modelling the spread of aids epidemic with vertical transmission abstract this paper considers a non linear mathematical model for hiv epidemic that spreads in a variable size population through both horizontal and vertical transmission the equilibrium points of the model system are found and their stability is investigated the model exhibits two equilibria namely the disease free and the endemic equilibrium it is found that if the basic reproduction number r0 1 a unique endemic equilibrium exists which is locally asymptotically stable and becomes globally asymptotically stable under certain conditions showing that the disease becomes endemic due to constant immigration of population into the community using stability theory and computer simulation it is shown that by controlling the rate of vertical transmission the spread of the disease can be reduced significantly and consequently the equilibrium values of infective and aids population can be maintained at desired levels a numerical study of the model is also used to investigate the influence of certain other key parameters on the spread of the disease

a search for bilinear equations passing hirota s three soliton condition ii mkdv type bilinear equations in this paper the results of a search for complex bilinear equations with two soliton solutions are presented the following basic types are discussed a the nonlinear schrodinger equation b dx g f 0 a dx dt f f gg and b the benjamin ono equation p dx f f 0 it is found that the existence of two soliton solutions is not automatic but introduces conditions that are like the usual three and four soliton conditions the search was limited by the degree of a 2 and by degree of p 4 the main results are the following 1 iad3x dxdt idy b g f 0 d2xf f gg 2 d2x ad2y idt b g f 0 dxdy f f gg 3 iad3x d2x idt f f 0 and 4 dxdt i adx bdt f f 0

traveling waves in a nonlocal dispersal sirh model with relapse this paper is concerned with traveling wave solutions of a nonlocal dispersal susceptibleinfectiveremovalhealing for short sirh model with relapse it is found that the existence and nonexistence of traveling waves of the system are not only determined by the critical wave speed c but also by the basic reproduction number r0 of the corresponding system of ordinary differential equations more precisely we use schauders fixed point theorem to obtain the existence of traveling waves for r0 1 and c c and the nonexistence of traveling waves for r0 1 and 0

the hirota s direct method and the tanh coth method for multiple soliton solutions of the sawada kotera ito seventh order equation abstract in this work the sawada kotera ito seventh order equation is studied the tanh coth method is applied to obtain soliton solution of this equation the hirota s direct method combined with the simplified hereman s method are applied to determine the n soliton solutions for this equation the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

regular soliton solutions and singular soliton solutions for the modified kadomtsev petviashvili equations abstract in this paper we study the positive and the negative modified kadomtsev petviashvili mkp equations the mkp equations are completely integrable equations that admit n soliton solutions and an infinite number of conserved densities multiple soliton solutions are obtained for the positive model whereas multiple singular soliton solutions are established for the negative model the analysis depends mainly on the hirota s bilinear method

multiple front solutions for the burgers kadomtsev petviashvili equation abstract in this work we study a completely integrable dissipative equation the burgers equation is extended by using the sense of the kadomtsev petviashvili kp equation the new established burgers kp equation is studied by using the tanh coth method to obtain kink solutions and periodic solutions we also apply the powerful hirota s bilinear method to establish exact n soliton solutions for the derived integrable equation

solitons and singular solitons for the gardner kp equation abstract we present in this work a reliable treatment for the gardner kp gkp equation the hirota s bilinear method is used to obtain multiple soliton solutions for this completely integrable equation multiple singular soliton solutions are obtained as well the study highlights a variety of multi solitary wave and multi singular solitary solutions of the gardner kp equation

global dynamics of an seir epidemic model with vertical transmission we study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission the total host population is assumed to have constant density and the incidence term is of the bilinear mass action form we prove that the global dynamics are completely determined by the basic reproduction number r0 p q where p and q are fractions of infected newborns from the exposed and infectious classes respectively if r 0 p q le 1 the disease free equilibrium is globally stable and the disease always dies out if r0 p q 1 a unique endemic equilibrium exists and is globally stable in the interior of the feasible region and the disease persists at an endemic equilibrium state if it initially exists the contribution of the vertical transmission to the basic reproduction number is also analyzed

analysis of a delayed chlamydia epidemic model with pulse vaccination abstract in this paper we have considered a dynamical model of chlamydia disease with varying total population size saturation incidence rate and discrete time delay to become infectious it is assumed that there is a time lag to account for the fact that an individual infected with bacterium chlamydia trachomatis is not infectious until after some time after exposure the probability that an individual remains in the latency period exposed class at least t time units before becoming infectious is given by a step function with value 1 for 0 t and value zero for t the probability that an individual in the latency period has survived is given by e where denotes the natural mortality rate in all epidemiological classes pulse vaccination is an effective and important strategy for the elimination of infectious diseases and so we have analyzed this model with pulse vaccination we have defined two positive numbers r 1 and r 2 it is proved that there exists an infection free periodic solution which is globally attractive if r 1 1 and the disease is permanent if r 2 1 our analytical findings are illustrated through computer simulation using matlab which show the reliability of our model from the epidemiological point of view

multiple front solutions for the burgers equation and the coupled burgers equations abstract in this work multiple front solutions for the burgers equation and the coupled burgers equations are examined the tanh coth method and the cole hopf transformation are used the work highlights the power of the proposed schemes and the structures of the obtained multiple front solutions

