Export file:


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


  • Citation Only
  • Citation and Abstract

Epidemic models with nonlinear infection forces

1. Department of Mathematics, Southwest Normal University, Chongqing, 400715, PR

Epidemic models with behavior changes are studied to consider effects of protection measures and intervention policies. It is found that intervention strategies decrease endemic levels and tend to make the dynamical behavior of a disease evolution simpler. For a saturated infection force, the model may admit a stable disease-free equilibrium and a stable endemic equilibrium at the same time. If we vary a recovery rate, numerical simulations show that the boundaries of the region for the persistence of the disease undergo the changes from the separatrix of a saddle to an unstable limit cycle. If the inhibition effect from behavior changes is weak, we find two limit cycles and obtain bifurcations of the model as the population size changes. We also find that the disease may die out although there are two endemic equilibria.
  Article Metrics

Keywords basic reproduction number; epidemic; cycles.; stability; nonlinear incidence

Citation: Wendi Wang. Epidemic models with nonlinear infection forces. Mathematical Biosciences and Engineering, 2006, 3(1): 267-279. doi: 10.3934/mbe.2006.3.267


This article has been cited by

  • 1. Muhammad Ozair, Analysis of Pine Wilt Disease Model with Nonlinear Incidence and Horizontal Transmission, Journal of Applied Mathematics, 2014, 2014, 1, 10.1155/2014/204241
  • 2. Li-Ming Cai, Xue-Zhi Li, Global analysis of a vector-host epidemic model with nonlinear incidences, Applied Mathematics and Computation, 2010, 217, 7, 3531, 10.1016/j.amc.2010.09.028
  • 3. Zhaoyang Zhang, Honggang Wang, Chonggang Wang, Hua Fang, Modeling Epidemics Spreading on Social Contact Networks, IEEE Transactions on Emerging Topics in Computing, 2015, 3, 3, 410, 10.1109/TETC.2015.2398353
  • 4. Weiming Wang, Yun Kang, Yongli Cai, Global stability of the steady states of an epidemic model incorporating intervention strategies, Mathematical Biosciences and Engineering, 2017, 14, 5/6, 1071, 10.3934/mbe.2017056
  • 5. Yakui Xue, Xiaoming Tang, Xinpeng Yuan, Bifurcation Analysis of an SIV Epidemic Model with the Saturated Incidence Rate, International Journal of Bifurcation and Chaos, 2014, 24, 05, 1450060, 10.1142/S0218127414500606
  • 6. Guihua Li, Wendi Wang, Bifurcation analysis of an epidemic model with nonlinear incidence, Applied Mathematics and Computation, 2009, 214, 2, 411, 10.1016/j.amc.2009.04.012
  • 7. GIUSEPPE MULONE, BRIAN STRAUGHAN, MODELING BINGE DRINKING, International Journal of Biomathematics, 2012, 05, 01, 1250005, 10.1142/S1793524511001453
  • 8. W. Wang, Modeling Adaptive Behavior in Influenza Transmission, Mathematical Modelling of Natural Phenomena, 2012, 7, 3, 253, 10.1051/mmnp/20127315
  • 9. Guihua Li, Gaofeng Li, Bifurcation Analysis of an SIR Epidemic Model with the Contact Transmission Function, Abstract and Applied Analysis, 2014, 2014, 1, 10.1155/2014/930541
  • 10. G. Rozhnova, A. Nunes, Fluctuations and oscillations in a simple epidemic model, Physical Review E, 2009, 79, 4, 10.1103/PhysRevE.79.041922
  • 11. Jinhui Li, Zhidong Teng, Bifurcations of an SIRS model with generalized non-monotone incidence rate, Advances in Difference Equations, 2018, 2018, 1, 10.1186/s13662-018-1675-y
  • 12. Muhammad Ozair, Abid Ali Lashari, Il Hyo Jung, Kazeem Oare Okosun, Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence, Discrete Dynamics in Nature and Society, 2012, 2012, 1, 10.1155/2012/595487
  • 13. Yukihiko Nakata, Yoichi Enatsu, Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate, Mathematical Biosciences and Engineering, 2014, 11, 4, 785, 10.3934/mbe.2014.11.785
  • 14. Mingju Ma, Sanyang Liu, Jun Li, Traveling waves of a diffusive epidemic model with nonlinear infection forces, Advances in Difference Equations, 2016, 2016, 1, 10.1186/s13662-016-0972-6
  • 15. Xiaoyan Gao, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang, Positive steady states in an epidemic model with nonlinear incidence rate, Computers & Mathematics with Applications, 2018, 75, 2, 424, 10.1016/j.camwa.2017.09.029
