Eunice W. Mureithi

Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model

When both human and mosquito populations vary, forward bifurcation occurs if the basic reproduction number R 0 is less than one in the absence of disease-induced death. When the disease-induced death rate is large enough, R 0 = 1 is a subcritical backward bifurcation point. The domain for the study of the dynamics is reduced to a compact and feasible region, where the system admits a specific algebraic decomposition into infective and non-infected humans and mosquitoes. Stability results are extended and the possibility of backward bifurcation is clarified. A dynamically consistent nonstandard finite difference scheme is designed.

Weakly nonlinear wave motions in a thermally stratified boundary layer

We consider weakly nonlinear wave motions in a thermally stratified boundary layer. Attention is focused on the upper branch of the neutral stability curve, corresponding to small wavelengths and large Reynolds number. In this limit the motion is governed by a first harmonic/mean flow interaction theory in which the wave-induced mean flow is of the same order of magnitude as the wave component of the flow. We show that the flow is governed by a system of three coupled partial differential equations which admit finite-amplitude periodic solutions bifurcating from the linear, neutral points.

Analysis and Dynamically Consistent Numerical Schemes for the SIS Model and Related Reaction Diffusion Equation

The classical SIS epidemiological model is extended in two directions: (a) The number of adequate contacts per infective in unit time is assumed to be a function of the total population in such a way that this number grows less rapidly as the total population increases; (b) A diffusion term is added to the SIS model and this leads to a reaction diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the model (a), with the disease‐free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, traveling wave solutions are found for the model (b). Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous models are presented. In particular, for the model (a), a nonstandard version of the Runge‐Kutta method having high order of convergence is investigated. Numerical experiments that support the th...

Absolute-convective instability of mixed forced-free convection boundary layers

A spatio-temporal inviscid instability of a mixed forced-free convection boundary layer is investigated. The base flow considered is the self-similar flow with free-stream velocity ue ~ xn. Such a boundary-layer flow presents the unusual behaviour of generating a region of velocity overshoot, in which the streamwise velocity within the boundary layer exceeds the free-stream speed. A linear stability analysis has been carried out. Saddle points have been located and a critical value for the buoyancy parameter, G0c ≈ 3.6896, has been determined below which the flow is convectively unstable and above which the flow becomes absolutely unstable. Two spatial modes have been obtained, one mode being convective in nature and the other absolute. The convective-type spatial mode shows mode crossing behaviour at lower frequencies. Thermal buoyancy is shown to be destabilizing to the absolutely unstable spatial mode.