Positive Solutions Research Articles

Mathematical Model of Counteract Bandits Terrorism in Nigeria

This paper explores a mathematical model for analyzing the dynamics of banditry and terrorism, focusing on equilibrium states and stability conditions. Key concepts include the reproduction number Rb, which serves as a threshold parameter indicating the potential for spread or decline of these activities. The study finds that a banditry-free equilibrium is locally asymptotically stable when Rb < 1, suggesting that the activities will decline over time. In contrast, Rb>0 indicates the possibility of globally stable coexistence equilibrium, where banditry and terrorism persist at endemic levels. The model also identifies invariant regions, ensuring that the system's trajectories remain within certain bounds, and guarantees unique and positive solutions, essential for real-world applicability. These findings provide critical insights for developing comprehensive strategies to mitigate banditry and terrorism, highlighting the importance of addressing both root causes and immediate symptoms

Spatiotemporal analysis of Zika virus transmission dynamics incorporating human mobility and seasonal variations using modified homotopy perturbation method

AbstractThis study employed a mathematical model to evaluate how seasonal variations, vector dispersal, and mobility of people affect the spread of the Zika virus. The model's positive solutions, invariant zones, and solution existence and uniqueness were validated through proved theorems. The equilibria points were identified, and the basic reproduction number was calculated. The model was semi-analytically solved using a modified homotopy perturbation approach, and an applied convergence test proved that the solution converges. The simulation results indicated that under optimal breeding conditions, the density of healthy mosquitoes peaked in the fourth month. Two months later, increased mosquito dispersal and human carriers facilitated by favorable weather led to a rise in mosquito infectiousness, peaking between the fourth and eighth months due to significant seasonal effects, resulting in high Zika transmission. To effectively control mosquito populations and reduce Zika transmission, it is recommended that public health interventions focus on the critical periods spanning the third to eighth months.

On the large time behaviour of the solutions of an evolutionary-epidemic system with spatial dispersal.

This paper investigates some properties of the large time behaviour of the solutions of a spatially distributed system of equations modelling the evolutionary epidemiology of a plant-pathogen system. The model takes into account the phenotypic trait and the mutation of the pathogen, which is described by a non-local operator. We roughly speaking prove that the solutions separate the phenotype trait from the spatio-temporal evolution in the large time asymptotic. This feature is obtained by investigating the positive and bounded entire solutions of the problem, that are shown to exhibit such a separation of the variables property, by reformulating them as the positive solutions of suitable integral equations in some ordered Banach space. In addition, some numerical simulations are performed to support our theoretical results.

On the positive solutions for some diffusive logistic equations

In this paper, we study the positive solutions of the diffusive logistic equations. It is shown that the asymptotic profiles of the positive solutions display differences between nonlocal dispersal equation and semilinear elliptic equation. When the degeneracy domain becomes small, we find that the positive solution of semilinear elliptic equation admits a blow-up profile. However, when the non-degeneracy domain becomes large, the positive solution of nonlocal equation exhibits a blow-up profile.

Dynamics of a single population model with memory and nonlinear boundary condition

In this paper, a heterogeneous single population model with memory effect and nonlinear boundary condition is investigated. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially nonconstant positive steady state solutions appear from two trivial solutions, respectively. Through the distribution of the eigenvalues and bifurcation analysis, sufficient conditions for the occurrence of the Hopf bifurcation associated with one spatially nonconstant positive steady state are obtained. Results about the stability associated with the other one are also yielded. It is found that with the interaction of memory delay, spatial heterogeneity and nonlinear boundary condition, the Hopf bifurcation will happen in such diffusive model. To be specific, the memory delay could induce a single stability switch from stability to instability. As contrast to the diffusive model only with memory effect, the stability switch is independent of memory delay. The obtained results are applied to some specific models, from which we can find that the theoretical analysis is verified and the results enrich the existing ones.

Countably many positive solutions for iterative system of boundary value problems on time scales

This paper establishes the existence of countably many positive solutions for a second order iterative system of two-point boundary value problems by using Holder’s inequality and Guo–Krasnoselskii’s fixed point theorem for operator on a cone.

