mathrm{0}}$, $Q_{\mathrm{0}}^v$, and $R_{\mathrm{0}}^c$ to below unity are viable control strategies.

Acute myeloid leukemia (AML) is a malignant hematopoietic disorder characterized by uncontrolled proliferation of immature myeloid cells. In the AML cases, the phosphoinositide 3-kinases (PI3K)/AKT signaling pathways are frequently activated and strongly contribute to proliferation and survival of these cells. In this paper, a mathematical model of the PI3K/AKT signaling pathways in AML is constructed to study the dynamics of the proteins in these pathways. The model is a 5-dimensional system of the first-order ODE which describes the interaction of the proteins in AML. The interactions between those components are assumed to follow biochemical reactions, which are modelled by Hill’s equation. From the numerical simulations, there are three potential components targets in PI3K/AKT pathways to therapy in the treatment of AML patient.

A mathematical model is formulated that captures the essential dynamics of waterborne disease transmission under the assumption of a homogeneously mixed population. The important mathematical features of the model are determined and analysed. The model is extended by introducing control intervention strategies such as vaccination, treatment, and water purification. Mathematical analyses of the control model are used to determine the possible benefits of these control intervention strategies. Optimal control theory is utilized to determine how to reduce the spread of a disease with minimum cost. The model is validated using a cholera outbreak in Haiti.

36 years ago, Thomas Saaty introduced a new mathematical methodology, called Analytic Hierarchy Process (AHP), regarding the decision-making processes. The methodology was widely applied by Saaty and by other authors in the different human activity areas, like planning, business, education, healthcare, etc. but, in general, in the area of management. In this paper, we provide two new proofs for well-known statement that the maximal eigenvalue ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is equal to $n$ for the eigenvector problem $Aw=\lambda w$, where $A$ is, so-called, the consistent matrix of pairwise comparisons of type $n\times n$ ($n$$\ge$ 2) with the solution vector $w$ that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution $Aw=nw$ as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix $A$ can be simply programmed and used.

In this paper, a mathematical model of a prey-predator system with infectious disease in the prey population is proposed and studied. It is assumed that there is a constant refuge in prey as a defensive property against predation and harvesting from the predator. The proposed mathematical model is consisting of three first-order nonlinear ordinary differential equations, which describe the interaction among the healthy prey, infected prey, and predator. The existence, uniqueness, and boundedness of the system’ solution are investigated. The system's equilibrium points are calculated with studying their local and global stability. The persistence conditions of the proposed system are established. Finally the obtained analytical results are justified by a numerical simulation.

In this paper a within host mathematical model for Human Immunodeficiency Virus (HIV) transmission incorporating treatment is formulated. The model takes into account the efficacy of combined antiretroviral treatment on viral growth and T cell population in the human blood. The existence of an infection free and positive endemic equilibrium is established. The basic reproduction number $R_{\mathrm{0}}$ is derived using the method of next generation matrix. We perform local and global stability analysis of the equilibria points and show that if , then the infection free equilibrium is globally asymptotically stable and theoretically the virus is cleared and the disease dies out and if $R_{\mathrm{0}}>\mathrm{1}$, then the endemic equilibrium is globally asymptotically stable implying that the virus persists within the host. Numerical simulations are carried out to investigate the effect of treatment on the within host infection dynamics.

Artificial Bee Colony (ABC) algorithm is one of the efficient nature-inspired optimization algorithms for solving continuous problems. It has no sensitive control parameters and has been shown to be competitive with other well-known algorithms. However, the slow convergence, premature convergence, and being trapped within the local solutions may occur during the search. In this paper, we propose a new Modified Artificial Bee Colony (MABC) algorithm to overcome these problems. All phases of ABC are determined for improving the exploration and exploitation processes. We use a new search equation in employed bee phase, increase the probabilities for onlooker bees to find better positions, and replace some worst positions by the new ones in onlooker bee phase. Moreover, we use the Firefly algorithm strategy to generate a new position replacing an unupdated position in scout bee phase. Its performance is tested on selected benchmark functions. Experimental results show that MABC is more effective than ABC and some other modifications of ABC.

The paper is devoted to several examples of control algorithm development for two-wheeled double-track robot and low-tonnage vessel-catamaran with two Azipods that show practical aspects of the application of one nonlinear system control method — terminal state method. This method, developed by the authors of the present paper, belongs to the class of methods for inverse dynamics problem solving. Mathematical models of control objects in the form of normal systems of third-order nonlinear differential equations for the wheeled robot and seventh-order ones for the vessel are presented. Design formulas of the method in general form for terminal and stabilizing controls are shown. A routine of obtaining calculation expressions for control actions is shown. Results of computer simulation of bringing the robot to a given point in a given time, as well as bringing the vessel to a given course during a “strong” maneuver, are described.