Periodic solutions and bifurcation in an S I S epidemic model with birth pulses.

The dynamical behavior of an S I S epidemic model with birth pulses and a varying population is discussed analytically and numerically. This paper investigates the existence and stability of the infection-free periodic solution and the endemic periodic solution. By using discrete maps, the center manifold theorem, and the bifurcation theorem, the conditions of existence for bifurcation of the positive periodic solution are derived. Moreover, numerical results for phase portraits, periodic solutions, and bifurcation diagrams, which are illustrated with an example, are in good agreement with the theoretical analysis.

Introduction

Infectious diseases have tremendous influence on human and animal population sizes. For example, severe acute respiratory syndrome (SARS) affected China in 2003, myxomatosis caused enormous decreases in the rabbit population in Australia in the 1950s, the Black Death in Europe in the 14th century killed up to a quarter of the human population. Mathematical modelling is of considerable importance in epidemiology because it may provide understanding of the underlying mechanisms which influence the spread of disease and may suggest control strategies. In the case where the infectious lose immunity and become susceptible immediately after recovering, an epidemic model is used to describe the dynamics of the population [1]. Recent years have also seen wide-scope potential applications of the epidemic model in various scientific fields, such as that of complex networks [2].

The study of the epidemic model mainly concerns global asymptotic stability. For example, Rass [3] obtained asymptotic stability for a multi-type model, and Iannelli et al. [4] considered an age-structured epidemic model of type and analyzed the global dynamical behavior for the model when the population density converges uniformly to a steady state. Global analysis of discrete-time and epidemic models was given in [5]. Further, the complex dynamical behavior of the epidemic model is discussed by using bifurcation theory. For example, Zhang et al. [6] obtained the conditions of backward bifurcation and the existence of bistable endemic equilibria. Zhang et al. [7] investigated Hopf bifurcation in a delayed epidemic model with stage structure by using the normal form theory and the center manifold argument.

In the above cited papers, ordinary differential equations (ODEs) were used to build an epidemic model. Impulsive differential equations (IDEs) are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change in their values. The study of IDEs mainly concerns the properties of their solutions, such as the existence, uniqueness, stability, boundedness, and periodicity; see [8], [9]. Sufficient conditions for the local and global stability of the susceptible pest-eradication periodic solution are found by means of Floquet theory and comparison methods in [10]. Nieto and Regan [11] present a new approach via variational methods and critical point theory for obtaining the existence of solutions to impulsive problems. The dynamics of an impulsively controlled three-trophic food chain system with general nonlinear functional responses for the intermediate consumer and the top predator are analyzed using the Floquet theory and comparison techniques in [12]. IDEs are widely used in epidemiology. For example, an epidemic model with a pulse vaccination was investigated in [13] and some results are obtained for the global stability of the disease-free periodic solution as well as the existence and stability of the endemic periodic solution are investigated analytically and numerically.

In most models of population dynamics, increases in population due to births are assumed to be time independent. However, many species give birth only during a single period of the year and Caughley [14] termed this growth pattern a birth pulse. Roberts and Kao [15] proposed a model for the dynamics of a fatal infectious disease and discussed the existence and stability of periodic solutions. Tang and Chen [16] obtained an exact periodic solution and the threshold conditions for its stability by using the stroboscopic map in a single-species model. In this paper, the birth pulse is used to build an epidemic model.

The existence and bifurcation of periodic solutions for an epidemic model built from ODEs have been discussed by many authors; see [6], [7] for example. However, little is known about the bifurcation theory of an epidemic model built from IDEs. Periodic boundary value problems for non-Lipschitzian impulsive functional differential equations were considered in [17]. Using a projection method, the bifurcation of positive periodic solutions for an impulsively controlled pest management mode was discussed in [18]. The complex dynamics of a Holling type II prey–predator system with state feedback control was investigated in [19]. Zhou and Liu [13] discussed the existence of a disease-free periodic solution and an endemic periodic solution by using the explicit solution. Lakmeche et al. [20] transformed the problem of periodic solution into a fixed-point problem and obtained conditions for the stability of the trivial solution and the existence of a positive period-1 solution. Many authors (for example, see [21]) discussed only the bifurcation of a non-trivial periodic solution of an epidemic model by using the results obtained in [20]. By using the explicit solution, the stroboscopic map is obtained and used to discuss the bifurcation of the periodic solution in [16]. However, the explicit solution is not easy to obtain in an epidemic model and the bifurcation of the endemic periodic solution is difficult to discuss. Some authors investigated the complex dynamics of epidemic models, such as period-doubling bifurcation, chaos and crisis etc., by means of numerical simulations; see [22] for example. Therefore, theoretically analyzing the bifurcation theory of the model is a challenging task.