Positive solutions to quasilinear p(y)-Laplacian equation with Robin boundary condition

This article addresses the following Robin boundary value problem: { − div ( b ( y ) | ∇w | p ( y ) − 2 ∇w ) − F ( y ) | w | η ( y ) − 2 w = λG ( y ) | w | γ ( y ) − 2 w in Ω , b ( y ) | ∇w | p ( y ) − 2 ∂ w ∂ ν + H ( y ) | w | p ( y ) − 2 w = 0 on ∂Ω , here Ω ⊂ R N ( N ≥ 2 ) and p ( y ) , η ( y ) , γ ( y ) ∈ C ( Ω ¯ ) . We establish the existence and multiplicity results of the given problem in generalized Sobolev spaces W 1 , p ( y ) ( Ω ) . The main results of this article are derived using the Nehari manifold method under some appropriate assumptions on functions b, F, G and H.

Negative flux fixups using positivity-preserving limiters for SN-discontinuous finite element scheme of neutron transport equation on unstructured triangular meshes

One significant challenge of spatial discretization for the SN transport equation is the appearance of negative solutions, resulting in numerical algorithms to be unstable or slow iterative convergence in some problems. The discontinuous Galerkin finite element (DGFEM) spatial scheme on unstructured meshes has attracted much attention in recent years, but the issue of negative solutions is still a long-standing problem. This paper studies a negative flux fixup method using positivity-preserving limiters for solving the SN-DGFEM neutron transport equation. This method uses the hierarchical basis functions and triangular reference elements to obtain arbitrary high order DGFEM scheme. In the element where appears negativities, a scaling or a rotation positivity-preserving limiter is used to scale or rotate the polynomial distribution to ensure positive solutions. Five different types of problems are selected to verify the accuracy and convergence of the method. Numerical results demonstrate that the method can produce non-negative solutions and maintain the convergence order of the original scheme, as well as not introduce too much additional computational effort.

ON INEXACT NEWTON METHODS FOR SOLVING TWO NONLINEAR MATRIX EQUATIONS

In this paper we consider inexact Newton methods for finding the largest positive definite solutions of two nonlinear matrix equations X+ A∗X−1A = Q and X − A∗X−1A = Q, respectively. Using Newton’s method for considered equations requires solving a Stein’s equation at each iteration. For solving the Stein’s equation, we use Smith type iterations instead of exact methods. Nonlocal convergence of the process is shown. Numerical experiments are included to illustrate the theory.

Two Positive Solutions for Elliptic Differential Inclusions

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

Existence of Nontrivial Solutions for Boundary Value Problems of Fourth-Order Differential Equations

This article investigates the solvability problem of fourth-order differential equations with two-point boundary conditions; specifically, conclusions regarding sign-changing solutions are obtained. The methods used in this article are fixed-point theorems on lattices. Firstly, under some sublinear conditions, the existence of three nontrivial solutions is demonstrated, including a sign-changing solution, a negative solution and a positive solution. Secondly, under some unilaterally asymptotically linear and superlinear conditions, the existence of at least one sign-changing solution is proved. Finally, this article provides several specific examples to illustrate the obtained conclusions.

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities

Abstract We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by S u ≔ − div ϕ u 2 + ∣ ∇ u ∣ 2 2 ∇ u + ϕ u 2 + ∣ ∇ u ∣ 2 2 u , Su:= -\hspace{0.1em}\text{div}\hspace{0.1em}\left\{\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)\nabla u\right\}+\phi \left(\frac{{u}^{2}+{| \nabla u| }^{2}}{2}\right)u, where ϕ : [ 0 , + ∞ ) → R \phi :\left[0,+\infty )\to {\mathbb{R}} is a positive continuous function. This operator is introduced by Stuart [Two positive solutions of a quasilinear elliptic Dirichlet problem, Milan J. Math. 79 (2011), 327–341] and depends on not only ∇ u \nabla u but also u u . This particular quasilinear term generally appears in the study of nonlinear optics model, which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution, which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator S S : (i) the almost everywhere convergence of the gradient for the minimizing sequence { u n } \left\{{u}_{n}\right\} ; (ii) seeking the reasonable restrictions about S S .

