A mathematical model has been developed to explore the co-infection of rotavirus and malaria. A qualitative and comprehensive mathematical analysis has been carried out. A rotavirus only model is considered first. In its analysis, the basic reproduction number (R0) is calculated. The existence of a disease free and a unique positive endemic equilibria is established and are proved to be both globally stable when R0 < 1 and R0 > 1 respectively. The basic reproduction number (Rmr) of the co-infection model has also been established. The disease free equilibrium is proved to be locally stable whenever Rmr < 1 but not globally stable due to coinfection. However, it has been observed that if maximum protection is given against co-infection, then global stability may be achieved. Analysis of co-infection model indicates that it may undergo a forward bifurcation. Numerical simulations using reasonable parameter values indicate that the co-infection persists whenever Rmr is greater than unity and dies out when Rmr is less than unity.
A new collocation method for the solution of a class of second-order two-point boundary value problems associated with physiology and other areas with a singular point at one endpoint is constructed. The singularity of the differential equation is modified by L’Hôpital’s rule and the boundary condition y(0) = 0. Quintic B-spline functions on equidistant collocation points are used to approximate the solution. The quasi-linearization technique is used to reduce a non-linear problem to a sequence of linear problems. The system obtained on discretization is transformed to the system of linear algebraic equations which is easy to be solved. It is proved that the proposed algorithm converges to a smooth approximate solution of the singular boundary value problems and the error estimates are given. To check the theory and to demonstrate the efficiency of the proposed method, several numerical illustrations from physical model problems have been carried out. To show the effectiveness of the proposed method comparisons with several existing methods has also been done. AMS Subject Classification: 65L10, 65L12, 34B16 Received: January 26, 2018 ; Accepted: April 23, 2018 ; Published: May 12, 2018. doi: 10.12732/npsc.v26i1.6 Dynamic Publishers, Inc., Acad. Publishers, Ltd. https://acadsol.eu/npsc
In this note we gave new interpretation of Tembhurne-Sathe modification of Euclidean algorithm for calculation of greatest common divisor (GCD). Our results are different optimized ways of approaches presented in [1]-[26], [44]-[68]. Our approach is about 9% and 69% faster than Tembhurne-Sathe algorithm in iterative and recursive implementations respectively. For computer implementation Visual C# 2017 programming environment is used.
