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This paper develops a second-order explicit predictor-corrector numericalapproach for solving a mathematical model on the dynamic of cytokineexpressions and human immune cell activation in response to the bacteriumstaphylococcus aureus (S. aureus). The proposed algorithm is at leastzero-stable and second-order accurate. Mathematical modeling works that analyzethe human body in response to some antigens have predicted concentrations of abroad range of cells and cytokines. This study deals with a coupledcellular-cytokine model which predicts cytokine expressions in response togram-positive bacteria S. aureus. Tumor necrosis factor alpha, interleukin 6,interleukin 8 and interleukin 10 are included to assess the relationshipbetween cytokine release from macrophages and the concentration of the S.aureus antigen. Ordinary differential equations are used to model cytokinelevels while the cellular responses are modeled by partial differentialequations. Interactions between both components provide a more robust andcomplete systems of immune activation. In the numerical simulations, a lowconcentration of S. aureus is used to measure cellular activation and cytokineexpressions. Numerical experiments indicate how the human immune systemresponds to infections from different pathogens. Furthermore, numericalexamples suggest that the new technique is faster and more efficient than alarge class of statistical and numerical schemes discussed in the literaturefor systems of nonlinear equations and can serve as a robust tool for theintegration of general systems of initial-boundary value problems.

A Fast Second-Order Explicit Predictor-Corrector Numerical Technique To  Investigating And Predicting The Dynamic Of Cytokine Levels And Human Immune  Cells Activation In Response To Gram-Positive Bacteria: Staphylococcus Aureus