Research Article Received 12 April 2012 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.2620 MOS subject classification: 92D30; 34B60; 34D05; 34D23; 34K45 Qualitative analysis of the SICR epidemic model with impulsive vaccinations Meihong Qiao a , Anping Liu a and Urszula Fory´ s a,b * † Communicated by M. A. Lachowicz 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. Copyright © 2012 John Wiley & Sons, Ltd. Keywords: impulsive vaccinations; global asymptotic stability; persistence; HBV infection; CHB infection 1. Introduction Viral infectious diseases are one of the severest public health problems world wide. Virus infections, including Hepatitis B virus (HBV) infection that is of our main interest, can have different outcomes ranging from acute to fatal fulminant infection, as well as chronic infection (like in case of Chronic Hepatitis B, CHB), which may result in severe complications as carcinoma. Because of the possible complications development and increasing possibilities of spreading, curing virus infections is urgent, and vaccinations are often used to prevent and control of spreading. However, in the long run vaccinations should be paid more attention to because of the low cure rate of virus infection in case of some viruses. On the other hand, in China, the preventive vaccinations are specially effective to some virus prevention. After such preventing vaccination, in most healthy person’s body, some number of antibodies appear, keeping the body away from virus infection. In the literature, many models of epidemic diseases and qualitative results on such models can be found. However, most models involved are ordinary differential equations or time-delay differential equations. Under the usual conditions, to effectively control the epidemic occurrence and prevalence, vaccinations should be performed at given times. Therefore, it seems to be more reasonable to use differential equations with pulses to describe some epidemic diseases in the context of vaccinations against it. Recently, Roberts and Kao [1], Stone et al. [2], Jin and Ma [3] studied epidemic Susceptible-Infected (SI) or Susceptible-Infected-Removed (SIR) models with impulsive vaccinations. In that context, we use the standard notation for epidemic models where S, I and R denote the susceptible, infective and recover group in the population, respectively. Roberts and Kao [1] investigated the SI epidemic model with impulses at birth and obtained the existence and local stability of the disease-free periodic solution (DFPS). Stone et al. [2] investigated the SIR model with impulsive vaccinations, derived the basic reproduction number and proved that the DFPS is globally asymptotically stable. Moreover, in [3], the DFPS in the SIRS epidemic model with standard epidemic rate and impulsive vaccinations was proved to be globally asymptotically stable. Compartmental models were also considered in that context, for example, by van den Driessche and Watmough [4]. Considering the importance and urgency of virus infection prevention, control and treatment, and the characteristic of virus infection, in this paper, the SICR model with impulsive vaccinations is proposed based on the model presented by Keeling and Rohani [5]. a School of Mathematics and Physics, China University of Geoscience, Wuhan 430074, Hubei Province, China b University of Warsaw Fac. Math. Inf. Mech., Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland *Correspondence to: Urszula Fory´ s, University of Warsaw, Fac. Math. Inf. Mech., Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland. † E-mail: urszula@mimuw.edu.pl Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012