2006, 3(1): 161-172. doi: 10.3934/mbe.2006.3.161

Epidemic threshold conditions for seasonally forced SEIR models

1. 

Department of Mathematics and Statistics, McMaster University, Hamilton, ON Canada L8S 4K1, Canada

2. 

Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China

Received  January 2005 Revised  April 2005 Published  November 2005

In this paper we derive threshold conditions for eradication of diseases that can be described by seasonally forced susceptible-exposed-infectious-recovered (SEIR) models or their variants. For autonomous models, the basic reproduction number $\mathcal{R}_0 < 1$ is usually both necessary and sufficient for the extinction of diseases. For seasonally forced models, $\mathcal{R}_0$ is a function of time $t$. We find that for models without recruitment of susceptible individuals (via births or loss of immunity), max$_t{\mathcal{R}_0(t)} < 1$ is required to prevent outbreaks no matter when and how the disease is introduced. For models with recruitment, if the latent period can be neglected, the disease goes extinct if and only if the basic reproduction number $\bar{\mathcal{R}}$ of the time-average systems (the autonomous systems obtained by replacing the time-varying parameters with their long-term time averages) is less than 1. Otherwise, $\bar{\mathcal{R}} < 1$ is sufficient but not necessary for extinction. Thus, reducing $\bar{\mathcal{R}}$ of the average system to less than 1 is sufficient to prevent or curtail the spread of an endemic disease.
Citation: Junling Ma, Zhien Ma. Epidemic threshold conditions for seasonally forced SEIR models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 161-172. doi: 10.3934/mbe.2006.3.161
[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) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[2]

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

[3]

Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073

[4]

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) : 1455-1474. doi: 10.3934/mbe.2013.10.1455

[5]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166

[6]

Jinhuo Luo, Jin Wang, Hao Wang. Seasonal forcing and exponential threshold incidence in cholera dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2261-2290. doi: 10.3934/dcdsb.2017095

[7]

Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz, Huaiping Zhu. How seasonal forcing influences the complexity of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 785-807. doi: 10.3934/dcdsb.2018043

[8]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239

[9]

E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229

[10]

Sukhitha W. Vidurupola, Linda J. S. Allen. Basic stochastic models for viral infection within a host. Mathematical Biosciences & Engineering, 2012, 9 (4) : 915-935. doi: 10.3934/mbe.2012.9.915

[11]

Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565

[12]

Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457

[13]

Fred Brauer. Some simple epidemic models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 1-15. doi: 10.3934/mbe.2006.3.1

[14]

Fred Brauer, Zhilan Feng, Carlos Castillo-Chávez. Discrete epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (1) : 1-15. doi: 10.3934/mbe.2010.7.1

[15]

James M. Hyman, Jia Li. Differential susceptibility and infectivity epidemic models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 89-100. doi: 10.3934/mbe.2006.3.89

[16]

Julien Arino, Fred Brauer, P. van den Driessche, James Watmough, Jianhong Wu. A final size relation for epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 159-175. doi: 10.3934/mbe.2007.4.159

[17]

Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 333-345. doi: 10.3934/dcdsb.2007.8.333

[18]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445

[19]

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

[20]

Zhen Jin, Guiquan Sun, Huaiping Zhu. Epidemic models for complex networks with demographics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1295-1317. doi: 10.3934/mbe.2014.11.1295

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]