Pulse vaccination of an epidemic model with two parallel infectious stages and time delays.

An epidemic model with two parallel infectious stages and time delays and pulse vaccination is proposed. We introduce four thresholds and further obtain the conditions that the disease will be extinct or not. Corollaries show that under condition that θ > max { θ ∗ 1 , θ ∗ 2 } the disease will fade out, and if θ < min { θ 1 ∗ , θ 2 ∗ } , the disease will be endemic. Our results indicate that a larger pulse vaccination rate will lead to the eradication of a disease. Furthermore, two thresholds ℜ ∗ 1 and ℜ ∗ 2 show that the diversity of the contagiousness affects the basic properties of these models. In addition, numerical results indicate that the probability for an infected individual to enter different infective compartments greatly affects two infective compartments.

Introduction

Infectious diseases have tremendous influence on human life. Every year, millions of people die of various infectious diseases. Controlling infectious diseases has been an increasingly complex issue in recent years. Vaccination is an important strategy for the elimination of infectious diseases  [7], [11], [16], [18]. Recently, pulse vaccination epidemic models have been the subject of intense theoretical analysis  [1], [10], [15], [19]. Theoretical results show that the pulse vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination  [1].

For many infectious diseases, and in particular for the diseases with a prolonged infectious period, the infection has different consequences. For instance, some infected hosts can be properly detected and treated, isolated or removed, whereas others remain undetected and untreated. It is obvious that in such circumstances the length of the infectious periods and the levels of contagiousness for these detected and undetected hosts can significantly differ. Moreover, the behavior of a virus within a host may be different. Some infections, such as Hepatitis B, can have either a highly contagious and comparatively short acute infectious stage, or can persist within a host for a long time without apparent symptoms and having comparatively low level of contagiousness. This leads to the questions whether this diversity of the levels of contagiousness and the lengths of the infective period affects the system dynamics, and whether the classic compartmental models in mathematical epidemiology, which postulate that all the hosts are identical, are adequate to describe the spread of an infection  [3].

In the study of infectious disease, a problem of interest is the coexistence of multiple strains. Examples of persistent co-circulating multistrain diseases include influenza  [2], malaria  [9], and dengue fever  [6]. More recently, the avian flu viruses, including H5N1, were reported to coexist in several genotypes until 2004  [21]. Other examples of viruses possessing multiple strains may be seen in corona viruses, such as severe acute respiratory syndrome (SARS)  [22]. The presence of multiple strains adds greater complication to models of disease dynamics due to an increasing number of stages through different infection–recovery combinatorics. So in this paper, we consider the coexistence of two strains.

Standard epidemiological models use a bilinear incidence rate based on the law of mass action  [5], [1]. However, as the number of susceptible populations is large, it is unreasonable to consider the bilinear incidence rate because the number of susceptible populations with every infective contact within a certain time is limited. If the population is saturated with infective population, there are kinds of incidence forms that are used in epidemiological model: the proportionate mixing incidence   [4], [20], nonlinear incidence   [13], [14] or   [17].

However, epidemic models with nonlinear incidence, constant infectious period and vaccination have never been seen by now. Time delay and vaccination are introduced into epidemic disease models, which greatly enriches biologic background. Therefore, in the present paper, we create a new two-delay epidemic model with nonlinear incidence and study its dynamic behavior.

The main purpose of this paper is to establish sufficient condition for the disease to die out, and we obtain that a large pulse vaccination rate will lead to eradication of the disease. The second purpose of this paper is to establish a sufficient condition for the permanence of the epidemic model. The organization of this paper is as follows. In the next section, we propose a two-delay disease Model with pulse vaccination and infectious Period. To prove the main results, we also give two lemmas. In Section  3, using the discrete dynamical system determined by the stroboscopic map, we establish sufficient conditions for the global attractivity of infection-free periodic solution. The sufficient conditions for the permanence of the model are obtained in Section  4. In particular, some results are obtained in Section  5 if there is only one infective compartment. Finally, we give a brief conclusion.

