##
**On a generalized time-varying SEIR epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls.**
*(English)*
Zbl 1219.34104

Summary: This paper discusses a generalized time-varying SEIR disease propagation model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growth and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the loss of immunity of newborns. The presence of outsider infections is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-by-immunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and a suitable reference one.

### MSC:

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

92D30 | Epidemiology |

92C60 | Medical epidemiology |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

PDF
BibTeX
XML
Cite

\textit{M. De La Sen} et al., Adv. Difference Equ. 2010, Article ID 281612, 42 p. (2010; Zbl 1219.34104)

### References:

[1] | De La Sen, M; Alonso-Quesada, S, A control theory point of view on beverton-Holt equation in population dynamics and some of its generalizations, Applied Mathematics and Computation, 199, 464-481, (2008) · Zbl 1137.92034 |

[2] | De La Sen, M; Alonso-Quesada, S, Control issues for the beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: non-adaptive and adaptive cases, Applied Mathematics and Computation, 215, 2616-2633, (2009) · Zbl 1179.92069 |

[3] | De La Sen, M; Alonso-Quesada, S, Model-matching-based control of the beverton-Holt equation in ecology, 21, (2008) · Zbl 1149.92029 |

[4] | De La Sen, M, About the properties of a modified generalized beverton-Holt equation in ecology models, 23, (2008) · Zbl 1148.92031 |

[5] | De La Sen, M, The generalized beverton-Holt equation and the control of populations, Applied Mathematical Modelling, 32, 2312-2328, (2008) · Zbl 1156.39301 |

[6] | Mollison D (Ed): Epidemic Models: Their Structure and Relation to Data. Newton Institute, Cambridge University Press, New York, NY, USA; 1995. · Zbl 0831.00011 |

[7] | Keeling MJ, Rohani P: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton, NJ, USA; 2008:xvi+368. · Zbl 1279.92038 |

[8] | Yildirim, A; Cherruault, Y, Anaytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method, Kybernetes, 38, 1566-1575, (2009) · Zbl 1192.65115 |

[9] | Erturk, VS; Momani, S, Solutions to the problem of prey and predator and the epidemic model via differential transform method, Kybernetes, 37, 1180-1188, (2008) · Zbl 1180.49041 |

[10] | Ortega, N; Barros, LC; Massad, E, Fuzzy gradual rules in epidemiology, Kybernetes, 32, 460-477, (2003) · Zbl 1040.92038 |

[11] | Khan, H; Mohapatra, RN; Vajravelu, K; Liao, SJ, The explicit series solution of SIR and SIS epidemic models, Applied Mathematics and Computation, 215, 653-669, (2009) · Zbl 1171.92033 |

[12] | Song, X; Jiang, Y; Wei, Hg, Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays, Applied Mathematics and Computation, 214, 381-390, (2009) · Zbl 1168.92326 |

[13] | Zhang, T; Liu, J; Teng, Z, Dynamic behavior for a nonautonomous SIRS epidemic model with distributed delays, Applied Mathematics and Computation, 214, 624-631, (2009) · Zbl 1168.92327 |

[14] | Mukhopadhyay, B; Bhattacharyya, R, Existence of epidemic waves in a disease transmission model with two-habitat population, International Journal of Systems Science, 38, 699-707, (2007) · Zbl 1160.93349 |

[15] | Barreiro, A; Baños, A, Delay-dependent stability of reset systems, Automatica, 46, 216-221, (2010) · Zbl 1214.93072 |

[16] | De La Sen, M, On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays, Applied Mathematics and Computation, 190, 382-401, (2007) · Zbl 1117.93034 |

[17] | De La Sen, M, Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays, Applied Mathematics and Computation, 185, 508-526, (2007) · Zbl 1108.93062 |

[18] | Daley DJ, Gani J: Epidemic Modelling: An Introduction, Cambridge Studies in Mathematical Biology. Volume 15. Cambridge University Press, Cambridge, UK; 1999:xii+213. · Zbl 0922.92022 |

[19] | Piccardi, C; Lazzaris, S, Vaccination policies for chaos reduction in childhood epidemics, IEEE Transactions on Biomedical Engineering, 45, 591-595, (1998) |

[20] | Zhang, T; Liu, J; Teng, Z, Dynamic behavior for a nonautonomous SIRS epidemic model with distributed delays, Applied Mathematics and Computation, 214, 624-631, (2009) · Zbl 1168.92327 |

[21] | Gao, S; Teng, Z; Xie, Dehui, The effects of pulse vaccination on SEIR model with two time delays, Applied Mathematics and Computation, 201, 282-292, (2008) · Zbl 1143.92024 |

[22] | Khan, QJA; Krishnan, EV, An epidemic model with a time delay in transmission, Applications of Mathematics, 48, 193-203, (2003) · Zbl 1099.92062 |

[23] | Boichuk, A; Langerová, M; Škoríková, J, Solutions of linear impulsive differential systems bounded on the entire real axis, 10, (2010) · Zbl 1204.34040 |

[24] | Nieto, JJ; Tisdell, CC, On exact controllability of first-order impulsive differential equations, 9, (2010) · Zbl 1193.34125 |

[25] | Yu, HG; Zhong, SM; Agarwal, RP; Xiong, LL, Species permanence and dynamical behavior analysis of an impulsively controlled ecological system with distributed time delay, Computers & Mathematics with Applications, 59, 3824-3835, (2010) · Zbl 1198.34171 |

[26] | Marchenko, VM; Zachkevich, Z, Representation of solutions of control hybrid differential-difference impulse systems, Differential Equations, 45, 1811-1822, (2009) · Zbl 1190.34100 |

[27] | Marchenko, VM; Luazo, ZZ, On the stability of hybrid differential-difference systems, Differential Equations, 45, 1811-1822, (2009) · Zbl 1190.34100 |

[28] | De La Sen, M, A method for general design of positive real functions, IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, 45, 764-769, (1998) · Zbl 0952.94025 |

[29] | De La Sen, M; Alonso-Quesada, S, On vaccination control tools for a general SEIR-epidemic model, Proceedings of the 18th Mediterranean Conference on Control and Automation (MED ’10), 1, 1322-1328, (2010) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.