In this paper, an ordinary differential system is used to build an epidemic model with a birth pulse and a varying population. To study this epidemic model, we construct discrete maps and present analytical results about the complex dynamical behavior. The rest of the paper is organized as follows. In the next section, an epidemic model with a birth pulse and a varying population is introduced. The conditions for the existence and stability of the periodic infection-free solution are derived in Section 3. In Section 4, the existence and stability of the positive periodic solution are discussed. Bifurcation analysis of the epidemic model is given in detail in Section 5. The parameter value at which the epidemic periodic solution bifurcates from the infection-free periodic solution is calculated. The conditions of existence for flip bifurcation are derived by using the center manifold theorem and the bifurcation theorem. The numerical results are presented in Section 6, to verify the theoretical analysis, and the conclusion is presented in Section 7, finally.

Model description

A classical epidemic model was introduced by Kermack and McKendrick [1]. On the assumption that recovery from the nonfatal infective disease does not confer immunity, one particular case of this classical epidemic model is where the overdot denotes differentiation with respect to time, is the susceptible component of the population, is the infected component of the population, represents a constant birth rate, is the average number of adequate contacts with susceptibles for an infective individual per unit time, is the natural death rate and is the rate at which infective individuals lose immunity and return to the susceptible class.

The dynamics of system (1) is simple. There exists a unique positive equilibrium in (1) for . Nucci and Leach [23] presented an explicit solution of (1) by means of the Painlevé analysis and the Lie theory of transformation groups.

In system (1), represents a constant birth rate, which means that the population is born throughout the year. However, many species give birth only during a single period of the year and Caughley termed this growth pattern a birth pulse. In most cases, the birth pulse is assumed to be the linear birth pulse [24], where . Roberts and Kao [15] considered the birth pulse , where . In paper [16], and .

In this paper, the birth pulse is taken as , where , is the maximum birth rate, is the maximum death rate, is a parameter reflecting the relative importance of density-dependent population regulation through births and deaths. If all density dependence acts through the death rate, and if all density dependence acts through the birth rate. The newborn population is assumed to be susceptible to disease, that is, and . Now, on the basis of the impulsive differential equations, we develop system (1) by introducing periodic birth pulses and obtain the following epidemic model with birth pulses: where , the meanings of parameters , and are the same as in model (1), , is the time between two consecutive birth pulses, , are the quantities of susceptible components of the population after the birth pulse and , , . For more details about impulsive systems see [8], [9].

The epidemic model (2) is considered in the region in this paper. On the boundary line , and while on the boundary line , and . Set the initial point of system (2) as , where , . It follows from (2) that where . Then and . The trajectory originating from this initial point remains in region for . This trajectory reaches the point at time , and next jumps to the point with the effect of the birth pulse, where , . In view of and the function being a strictly monotone increasing function on , . Thus and the region is invariant for (2).

Existence and stability of the infection-free periodic solution

In this section, infectious individuals are entirely absent from the population permanently, i.e., . System (2) yields Suppose the value of is for ; then the solution for the first equation of system (3) is and for the effect of the impulse. Thus we obtain the one-dimensional discrete map For each fixed point of the map (4) there is an associated periodic solution of system , and vice versa. The fixed points of map (4) are

In the case of the fixed point , For , , then the fixed point is stable. Hence the trivial solution of system (2) is stable for .

In the case of the fixed point , For , the fixed point is stable, that is the infection-free periodic solution of system (2) is stable.