Model formulation and preliminaries

Motivated by Andrei  [3] and Gao  [8], in this paper, we assume that there are two alternative infectious pathways and two noninteracting infective subclasses. We denote the fractions of the susceptible individuals, the infectious individuals that belong to two different subclasses, and the recovered individuals in the population by , , and , respectively, that is, . Here is the size of the total variable population at time . We assume that after infection an individual immediately moves from the susceptibles compartment into one of the infective compartments and after a constant infectious period  [8] then into the recovered compartment. Furthermore, if the transmission of the infection is governed by the nonlinear incidence and pulse vaccination is incorporated into this model, then the corresponding equations of the SIR model are: Here the constant is the recruitment rate into the susceptible class (that includes immigrants and the newborns that are assumed to be susceptible); parameters and are the susceptible, infectious and recovered natural mortalities, and and are the disease transmission coefficient; is the length of the infectious period. The term reflects the fact that an individual has recovered from the th infective compartments and is still alive after infectious period . Parameter is the probability for an infected individual to enter the th infective compartment with . Pulse vaccination can be defined as the repeated application of vaccine across an age range. Assume the pulse scheme proposes to vaccinate a fraction of the entire susceptible population in a single pulse, applied every years.

Adding all the equations in model (2.1), the total variable population size is given by the differential equation from which, we have where . It follows that The first three equations in system (2.1) do not depend on the fourth and the fifth equation. Therefore, we restrict our attention to the following reduced system: The initial condition of (2.3) is given as Here and is the space of all piecewise functions with points of discontinuity at of the first kind and which are continuous from the left, i.e., , where . Let be the following subset of From biological considerations, we discuss system (2.3) in the closed set . It can be verified that is positively invariant with respect to system (2.3).

Global attractivity of infection-free periodic solution

To prove our main results, we state some notations and lemmas which will be essential to our proofs.

[8]

Consider the following impulsive differential equations where . Then the above system exists a unique positive periodic solution given by which is globally asymptotically stable, where

[12]

Let , then is said to belong to class if

is continuous in and for each , exists.

is locally Lipschitzian in .

Let . Assume that where is continuous in and for exists, is non-decreasing. Let be the maximal solution of the scalar impulsive differential equation existing on . Then implies that , where is any solution of   (2.3) .

Now we shall prove the disease-free periodic solution is the global attractivity. We first demonstrate the existence of the disease-free periodic solution, in which infectious individuals are entirely absent from the population permanently, i.e.  for all . Under this condition, the growth of susceptible individuals must satisfy According to Lemma 3.1, we know that the periodic solution of system (3.1) is globally asymptotically stable. Hence system (2.3) has a disease-free periodic solution .

Denote

If and , then disease-free periodic solution of system   (2.3)   is globally attractive.

Since and , we can choose sufficiently small such that From the first equation of system (2.3), we have , then we consider the following comparison system with pulse: By Lemma 3.1, we obtain the periodic solution of system (3.3): By the comparison theorem  [12], and from the fact that , there exists such that Furthermore, from the second and the third equation, we have, for and ,

According to (3.2), (3.3), we have , then . Therefore, and , i.e. for any sufficiently small , there exists an integer such that for all . From the first equation of system (2.3), we have then we consider the following comparison system with pulse: From Lemma 3.1, we obtain the periodic solution of system (3.5): By the comparison theorem  [12], there exists an integer such that Since is sufficiently small, from (3.4), (3.6), we know that Hence, infection-free periodic solution of system (2.3) is globally attractive. The proof is completed.

Next, we give equivalent conditions of Theorem 3.1 for a well biological meaning. They are given in the following corollaries.

Set and

Suppose that and . Then disease-free periodic solution of system   (2.3)   is globally attractive provided that .

If and , then the disease-free periodic solution of system   (2.3)   is globally attractive provided that .

Theorem 3.1 determines the global attractivity of the disease-free periodic solution of system (2.3) in for the case and , Its epidemiology implication is that the infectious population vanishes in time so the disease dies out. Corollary 3.1, Corollary 3.2 imply that a larger pulse vaccination rate or a shorter pulse period of immune vaccination will lead to the eradication of a disease.

Permanence

In this section, we state the disease is endemic if the infectious population persists above a certain positive level for sufficiently large time. The endemicity of the disease can be well captured and studied through the notation of uniform persistence.

System (2.3) is said to be uniformly persistent if there exist positive constants (Both are independent of the initial values), such that every solution with positive initial conditions of system (2.3) satisfies

Denoted by we have

If and , then the disease is uniformly persistent.

Let be any solution with initial values of system (2.3), then it is obvious that for all . We are left to prove there exist positive constants and such that for all .

Firstly, from the first equation of system (2.3), we have Considering the following comparison equations: by Lemma 3.1 and the comparison theorem  [3], we know that for any sufficiently small , there exists a ( is sufficiently large) such that Now, we shall prove there exists an such that for all sufficiently large . For clarity, we prove it in the following two steps:

Step 1. Since and there exist sufficiently small and such that where

For , there must exist a such that . Otherwise, for all . From the first equation of system (2.3), we have Consider the following comparison equations: By the same method, there exists , for , we have The second and the third equations of system (2.3) can be translated into the following form:

Define a function as follows: For , the derivative of along the solution of system (2.3) is From (4.3), we obtain for , which implies that . This contradicts the boundedness of . Hence, there exists a such that .

