# Stability Analysis of an Age-Structured SEIRS Model with Time Delay

^{*}

## Abstract

**:**

## 1. Introduction

## 2. An Age-Structured SEIRS Model with Time Delay

## 3. Traveling Wave Solution

**Theorem**

**1.**

**Proof.**

## 4. Stability Analysis of System

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 5. Simulation

## 6. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kermack, W.O.; McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A
**1927**, 115, 700–721. [Google Scholar] [CrossRef] [Green Version] - Brauer, F.; Castillo-Chavez, C. Mathematical Models in Population Biology and Epidemiology; Springer: New York, NY, USA, 2012. [Google Scholar]
- Zhou, X.; Cui, J.A. Analysis stability and bifurcation for an SEIR epidemic model with saturated recovery rate. Commun. Nonliear Sci. Numer. Simul.
**2011**, 16, 4438–4450. [Google Scholar] [CrossRef] - Chauhan, S.; Misra, O.P.; Dhar, J. Stability Analysis of SIR Model with Vaccination. J. Comput. Appl. Math.
**2014**, 4, 17–23. [Google Scholar] - Vargas-De-Len, C. On the global stability of SIS, SIR and SIRS epidemic model with standard incidence. Chaos Solitons Fractals
**2011**, 44, 1106–1110. [Google Scholar] [CrossRef] - Shaikhet, L. Stability of SIR Epidemic Model Equilibrium Points; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Chen, L.J.; Sun, J.T. Global stability and optimal control of an SIRS epidemic model on heterogeneous networks. Physica A
**2014**, 15, 196–204. [Google Scholar] [CrossRef] - Lakshmanan, M.; Senthilkumar, D.V. Dynamics of Nonlinear Time-Delay Systems; Springer: Berlin/Heidelberg, Germany, 2010; p. 313. [Google Scholar]
- Smith, H. An Introduction to Delay Differential Equations with Application to the Life Sciences; Springer: Berlin/Heidelberg, Germany, 2011; p. 172. [Google Scholar]
- Sharma, S.; Mondal, A.; Pal, A.K.; Samanta, G.P. Stability analysis and optimal control of avian influenza virus A with time delays. Int. J. Dyn. Control
**2018**, 6, 1351–1366. [Google Scholar] [CrossRef] - Xu, R.; Ma, Z.E.; Wang, Z.P. Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity. J. Comput. Appl. Math.
**2010**, 59, 3211–3221. [Google Scholar] [CrossRef] [Green Version] - Shu, H.Y.; Fan, D.J.; Wei, J.J. Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission. Nonlinear Anal. RWA
**2012**, 13, 1581–1592. [Google Scholar] [CrossRef] - De la Sena, M.; Alonso-Quesadaa, S.; Ibeasb, A. On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules. Appl. Math. Comput.
**2015**, 270, 953–976. [Google Scholar] [CrossRef] - Cooke, K.L.; Driessche, P. Analysis of an SEIRS epidemic model with two delays. J. Math. Biol.
**1996**, 35, 240–260. [Google Scholar] [CrossRef] - Gao, S.J.; Chen, L.S.; Teng, Z.d. Impulsive Vaccination of an SEIRS Model with Time Delay and Varying Total Population Size. Bull. Math. Biol.
**2007**, 69, 731–745. [Google Scholar] [CrossRef] [PubMed] - Wang, W.D. Global behavior of an SEIRS epidemic model with time delays. Appl. Math. Lett.
**2002**, 15, 423–428. [Google Scholar] [CrossRef] - Schenzle, D. An age-structured model of pre-and post-vaccination measles transmission. IMA J. Appl. Math.
**1984**, 1, 169–191. [Google Scholar] [CrossRef] [PubMed] - Metz, J.A.J.; Diekmann, O. The Dynamics of Physiologically Structured Populations. In Lecture Notes in Biomathematics; Springer: Berlin, Germany, 1986; p. 68. [Google Scholar]
- Iannelli, M. Mathematical Theory of Age-Structured Population Dynamics; Giardini Editorie Stampatori: Pisa, Italy, 1995. [Google Scholar]
- Hethcote, H.W. The mathematics of infectious diseases. SIAM Rev.
**2000**, 42, 599–653. [Google Scholar] [CrossRef] [Green Version] - Mccluskey, C. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Math. Biosci. Eng.
**2012**, 9, 819–841. [Google Scholar] - Zou, L.; Ruan, S.; Zhang, W. An age-structured model for the transmission dynamics of hepatitis B. SIAM J. Math. Appl.
**2010**, 70, 3121–3139. [Google Scholar] [CrossRef] [Green Version] - Gandolfi, A.; Iannelli, M.; Marinoschi, G. An age-structured model of epidermis growth. J. Math. Biol.
**2011**, 62, 111–141. [Google Scholar] [CrossRef] - Alexanderian, A.; Gobbert, M.K.; Fister, K.R.; Gaff, H.; Lenhart, S.; Schaefer, E. An age-structured model for the spread of epidemic cholera. Nonlinear Anal. RWA
**2011**, 12, 3483–3498. [Google Scholar] [CrossRef] [Green Version] - Akimenko, V.V. Asymptotically stable states of non-linear age-structured monocyclic cell population model I. Travelling wave solution. Math. Comput. Simul.
**2017**, 133, 2–23. [Google Scholar] [CrossRef] - Akimenko, V.V. Asymptotically stable states of non-linear age-structured monocyclic cell population model II. Numerical simulation. Math. Comput. Simul.
**2017**, 133, 24–38. [Google Scholar] [CrossRef] - Akimenko, V.V. Nonlinear age-structured models of polycyclic population dynamics with death rates as a power functions with exponent n. Math. Comput. Simul.
**2017**, 133, 175–205. [Google Scholar] [CrossRef] - Akimenko, V.V. An age-structured SIR epidemic model with fixed incubation period of infection. Math. Comput. Appl.
**2017**, 73, 1485–1504. [Google Scholar] [CrossRef] - Guo, Z.K.; Huo, H.F.; Xiang, H. Global dynamics of an age-structured malaria model with prevention. Math. Biosci. Eng.
**2019**, 16, 1625–1653. [Google Scholar] [CrossRef] [PubMed] - Huo, H.F.; Yang, P.; Xiang, H. Dynamics for an SIRS epidemic model with infection age and relapse on a scale-free network. J. Frankl. Inst.
**2019**, 356, 7411–7443. [Google Scholar] - Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J. On the definition and the computation of the basic reproduction ratio R
_{0}in models for infectious diseases in heterogeneous populations. J. Math. Biol.**1990**, 28, 365. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hassard, B.; Kazarinoff, N.; Wan, Y.H. Theory and Applications of Hopf Bifurcation; Cambridge University Press: London, UK, 1981. [Google Scholar]

