Qualitative analysis of the SICR epidemic model with impulsive vaccinations. (English) Zbl 1318.92049

Summary: Control of epidemic infections is a very urgent issue today. To develop an appropriate strategy for vaccinations and effectively prevent the disease from arising and spreading, we proposed a modified susceptible-infected-removed model with impulsive vaccinations. For the model without vaccinations, we proved global stability of one of the steady states depending on the basic reproduction number \(R_{0}\). As typically in the epidemic models, the threshold value of \(R_{0}\) is 1. If \(R_{0}\) is greater than 1, then the positive steady state called endemic equilibrium exists and is globally stable, whereas for smaller values of \(R_{0}\), it does not exist, and the semi-trivial steady state called disease-free equilibrium is globally stable. Using impulsive differential equation comparison theorem, we derived sufficient conditions under which the infectious disease described by the considered model disappears ultimately. The analytical results are illustrated by numerical simulations for Hepatitis B virus infection that confirm the theoretical possibility of the infection elimination because of the proper vaccinations policy.


92D30 Epidemiology
34A37 Ordinary differential equations with impulses
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Roberts, The dynamics of an infectious disease in a population with birth pulse, Mathematical Biosciences 149 (1) pp 23– (1998) · Zbl 0928.92027
[2] Stone, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Mathematical and Computer Modelling 31 (4-5) pp 207– (2000) · Zbl 1043.92527
[3] Jin, The SIR Epidemical Models with Continuous and Impulsive Vaccinations, Journal of North China Institute of Technology 24 (4) pp 235– (2003)
[4] Driessche, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180 pp 29– (2002) · Zbl 1015.92036
[5] Keeling, Modeling Infectious Diseases in Humans and Animals (2008) · Zbl 1279.92038
[6] Marinone, HBV disease: HBsAg carrier and occult B infection reactivation in haematological setting, Digestive and Liver Disease 43 pp 49– (2011)
[7] Kamgang, Computation of threshold conditions for epidemiological model sand global stability of the disease-free equilibrium (DFE), Mathematical Biosciences 213 pp 1– (2008) · Zbl 1135.92030
[8] Li, Global stability of SEIRS models in epidemiology, Canadian Applied Mathematics Quarterly 7 pp 409– (1999) · Zbl 0976.92020
[9] Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain Journal of Mathematics 20 pp 857– (1990) · Zbl 0725.34049
[10] Li, Dulac criteria for autonomous systems having an invariant affine manifold, Journal of Mathematical Analysis and Applications 199 pp 374– (1996) · Zbl 0851.34031
[11] Coppel, Stability and Asymptotic Behavior of Differential Equations (1965)
[12] Martin, Logarithmic norms and projections applied to linear differential equations, Journal of Mathematical Analysis and Applications 45 pp 432– (1974) · Zbl 0293.34018
[13] Freedman, Uniform persistence and flows near positively invariant sets, Journal of Dynamics and Differential Equations 6 pp 583– (1994) · Zbl 0811.34033
[14] Lakshmikantham, Theory of Impulsive Differential Equations (1989)
[15] Gao, Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear Analysis: Real World Applications 9 pp 599– (2008) · Zbl 1144.34390
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