parameter uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution in this paper we present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection diffusion two point boundary value problem the solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc length monitor function it is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter in addition we prove the uniform first order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain numerical results are presented that demonstrate the sharpness of our results

algebraic geometrical solutions of some multidimensional nonlinear evolution equations the known 2 1 dimensional breaking soliton equation the coupled kp equation with three potentials and a new 3 1 dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the 1 1 dimensional akns equations the abel jacobi coordinates are introduced to straighten out the associated flows from which algebraic geometrical solutions of the 2 1 dimensional breaking soliton equation the coupled kp equation and the 3 1 dimensional evolution equation are explicitly given in terms of the riemann theta functions

the hirota s direct method for multiple soliton solutions for three model equations of shallow water waves abstract multiple soliton solutions for three model equations for shallow water waves are determined the three models are completely integrable the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these equations the tanh coth method is used to obtain single soliton solutions and other solutions for these three models the three models have different linear dispersion relations but possess the same coefficients for the polynomials of exponentials

multiple soliton solutions for the lax kadomtsev petviashvili lax kp equation a new completely integrable dispersive equation is derived the new equation is obtained by extending the lax fifth order equation using the sense of the kadomtsev petviashvili kp equation in extending the kdv equation the newly derived lax kadomtsev petviashvili lax kp equation is investigated by using the tanh coth method and the hirota bilinear method to derive single soliton solution and n soliton solutions respectively the study highlights the multiple soliton solutions of the derived completely integrable equation

analysis of a nonautonomous dynamical model of diseases through droplet infection and direct contact in this paper we have considered a nonautonomous dynamical model of diseases that spread by droplet infection and also through direct contact with a lower risk with varying total population size and distributed time delay to become infectious it is assumed that there is a time lag due to incubation period of pathogens i e the development of an infection from the time the pathogen enters the body until signs or symptoms first appear here we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique we have obtained the explicit formula of the eventual lower bounds of infected persons we have introduced some new threshold values by lyapunov functional method we have also obtained some sufficient conditions for global asymptotic stability of this model computer simulations are carried out to explain the analytical findings the aim of the analysis of this model is to trace the parameters of interest for further study with a view to informing and assisting policy maker in targeting prevention and treatment resources for maximum effectiveness

quasi periodic solutions for some 2 1 dimensional integrable models generated by the jaulent miodek hierarchy some 2 1 dimensional integrable models including the modified kadomtsev petviashvili equation generated by the jaulent miodek hierarchy are investigated with the help of the jaulent miodek eigenvalue problem these 2 1 dimensional integrable models are separated into compatible hamiltonian systems of ordinary differential equations using the generating function flow method the involutivity and the functional independence of the integrals are proved the abel jacobi coordinates are introduced from which the quasi periodic solutions for these 2 1 dimensional integrable models are derived by resorting to the riemann theta functions pacs number 0230j

four 2 1 dimensional integrable extensions of the kdv equation multiple soliton and multiple singular soliton solutions in this work four 2 1 dimensional nonlinear completely integrable equations generated by extending the kdv equation are developed the necessary condition for the complete integrability of these equation are formally derived multiple soliton solutions and multiple singular soliton solutions are determined to emphasize the compatability of these models the dispersion relations of these models are characterized by distinct physical structures the resonance phenomenon for these equations does not exist for any model

multiple soliton solutions of two extended model equations for shallow water waves abstract multiple soliton solutions for two extensions of model equations for shallow water waves presented in m j ablowitz d j kaup a c newell h segur the inverse scattering transform fourier analysis for nonlinear problems stud appl math 53 1974 249 315 r hirota j satsuma n soliton solutions of model equations for shallow water waves j phys soc jpn 40 2 1976 611 612 are obtained the two extensions possess identical dispersion relations but with different structures for the n soliton solutions for n 1 the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these extended equations the tanh coth method is used to obtain single soliton solutions

multiple kink solutions and multiple singular kink solutions for the 2 1 dimensional burgers equations abstract in this work we study the 2 1 dimensional painleve integrable burgers equation multiple kink solutions and multiple singular kink solutions are formally derived for this equation the analysis depends mainly on the hirota s bilinear method

single and multiple soliton solutions for the 2 1 dimensional kdv equation abstract the 2 1 dimensional kdv equation is studied by four distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for this completely integrable equation the tanh coth method and the cosh ansatz are used to obtain single soliton solutions and travelling wave solutions the exp function method gives travelling wave solutions with some free parameters the work highlights the reliability of the four methods

multiple soliton solutions for the kp equation by hirota s bilinear method and by the tanh coth method abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions n 1 2 3 4 5 for the kadomtsev petviashvili kp equation the one soliton solutions will be handled by the tanh coth method the work emphasizes the power of the two methods that will be used

a numerical solution and an exact explicit solution of the nls equation we consider traveling wave solutions of the nonlinear schrodinger nls for short equation in this paper by considering the decomposition scheme we first obtain the exact solutions of the nls equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some initial values of the nls equation