Supplementary Text: Derivation of and the proof of the main result.
Model (2.1) has a disease-free periodic solution . To estimate the basic reproduction ratio of model (2.1),according to the general procedure established by Wang and Zhao[[1]], we first introduce two matrices,
It is easy to see that is non-negative, and is cooperative in the sense that the off-diagonal elements of are non-negative. Let and be the monodromy matrix of the linear -periodic system and the spectral radius of , respectively. For the linear -periodic system,
,(3.1)
assume that , is the evolution operator of this system. That is, for each, the matrix satisfies
where I is the identity matrix. Thus, the monodromy matrix of (3.1) is equal to, .
In view of the periodic environment, we assume that, -periodic ins, is the initial distribution of infectious individuals. Then is the rate of new infections produced by the infected individuals who were introduced at times. Given, then gives the distribution of those infected individuals who were newly infected at time sand remain in the infected compartments at timet. It follows that
is the distribution of accumulative new infections at time tproduced by all those infected individuals introduced at time previous tot.
Let be the ordered Banach space of all -periodic functions from RtoR2, which is equipped with the maximum norm and the positive cone. Then we can define a linear operator by
where L is called the next infection operator. The basic reproduction ratio of system (2.1) is defined as the spectral radius of L, i.e..
In order to characterize, we consider the following linear -periodic equation
(3.2)
with parameter. Let, be the evolution operator of the system (3.2) onR2. We have
It is easy to verify that system (2.1) satisfies assumptions (A1)-(A7) in [1]. Thus, we have the following two results, which will be used in our numerical computation of the basic reproduction ratio and the proof of our main result, respectively.
Lemma 3.1. The following statements are valid:
(i) If has a positive solution , then is an eigenvalue ofL, and hence.
(ii) If, then is the unique solution of .
(iii)if and only if for all.
Lemma 3.2. The following statements are valid:
(i) if and only if.
(ii) if and only if.
(iii) if and only if.
Thus, the disease-free equilibrium is locally asymptotically stable if , and unstable if .
By Lemma 3.1(ii), we know that the basic reproduction ratio is determined by parameter of . We can calculate the basic reproduction ratio using the numerical method.
In the autonomous case, i.e. and for any , we obtain , and for any , the basic reproduction ratio of the disease iswhich corresponds to the result of Barbour (1996) [[2]].
In the following, we prove the main result, which shows that is a threshold parameter for the extinction and the uniform persistenceofthe schistosomiasis model (2.1).
Wefirstdefine for model (2.1). It is easy to prove the following theorem.
Theorem3.1. Model (2.1) has a unique solution with the initial value
,
and this compact set is positively invariant.
Define
.
Let be the Poincare map associated with model (2.1), that is,
,
where is the unique solution of model (2.1) with . It is easy to see that
.
We establish the following lemma which will be useful in subsequent main result.
Lemma 3.3. If the basic reproduction ratio , then there exists a , such that for any with , we have
(3.3)
Proof. Since , Lemma 3.2 implies . It follows that holds for sufficiently small , where
By the continuity of the solutions with respect to the initial values, there exists a such that for all with , there holds , for all . Next, we claim that . Assume, by contradiction, that (3.3) does not hold. Then we have
for some . Without loss of generality, we assume that , for all. It follows that
.
For any , let , where , and is the largest integer less than or equal to . Therefore, we have
.
Note that . It then follows that ,, for all . From model(2.1), we obtain
(3.4)
We then consider the following system
(3.5)
By Zhang and Zhao ([[3]], Lemma 2.1), we know that there exists a positive, -periodic function such that is a solution of system (3.5), where. Since , is a positive constant. Let , and be any nonnegative integer, and we get
as , since and . For any nonnegative initial condition of system (3.4), there exists a sufficiently small such that . By the comparison principle ([[4]], Theorem B.1), we have , for all.Thus, we obtain and , as , a contradiction.
Then we have the followingTheorem.
Theorem 3.2.If the basic reproduction ratio , then the unique disease-free equilibrium is globally asymptotically stable. If the basic reproduction ratio , then there exists a constant such that any solution of system (2.1) with initial value
()
satisfies
and .
Proof: By Lemma 3.2, we know that if , then is locally asymptotically stable. It is sufficient to prove that is globally attractive if . From system (2.1), we have
(3.6)
Consider the following comparison system
(3.7)
Applying Lemma 3.2, we know that if and only if . By Zhang and Zhao ([3], Lemma 2.1), it follows that there exists a positive, -periodic function such that is a solution of system (3.7), where . Since , is a negative constant. Therefore, we have as . This implies that the zero solution of system (3.7) is globally asymptotically stable. For any nonnegative initial value of system (3.6), there is a sufficiently large such that holds. Applying the comparison principle ([4], Theorem B.1), we have , for all , where is also the solution of system (3.7). Therefore, we get , and as. We finish the proof of the first part of the theorem.
By Theorem 3.1, the discrete-time system admits a global attractor in . Now we prove that is uniformly persistent with respect to . Clearly, there is exactly one fixed point of in . Lemma 3.3 implies that is an isolated invariant set in and . By Zhao ([[5]],Theorem 1.3.1), it follows that is uniformly persistent with respect to . By Zhao ([5], Theorem 3.1.1), the solutions of system (2.1) are uniformly persistent with respect to , that is, there exists a constant such that any solution of system (2.1) with initial value satisfies
and.
[[1]]Wang WD, Zhao XQ: Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Diff Equat 2008, 20:699-717.
[[2]] Barbour AD: Modelling the transmission of schistosomiasis an introductory view.Am J Trop Med Hyg1996, 55 (5 Suppl):135-143.
[[3]] Zhang F, Zhao XQ: A periodic epidemic model in a patchy environment.J Math Anal Appl 2007,325: 496-516.
[[4]] SmithHL, WalmanP:The Theory of the Chemostat.Cambridge University Press, Cambridge, 1995.
[[5]] Zhao XQ:Dynamical Systems in Population Biology.Springer, New York 2003.