There are three ways in which a fixed point of a discrete map may fail to be hyperbolic, that is, has an eigenvalue +1, an eigenvalue −1 or a pair of complex eigenvalues with  [25]. The bifurcation associated with the appearance of eigenvalue 1 is called a fold (or tangent) bifurcation. This bifurcation is also referred to as a limit point, saddle-node bifurcation, and turning point, among others. The bifurcation associated with the appearance of eigenvalue −1 is called a flip (or period-doubling) bifurcation.

From the above, the trivial solution of system (2) is stable for and the infection-free periodic solution (6) of system (2) is stable for ; then a bifurcation occurs at . Moreover, and for ; hence this bifurcation is a fold bifurcation. Then the following result is obtained.

A fold bifurcation occurs at in system (2) . The trivial solution of system (2) is stable for while the infection-free periodic solution (6) of system (2) is stable for .

Existence and stability of the positive periodic solution

In this section, we discuss the existence and stability of the positive periodic solution , where and . To obtain the explicit solution of system (2), let It follows from (2) that

Set the initial point of system (8) as . The trajectory originating from this initial point reaches the point at time , and next jumps to the point with the effect of the impulse. Like (4), it follows from the first equation of system (8) and the birth pulse at time that and

It follows from the second equation of (8) and that and , that is

From (9), (10), we obtain the following discrete map:

For each fixed point of the map (11) there is an associated periodic solution of system (2), and vice versa. For the facts that and , the discussion of equilibrium of map (11) is meaningless; we omit it here. Now suppose that the positive fixed point of map (11) is ; then and

For and , the following conditions are needed: that is, Hence there exists a nonlinear periodic solution in system (8) under condition (14).

Now we discuss the stability of this positive periodic solution. The associated characteristic polynomial of the fixed point is given by and

It is easy to calculate that for . Further, it follows from (14) that and and then .

Thus, under condition (14) and , that is, and , which means that the periodic solution of system (8) is stable. Thus the following proposition about the existence and stability of an epidemic periodic solution of system (2) is obtained.

Suppose the following conditions hold:  System (2) has a stable positive periodic solution.

,

.

Bifurcation analysis

Bifurcation of the positive periodic solution near the infection-free periodic solution

In this subsection, we deal with the problem of the bifurcation of the positive periodic solution of system (2) near the infection-free periodic solution. In the following, is viewed as a parameter. As shown in Section 3, the infection-free periodic solution of system (2) is It follows from (7), (11) that the positive periodic solution of system (2) is where is shown in (12).

Let ; system (2) may be rewritten as which are satisfied with , , for and for .

Let be the flow associated with (18), , , where , , . The following notation is given in paper [17]:

The result concerning the bifurcation of the positive periodic solution is the following Lemma:

If and , then we have:

if , then we have a bifurcation. Moreover, we have a bifurcation of a positive periodic solution if and a subcritical case if ;

if , then we have an undetermined case.

For more details see Ref. [17]. The semi-trivial periodic solution and the positive periodic solutions are given in (16), (17); then the bifurcation of the positive periodic solution does occur in our case. In what follows, we just calculate the value of the parameter at which the bifurcation occurs. In our case, The condition gives that is,

Then we get the following result.

In system (2) , the positive periodic solution (17) bifurcates from the infection-free periodic solution (16) at , where is shown in (19) .

Flip bifurcation

For , one of the eigenvalues of the fixed point (12) is . As mentioned in Section 3, the bifurcation associated with the appearance of the eigenvalue −1 is called a flip (or period-doubling) bifurcation. Hence is a candidate for a flip bifurcation point. Viewing as a parameter, we discuss flip bifurcation of system (2) by using the map (11) and the following lemma.

Let be a one-parameter family of a map such that has a fixed point with eigenvalue −1. Assume the following conditions: Then there is a smooth curve of fixed points of passing through , the stability of which changes at . There is also a smooth curve passing through such that is a union of hyperbolic period-2 orbits.

at ;

at .

For the proof of Lemma 5.2, refer to Ref. [25]. In (F2) the sign of determines the stability and the direction of bifurcation of the orbits of period 2. If is positive, the orbits are stable; if is negative they are unstable.

A flip bifurcation occurs at in system (2) . For some , system (2) has a stable -periodic solution for .

It follows from Section 4 that one of the eigenvalues of the fixed point is at , and is a candidate for a flip bifurcation point. Map (11) may be rewritten as where The map in Lemma 5.2 is one-dimensional while the map (20) is two-dimensional in our case. The center manifold theorem is used here to obtain a one-dimensional map; hence, flip bifurcation is discussed.

