A Simple Epidemic Model with Surprising Dynamics

F. Berezovskaya; G. Karev; Baojun Song; Carlos Castillo-Chavez; F. Berezovskaya; G. Karev; Baojun Song; Carlos Castillo-Chavez

doi:10.3934/mbe.2005.2.133

Mathematical Biosciences and Engineering

2005, Volume 2, Issue 1: 133-152. doi: 10.3934/mbe.2005.2.133

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A Simple Epidemic Model with Surprising Dynamics

1.
Department of Mathematics, Howard University, Washington D.C., 20059
2.
Oak Ridge Institute for Science and Education (ORISE) 8600 Rockville Pike, Bldg. 38A, Rm. 5N511N, Bethesda, MD 20894
3.
Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043
4.
Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287

Received: 01 July 2004 Accepted: 29 June 2018 Published: 01 November 2004
MSC : 92D30, 34C37, 37G35.

A simple model incorporating demographic and epidemiological processes is explored. Four re-parameterized quantities the basic demographic reproductive number ($\R_d$), the basic epidemiological reproductive number ($\R_0$), the ratio ($\nu$) between the average life spans of susceptible and infective class, and the relative fecundity of infectives ($\theta$), are utilized in qualitative analysis. Mathematically, non-analytic vector fields are handled by blow-up transformations to carry out a complete and global dynamical analysis. A family of homoclinics is found, suggesting that a disease outbreak would be ignited by a tiny number of infectious individuals.
- dynamical system,
- bifurcation analysis,
- global stability.,
- Epidemic model
Citation: F. Berezovskaya, G. Karev, Baojun Song, Carlos Castillo-Chavez. A Simple Epidemic Model with Surprising Dynamics[J]. Mathematical Biosciences and Engineering, 2005, 2(1): 133-152. doi: 10.3934/mbe.2005.2.133

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Abstract

A simple model incorporating demographic and epidemiological processes is explored. Four re-parameterized quantities the basic demographic reproductive number ($\R_d$), the basic epidemiological reproductive number ($\R_0$), the ratio ($\nu$) between the average life spans of susceptible and infective class, and the relative fecundity of infectives ($\theta$), are utilized in qualitative analysis. Mathematically, non-analytic vector fields are handled by blow-up transformations to carry out a complete and global dynamical analysis. A family of homoclinics is found, suggesting that a disease outbreak would be ignited by a tiny number of infectious individuals.
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