# A mathematical model for the spread of Streptococus Pneumoniae with transmission due to sequence type

Research output: Contribution to journalArticle

### Abstract

This paper discusses a simple mathematical model to describe the spread of Streptococcus pneumoniae. We suppose that the transmission of the bacterium is determined by multi-locus sequence type. The model includes vaccination and is designed to examine what happens in a vaccinated population if MLSTs can exist as both vaccine and non vaccine serotypes with capsular switching possible from the former to the latter. 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 carriage-free 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 multi-locus sequence type with the largest transmission parameter. In the case where both multi-locus 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.
Language English 553-567 15 Discrete and Continuous Dynamical Systems - Supplement 2011 Special Issue 10.3934/proc.2011.2011.553 Published - Oct 2011

### Fingerprint

Vaccines
Mathematical Model
Mathematical models
Locus
Vaccine
Bacteria
Reproduction number
Vaccination
Line
Global Analysis
Less than or equal to
Global Stability
Stability Analysis
Die
Tend
Generalise
Model
Simulation

### Keywords

• streptococcus pneumoniae
• pneumoniae
• pneumococcal colonization
• mathematical analysis
• simulation
• global stability
• equilibrium and stability analysis
• basic reproduction number
• serotype
• multi-locus sequence type

### Cite this

@article{a40f2bea0719416ba39d49496d3a33b9,
title = "A mathematical model for the spread of Streptococus Pneumoniae with transmission due to sequence type",
abstract = "This paper discusses a simple mathematical model to describe the spread of Streptococcus pneumoniae. We suppose that the transmission of the bacterium is determined by multi-locus sequence type. The model includes vaccination and is designed to examine what happens in a vaccinated population if MLSTs can exist as both vaccine and non vaccine serotypes with capsular switching possible from the former to the latter. 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 carriage-free 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 multi-locus sequence type with the largest transmission parameter. In the case where both multi-locus 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.",
keywords = "streptococcus pneumoniae, pneumoniae , pneumococcal colonization , mathematical analysis, simulation , global stability, equilibrium and stability analysis, basic reproduction number, serotype, multi-locus sequence type",
author = "David Greenhalgh and Lamb, {Karen Elaine} and Christopher Robertson",
year = "2011",
month = "10",
doi = "10.3934/proc.2011.2011.553",
language = "English",
volume = "2011",
pages = "553--567",
journal = "Discrete and Continuous Dynamical Systems - Series A",
issn = "1078-0947",
number = "Special Issue",

}

In: Discrete and Continuous Dynamical Systems - Supplement, Vol. 2011, No. Special Issue, 10.2011, p. 553-567.

Research output: Contribution to journalArticle

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AU - Greenhalgh, David

AU - Lamb, Karen Elaine

AU - Robertson, Christopher

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N2 - This paper discusses a simple mathematical model to describe the spread of Streptococcus pneumoniae. We suppose that the transmission of the bacterium is determined by multi-locus sequence type. The model includes vaccination and is designed to examine what happens in a vaccinated population if MLSTs can exist as both vaccine and non vaccine serotypes with capsular switching possible from the former to the latter. 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 carriage-free 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 multi-locus sequence type with the largest transmission parameter. In the case where both multi-locus 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.

AB - This paper discusses a simple mathematical model to describe the spread of Streptococcus pneumoniae. We suppose that the transmission of the bacterium is determined by multi-locus sequence type. The model includes vaccination and is designed to examine what happens in a vaccinated population if MLSTs can exist as both vaccine and non vaccine serotypes with capsular switching possible from the former to the latter. 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 carriage-free 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 multi-locus sequence type with the largest transmission parameter. In the case where both multi-locus 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.

KW - streptococcus pneumoniae

KW - pneumoniae

KW - pneumococcal colonization

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KW - simulation

KW - global stability

KW - equilibrium and stability analysis

KW - basic reproduction number

KW - serotype

KW - multi-locus sequence type

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