For , it follows from (12) that the fixed point of map (20) is

Letting , and , we transform the fixed point of map (20) to the origin; then the map (20) becomes where .

On using (23) becomes where If we let and use the translation , then map (24) becomes where

Now the center manifold theorem is used to determine the nature of the bifurcation of the fixed point at . There exists a center manifold for (26) which can be represented as follows: In view of the form of , it is not necessary to calculate in our case, and the map restricted to the center manifold is given by Thus and at . Then conditions and hold. So a flip bifurcation occurs in view of Lemma 5.2. A positive -periodic solution bifurcates from the positive -periodic solution at , that is, Since in , then the positive -periodic solution is stable. This also means that for some , system (2) has a positive stable -periodic solution for . Thus we complete the proof of Proposition 5.2.

Numerical results

Now consider the following example:

In our case, , , . In the following, the infection-free periodic solution and epidemic periodic solution, bifurcation diagram, and detailed results about the existence of chaos are given to illustrate the theoretical analysis.

Set , , , and the initial point , . The time series of of system (29) for and are shown in Fig. 1 . It is seen that the infection-free periodic solution is stable for and the trivial solution is unstable for , which verifies Proposition 3.1.

Fig. 1

The time series of for system (29) with , , the initial point , and (a) , (b) .

Now set , , ; then , , Thus conditions (H1) and (H2) hold. From Proposition 4.1, system (29) has a stable positive periodic solution (see Fig. 2 (a)). It is seen from Fig. 2(b) that the solution with the initial point of system (29) tends to this positive periodic solution with increasing.

Fig. 2

The solutions for system (3) with , , . (a) Positive periodic solution. (b) Stability of the positive periodic solution.

Now set , . From (19), (27), Fig. 3 shows the bifurcation diagram of stable periodic solutions of system (29) with respect to parameter . In Fig. 3(a), the epidemic periodic solution bifurcates from the infection-free periodic solution at . For , the infection-free periodic solution is unstable and the epidemic periodic solution is stable. At , flip bifurcation, that is, period-adding bifurcation, occurs. A period-2 solution bifurcates from the epidemic periodic solution. It is seen from Fig. 3(a) that the numerical results are in good agreement with the theoretical results and system (29) possesses rich dynamics including different kinds of bifurcation and periodic windows. Fig. 3(b) shows the period-3 window. The period-3 solution is stable for and another flip bifurcation occurs at .

Fig. 3

The bifurcation diagrams of system (29) with respect to parameter ; (a) , (b) .

It is seen from the bifurcation diagram Fig. 3 that there exist periodic solutions. Fig. 4 shows the -periodic, -periodic, and -periodic solutions in system (29) with , and , respectively. Both theoretical and experimental investigation have revealed that the three main routes to chaos are the route via torus bifurcation, the period-doubling route, and intermittency. From Fig. 3(a), we know that the route to chaos is period-doubling bifurcation in our case. Fig. 4(d) shows that chaos does exist in system (29) for .

Fig. 4

Phase portraits of system (29). (a) -periodic solution with ; (b) -periodic solution with ; (c) -periodic solution with 3.46; (d) strange attractor with .

Conclusion

The dynamics of an epidemic model with birth pulses and a varying population was studied in this paper. It was seen that the dynamics of impulsive system (2) is very rich and interesting, although the corresponding system (1) without impulses is very simple. The existence and stability of the infection-free periodic solution and positive periodic solution (S(t), I(t)) are investigated. The conditions of existence for the bifurcation of the positive periodic solution are derived by virtue of the bifurcation theorem.

It is shown that the trivial solution of the system is stable for , the infection-free periodic solution is stable for , and the time period between two consecutive births is important for the population . For avoiding extinction, should be less than . A positive periodic solution (period 1) bifurcates from this infection-free periodic solution at (19) through a supercritical bifurcation; a period-2 solution bifurcates from this positive periodic solution (period 1) at through flip bifurcation (period-doubling bifurcations). The numerical results show that the chaotic solution is generated via a cascade of period-doubling bifurcations.