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A longtime stable fully discrete approximation of the CahnHilliard equation with inertial term
A mathematical model for the spread of streptococcus pneumoniae with transmission due to sequence type
1.  Department of Statistics and Modelling Science, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH 
2.  Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26, Richmond Street, Gasgow G1 1XH, United Kingdom, United Kingdom 
We start off with a discussion of Streptococcus pneumoniae and a review of previous work. We propose a simple mathematical model with two sequence types and then perform an equilibrium and (global) stability analysis on the model. We show that in general there are only three equilibria, the carriagefree equilibrium and two carriage equilibria. If the effective reproduction number $R_e$ is less than or equal to one, then the carriage will die out. If $R_e$ > 1, then the carriage will tend to the carriage equilibrium corresponding to the multilocus sequence type with the largest transmission parameter. In the case where both multilocus sequence types have the same transmission parameter then there is a line of carriage equilibria. Provided that carriage is initially present then as time progresses the carriage will approach a point on this line. The results generalize to many competing sequence types. Simulations with realistic parameter values confirm the analytical results.
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