Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

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.
  Article Metrics

Keywords 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. Mathematical Biosciences and Engineering, 2005, 2(1): 133-152. doi: 10.3934/mbe.2005.2.133


This article has been cited by

  • 1. Junyuan Yang, Xiaoyan Wang, Threshold Dynamics of an SIR Model with Nonlinear Incidence Rate and Age-Dependent Susceptibility, Complexity, 2018, 2018, 1, 10.1155/2018/9613807
  • 2. Yongli Cai, Jianjun Jiao, Zhanji Gui, Yuting Liu, Weiming Wang, Environmental variability in a stochastic epidemic model, Applied Mathematics and Computation, 2018, 329, 210, 10.1016/j.amc.2018.02.009
  • 3. Xinxin Li, Yongli Cai, Kai Wang, Shengmao Fu, Weiming Wang, Non-constant positive steady states of a host-parasite model with frequency- and density-dependent transmissions, Journal of the Franklin Institute, 2020, 10.1016/j.jfranklin.2020.02.058
  • 4. Faina S. Berezovskaya, Artem S. Novozhilov, Georgy P. Karev, Population models with singular equilibrium, Mathematical Biosciences, 2007, 208, 1, 270, 10.1016/j.mbs.2006.10.006
  • 5. John E. Franke, Abdul-Aziz Yakubu, Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models, Journal of Mathematical Biology, 2008, 57, 6, 755, 10.1007/s00285-008-0188-9
  • 6. Jean M. Tchuenche, Alexander Nwagwo, Local stability of anSIRepidemic model and effect of time delay, Mathematical Methods in the Applied Sciences, 2009, 32, 16, 2160, 10.1002/mma.1136
  • 7. Hai-Feng Huo, Zhan-Ping Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Communications in Nonlinear Science and Numerical Simulation, 2010, 15, 2, 459, 10.1016/j.cnsns.2009.04.018
  • 8. Yuqin Zhao, Daniel T. Wood, Hristo V. Kojouharov, Yang Kuang, Dobromir T. Dimitrov, Impact of Population Recruitment on the HIV Epidemics and the Effectiveness of HIV Prevention Interventions, Bulletin of Mathematical Biology, 2016, 78, 10, 2057, 10.1007/s11538-016-0211-z
  • 9. Yu. V. Tyutyunov, L. I. Titova, From Lotka–Volterra to Arditi–Ginzburg: 90 Years of Evolving Trophic Functions, Biology Bulletin Reviews, 2020, 10, 3, 167, 10.1134/S207908642003007X
  • 10. T.K. Kar, Ashim Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems, 2011, 104, 2-3, 127, 10.1016/j.biosystems.2011.02.001
  • 11. Yongli Cai, Weiming Wang, Spatiotemporal dynamics of a reaction–diffusion epidemic model with nonlinear incidence rate, Journal of Statistical Mechanics: Theory and Experiment, 2011, 2011, 02, P02025, 10.1088/1742-5468/2011/02/P02025
  • 12. T.K. Kar, Prasanta Kumar Mondal, Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear Analysis: Real World Applications, 2011, 12, 4, 2058, 10.1016/j.nonrwa.2010.12.021
  • 13. , Periodically forced discrete-time SIS epidemic model with disease induced mortality, Mathematical Biosciences and Engineering, 2011, 8, 2, 385, 10.3934/mbe.2011.8.385
  • 14. Joseph J. Crivelli, Juraj Földes, Peter S. Kim, Joanna R. Wares, A mathematical model for cell cycle-specific cancer virotherapy, Journal of Biological Dynamics, 2012, 6, sup1, 104, 10.1080/17513758.2011.613486
  • 15. Chengjun Yuan, Daqing Jiang, Donal O’Regan, Ravi P. Agarwal, Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation, Communications in Nonlinear Science and Numerical Simulation, 2012, 17, 6, 2501, 10.1016/j.cnsns.2011.07.025
  • 16. Weiming Wang, Yongli Cai, Mingjiang Wu, Kaifa Wang, Zhenqing Li, Complex dynamics of a reaction–diffusion epidemic model, Nonlinear Analysis: Real World Applications, 2012, 13, 5, 2240, 10.1016/j.nonrwa.2012.01.018
  • 17. , Global analysis of a simple parasite-host model with homoclinic orbits, Mathematical Biosciences and Engineering, 2012, 9, 4, 767, 10.3934/mbe.2012.9.767
