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From the lifecycle of schistosome, the phenomenon of time delay is widespread. In this paper, a two-dimensional system is studied that incorporates two time delays which are the incubation period of human and snail, respectively. Our purpose is to demonstrate that the time delays are harmless for stability of equilibria of the system. Further, sufficient conditions of stability of equilibria are obtained.

Mathematical models ([

where

In [

where

From biological view, we assume that system (2) holds for the time

where

into

In the following, we focus on dynamics of system (2) in a nonnegative cone

It is well known by the fundamental theory of functional differential equations [

The remainder of the paper is organized as follows. In the next section, the stability of the disease-free equilibrium of system (2) is obtained. In Section 3, we investigate the stability of the endemic equilibrium. Some dynamical behaviors are given by numerical simulations in Section 4. This paper is ended with a brief discussion.

In this section, the stability of the disease-free equilibrium of system (2) is investigated.

Using standard methods, it is easy to see that the disease-free equilibrium

Define the basic reproductive number by

Then for system (2), it is easy to obtain the following result:

(i) If

(ii) If

In the following, we study the global stability of the disease-free equilibrium

Theorem 2.1. If

Proof. First, according to [

Then the characteristic equation of system (2) at

where

When

If

Assume that there exists a

Separating real and image parts:

Adding up the squares of both equations, we obtain that

Note that

and

Next, the global stability of the disease-free equilibrium of system (2) is analyzed. And the strategy of proof is to use Lyapunov functionals and the LaSalle invariance principle.

Theorem 2.2. If

Proof. Let

Define

where

Calculating the derivative of

Define

We derive from (8) and (9) that

Define

It follows from (10) and (11) that

On substituting

If

which yields

It is obtained that the endemic equilibrium

Similar to the proof of Theorem 2.1, the following result is obtained.

Theorem 3.1. If

Proof. First, according to [

Then the characteristic equation of system (2) at

where

When

If

Assume that there exists a

Separating real and image parts:

Adding up the squares of both equations, we obtain that

We know that

Now, we are interested in the global stability of

Theorem 3.2. If

Proof. It is easy to check that equilibrium

When

It follows that

which leads to the nonexistence of periodic orbits by Bendixson theorem, therefore,

It is reported that cercariae are produced about 44 - 159 days after the miracidium penetration in snails. And the time from parasite eggs to miracidia to infect snail is about 21 days. Therefore, we choose

Thus, we can obtain

From the above theorems, we know that the two time delays are harmless. According to the expression of

the impact of C and H on schistosomiasis transmission is discussed. Fixing

From the formula of the basic reproductive number, we know that the basic reproductive number is a decrease function of the rates of chemotherapy and predation or harvesting. This means chemotherapy and predation or harvesting can influence the system.

However, to find out the most influential control measure, we need sensitivity analysis. Now we carry out the sensitivity analysis by calculating the derivation of

From

In brief, the basic reproductive number

By sensitivity analysis of the basic reproductive number on the rates of chemotherapy and predation or harvesting, we know that the basic reproductive number is a decrease function of the rates of chemotherapy and predation or harvesting. In numerical simulations, we also find that the smaller of values of the rate of chemotherapy, the more sensitive of the basic reproductive number

Although the two time delays are harmless, all of these results imply that the rates of chemotherapy and predation or harvesting can influence the dynamic behaviors. