Abstract
We formulate and discuss models for the spread of infectious diseases with variable population sizes and vaccinations on the susceptible individuals. First, we assume that the susceptible individuals are vaccinated continuously. We establish the threshold-like results for the existence and global stability of the disease-free and the endemic equilibriums for these systems. Especially, we prove the global stability of the endemic equilibriums by converting the systems into integrodifferential equations. Second, we suppose that vaccinations occur once per time period. We obtain the existence and global stability of the disease-free periodic solutions for such systems with impulsive effects. By a useful bifurcation theorem, we acquire the existence of the endemic periodic solutions when the disease-related deaths do not occur. At last, we compare the results with vaccinations and without vaccinations and illustrate our results by numerical simulations.
1. Introduction
Confidence that the infectious diseases would soon be eliminated was created by the improved sanitation, effective antibiotics and vaccination programs in the 1960s, but it collapsed now. Human and animal invasions of new ecosystems, global warming, environmental degradation, and increased international travels provide many opportunities for the spread and the eruption of infectious diseases. It is clear that new infectious diseases are emerging, and some eliminated diseases are reemerging since the infectious agents’ evolvement and adaptation to the environment. Moreover, these infectious diseases lead to terrible suffering and mortality. Consequently, infectious diseases are receiving more and more attention in developing countries, even in the developed countries.
The emerging and reemerging of infectious diseases have been studied by many scientists in different fields. Mathematical models are important tools to analyze and control the spread of infectious diseases. Hethcote [1] gives a review on the mathematics of infectious diseases. Most models for the transmission of infectious diseases descend from the pioneering work of Kermark-Mckerdrick on SIR (susceptible-infectious-removal), in which vital dynamics (birth and death) is negligible for the short incubation of infectious diseases. The possible and realistic situations would be to discuss epidemic models with varying population size, which may refer to Mena-Lorca and Hethcote [2], Anderson and May [3], Gao and Hethcote [4], Li and Graef [5], and Brauer and Driessche [6]. Thresholds are obtained which determine whether the diseases die out or break out. The existence and stability of equilibrium points are investigated for each model, but there is still some work to do. Such as in [4], the global stability of the endemic equilibrium points was not obtained. We complete this in the present paper.
Vaccination programs have been applied to prevent and control the yield and spread of infectious diseases, which achieved a lot. Models with vaccination are constructed and analyzed by Shulgin et al. [7], Stone et al. [8], Li and Ma [9], Greenhalgh and Das [10] and Greenhalgh [11]. Some useful results are obtained, and results with vaccination and without vaccination are compared. But they assumed that the susceptible is vaccinated continuously. In fact, it would be more realistic and reasonable that the susceptible is assumed to be vaccinated in a single pulse or at fixed moments. In this paper, we consider not only a constant flow of new members into the susceptible but also vaccinating continuously and impulsively on the susceptible. We will investigate the dynamical behaviors of these epidemic models, which are described by continuous or impulsive differential equations. The models of infectious diseases with impulsive effects had been discussed in [12], where the birth, rather than the vaccination, is assumed to be impulsive.
We denote by the number of members of a population who are susceptible to an infection at time , the number of members who are infective at time , and the number of members at time who have been removed as the result of recovery from the infection with temporary immunity against reinfection. The total population size at time is represented by with . In addition, basic hypotheses are needed to formulate our models:(1)there is a constant flow of new members into the susceptible in unit time;(2)a fraction of the susceptible is vaccinated in unit time or in a single pulse once per time period, which will enter directly into the removal owing to obtaining the immunity;(3)there is a constant per capita natural death rate in each group;(4)a fraction of the infective dies from the infection, and a fraction of the infective recovers in unit time;(5)a fraction of the removal loses their immunity and becomes the susceptible in unit time;(6)the force of the infection is , where is the effective per capita contact rate of the infective individuals and the incidence rate is .
In the next section, an SIR model with variable population size and continuous vaccination is analyzed. The existence and stability of the equilibrium points for this model are discussed. The global stability of the disease-free equilibrium is proved by differential comparison theorem, and the global stability of the endemic equilibrium is obtained by converting the system into an integrodifferential equation. In Section 3, we consider an impulsive differential epidemic model, of which the stability and the existence of disease-free periodic solution are discussed. Further, the existence of endemic periodic solution is also studied for such a system with , which implies that the disease-related deaths are not considered. At last, we give some examples to illustrate our results by numerical simulation.
2. An SIR Model with Continuous Vaccination
In this section, we discuss the disease transmission model, which is described by Denoting , and adding these three equations, we have Therefore, we may obtain system (2.3), which is equivalent to system (2.1), and consider the following system: For nonnegative initial conditions , it is easily known that , and remain nonnegative, and the total population size is ultimately upper bounded by . Moreover, we have Substituting them into the second equation of (2.3), then With respect to equilibriums of (2.3), we have the following.
Theorem 2.1. There only exists the disease-free equilibrium , which is globally asymptotically stable if . Here, and .
