Journal of Mathematical Analysis and Modeling

Numerical solution of systems of fractional order integro-differential equations with a Tau method based on monic Laguerre polynomials

2022-11-14T18:16:02+00:00

In this paper, numerical technique based on monic Laguerre polynomials is proposed to obtain approximate solutions of initial value problems for systems of fractional order integro-differential equations (FIDEs). Operational fractional integral matrix is constructed. This operational matrix is applied together with the monic Laguerre Tau method to solve systems of FIDEs. This systems of FIDEs will be transformed into a system of algebraic equations which can be solved easily. Numerical results and comparisons with other methods are also presented to show the efficiency and applicability of the proposed method.

Positivity results on the solutions for nonlinear two-term boundary value problem involving the ψ-Caputo fractional derivative

2022-10-11T06:51:12+00:00

In this work, we consider a nonlinear two-term boundary value problem involving the ψ-Caputo fractional derivative with integral boundary conditions. By the construction of its associated Green function and application of the upper and lower solutions method together with some fixed point theorems due to Banach and Schauder, we establish the existence and uniqueness of positive solutions to our considered main problem. In the end some illustrative examples are provided to validate our theoretical results.

Modelling the Impact of Sexual Activities and Inflow of Sexually Active Infected Immigrants on HIV/AIDS Epidemics

2022-07-01T16:33:43+00:00

Abstract
In this paper mathematical modeling on the impact of sexual activities and inflow of sexually active infected immigrants on the dynamics of HIV/AIDS epidemics were proposed and analyzed. The equilibrium points of the model were calculated and
the stability analysis of the model around these equilibrium points was studied using stability theory of differential equations and numerical simulation. Model dynamics is discussed with impacts of the direct inflow of high sexually active invective immigrants. On analyzing these situations, it is found that the disease is always persistent if the direct immigration of invective is allowed in the community. We investigate the local asymptotic stability of the deterministic epidemic model and similar properties
in terms of the basic reproduction number. The disease free and endemic equilibrium points are determined and the stability of both was investigated . The disease free equilibrium point is locally and globally asymptotically stable when R0 < 1 and unstable when R0 > 1. Numerical simulation of the model is carried out to assess the impact of heterosexuality with high sexual activities and inflow of sexually active infected immigrants on the dynamics of HIV/AIDS epidemics. The result of our model
showed that as the rate for inflow of high sexual active infected immigrants increases and as the number of sexual partners increase, then the infected populations also increase. Our Sensitivity analysis supports the results of the numerical analysis of
the model.

Keywords: HIV/AIDS, Infected Immigrants, Sexual partners, Reproduction Number