**Figure 2.**Local asymptotic stability of the endemic equilibrium point ${H}^{*}=({N}_{s}^{*},{N}_{e}^{*},{N}_{i}^{*},{N}_{r}^{*})$ when $\tau =0$.

**Figure 3.**Local asymptotic stability of the endemic equilibrium point ${H}^{*}=({N}_{s}^{*},{N}_{e}^{*},{N}_{i}^{*},{N}_{r}^{*})$ when $\tau =4.3526<{\tau}_{0}$. (

**a**) Time response diagram of ${N}_{s}$, ${N}_{e}$, ${N}_{i}$ and ${N}_{r}$; (

**b**) Phase diagram of ${N}_{s}$, ${N}_{e}$ and ${N}_{i}$.

**Figure 4.**Instability of the endemic equilibrium point ${H}^{*}=({N}_{s}^{*},{N}_{e}^{*},{N}_{i}^{*},{N}_{r}^{*})$ when $\tau =6.3526>{\tau}_{0}$. (

**a**) Time response diagram of ${N}_{s}$; (

**b**) Time response diagram of ${N}_{e}$; (

**c**) Time response diagram of ${N}_{i}$; (

**d**) Time response diagram of ${N}_{r}$; (

**e**) Phase diagram of ${N}_{s}$, ${N}_{e}$ and ${N}_{i}$.

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Yin, Z.; Yu, Y.; Lu, Z.
Stability Analysis of an Age-Structured SEIRS Model with Time Delay. *Mathematics* **2020**, *8*, 455.
https://doi.org/10.3390/math8030455

**AMA Style**

Yin Z, Yu Y, Lu Z.
Stability Analysis of an Age-Structured SEIRS Model with Time Delay. *Mathematics*. 2020; 8(3):455.
https://doi.org/10.3390/math8030455

**Chicago/Turabian Style**

Yin, Zhe, Yongguang Yu, and Zhenzhen Lu.
2020. "Stability Analysis of an Age-Structured SEIRS Model with Time Delay" *Mathematics* 8, no. 3: 455.
https://doi.org/10.3390/math8030455