Liouville-type theorem for higher order Hardy-Hénon type systems on the sphere

In this paper, we study Liouville type theorems for the positive solutions to the following higher order Hardy-Hénon type system involving the conformal GJMS operator on the sphere Sn. In order to study this we first employ the Mobius transform to transform the above Hardy-Hénon type system on the sphere Sn into a higher order elliptic system on Rn. Then, we show that every positive solution of the higher order elliptic system on Rn is a solution to the associated integral system on Rn by using polyharmonic average and iteration arguments. We use the method of moving planes in integral form to prove that there are no positive solutions for the integral system on Rn. Finally, together with the symmetry of the sphere Sn, we obtain the Liouville type theorem of the higher order Hardy-Hénon type system involving the GJMS operator on the sphere. The results of this paper are also new even for the Lane-Emden system on the sphere.

Global dynamics of a periodically forced SI disease model of Lotka–Volterra type

In this paper, we investigate the dynamics of an SI disease model of Lotka–Volterra type in the presence of a periodically fluctuating environment. We give a global analysis of the dynamical behavior of the model. Interestingly, our results show that the permanence guarantees the existence of a unique positive harmonic time-periodic solution which is globally attracting when the horizontal disease transmission has a weaker impact than the intraspecific competition. While for the case when the horizontal disease transmission has a stronger impact than the intraspecific competition, we numerically show that complex dynamics such as chaos can occur in a permanent system. Nonetheless, we provide sufficient conditions for the existence and uniqueness of the positive harmonic time-periodic solution for the latter case. The impact of the environment on the spread of disease is studied by using a bifurcation analysis. We show that in each of the qualitatively different cases of the associated autonomous SI model in a constant environment, an alternative possibility can appear in the periodic model.

Antibody titer after anti-idiotype rabies vaccination with nano-chitosan adjuvant

Background.The rabies virus neutralizing antibodies titers is a public health problem in the world, including Indonesia. Rabies is zoonotic and causes death in humans with a case fatality rate of 100%. Anti-idiotype antibody (Ab2) from chicken immunoglobulin (IgY) can be a substitute antigen for the rabies virus. This research aims to study the potential of rabies vaccine based on anti-idiotype antibody (Ab2) with nano-chitosan as an adjuvant.Materials and methods.The production of Ab2 is derived from purified and characterized chicken immunoglobulins. Nano-chitosan is made from chitosan from shrimp shell waste. Vaccination tests were carried out on rats compared with commercial vaccines as a positive control and physiological solutions as a negative control. The characterization results of IgY Ab2 were IgY rabies with BM ~180 kDa, heavy chains (BM ~60 kDa), and light chains (BM ~30 kDa) of IgY. Nano-chitosan is less than 100 nm in size, can dissolve well, and does not cause side effects in experimental animals.Results.The vaccine formula uses a concentration of 0.5%, where the ratio of nano-chitosan and Ab2 is 1:1. The Ab2 concentration used was about 1000 units per mL of the Ab2 suspension.Conclusion.This study concludes that anti-idiotype antibodies dissolved in nano-chitosan adjuvant can induce the formation of antibodies with titers that are not statistically different from the antibody titers induced by commercial rabies vaccines. Nano-chitosan has a potential vaccine adjuvant candidate, safe to use, and can enhance the immunogenicity of an antigen applied subcutaneously. According to the results of this study, it is recommended that post-vaccination antibody titers be measured regularly: for example, every one week for six weeks after vaccination. This measurement determines the time of the peak of the antibody titer so that the time for revaccination can be determined.

A Study on Positive Solutions to Nonlinear Fractional Differential Equations

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are obtained.

