Jeremy J. Thibodeaux

Modeling erythropoiesis subject to malaria infection

A mathematical model of erythropoiesis subject to malaria infection is developed by combining ideas from previous models that addressed only one of the two phenomena. The nature of the model allows one to account for suppression of erythropoiesis by the toxin hemozoin, which is a by-product of digested hemoglobin. Following the derivation of the model, numerical simulations are performed and show that the number of parasites produced per bursting erythrocyte has the most significant effect of erythropoiesis. It is also shown that removing hemozoin may be an effective method for aiding the recovery of the erythrocyte population, but is not effective in maintaining a healthy population in the early stages of infection. The second half of the paper introduces an implicit finite difference scheme that was used to perform the simulations previously mentioned. An existence-uniqueness result is then provided via the numerical method.

A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection.

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented.

A monotone approximation for a size-structured population model with a generalized environment

We study a nonlinear size-structured population model with an environment general enough to include hierarchy. We also remove the standard requirement that individuals have nonnegative growth rates, which allows the modeling of populations in which individuals may experience a reduction in size. To show existence and uniqueness of the solution to the model, we establish a comparison principle and construct monotone sequences. A fully discretized numerical scheme based on these monotone sequences is presented and utilized to provide some numerical examples.

Zero Divisor Graphs of Finite Direct Products of Finite Rings

We determine the number of edges of the finite direct product of finite rings. We apply this result to finite rings without idempotents, in particular direct products of ℤ m .