The Effect of Impulsive Vaccination on Delayed SEIRS Epidemic Model Incorporating Saturation Recovery

A delay SEIRS model with pulse vaccination incorporating a general media coverage function and saturation recovery is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the diseasefree periodic solution and its exact expression. Further, using the comparison theorem, we establish the sufficient conditions of global attractivity of the disease-free periodic solution. Moreover, we show that the disease is uniformly persistent if vaccination rate is less than θ ∗ . Finally, we discuss the effect of media coverage on controlling disease.


Introduction
In recent years, controlling infectious disease is a very important issue; vaccination is a commonly used method for controlling disease; the study of vaccines against infectious disease has been a boon to mankind.There are now vaccines that are effective in preventing such viral infections as rabies, yellow fever, poliovirus, hepatitis B, parotitis, and encephalitis B. Eventually, vaccines will probably prevent malaria, some forms of heart disease, and cancer.Vaccines have been very important to people.Theoretical results show that pulse vaccination strategy can be distinguished from the conventional strategies leading to disease eradication at relatively low values of vaccination [1].Theories of impulsive differential equations are found in the books [2,3].In recent years, their applications can be found in the domain of applied sciences [4][5][6][7].In this paper, we consider impulsive vaccination to susceptible individuals.
In the classical endemic models, the incidence rate is assumed to be mass action incidence with bilinear interactions given by , where  is the probability of transmission per contact, and  and  represent the susceptible and infected populations, respectively.If the population is saturated with infective individuals, there are three kinds of incidence forms that are used in epidemiological model: the proportionate mixing incidence / [8], nonlinear incidence     [9], and saturation incidence /(1 + ) [10] or   /(1 +  ℎ ) [11].However, some factors such as media coverage, manner of life, and density of population may affect the incidence rate directly or indirectly; nonlinear incidence rate can be approximated by a variety of forms, such as (1 − ), ( > 0), ( 1 −  2 (/( + ))), ( 1 >  2 > 0,  > 0) which were discussed by [12][13][14].
In this paper, we suggest a general nonlinear incidence rate ( 1 − 2 ( ℎ /(+ ℎ ))), ( 1 >  2 > 0,  > 0, ℎ ≥ 1) which reflects some characters of media coverage, where  1 =  1 ,  2 =  2 ,  is the transmission probability under contacts in unit time,  1 is the usual contact rate,  2 is the maximum reduced contact rate through actual media coverage, and  is the rate of the reflection on the disease.Again, media coverage can not totally interrupt disease transmission, so we have  1 >  2 .We use  2 ( ℎ /( +  ℎ )) to reflect the amount of contact rate reduced through media coverage.When infective individuals appear in a region, people reduce their contact with others to avoid being infected, and the more infective individuals being reported, the less contact with others; hence, we take the above form.Few studies have appeared on this aspect.
In the classical disease transmission models, the recovery from infected class per unit of time is assumed to be proportional to the number of infective individuals (denoted by ); say , where  > 0 is the removal rate.This is a reasonable approximation to the truth when the number of the infectious individuals is not too large and below the capacity of health care settings.If the number of illness exceeds a fixed large size, then the number of recovered is independent of further changes in infectious size.We adopt the Verhulst-type function () = /( + ) to model the recovered part which increases for small infectives and approaches a maximum for large infectives.Here,  gives the maximum recovery per unit of time, and , the infected size at which is 50% saturation (() = /2), measures how soon saturation occurs.Cui et al. studied this removal rate [15].
Cooke and Van den Diressche [16] investigated an SEIRS model with the latent period and the immune period; the model is as follows: where  is the natural birth and death rate of the population,  is average number of adequate contacts of an infectious individuals per unit time,  is the recovery rate of infectious individuals,  is the latent period of the disease, and  is immune period of the population.All coefficients are positive constants.It is easy to obtain from system (1) that the total population is constant.For convenience, we assume that () = () + () + () + () = 1.Based on the above assumptions, we have the following SEIRS epidemic model with vaccination: where  ∈  + ,  + = {0, 1, 2, . ..}, () = () + () + () + () = 1.Note that the variables  and  do not appear in the first and third equations of system (2); this allows us to attack (2) by studying the subsystem: The main purpose of this paper is to establish sufficient conditions that the disease dies out and show that the disease is uniformly persistent under some conditions.

Global Attractivity of Infection-Free Periodic Solution
In this section, we analyse the attractivity of infection-free periodic solution of system (3).If we let () = 0, then the growth of susceptible individuals must satisfy By Lemma 2, we know that periodic solution of system ( 9) is globally asymptotically stable.About the global attractivity of infection-free periodic solution ( S (), 0) of system (3), we have the following theorem.
Theorem 6.The infection-free periodic solution ( S (), 0) of system (3) is globally attractivity provided that  * < 1, where Proof.Since  * < 1, we can choose  0 > 0 sufficiently small such that where From the first equation of system (3), we have Then, we consider the following comparison system with pulses: By Lemma 2, we know that there is a unique periodic solution of system ( 13) which is globally asymptotically stable.
From the first equation of system (3), we have Consider the following comparison impulsive differential equations for  >  2  +  and  >  2 : By Lemma 2, we have that the unique periodic solution of system ( 19) and the unique periodic solution of system (20) are globally asymptotically stable.Let ((), ()) be the solution of system (3) with initial values (4) and (0 + ) =  0 > 0,  1 () and let  2 () be the solutions of system ( 19) and ( 20) with initial values  1 (0 + ) =  2 (0 + ) =  0 , respectively.By the comparison theorem in impulsive differential equation, there exists an integer  3 >  2 such that  3  >  2  +  and Because  0 is arbitrarily small, it follows from (23) that is globally attractive.The proof is complete.

Denote that 𝜃
According to Theorem 6, we can obtain the following result.

Corollary 7.
The infection-free periodic solution ( S (), 0) of system (3) is globally attractivity provided that  >  * or  2 <  * 2 or  >  * .From Corollary 7, we know that the disease will disappear if the vaccination rate is larger than  * .

Permanence
In this section, we say the disease is endemic if the infectious population persists above a certain positive level for sufficiently large time.
Denote that then there is a positive constant  such that each positive solution ((), ()) of system (3) satisfies () >  for  large enough.
According to this solution, we define According to (30), we calculate the derivative of  along the solutions of system (3) ) .