How vector feeding preference through an infectious host relates to the seasonal transmission rates in a mathematical vector‐host model

Vector‐borne diseases, such as leishmaniasis, dengue, malaria, and yellow fever, transmitted by microparasites show periodic fluctuations in their prevalence. The novelty of this research is to assess the relationship between the vector feeding preference for an infectious host and the annual seasonal transmission through a vector‐host mathematical model. For the first time, numerical simulations illustrate that by increasing the vector feeding preference value in the transmission dynamics, periodic fluctuations accentuate and the endemic equilibrium average increases in vector and host populations. Moreover, increasing the vector feeding preference value, the amplitude strengthens for the infectious host and vector populations. This periodic behavior shows a similar pattern with the Peruvian incidence data from 2000 to 2016 for Andean cutaneous leishmaniasis provided by the Ministry of Health of Peru (MINSA). In addition, using the Floquet theory, the time average method and the linear operator method provides for the first time that the basic reproduction number for a nonautonomous system depends explicitly on the vector feeding preference for the infectious host. The nonautonomous model system shows that is a threshold parameter for the local stability of the disease‐free periodic solution. Therefore, the vector feeding preference is an important factor that should be considered and attended to for future research. Public and veterinary health in Peru and other countries should consider the vector feeding preference for specific host to vector control.     

Residue formulae, vector partition functions and lattice points in rational polytopes

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition functions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational convex polytopes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

     

Invariant theory and reversible-equivariant vector fields

In this paper we present results for the systematic study of reversible-equivariant vector fields-namely, in the simultaneous presence of symmetries and reversing symmetries-by employing algebraic techniques from invariant theory for compact Lie groups. The Hilbert-Poincaré series and their associated Molien formulae are introduced, and we prove the character formulae for the computation of dimensions of spaces of homogeneous anti-invariant polynomial functions and reversible-equivariant polynomial mappings. A symbolic algorithm is obtained for the computation of generators for the module of reversible-equivariant polynomial mappings over the ring of invariant polynomials. We show that this computation can be obtained directly from a well-known situation, namely from the generators of the ring of invariants and the module of the equivariants.     

Inversion of vector delay systems

We consider stationary linear vector systems with commensurable delays. We obtain sufficient conditions for the reducibility of such systems to canonical form with the extraction of null dynamics. A constructive algorithm for the reduction of a system to that form is presented. We suggest a method for estimating the unknown input for vector delay systems with given accuracy.     

Vector acoustic solitons from the coupling of long and short waves in a paramagnetic crystal

We investigate the propagation of a longitudinal-transverse elastic pulse in a statically deformed crystal containing paramagnetic impurities and placed in an external magnetic field. We derive a system of three nonlinear wave equations describing the interaction of the pulse with the paramagnetic impurities in the quasiresonance approximation in the Faraday geometry. We assume that the transverse components of the pulse, which cause quantum transitions, have carrier frequencies and are short-wave (acoustic), while the longitudinal component has no carrier frequency and is long-wave. We show that in the case of an equilibrium initial distribution of populations of quantum levels of paramagnetic impurities, the coupling between the longitudinal and transverse components is weak, the pulse is therefore strictly transverse, and its dynamics are described by the Manakov system. With a nonequilibrium initial distribution of populations, conditions of effective interaction between all components of the elastic pulse can be reached, and their nonlinear dynamics are described by a vector generalization of the Zakharov equations. In the case of a unidirectional propagation of the pulse, these equations reduce to the Yajima-Oikawa vector system. We show that the obtained system of equations and its version with an arbitrary number of short-wave components can be integrated using the inverse scattering transform. We construct infinite hierarchies of solutions of the Yajima-Oikawa vector system (including a solution on a nontrivial background). We consider stationary (complex-valued Garnier system) and self-similar reductions of that system, also admitting a representation in the form of compatibility conditions.     

从支持向量机到非平行支持向量机

非平行支持向量机是支持向量机的延伸,受到了广泛的关注.非平行支持向量机构造允许非平行的支撑超平面,可以描述不同类别之间的数据分布差异,从而适用于更广泛的问题.然而,对非平行支持向量机模型与支持向量机模型之间的关系研究较少,且尚未有等价于标准支持向量机模型的非平行支持向量机模型.从支持向量机出发,构造出新的非平行支持向量机模型,该模型不仅可以退化为标准支持向量机,保留了支持向量机的稀疏性和核函数可扩展性.同时,可以描述不同类别之间的数据分布差异,适用于更广泛的非平行结构数据等.最后,通过实验初步验证了所提模型的有效性.     

Fréchet subdifferentials of efficient point multifunctions in parametric vector optimization

In this paper, we establish new formulae for computing and/or estimating the Fréchet subdifferential of the efficient point multifunction of a parametric vector optimization problem. These formulae are presented in a broad class of conventional vector optimization problems with the presence of geometric, operator and (finite and infinite) functional constraints.     

