##### A Measure for Evaluating the Degree of Slope Rotatability in Three Level Second Order Slope Rotatable Designs

Chepkemoi Ednah, John Muindi Mutiso, Fredrick Oluoch Nyamwala

Journal of Advances in Mathematics and Computer Science, Page 1-12
DOI: 10.9734/jamcs/2019/v30i630090

Response surface methodology (RSM) often deals with a natural and desirable property rotatability, which requires that, the variance of the predicted response at a point remains constant at all such points that are equidistant from the design center. To achieve stability in prediction variance, this important property of rotatability was developed. Analogous to rotatability, the concept of slope-rotatability has been progressed. The idea of slope - rotatability is an important design criterion for response surface design. Recently, in the design of experiments for response surface analysis, attention has been focused on the estimation of dierences in response rather than absolute value of the response mean itself. The slope-rotatable design is that of which the variance of partial derivative is only a functions of : distance from the design center. If circumstances are such that exact slope rotatability is unattainable because of more cost and time, and more important restrictions such as orthogonal blocking it is still a good idea to make the design as slope rotatable as possible. Thus, it is important to measure the extent of deviation from slope rotatability. In this study, a new measure of the degree of slope-rotatability for three level second-order slope rotatable designs using a pair of a partially balanced incomplete block design is suggested that enables us to assess the degree of slope-rotatability for a given response surface design. This determines the degree slope rotatability for the design when subjected to existing conditions of measure. The measure takes the value zero when the design is exact slope-rotatable, and becomes larger as the design deviates from being slope-rotatable design.

##### Asymptotic Analysis of the Static and Dynamic Buckling of a Column with Cubic - Quintic Nonlinearity Stressed by a Step Load

A. M. Ette, J. U. Chukwuchekwa, I. U. Udo–Akpan, W. I. Osuji

Journal of Advances in Mathematics and Computer Science, Page 1-35
DOI: 10.9734/jamcs/2019/v30i630091

In this paper, the static and dynamic buckling loads of a viscously damped imperfect finite column lying on an elastic foundation with cubic – quintic nonlinearity but trapped by a step load (in the dynamic case) is investigated analytically. The main objective is to determine analytically both the static and dynamic buckling loads by means of perturbation and asymptotic procedures and relate both buckling loads in one single formula. The formulation contains small perturbations particularly in the viscous damping and imperfection amplitude. Multi – timing perturbation techniques and asymptotics are easily utilized in analyzing the problem. The results, which are nontrivially obtained, are implicit in nature and are valid as long as the magnitudes of the small perturbations become asymptotically small compared to unity.

##### Global Existence and Blow-up of Classical Solution for an Attraction-repulsion Chemotaxis System with Logistic Source

Lijun Yan, Zuodong Yang

Journal of Advances in Mathematics and Computer Science, Page 1-16
DOI: 10.9734/jamcs/2019/v30i630092

We consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic-elliptic type with logistic source

under homegeneous Neumann boundary conditions in a bounded domain \Omega\subset R^{n}(n\geq2) with smooth boundary, where
D(u)\geq c_{D}(u+1)^{m-1} with m\geq1and c_{D}>0, f(u)\leq a-bu^{\eta} with \eta>1.{ We show two cases that the system admits a unique
global bounded classical solution depending on 0\leq S(u)\leq C_{s}(u+1)^{q}, 0\leq F(u)\leq C_{F}(u+1)^{g} by Gagliardo-Nirenberg inequality.
For specific D(u),S(u),F(u) with logistic source for \eta>1 and n=2, we establish the finite time blow-up conditions for
solutions that the finite time blow-up occurs at x_{0}\in\Omega whenever \int_{\Omega}u_{0}(x)dx>\frac{8\pi}{\chi\alpha-\xi\gamma}
with \chi\alpha-\xi\gamma>0, under \int_{\Omega}u_{0}(x)|x-x_{0}|^{2}dx sufficiently small.

##### Optimal Control of an ”SIR” Epidemic Model in a Chemostat Using Some Suitable Protein Doses

Miled El Hajji

Journal of Advances in Mathematics and Computer Science, Page 1-15
DOI: 10.9734/jamcs/2019/v30i630095

A modified ”SIR” epidemic model is proposed taking into account of suitable protein doses that are applied on the total population as a control to manage a disease outbreak when treatments are not available. The proteins cause a change in behavior resulting in three susceptible classes. The stability analysis is studied and the optimal control theory is applied to the system of differential equations to achieve the goal of minimizing the infected population (while minimizing the cost). Some numerical simulations are given in order to illustrate the obtained results.

##### Optimal Control Analysis of an Age-Structured Malaria Model Incorporating Children under Five Years and Pregnant Women

George Theodore Azu-Tungmah, Francis T. Oduro, Gabriel A. Okyere

Journal of Advances in Mathematics and Computer Science, Page 1-23
DOI: 10.9734/jamcs/2019/v30i630096

In this article, we apply the optimal control theory to a new age-structured malaria model with three infectious compartments for people under five years, over five years and pregnant women. The model is formulated for malaria endemic areas in the world and the following malaria control strategies ITN, IRS, Chemoprophylaxis and Improved Clinical Treatment were examined and analysed on the mode. The Cost-effectiveness Analysis points out that more attention should be given Insecticide -Treated bed nets (ITNs) in order to eliminate the malaria disease globally because the female Anopheles mosquitoes need human blood to lay their eggs. The expression for the effective reproduction number  has been derived by using the next-generation method. The impact of the controls on the was studied and it came out that all the four controls have a positive impact such that the ITNs can reduce to zero as the value of ITNs approaches one. Pontryagin’s Maximum Principle was applied to analyse the optimal control model theoretically and the optimality system was solved numerically through an iterative scheme.

The optimal plots (Figs. 4-8) reveal that best control strategies for malaria elimination is the combination of ITN, Chemoprophylaxis and Improved Clinical Treatment. However, the Cost-effectiveness Analysis points out that ITN is economically best solution for fighting malaria in poor malaria endemic areas.