Proof. The existence and uniqueness of the disease-free equilibrium for system (2.3) are easily obtained if inequality holds. Next, we first discuss the local stability of . The Jacobian matrix of system (2.3) at a point is
Thus,
and has three eigenvalues with negative real part if inequality is true, which shows that the disease-free equilibrium is locally stable.
Further, we assume . It is easily known that for any , that there exists such that
Hence, we obtain from the third equation of (2.3) that
Then, for any , there exists such that
Therefore, we have
This shows that for any , there exists such that
Moreover, we have
By the assumption of , we may choose , and , are small enough such that
Therefore, we obtain
This leads to
If , we obtain from (2.5)
Since , , and , then there exists and such that
Thus, we prove that (2.15) is held. Further, (2.16) is obtained.
Theorem 2.2. The unstable disease-free equilibrium and the local stable endemic equilibrium of system (2.3) coexist if holds. Here,
Proof. It is easily known that system (2.3) has unique positive equilibrium except the disease-free equilibrium if inequality holds.
Moreover, we may obtain the Jacobian matrix of (2.3) at equilibrium as
The characteristic equation of is given by
and the coefficients are
Clearly, and . By Routh-Hurwitz criterion, there exist three eigenvalues with negative real part for Jacobian matrix , which shows that equilibrium is locally stable.
Further, we study the global stability of .
Theorem 2.3. If is held besides , then the positive equilibrium of (2.3) is globally asymptotically stable.
Proof. Since , we may make the change of variable , thus
and (2.5) is equivalent to
Further, we define and
so that has a jump at and for if , then
Hence, system (2.3) is reduced to a single integrodifferential equation
Here, and
With the equilibrium condition of (2.3), can be simplified as
which is negative exponential.
Obviously, the equilibrium of (2.5) corresponds to the equilibrium of (2.27). According to Theorem 18.2.3 of Gripenberg et al. [13], since is of strong positive type ([14, 15]), is continuous and as , is also continuous and for , , and are in , it follows that every bounded solution of (2.27) satisfies . Owing to only for , this implies that every solution of (2.27) tends to zero as and therefore the equilibrium of (2.5) is globally asymptotically stable. As (2.5) is equivalent to system (2.3), the unique positive equilibrium of (2.3) is globally asymptotically stable.
It is similar to the classical SIR models that there exists the threshold quantity and it is given by for the model (2.1). If , then system (2.1) has only the globally asymptotically stable disease-free equilibrium. This shows that the epidemic disease will die out. Otherwise, if , system (2.1) has a locally stable positive equilibrium except disease-free equilibrium. Moreover, the global asymptotical stability of positive equilibrium is obtained under the assumption of . This reveals a fact that the disease may be “invaded” or always exists in the population forever. According to the expression of , it is the convenient and important policy to control the occurrence of the disease that the flow of the members decreases, the vaccinated members increase, and the infectious period shortens.
3. An SIR Model with Impulsive Vaccinations
In this section, the assumption that the susceptible is vaccinated continuously is replaced by the assumption that the susceptible is vaccinated impulsively, that is, the susceptible is vaccinated at the fixed moments. At the moment , the vaccinated susceptible will enter directly to the removal owing to acquiring the temporary immunity, which leads to the following system: where is the vaccinated period. Clearly, system (3.1) is equivalent to the system Let , then system (3.2) is simplified as
Lemma 3.1. Consider the following system: Then there exists a positive periodic solution , which is globally attractive. Here for .
The proof is so simple that we omit it.
By Lemma 3.1, we can easily know that there exists a positive periodic solution for system (3.3), and their expression is This indicates that system (3.2) has a disease-free periodic solution with period , and its local stability may be determined by considering the behavior of small-amplitude perturbation of the solution. Define , than these may be written where satisfies with , the identity matrix. The resetting impulsive conditions of (3.2) become Hence, if all eigenvalues of the monodromy matrix , have absolute value less than one, then the -periodic solution is locally stable. Actually, , and need not to be solved out, and we have , and . Therefore, if , which is equivalent to , then holds. And we have the following.
Theorem 3.2. If inequality holds, then there exists a disease-free periodic solution for (3.2), which is locally stable. Moreover, it is globally asymptotically stable.
Proof. Denote that . In fact, we only prove that , and for .
For every solution of (3.2) with positive initial value , it is clear that , which leads to
Here, may be arbitrary small, and a positive integer may be large enough. Hence, we have
Furthermore, we consider the system
Therefore, we have , and is the solution of (3.12) with the initial value . In addition, for system (3.12), it is easily known that there exists a positive periodic solution , and every solution with positive initial value is globally attractive by Lemma 3.1. Here,
Thus, for an arbitrary small , there exists a positive integer such that
As a result, we have
So,
Since holds, we may choose and small enough such that , which leads to
And since
by (3.17), we have
Therefore, for an arbitrary small , there exists a positive integer such that
Then, we have
Considering the system
we have , where is the solution of (3.22) with initial value . Denote that is the globally asymptotically attractive and positive periodic solution; thus, for an arbitrary small number , there is a positive integer such that
and since , we may get
Let and , then for , we have
And since and will approach the positive periodic solution of system (3.3) for , therefore, we have
Thus, the global stability of the boundary periodic solution has been proven. This indicates that the disease-free periodic solution of (3.1) is also globally asymptotically stable.
