Mathematical Biosciences and Engineering, 2014, 11(4): 679-721. doi: 10.3934/mbe.2014.11.679

Primary: 92-08; Secondary: 92B05.

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A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme

1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
2. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010
3. Department of Biology, University of Louisiana at Lafayette, Lafayette, LA 70504-2451

We develop a finite difference scheme to approximate the solution of a novel size-structured mathematical model of the transmission dynamics of Mycobacterium marinum (Mm) in an aquatic environment. The model consists of a system of nonlinear hyperbolic partial differential equations coupled with three nonlinear ordinary differential equations. Existence and uniqueness results are established and convergence of the finite difference approximation to the unique bounded variation weak solution of the model is obtained. Numerical simulations demonstrating the accuracy of the method are presented. We also conducted preliminary studies on the key features of this model, such as various forms of growth rates (indicative of possible theories of development), and conditions for competitive exclusion or coexistence as determined by reproductive fitness and genetic spread in the population.
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