Step 2. According to step 1, for any positive solution of system (2.3), we are left to consider two cases. First, If for all , then our aim is obtained. Second, if for some , we can choose and ( is demanded be sufficiently large) such that and for . Therefore, there exists a such that for In this case, we shall discuss three possible cases in terms of the sizes of :

Case I. If , then it is obvious that , for .

Case II. If , then from the second equation of system (2.3), we can deduce By (4.3), (4.4), (4.5), we have for .

Case III. If , we shall consider the following two cases, respectively.

Case IIIa. For , it is easy to obtain from (4.6).

Case IIIb. For , we claim . Otherwise, we suppose there is a , such that for , and . From (4.2), (4.3), (4.5), we have which is contradictory to . Therefore, the claim holds true.

Since is arbitrary, there exists , such that for all . The proof is completed.

Denoted by by Theorem 4.1, Parallel conclusions are given in the following corollaries.

If , then the disease is uniformly persistent in model   (2.3) .

If and , then the disease is uniformly persistent in model   (2.3)   provided that and .

Corollary 4.1, Corollary 4.2 indicate that a less pulse vaccination rate or a longer infectious period will not lead to the eradication of the two diseases at the same time.

Next, we present some numerical simulations to illustrate the effects of different probability on population. Consider the following choice of parametric values:

.

System (2.3) is numerically solved for the above choice of parameters and time series are displayed in Fig. 1 (a)–(c) with initial value for 50 pulsing cycles. The last 10 pulsing cycles are plotted in Fig. 1(d). The solid line denotes time series of the variables when . The dotted line denotes time series of the variables when . The dashed line denotes time series of the variables when . Fig. 1 displays the effects of the probability for an infected individual to enter different infective compartments. For the above cases, the levels of the total infective are displayed in Fig. 2 (a).

Fig. 1

Dynamic behaviors under different probability for an infected individual to enter different infective compartments. (a), (b) and (c) are time series. (d) Phase portrait.

Fig. 2

(a) The levels of the total infective under the different probability ratios of an infected individual to enter the two infective compartments. (b) The effect of parameter on whether the disease will be extinct or not.

Finally, we investigate the effect of parameter on the extinction of the disease. Let . The levels of the total infective are displayed in Fig. 2(b) as parameter takes 0.6, 1 and 1.2, respectively.

An epidemic model with pulse vaccination and time delay

If there is only an infective compartment , then system (2.1) becomes

Denoted by and by Theorem 3.1, Theorem 4.1, we get the corollaries as follows.

If , then disease-free periodic solution of system   (5.1)   is globally attractive, if , then the disease is permanent in model   (5.1) .

Suppose that . If , then disease-free periodic solution of system   (5.1)   is globally attractive.

Denoted by and by Corollary 4.1, Corollary 4.2, we get the corollaries as follows.

If hold, then the disease is permanent in model   (5.1) .

If , then the disease will be endemic and system   (5.1)   is permanent provided that .

Conclusions

Some infectious diseases possess multiple parallel infectious stages such as Hepatitis B. This leads to the diversity of the levels of contagiousness and the lengths of the infective period which affect the system dynamics. In view of these facts and based on pulse vaccination, an epidemic model with two parallel infectious stages and time delays and pulse vaccination is proposed. We introduce four thresholds and and further obtain that the disease will be extinct if and , and persistent if and . Corollary 3.1, Corollary 4.1 show that under condition that the disease will fade out, and if , the disease will be uniformly persistent. Our results indicate that a larger pulse vaccination rate will lead to the eradication of a disease. Furthermore, two thresholds and show that the diversity of the contagiousness affects the basic properties of these models.

Finally, we illustrate how probability for an infected individual to enter different infective compartments affects two infective compartments. (1) Corollary 3.1, Corollary 5.2 display the conditions of the eradication of the disease when there are two infective compartments and only one infective compartment, respectively. Obviously, the threshold is larger than , which implies that a larger fraction of susceptibles should be vaccinated against the disease if multiple strains coexist. (2) Fig. 2(a) indicates that the number of the infected decreases as the probability ratio () of an infected individual to enter the two infective compartments increases when the other parameters are the same as those in Fig. 1. (3) During the above investigation, we find that whether the disease will be extinct or not is dependent on the parameter in the nonlinear incidence which governs the transmission of the infection (see Fig. 2(b)).