Chlamydia infection with vaccination asymptotic for qualitative and chaotic analysis using the generalized fractal fractional operator

In this work, we solve a system of fractional differential equations utilizing a Mittag-Leffler type kernel through a fractal fractional operator with two fractal and fractional orders. A six-chamber model with a single source of chlamydia is studied using the concept of fractal fractional derivatives with nonsingular and nonlocal fading memory. The fractal fractional model of the Chlamydia system can be solved by using the characteristics of a non-decreasing and compact mapping. A suggested model with the Lipschitz criteria and linear growth is studied both qualitatively and quantitatively, taking into account boundedness, uniqueness, and positive solutions at equilibrium points with Leray-Schauder results under time scale concepts. We examined the framework of local and global stability and insight into Lyapunov function properties for the infectious disease model. Chaos Control will employ the regulate for linear responses approach to stabilize the system following its equilibrium points. This will take into consideration a fractional order framework with a managed design, where solutions are bounded in the feasible domain and have a greater impact at the lower minimum infectious rate. To illustrate the implications of fractional and fractal dimensions with varying interest rate values through simulations with Newton’s polynomial method under the Mittag-Lefller kernel. Additionally, a comparative analysis of results is also derived by employing power and exponential decay kernels at various fractional orders.

Infinitely many positive solutions for p-Laplacian equations with singular and critical growth terms

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p-Laplacian type involving a singularity and a critical Sobolev exponent {−Δpu=up∗−1+λ|u|γ−1u,inΩ,u=0,on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta _{p}u=u^{p^{*}-1}+\\frac{\\lambda}{|u|^{\\gamma -1}u}, & \ ext{in} ~\\Omega , \\\\ u=0, &\ ext{on}~\\partial \\Omega , \\end{cases} $$\\end{document} where Ω is a bounded domain, p∗=NpN−p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p^{\\ast}=\\frac{Np}{N-p}$\\end{document} (N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N \\geq 3$\\end{document}) is the critical Sobolev exponent and λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda > 0$\\end{document}. Based on the cutoff technique, we prove that the above problem possesses infinitely many positive solutions with 0<γ<1,1<p<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0 < \\gamma <1, 1< p < N$\\end{document}, and λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda > 0$\\end{document} small enough.

Capture and catalytic conversion of CO2 from marine and offshore applications – A review

Abstract This contribution examines recent advances in modeling fluid dynamics and capture/conversion of CO2 from marine and offshore applications in packed-bed columns/trickle-bed reactors subjected to static inclination, externally-induced rolling and heaving motions. Breaking the axial symmetry of the two-phase flow once the packed bed system is tilted results in a significant decrease in process performance, the extent of which is determined by the magnitude of the tilt and the column/reactor diameter. Only the conversion of CO2 from marine engine emissions on-board large cargo ships via catalytic cycloaddition of CO2 to styrene oxide in multiphase large-diameter trickle-bed reactors increases slightly with increasing reactor inclination. Under the externally induced column/reactor oscillations, CO2 capture/conversion performance shifts toward the steady-state solution of the vertical column/reactor as the asymmetry between the two inclined positions decreases. Oscillating (between two inclined symmetrical positions) and heaving packed-bed columns/trickle-bed reactors generate an oscillating performance around the steady-state solution of the vertical static position, which is driven by the amplitude and period of the angular and heaving motions via the continuous evolution of the intensity of the reverse secondary flow and by the magnitude of the reactor/column diameter. The catalytic conversion of CO2 captured from marine emissions via an integrated process coupling reverse water-gas shift reaction and methanol synthesis in a fixed-bed reactors network system shows a remarkable enhancement with the addition of H2O adsorbent to the reaction systems in the sorption-enhanced process periods.

Investigation of an optimal control strategy for a cholera disease transmission model with programs

Cholera is a disease of poverty affecting people with inadequate access to safe water and basic sanitation. Conflict, unplanned urbanization and climate change all increase the risk of cholera. In this article, an optimal control deterministic mathematical model of cholera disease with cost-effectiveness analysis is developed and analyzed considering both direct and indirect contact transmission pathways. The model qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction number of the model is obtained. We also performed sensitivity analysis of the basic parameters of the model. Then an optimal control problem is designed with a control functional having five controls: vaccination, treatment, environment sanitation and personal hygiene, and water quality improvement program. We examined the existence and uniqueness of the optimal controls of the system. Through the implementation of Pontryagin's maximum principle, the characterization of the optimal controls optimality system is established. The numerical simulation results the integrated control strategies demonstrated that strategy 2, 7, and 12 are effective programs to combat cholera disease from the community. Based on the local circumstances, available funds, and resources, it is recommended to the government stakeholders and policymakers to execute any one of the three integrated intervention programs.