  • 18. Fathalla A. Rihan, M. Naim Anwar, Qualitative Analysis of Delayed SIR Epidemic Model with a Saturated Incidence Rate, International Journal of Differential Equations, 2012, 2012, 1, 10.1155/2012/408637
  • 19. , The ratio of hidden HIV infection in Cuba, Mathematical Biosciences and Engineering, 2013, 10, 4, 959, 10.3934/mbe.2013.10.959
  • 20. Sheng Wang, Wenbin Liu, Zhengguang Guo, Weiming Wang, Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model, Abstract and Applied Analysis, 2013, 2013, 1, 10.1155/2013/216913
  • 21. A.K. Misra, S.N. Mishra, A.L. Pathak, P.K. Srivastava, Peeyush Chandra, A mathematical model for the control of carrier-dependent infectious diseases with direct transmission and time delay, Chaos, Solitons & Fractals, 2013, 57, 41, 10.1016/j.chaos.2013.08.002
  • 22. Baojun Song, Zhilan Feng, Gerardo Chowell, From the guest editors, Mathematical Biosciences and Engineering, 2013, 10, 5/6, 10.3934/mbe.2013.10.5i
  • 23. Yuan Yuan, Hailing Wang, Weiming Wang, The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model, Abstract and Applied Analysis, 2013, 2013, 1, 10.1155/2013/921401
  • 24. Gui-Quan Sun, Zhenguo Bai, Zi-Ke Zhang, Tao Zhou, Zhen Jin, Positive Periodic Solutions of an Epidemic Model with Seasonality, The Scientific World Journal, 2013, 2013, 1, 10.1155/2013/470646
  • 25. Yun Kang, Carlos Castillo-Chávez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness, Discrete & Continuous Dynamical Systems - B, 2014, 19, 1, 89, 10.3934/dcdsb.2014.19.89
  • 26. Tao Wang, Dynamics of an epidemic model with spatial diffusion, Physica A: Statistical Mechanics and its Applications, 2014, 409, 119, 10.1016/j.physa.2014.04.028
  • 27. Jing Li, Gui-Quan Sun, Zhen Jin, Pattern formation of an epidemic model with time delay, Physica A: Statistical Mechanics and its Applications, 2014, 403, 100, 10.1016/j.physa.2014.02.025
  • 28. Ranjit Kumar Upadhyay, Parimita Roy, Vikas Rai, Deciphering Dynamics of Epidemic Spread: The Case of Influenza Virus, International Journal of Bifurcation and Chaos, 2014, 24, 05, 1450064, 10.1142/S0218127414500643
  • 29. Yongli Cai, Weiming Wang, Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete and Continuous Dynamical Systems - Series B, 2015, 20, 4, 989, 10.3934/dcdsb.2015.20.989
  • 30. Pan-Ping Liu, Periodic solutions in an epidemic model with diffusion and delay, Applied Mathematics and Computation, 2015, 265, 275, 10.1016/j.amc.2015.05.028
  • 31. Yongli Cai, Yuan Yuan, Xinze Lian, Weiming Wang, Extinction in a Feline Panleukopenia virus model incorporating direct and indirect transmissions, Applied Mathematics and Computation, 2015, 258, 358, 10.1016/j.amc.2015.01.021
  • 32. Yongli Cai, Shuling Yan, Hailing Wang, Xinze Lian, Weiming Wang, Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission, International Journal of Bifurcation and Chaos, 2015, 25, 08, 1550099, 10.1142/S0218127415500996
  • 33. Wei Tan, Jianguo Gao, Wenjun Fan, Bifurcation Analysis and Chaos Control in a Discrete Epidemic System, Discrete Dynamics in Nature and Society, 2015, 2015, 1, 10.1155/2015/974868
  • 34. Li-Peng Song, Rong-Ping Zhang , Li-Ping Feng , Qiong Shi, Pattern dynamics of a spatial epidemic model with time delay, Applied Mathematics and Computation, 2017, 292, 390, 10.1016/j.amc.2016.07.013
  • 35. Yuting Liu, Meijing Shan, Xinze Lian, Weiming Wang, Stochastic extinction and persistence of a parasite–host epidemiological model, Physica A: Statistical Mechanics and its Applications, 2016, 462, 586, 10.1016/j.physa.2016.06.022
  • 36. Muhammad Altaf Khan, Ebenezer Bonyah, Shujaat Ali, Saeed Islam, Saima Naz Khan, Stability analysis of delay seirepidemic model, International Journal of ADVANCED AND APPLIED SCIENCES, 2016, 3, 7, 46, 10.21833/ijaas.2016.07.008

Reader Comments

your name: *   your email: *  

Copyright Info: 2005, F. Berezovskaya, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved