Dynamical Analysis of a Schistosomiasis japonicum Model with Time Delay

In this paper, a Schistosomiasis japonicum model incorporating time delay is proposed which represents the developmental time from cercaria penetration through skins of human hosts to egg laying. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equations, the local stability of the equilibria is investigated. And it proves that Hopf bifurcations occur when the time delay passes through a sequence of critical value. Furthermore, the explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using techniques from the normal form theory and Center Manifold Theorem. Some numerical simulations which support our theoretical analysis are also conducted.


Introduction
Schistosomiasis is one of the most prevalent parasitic diseases.In 2004, WHO suggested that there were 200 million individuals infected worldwide [1].However, today, more than 207 million people, 85% of whom live in Africa, are infected with schistosomiasis [2], and the estimated 700 million people are at risk of infection in 76 countries where the disease is considered endemic, as their agricultural work, domestic chores, and recreational activities expose them to infested water [2] [3].Globally, 200,000 deaths are attributed to schistosomiasis annually [4].Thus, controlling schistosomiasis is a long-term task in the developing nations, and mathematical modeling of Schistosoma japonicum transmission is beneficial for the development of new strategies for control.
where 1 0 β > and 2 0 β > are the disease transmission coefficients; 0 v > is the recovery rate of infected host.The birth and death rates have been scaled to 0 b > for the host and 0 c > for the vector.
Note that an infected snail cannot infect susceptible man (or an animal) directly and vice versa.Schistosomiasis has a complicated life cycle involving two free living stages, the miracidia and the cercariae; and two host populations, the human and the snail.The parasite eggs hatch into free-swimming larva called miracidia in water; the miracidium then penetrates an appropriate snail at suitable temperature.In the infected snail, the miracidium undergoes asexual multiplication through a series of stages called sporocyst; then thousands of free-swimming cercariae are released.Cercariae are shed from the infected snail and penetrate the skin of a definitive host (such as human) within a few minutes after exposure and transform into schistosomula, which travel through the bloodstream to the liver, where they mature into adults and start producing eggs [20].The eggs infiltrate through the tissues and are passed in the feces.That finishes schistosomiasis life cycle.Besides, the model (1.1) can be used to describe the transmission of schistosomiasis since schistosomiasis is a snail-vector disease.
It is known that there are prepatent periods of schistosoma.In fact, it is about five weeks from the time of cercaria penetration through skins of human hosts to the time when eggs are discharged [21].That is, a susceptible host becomes infection for some time and then excretes feces with parasite eggs.It is easy to see that the prepatent period of hosts is very important for Schistosome japonicum transmission.Hence, it is necessary to study the impact of the prepatent period on schistosomiasis transmission.The aim of this paper is to incorporate the prepatent period of infected hosts into (1.1), and estimate the impact of the prepatent period on the schistosomiasis transmission.In this paper, we incorporate effects of the prepatent period of infected hosts into the model (1.1) and propose a schistosomiasis model as follows: , , , : 0, 1, 2,3, 4 In system (1.2),A and Λ are the recruitment rates of hosts and snails, respectively.The constant 1 β is the per capita rate of infection of hosts by cercaria released by a infected snail, 2 β is the per capita rate of infection of snails by miracidia from the parasite eggs from a infected host.The constant v is the recovery rate of infected host.Constants i d and ( ) represent the natural death rate and disease inducing death rate of hosts and snails, respectively.τ is the prepatent period in host.We assume that all parameters are positive constants.
In fact, the reciprocal of the death rate, is infinitely close to 1. Therefore, we make a simplification common in system (1.2) with the survival rate and assume that the survival rate has negligible impact on dynamics.Thus system (1.2) can be written in the following form: The effects of time delays on the dynamical behaviors of schistosomiasis have been investigated in the literatures [21] [22] [23] [24].For example, Liang et al. [21] investigated the development period of worms in human hosts, they described temperature-dependent and precipitation-dependent effects on snail abundance and infection as well as seasonal aspects of local agricultural practice.
In [23], a fixed delay was inspired by the life history of schistosomes, they investigated the impact of the delay on the invasion and persistence of drug-resistant parasite strains as well as on multi-strain coexistence.The main purpose of this paper is to study the effects of the time delay on the dynamical behaviors of (1.3), and discuss the direction of the bifurcation and stability of the bifurcating periodic solutions.
The remainder of the paper is organized as follows.In Section 2, we obtain the stability of disease-free equilibrium and the existence of the endemic equilibrium.In Section 3, by analyzing the characteristic equation of the linearized system of system (1.3) at the endemic equilibrium, we discuss the stability of the endemic equilibrium and the existence of the Hopf bifurcations occurring at the endemic equilibrium.In Section 4, by using the normal form theory and the Center Manifold Theorem, the formulae determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions are obtained.Some numerical simulations are presented to illustrate our theoretical results in Section 5.This paper ends with a brief conclusion.

Equilibrium Analysis
In this section, we discuss the existence of equilibria and the stability of the disease free equilibrium.
When the infective hosts and the infective snails do not exist, i.e., 0

Λ
. This is the infection free equilibrium for schistosomiasis.The following theorem determines stability of 0 E and existence of endemic equilibrium in terms of a threshold parameter ( )( ) 3) has a unique equilibrium 3)   has an endemic equilibrium ( ) Proof Computing the nonnegative solutions of the following equations: we can easily obtain the existence of two equilibria 0 E and * E .
Next, we show the stability behavior of equilibrium 0 E by finding the eigenvalues of the corresponding Jacobian matrix obtained for system (1.3).
The Jacobian matrix for system (1.3) is as follows: e 0 e 0 0 0 0 Let ( ) 0 J E be the Jacobian matrix J evaluated at the equilibrium 0 E .From ( ) 0 J E , it is easy to calculate the associated characteristic equation of system It is obvious that Hence, we only need to discuss the roots of the following equation: Note that all characteristic roots of (2.5) are negative if 0 1 R < .This yields that all roots of (2.4) are negative if 0 1 R < .We complete the proof.

Endemic Equilibrium and Hopf Bifurcation
In this section, We investigate the stability of the endemic equilibrium where ( ) In the above expression of ( ) P λ and ( ) where Now it is easy to see that 1 0 a > .Thus for the local stability of the endemic equilibrium ( ) 3) without delay, we have the following result.
Theorem 3.1.When 0 1 R > , the endemic equilibrium ( ) , , , E X Y S I is locally asymptotically stable for 0 τ = if the following conditions are satisfied: , , a a a and 4 a are defined as above.
Then we turn to an investigation of local stability of the endemic equilibrium We know all roots of characteristic Equation (3.1) have negative real parts at 0 τ = when the conditions in Theorem 3.1 are satisfied.Next we will show that there is a unique pair of purely imaginary roots ( ) Assume that for some 0 τ > , ( ) is a root of (3.1), which implies ( ) Separating real and imaginary parts, we get the following equations: where = in above Equation (3.5), we have Now if the coefficients in ( ) f η satisfy the conditions of the Routh-Hurwitz criterion, then Equation (3.6) will not have any positive real root, thus we may not get any positive value of ω which satisfies the Equation (3.5).In this case the result may be written in the form of following theorem.
Theorem 3.2.Assume that the coefficients in ( ) the conditions of the Routh-Hurwitz criterion, then the endemic equilibrium 3) is asymptotically stable for all delay 0 τ > if it is stable in the absence of delay.
Assuming contrary that the values of ( ) ) ( ) Then k τ corresponding to this positive value of 0 ω is given as follows:


By using Butler's Lemma, we can say that the endemic equilibrium * E remains stable for 0 τ τ < .
Next we investigate whether there is a phenomenon of Hopf bifurcation as τ increases through 0 τ .For this the following lemma is needed.Proof Differentiating Equation (3.1) with respect to τ , we get where

P P P P
) ( ) From the above argument, we know Here it may be noted that ( ) τ .Next, in Section 5 the stability of bifurcating periodic solutions is investigated by analyzing higher order terms according to Hassard et al. [25].

Stability and Direction of Hopf Bifurcation
In this section, in term of the center manifold and normal form theory due to Hassard et al. [25], the direction of hopf bifurcation and the stability of periodic bifurcation solution are discussed.Without loss of generality, and k τ µ τ = + .So, 0 µ = is the Hopf value of system (1.3).

Let
In fact, we can choose where δ denote the Dirac delta function: ( ) where and a bilinear inner product where ( ) ( ) Supposed ( ) ( ) 1, , , e k i q q q q ωτ θ θ = is the eigenvectors of ( ) Then from the definition of ( ) and (4.2), (4.4) and (4.5), we have ) Similarly, we can obtain the eigenvector q s q θ = , we need to determine the value of D. By (4.7), we have q s q D q q q q q q D q q q q q q D q q q q q q q q q q q q D q q q q q q I q Next we will compute the coordinate to describe the center manifold 0 C at 0 µ = .Let t u be the solution of (4.6) when 0 On the center manifold 0 C , we have ( where z and z are local coordinates for center manifold 0 C in the direction of * q and * q .Note that W is real if t u is real.We only consider real solutions.For solution It follows from (4.8) and (4.9) that , , , , Then from the definition of ( ) Comparing the coefficients with (4.11), we have   From (4.11) and (4.13), we know that for H z z q f q g z z q g z z q g q g q z From the definition of ( )  where ( ) ( ) . By (4.13), we know when By (4.3), we have ( ) ( ) ( ) ( ) Similarly, substituting (4.23) and (4.29) into (4.25),we also get   According to [25], we can obtain the following result.
From Theorem 4.1, we know that the value of 2 µ determines the directions of the Hopf bifurcation, the values of 2 β and 2 T determine the stability and the period of the bifurcating periodic solutions, respectively.

A Numerical Example
In this section, we implement numerical simulations to testify the above theoretical results.Let

Conclusion
In this paper, we have investigated a delayed schistomiasis model, and studied  the local stability of the equilibria and Hopf bifurcation.We have shown that if 0 1 R < , the disease-free equilibrium is locally asymptotically stable.Further, the sufficient conditions for the stability of the endemic equilibrium are obtained.
That is, if 0 1 R > and the condition (3.7) hold, the endemic equilibrium * E is asymptotically stable for all [ ) 0 0, τ τ ∈ .As τ increases, the endemic equilibrium loses its stability and a sequence of Hopf bifurcations occurs at the endemic equilibrium; that is, urcates from the equilibrium.This shows that the density of thea family of periodic orbits bif susceptible human, snails and the infected human, snails may keep in an oscillatory mode near the endemic equilibrium.By the normal form theory and the Center Manifold Theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits have been discussed.

1 1 d
, is equivalent to the life expectancy of human.Assume that

7 )
is satisfied.This proves the Lemma 3.1.Thus we have the following result: Theorem 3.3.If 0 1 R > and the condition (3.7) hold, then the endemic equilibrium * E of system (1.3) remains stable for all bifurcation.Remark.It must be pointed out that Theorem 3.3 cannot determine the stability and direction of bifurcation periodic solutions.That is to say, the periodic solutions may exist for are also eigenvalues of * A .We first need to compute the eigenvectors of ( )

2 z
) Substituting (4.16) and (4.17) into (4.15) and comparing the coefficients of and zz , we have determine the qualities of bifurcation periodic solution in the center manifold at the critical values k τ .

Figures 2
Figures 2(a)-(f) suggest that Hopf bifurcation occurs from the endemic equilibrium * E when