Support vector quantile regression with varying coefficients

Quantile regression has received a great deal of attention as an important tool for modeling statistical quantities of interest other than the conditional mean. Varying coefficient models are widely used to explore dynamic patterns among popular models available to avoid the curse of dimensionality. We propose a support vector quantile regression model with varying coefficients and its two estimation methods. One uses the quadratic programming, and the other uses the iteratively reweighted least squares procedure. The proposed method can be applied easily and effectively to estimating the nonlinear regression quantiles depending on the high-dimensional vector of smoothing variables. We also present the model selection method that employs generalized cross validation and generalized approximate cross validation techniques for choosing the hyperparameters, which affect the performance of the proposed model. Numerical studies are conducted to illustrate the performance of the proposed model.     

On vector formulations of auction-type problems with applications

We propose an extension of the auction model with many divisible commodities for vector price (validity) functions. It can be viewed as a new general equilibrium model for complex systems with active elements. We give its sufficient vector variational inequality formulation and new general existence results for different ordering cones. We suggest vector extensions of network and spatial equilibrium problems with capacity bounds and show that they are particular cases of the general auction model. We also give new sufficient conditions for existence of solutions for these problems.     

On accuracy,robustness and tolerances in vector Boolean optimization

A Boolean programming problem with a finite number of alternatives where initial coefficients (costs) of linear payoff functions are subject to perturbations is considered. We define robust solution as a feasible solution which for a given set of realizations of uncertain parameters guarantees the minimum value of the worst-case relative regret among all feasible solutions. For the Pareto optimality principle, an appropriate definition of the worst-case relative regret is specified. It is shown that this definition is closely related to the concept of accuracy function being recently intensively studied in the literature. We also present the concept of robustness tolerances of a single cost vector. The tolerance is defined as the maximum level of perturbation of the cost vector which does not destroy the solution robustness. We present formulae allowing the calculation of the robustness tolerance obtained for some initial costs. The results are illustrated with several numerical examples.     

On vector optimization problems and vector variational inequalities using convexificators

In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points.     

Dynamics of a system of three interacting populations with Allee effects and stocking

A model of three interacting populations where two populations engage in competition and two populations are in predator–prey type interaction is proposed and analysed. One of the two competing populations is subject to Allee effects and is also a pest population. The other competing population is regarded as a control agent and is the host for the predator population. There is a constant level of the external control agents released into the interaction at each generation after parasitism. We provide asymptotic dynamics of the competition subsystem and prove that a Neimark–Sacker bifurcation occurs for the host–parasitoid subsystem when the unique interior steady state loses its stability. The three interacting populations are impossible to persist for all positive initial conditions. Sufficient conditions based on the initial population size of the population with Allee effects are derived for persistence of the three populations.     

Inhomogeneous maps and mathematical theory of selection

In this paper we develop a theory of general selection systems with discrete time and explore the evolution of selection systems, in particular, inhomogeneous populations. We show that the knowledge of the initial distribution of the selection system allows us to determine explicitly the system distribution at the entire time interval. All statistical characteristics of interest, such as mean values of the fitness or any trait can be predicted effectively for indefinite time and these predictions dramatically depend on the initial distribution. The Fisher Fundamental theorem of natural selection (FTNS) and more general the Price equations are the famous results of the mathematical selection theory. We show that the problem of dynamic insufficiency for the Price equations and for the FTNS can be resolved within the framework of selection systems. Effective formulas for solutions of the Price equations and for the FTNS are derived. Applications of the developed theory to some other problems of mathematical biology (dynamics of inhomogeneous logistic and Ricker model, selection in rotifer populations) are also given. Complex behavior of the total population size, the mean fitness (in contrast to the plain FTNS) and other traits is possible for inhomogeneous populations with density-dependent fitness. The temporary dynamics of these quantities can be investigated with the help of suggested methods.     

Dynamics of an age‐structured host‐vector model for malaria transmission

In this paper, we propose a host‐vector model for malaria transmission by incorporating infection age in the infected host population and nonlinear incidence for transmission from infectious vectors to susceptible hosts. One novelty of the model is that the recovered hosts only have temporary immunity and another is that successfully recovered infected hosts may become susceptible immediately. Firstly, the existence and local stability of equilibria is studied. Secondly, rigorous mathematical analyses on technical materials and necessary arguments, including asymptotic smoothness and uniform persistence of the system, are given. Thirdly, by applying the fluctuation lemma and the approach of Lyapunov functionals, the threshold dynamics of the model for a special case were established. Roughly speaking, the disease‐free equilibrium is globally asymptotically stable when the basic reproduction number is less than one and otherwise the endemic equilibrium is globally asymptotically stable when no reinfection occurs. It is shown that the infection age and nonlinear incidence not only impact on the basic reproduction number but also could affect the values of the endemic steady state. Numerical simulations were performed to support the theoretical results.     

