Primary: 92D30, 34A34; Secondary: 34D20, 34D23.

Export file:

Format

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

Content

• Citation Only
• Citation and Abstract

Global stability for epidemic model with constant latency and infectious periods

1. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074
2. CIMAB, University of Milano, via C. Saldini 50, I20133 Milano
3. Graduate School of Science and Technology, Shizuoka University, Hamamatsu, 4328561

## Abstract    Related pages

In recent years many delay epidemiological models have been proposed to study at which stage of the epidemics the delays can destabilize the disease free equilibrium, or the endemic equilibrium, giving rise to stability switches. One of these models is the SEIR model with constant latency time and infectious periods [2], for which the authors have proved that the two delays are harmless in inducing stability switches. However, it is left open the problem of the global asymptotic stability of the endemic equilibrium whenever it exists. Even the Lyapunov functions approach, recently proposed by Huang and Takeuchi to study many delay epidemiological models, fails to work on this model. In this paper, an age-infection model is presented for the delay SEIR epidemic model, such that the properties of global asymptotic stability of the equilibria of the age-infection model imply the same properties for the original delay-differential epidemic model. By introducing suitable Lyapunov functions to study the global stability of the disease free equilibrium (when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $\mathcal{R}_0>1$) of the age-infection model, we can infer the corresponding global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in [2] is globally asymptotically stable whenever it exists.
Furthermore, we also present a review of the SIR, SEIR epidemic models, with and without delays, appeared in literature, that can be seen as particular cases of the approach presented in the paper.
Figure/Table
Supplementary
Article Metrics

Citation: Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences and Engineering, 2012, 9(2): 297-312. doi: 10.3934/mbe.2012.9.297

• 1. Christian Selinger, Michael G Katze, Mathematical models of viral latency, Current Opinion in Virology, 2013, 3, 4, 402, 10.1016/j.coviro.2013.06.015
• 2. D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese, R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), Journal of Biological Dynamics, 2012, 6, sup2, 103, 10.1080/17513758.2012.716454
• 3. Bentout Soufiane, Tarik Mohammed Touaoula, Global analysis of an infection age model with a class of nonlinear incidence rates, Journal of Mathematical Analysis and Applications, 2016, 434, 2, 1211, 10.1016/j.jmaa.2015.09.066
• 4. P. van den Driessche, Zhisheng Shuai, Fred Brauer, Dynamics of an age-of-infection cholera model, Mathematical Biosciences and Engineering, 2013, 10, 5/6, 1335, 10.3934/mbe.2013.10.1335
• 5. Salih Djilali, Tarik Mohammed Touaoula, Sofiane El-Hadi Miri, A Heroin Epidemic Model: Very General Non Linear Incidence, Treat-Age, and Global Stability, Acta Applicandae Mathematicae, 2017, 152, 1, 171, 10.1007/s10440-017-0117-2
• 6. Xavier Bardina, Marco Ferrante, Carles Rovira, Stochastic Epidemic SEIRS Models with a Constant Latency Period, Mediterranean Journal of Mathematics, 2017, 14, 4, 10.1007/s00009-017-0977-8
• 7. Cruz Vargas-De-León, Global stability properties of age-dependent epidemic models with varying rates of recurrence, Mathematical Methods in the Applied Sciences, 2016, 39, 8, 2057, 10.1002/mma.3621
• 8. Gang Huang, Chenguang Nie, Yueping Dong, Global stability for an SEI model of infectious diseases with immigration and age structure in susceptibility, International Journal of Biomathematics, 2019, 10.1142/S1793524519500426