Clearly, if holds, then is true. So, we have the following.
Corollary 3.3. If holds, then system (3.2) has a unique disease-free periodic solution for arbitrary and , which is globally asymptotically stable.
In the following, one only considers the case: .
Denote . Notice that the function satisfies is continuous on , , ; moreover, . Then one has the following.
Lemma 3.4. If , then there exists and only exists a root in for function , and for and for .
Noting that is equivalent to , one has the following.
Corollary 3.5. For system (3.2), if , then there exists such that system (3.2) has a unique disease-free periodic solution for , which is globally asymptotically stable.
Now, we are in the position of studying the behaviors of (3.1) under the assumption of . Here, we only discuss the existence of endemic periodic solution of system (3.1) with . From the viewpoint of biology, we neglect the death-related disease. At this time, system (3.1) can be rewritten as follows: If (3.27) has a periodic solution, it is certain that . Then, we may change to consider the two-dimensional system (3.28), which is equivalent to (3.27). Note that there exists a positive periodic solution of system where then system (3.28) has a boundary periodic solution . With bifurcation technique and the important Theorem [16], we may obtain the positive periodic solution of (3.28). As and if we choose , the unique root of , to be the bifurcated parameter, then we can easily see that . Further, we have and . According to the theorem of Lakmeche and Arino [16], the supercritical bifurcation occurs for system (3.28).
Theorem 3.6. If and is close to enough, then there exists positive periodic solution for system (3.28). Here, is the root of .
Further, one has the following.
Corollary 3.7. For system (3.1), if and is close to enough, then there exists positive periodic solution . Here, is defined in Theorem 3.6.
Thus, we complete the discussion on the epidemic model with impulsive vaccination. And we know that if the period of vaccination is smaller than , the disease will die out forever, but once it is larger than , the infectious disease is going to be the endemic disease.
4. Discussion and Numerical Simulation
In this paper, we first assume that the susceptible is vaccinated continuously. The model is formulated by a continuous differential system. Similar to most models for the spread of infectious diseases, there is a threshold parameter . If , the disease-free equilibrium is approached by all solutions; if , the disease-free equilibrium is unstable and the endemic equilibrium exists, which is locally stable. Especially, we prove that the endemic equilibrium is globally asymptotically stable if . Obviously, there is a unique such that . If exceeds , then , and if is less than , then . We show that increasing the vaccinated members is helpful to control the spread of the infectious diseases.
Let , and , then we have . By Theorem 2.1, we know that system (2.1) has a unique disease-free equilibrium with , which is globally asymptotically stable. The time series of solutions with initial value are given in Figure 1.
If we choose , then and . By Theorem 2.2, there exists a globally asymptotically stable endemic equilibrium with and . Setting the initial values , and , we draw the time series graph of the solution of (2.1) in Figure 2.
We continue to analyze our results. Let , that is, without vaccination on the susceptible. The model was discussed, and a threshold parameter was obtained for such a model in [4]. It is clear that is identical to with . In addition, in [4], the global stability of the endemic equilibrium was not proved. By our conclusion, we know that the endemic equilibrium is globally asymptotically stable if , that is, we generalize the results in [4].
Second, the susceptible is supposed to be vaccinated at fixed moments. Such an epidemic model is described by an impulsive differential system, which also has a threshold parameter . If , the disease-free periodic solution is globally stable, while it is unstable for . Clearly, if , then . Hence, there exists a unique globally asymptotically disease-free periodic solution. If , then there exists a unique globally asymptotically disease-free periodic solution for , and is the root of . With respect to the existence of positive periodic solution, we only consider a simple model, in which , that is, the disease-related death is neglected for the infective. The positive periodic solution exists if the vaccinated period satisfies and is close to the critical value enough. It is implied that the epidemic disease may be controlled by shortening the vaccination period.
Let and . Then for impulsive system (3.1). Hence, there exists a disease-free periodic solution. Moreover, it is globally asymptotically stable. Figures 3, 4, and 5 are the time series of the solution with initial values and, and Figure 6 is the trajectory phase for it, which implies that this solution tends to the disease-free periodic solution with .
If we extend the period of the vaccination and choose , then . By Corollary 3.7, we know that there exists a positive periodic solution with period . We draw its time-series in Figures 7, 8, and 9. And the graph of its trajectory is drawn in Figure 10.
Acknowledgments
This work is supported by the Natural Science Foundation of Shanxi Province [no. 2008011002-2] and the Key Project of Chinese Ministry of Education [no. 210030], China.