Normal subdifferentials of efficient point multifunctions in parametric vector optimization

The normal subdifferential of a set-valued mapping with values in a partially ordered Banach space has been recently introduced in Bao and Mordukhovich (Control Cyber 36:531–562, 2007), by using the Mordukhovich coderivative of the associated epigraphical multifunction, which has proven to be useful in deriving necessary conditions for super efficient points of vector optimization problems. In this paper, we establish new formulae for computing and/or estimating the normal subdifferential of the efficient point multifunctions of parametric vector optimization problems. These formulae will be presented in a broad class of conventional vgector optimization problems with the presence of geometric, operator, equilibrium, and (finite and infinite) functional constraints.     

Existence of solutions of vector variational inequalities and vector complementarity problems

In this paper, we consider vector variational inequality and vector F-complementarity problems in the setting of topological vector spaces. We extend the concept of upper sign continuity for vector-valued functions and provide some existence results for solutions of vector variational inequalities and vector F-complementarity problems. Moreover, the nonemptyness and compactness of solution sets of these problems are investigated under suitable assumptions. We use a version of Fan-KKM theorem and Dobrowolski’s fixed point theorem to establish our results. The results of this paper generalize and improve several results recently appeared in the literature.     

Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan

In this paper we develop a mathematical model to study the dynamics of visceral leishmaniasis in the Sudan. To develop this model we consider the dynamics of the disease between three different populations, human, reservoir and vector populations. The model is analyzed at equilibrium and the stability of the equilibria is analyzed. The basic reproduction number is derived, and the threshold conditions for disease elimination established. Results show that the disease can be eliminated under certain conditions. Simulations of the model show that human treatment helps in disease control, and its synergy with vector control will more likely result in the elimination of the disease.     

Transformations of the equations of the first approximation of Chapman–Enskog methods and vector transport relations for mixtures of polyatomic gases

Transformations of the systems of equations of the first approximation of the classical and generalized Chapman–Enskog methods are proposed for those terms of the distribution functions by means of which one can calculate the vector transport relations for mixtures of polyatomic gases, connecting the diffusion and heat fluxes and gradients of the scalar gas-dynamic variables. The solutions of the transformed systems are expressed in terms of diffusion rates and gradients of the macroparameters. The derivation and formulae for the coefficients of the Stefan–Maxwell relations and their generalizations are simplified, and rigorous results for the matrices of the transport coefficients are established. Approximate vector transport relations are given for mixtures of non-equilibrium reacting polyatomic gases.     

Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector

A delay ordinary deterministic differential equation model for the population dynamics of the malaria vector is rigorously analysed subject to two forms of the vector birth rate function: the Verhulst-Pearl logistic growth function and the Maynard-Smith-Slatkin function. It is shown that, for any birth rate function satisfying some assumptions, the trivial equilibrium of the model is globally-asymptotically stable if the associated vectorial reproduction number is less than unity. Further, the model has a non-trivial equilibrium which is locally-asymptotically stable under a certain condition. The non-trivial equilibrium bifurcates into a limit cycle via a Hopf bifurcation. It is shown, by numerical simulations, that the amplitude of oscillating solutions increases with increasing maturation delay. Numerical simulations suggest that the Maynard-Smith-Slatkin function is more suitable for modelling the vector population dynamics than the Verhulst-Pearl logistic growth model, since the former is associated with increased sustained oscillations, which (in our view) is a desirable ecological feature, since it guarantees the persistence of the vector in the ecosystem.     

Mathematical assessment of a fractional-order vector–host disease model with the Caputo–Fabrizio derivative

Vector–host diseases outbreak is a major public health concern, and it has greatly affected human health and economy in various regions around the globe. Different approaches have been adopted to investigate the dynamical behavior and possible control of these diseases. In this study, we present a compartmental transmission model in order to explore the dynamics of vector–host infectious diseases. The saturated incidence rate instead of bilinear (or standard) and saturated treatment function is considered in model formulation which enhance the biological suitability of the proposed model. We first formulate the model based on nonlinear classical integer-order differential equations. Then, the proposed integer-order model is reformulated using the fractional-order operator in Caputo–Fabrizio sense with nonsingular kernel. We investigate the model equilibria and evaluate the expression for the most important threshold parameter known as the basic reproduction number. Furthermore, the existence and uniqueness are presented via the fixed point approach. Additionally, using an efficient numerical scheme, the iterative solution of the model is obtained. Finally, we present the model simulations to illustrate the impact of arbitrary fractional order and some of other important parameters involved in the model on the disease dynamics and minimization.