
Previous Article
A longtime stable fully discrete approximation of the CahnHilliard equation with inertial term
 PROC Home
 This Issue

Next Article
A natural differential operator on conic spaces
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.
[1] 
Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems  B, 2013, 18 (1) : 3756. doi: 10.3934/dcdsb.2013.18.37 
[2] 
Marc Briant. Stability of global equilibrium for the multispecies Boltzmann equation in $L^\infty$ settings. Discrete & Continuous Dynamical Systems  A, 2016, 36 (12) : 66696688. doi: 10.3934/dcds.2016090 
[3] 
Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete & Continuous Dynamical Systems  S, 2013, 6 (2) : 461478. doi: 10.3934/dcdss.2013.6.461 
[4] 
Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595607. doi: 10.3934/mbe.2007.4.595 
[5] 
Rolf Rannacher. A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis. Discrete & Continuous Dynamical Systems  S, 2012, 5 (6) : 11471194. doi: 10.3934/dcdss.2012.5.1147 
[6] 
Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 14551474. doi: 10.3934/mbe.2013.10.1455 
[7] 
Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure & Applied Analysis, 2011, 10 (5) : 14471462. doi: 10.3934/cpaa.2011.10.1447 
[8] 
Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multigroup SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems  B, 2014, 19 (4) : 11051118. doi: 10.3934/dcdsb.2014.19.1105 
[9] 
Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems  S, 2011, 4 (6) : 15331541. doi: 10.3934/dcdss.2011.4.1533 
[10] 
Ken Shirakawa. Stability analysis for phase field systems associated with crystallinetype energies. Discrete & Continuous Dynamical Systems  S, 2011, 4 (2) : 483504. doi: 10.3934/dcdss.2011.4.483 
[11] 
Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2007, 6 (1) : 6982. doi: 10.3934/cpaa.2007.6.69 
[12] 
Xiong Li. The stability of the equilibrium for a perturbed asymmetric oscillator. Communications on Pure & Applied Analysis, 2006, 5 (3) : 515528. doi: 10.3934/cpaa.2006.5.515 
[13] 
Zhiqi Lu. Global stability for a chemostattype model with delayed nutrient recycling. Discrete & Continuous Dynamical Systems  B, 2004, 4 (3) : 663670. doi: 10.3934/dcdsb.2004.4.663 
[14] 
Napoleon Bame, Samuel Bowong, Josepha Mbang, Gauthier Sallet, JeanJules Tewa. Global stability analysis for SEIS models with n latent classes. Mathematical Biosciences & Engineering, 2008, 5 (1) : 2033. doi: 10.3934/mbe.2008.5.20 
[15] 
Ramon Quintanilla, Reinhard Racke. Stability in thermoelasticity of type III. Discrete & Continuous Dynamical Systems  B, 2003, 3 (3) : 383400. doi: 10.3934/dcdsb.2003.3.383 
[16] 
Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discretetime coupled systems with multidiffusion by graphtheoretic approach and its application. Discrete & Continuous Dynamical Systems  B, 2016, 21 (1) : 253269. doi: 10.3934/dcdsb.2016.21.253 
[17] 
Anna Ghazaryan, Christopher K. R. T. Jones. On the stability of high Lewis number combustion fronts. Discrete & Continuous Dynamical Systems  A, 2009, 24 (3) : 809826. doi: 10.3934/dcds.2009.24.809 
[18] 
Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multigroup SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems  B, 2015, 20 (9) : 30573091. doi: 10.3934/dcdsb.2015.20.3057 
[19] 
Jinhu Xu, Yicang Zhou. Global stability of a multigroup model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems  B, 2016, 21 (3) : 977996. doi: 10.3934/dcdsb.2016.21.977 
[20] 
Jinhu Xu, Yicang Zhou. Global stability of a multigroup model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 10831106. doi: 10.3934/mbe.2015.12.1083 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]