  • 16. 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
  • 17. Nicolas Bacaër, Rachid Ouifki, Carel Pretorius, Robin Wood, Brian Williams, Modeling the joint epidemics of TB and HIV in a South African township, Journal of Mathematical Biology, 2008, 57, 4, 557, 10.1007/s00285-008-0177-z
  • 18. Florinda Capone, Valentina De Cataldis, Roberta De Luca, Influence of diffusion on the stability of equilibria in a reaction–diffusion system modeling cholera dynamic, Journal of Mathematical Biology, 2015, 71, 5, 1107, 10.1007/s00285-014-0849-9
  • 19. Yongli Cai, Yun Kang, Malay Banerjee, Weiming Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 2015, 259, 12, 7463, 10.1016/j.jde.2015.08.024
  • 20. Zigen Song, Jian Xu, Qunhong Li, Local and global bifurcations in an SIRS epidemic model, Applied Mathematics and Computation, 2009, 214, 2, 534, 10.1016/j.amc.2009.04.027
  • 21. Andrew J. Black, Alan J. McKane, Ana Nunes, Andrea Parisi, Stochastic fluctuations in the susceptible-infective-recovered model with distributed infectious periods, Physical Review E, 2009, 80, 2, 10.1103/PhysRevE.80.021922
  • 22. Alberto d’Onofrio, Piero Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, Journal of Theoretical Biology, 2009, 256, 3, 473, 10.1016/j.jtbi.2008.10.005
  • 23. Paul Georgescu, Gheorghe Moroşanu, Pest regulation by means of impulsive controls, Applied Mathematics and Computation, 2007, 190, 1, 790, 10.1016/j.amc.2007.01.079
  • 24. Yilei Tang, Deqing Huang, Shigui Ruan, Weinian Zhang, Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate, SIAM Journal on Applied Mathematics, 2008, 69, 2, 621, 10.1137/070700966
  • 25. Betty Nannyonga, Joseph Ssebuliba, Juliet Nakakawa, Betty Nabiyonga, J.Y.T. Mugisha, To apprehend or not to apprehend: A mathematical model for ending student strikes in a university, Applied Mathematics and Computation, 2018, 339, 607, 10.1016/j.amc.2018.07.034
  • 26. Weiming Wang, Xiaoyan Gao, Yongli Cai, Hongbo Shi, Shengmao Fu, Turing patterns in a diffusive epidemic model with saturated infection force, Journal of the Franklin Institute, 2018, 10.1016/j.jfranklin.2018.07.014
  • 27. Yongli Cai, Xinze Lian, Zhihang Peng, Weiming Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Analysis: Real World Applications, 2019, 46, 178, 10.1016/j.nonrwa.2018.09.006
  • 28. Feng Rao, Partha S. Mandal, Yun Kang, Complicated endemics of an SIRS model with a generalized incidence under preventive vaccination and treatment controls, Applied Mathematical Modelling, 2019, 67, 38, 10.1016/j.apm.2018.10.016
  • 29. Xiaojie Mu, Qimin Zhang, Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching, Mathematical Methods in the Applied Sciences, 2018, 10.1002/mma.5378
  • 30. Bin Yang, Yongli Cai, Kai Wang, Weiming Wang, Global threshold dynamics of a stochastic epidemic model incorporating media coverage, Advances in Difference Equations, 2018, 2018, 1, 10.1186/s13662-018-1925-z
  • 31. M Simões, M.M Telo da Gama, A Nunes, Stochastic fluctuations in epidemics on networks, Journal of The Royal Society Interface, 2008, 5, 22, 555, 10.1098/rsif.2007.1206
  • 32. Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete and Continuous Dynamical Systems - Series B, 2010, 15, 1, 93, 10.3934/dcdsb.2011.15.93
  • 33. Min Lu, Jicai Huang, Shigui Ruan, Pei Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, Journal of Differential Equations, 2019, 10.1016/j.jde.2019.03.005
  • 34. Qun Liu, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi, Stationary distribution of a stochastic within-host dengue infection model with immune response and regime switching, Physica A: Statistical Mechanics and its Applications, 2019, 121057, 10.1016/j.physa.2019.121057
  • 35. Rita Ghosh, Uttam Ghosh, , Applied Mathematics, 2015, Chapter 21, 219, 10.1007/978-81-322-2547-8_21
  • 36. Yan Wang, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi, Stationary distribution of an HIV model with general nonlinear incidence rate and stochastic perturbations, Journal of the Franklin Institute, 2019, 10.1016/j.jfranklin.2019.06.035

Reader Comments

your name: *   your email: *  

Copyright Info: 2006, Wendi Wang, 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