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 disease-free 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 . Finally, we discuss the effect of media coverage on controlling disease.
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 . 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 [4–7]. 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 , nonlinear incidence , and saturation incidence  or . 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, which were discussed by [12–14].
In this paper, we suggest a general nonlinear incidence rate which reflects some characters of media coverage, where , , is the transmission probability under contacts in unit time, is the usual contact rate, 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 . We use 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 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 , measures how soon saturation occurs. Cui et al. studied this removal rate .
Cooke and Van den Diressche  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 . Based on the above assumptions, we have the following SEIRS epidemic model with vaccination: where , , . 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.
2. Notations and Definitions
We introduce some notations and definitions and state some results which will be useful in subsequent sections.
Let . Denote and the map defined by the right hand of the first and second equations of systems (3). Set . Let be the space of continuous functions on with uniform norm. The initial conditions for (3) are
The solution of system (3) is a piecewise continuous function is continuous on , , and exists. Obviously, the smooth properties of guarantee the global existence and uniqueness of solution of system (3) (see , for details on fundamental properties of impulsive systems). Since , and whenever , for , . Moreover, and for . Therefore, we have the following lemma.
Lemma 3 (see ). Consider the following delayed differential equation: where ; for . We have(i)if , then ;(ii)if , then .
3. 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 , then the growth of susceptible individuals must satisfy
About the global attractivity of infection-free periodic solution of system (3), we have the following theorem.
Theorem 6. The infection-free periodic solution of system (3) is globally attractivity provided that , where .
Proof. Since , we can choose 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.
Let be the solution of system (3) with initial values (4) and let , be the solution of system (13) with initial values . By the comparison theorem in impulsive differential equation , there exists an integer such that for ; thus, Again, from the second equation of system (3), we know that (15) implies that where , .
Consider the following comparison system: According to (11) and Lemma 3, we have .
Let be the solution of system (3) with initial values (4) and let , be the solution of system (17) with initial values . By the comparison theorem, we have . Incorporating into the positivity of , we know that . Therefore, there exists an integer (where ) such that for all .
From the first equation of system (3), we have Consider the following comparison impulsive differential equations for and : 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 , and let be the solutions of system (19) and (20) with initial values , respectively. By the comparison theorem in impulsive differential equation, there exists an integer such that and Because is arbitrarily small, it follows from (23) that is globally attractive. The proof is complete.
Denote that , , and .
According to Theorem 6, we can obtain the following result.
Corollary 7. The infection-free periodic solution of system (3) is globally attractivity provided that or or .
From Corollary 7, we know that the disease will disappear if the vaccination rate is larger than .
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 and .
Theorem 8. If , then there is a positive constant such that each positive solution of system (3) satisfies for large enough.
Proof. From , we easily know that , and there exists sufficiently small such that
where . We claim that for any , it is impossible that for all . Suppose that the claim is not valid. Then, there is a such that for all . It follows from the first equation of (3) that, for ,
Consider the following comparison impulsive system for : By Lemma 2, we obtain that is the unique positive periodic solution of (27), which is globally asymptotically stable, where .
Let be the solution of system (3) with initial values (4) and let , be the solution of system (27) with initial values . By the comparison theorem for impulsive differential equation, there exists an integer such that for ; thus,
The second equation of system (3) can be rewritten as Let us consider any positive solution of system (3). According to this solution, we define According to (30), we calculate the derivative of along the solutions of system (3)By (25) and (29), for , we have
Let ; in the following, we will show that for . Suppose the contrary; then, there is a such that for and . However, the second equation of system (3) and (4) imply that This is a contradiction. Thus, for . So, (33) leads to which implies that as . This contradicts with . Hence, the claim is proved. From the claim, we will discuss the following two possibilities:(i) for large enough;(ii) oscillates about for large enough.
Evidently, we only need to consider the case (ii). Let and satisfy , and let for , where is sufficiently large such that for . Since is continuous and ultimately bounded and is not effected by impulses, we conclude that is uniformly continuous. Hence, there exists a constant (with and is independent of the choice of ) such that for . If , our aim is obtained. If , from the second equation of (3), we have that and ; then, we have for , where ,. The same arguments can be continued, and we can obtain for . Since the interval is chosen in an arbitrary way, we get that for large enough. In view of our arguments above, the choice of is independent of the positive solution of (3) which satisfies that for sufficiently large . This completes the proof.
Denote that , , and .
It follows from Theorem 8 that the disease is uniformly persistent provided that or or .
Theorem 9. If , then system (3) is permanent.
Proof. Suppose that is any positive solution of system (3) with initial conditions (4). From the first equation of system (3), we have . By similar arguments as those in the proof of Theorem 6, we have that
where , ( is sufficiently small).
Set . From Theorem 8 and (36), we know that the set is a global attractor in , and, of course, every solution of system (3) with initial conditions (4) will eventually enter and remain in region . Therefore, system (3) is permanent. The proof is complete.
From Theorem 9, we can obtain the following result.
Corollary 10. Assume that or or ; then, system (3) is permanent.
In this paper, we introduce media coverage and saturation recovery in the delayed SEIRS epidemic model with pulse vaccination and analyze detailedly in theory that media coverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease. We suggest the probability of transmission per contact , which reflects some characters of media coverage. We can get that , so is a monotone decreasing function on ; that is, if (the reduced valid contact rate through actual media coverage) is larger, then infection rate of disease is smaller. Again, , so is a monotone increasing function on ; that is, if is smaller (refection on the disease is quickly), then infection rate of disease is smaller. 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. From above analysis, we know that media coverage is very important on controlling disease, and media coverage should be considered in incidence rate.
From Corollaries 7 and 10, we can choose the proportion of those vaccinated successfully to all of newborns such that in order to prevent the epidemic disease from generating endemic, and the epidemic is permanent if . But for , the dynamical behavior of model (3) has not been studied, and the threshold parameter for the vaccination rate between the extinction of the disease and the uniform persistence of the disease has not been obtained. These issues will be considered in our future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the NNSF of China (11371048, 11201433), NSF of Henan Province (112300410156, 122300410117), NSF of the Education Department Henan Province (2011A110022), GGJS of Henan Province (2013GGJS-110), and XGGJS of Zhengzhou University of Light Industry (2012XGGJS003).
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