Competing species models with an infectious disease

Previous Article
The effect of global travel on the spread of SARS
MBE Home
This Issue
Next Article
An improved model of t cell development in the thymus and its stability analysis

2006, 3(1): 219-235. doi: 10.3934/mbe.2006.3.219

Competing species models with an infectious disease

Roberto A. Saenz ^1, and Herbert W. Hethcote ^1,

1.	Applied Mathematical and Computational Sciences, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242, United States, United States

Received January 2005 Revised April 2005 Published November 2005

Abstract
Related Papers

The frequency-dependent (standard) form of the incidence is used for the transmission dynamics of an infectious disease in a competing species model. In the global analysis of the SIS model with the birth rate independent of the population size, a modified reproduction number $\mathbf{R}_1$ determines the asymptotic behavior, so that the disease dies out if $\mathbf{R}_1 \leq 1$ and approaches a globally attractive endemic equilibrium if $\mathbf{R}_1 > 1$. Because the disease- reduced reproduction and disease-related death rates are often different in two competing species, a shared disease can change the outcome of the competition. Models of SIR and SIRS type are also considered. A key result in all of these models with the frequency-dependent incidence is that the disease must either die out in both species or remain endemic in both species.

Keywords: coexistence, mathematical models, epidemiology, dynamics, competing species, ecology, infectious disease, reproduction number..

Mathematics Subject Classification: 92D30, 92D4.

Citation: Roberto A. Saenz, Herbert W. Hethcote. Competing species models with an infectious disease. Mathematical Biosciences & Engineering, 2006, 3 (1) : 219-235. doi: 10.3934/mbe.2006.3.219

[1]	Linda J. S. Allen, Vrushali A. Bokil. Stochastic models for competing species with a shared pathogen. Mathematical Biosciences & Engineering, 2012, 9 (3) : 461-485. doi: 10.3934/mbe.2012.9.461
[2]	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) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[3]	Horst R. Thieme. Distributed susceptibility: A challenge to persistence theory in infectious disease models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 865-882. doi: 10.3934/dcdsb.2009.12.865
[4]	Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
[5]	Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[6]	Alexander V. Budyansky, Kurt Frischmuth, Vyacheslav G. Tsybulin. Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 547-561. doi: 10.3934/dcdsb.2018196
[7]	Julián López-Gómez. On the structure of the permanence region for competing species models with general diffusivities and transport effects. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 525-542. doi: 10.3934/dcds.1996.2.525
[8]	Sze-Bi Hsu, Chiu-Ju Lin. Dynamics of two phytoplankton species competing for light and nutrient with internal storage. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1259-1285. doi: 10.3934/dcdss.2014.7.1259
[9]	Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989
[10]	Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841
[11]	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) : 595-607. doi: 10.3934/mbe.2007.4.595
[12]	John D. Nagy. The Ecology and Evolutionary Biology of Cancer: A Review of Mathematical Models of Necrosis and Tumor Cell Diversity. Mathematical Biosciences & Engineering, 2005, 2 (2) : 381-418. doi: 10.3934/mbe.2005.2.381
[13]	David J. Gerberry. An exact approach to calibrating infectious disease models to surveillance data: The case of HIV and HSV-2. Mathematical Biosciences & Engineering, 2018, 15 (1) : 153-179. doi: 10.3934/mbe.2018007
[14]	S.M. Moghadas. Modelling the effect of imperfect vaccines on disease epidemiology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 999-1012. doi: 10.3934/dcdsb.2004.4.999
[15]	Carlos M. Hernández-Suárez, Oliver Mendoza-Cano. Applications of occupancy urn models to epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (3) : 509-520. doi: 10.3934/mbe.2009.6.509
[16]	Andreas Widder. On the usefulness of set-membership estimation in the epidemiology of infectious diseases. Mathematical Biosciences & Engineering, 2018, 15 (1) : 141-152. doi: 10.3934/mbe.2018006
[17]	Sara Y. Del Valle, J. M. Hyman, Nakul Chitnis. Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1475-1497. doi: 10.3934/mbe.2013.10.1475
[18]	Yang Kuang, John D. Nagy, James J. Elser. Biological stoichiometry of tumor dynamics: Mathematical models and analysis. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 221-240. doi: 10.3934/dcdsb.2004.4.221
[19]	Tsanou Berge, Samuel Bowong, Jean Lubuma, Martin Luther Mann Manyombe. Modeling Ebola Virus Disease transmissions with reservoir in a complex virus life ecology. Mathematical Biosciences & Engineering, 2018, 15 (1) : 21-56. doi: 10.3934/mbe.2018002
[20]	Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571

2018 Impact Factor: 1.313

American Institute of Mathematical Sciences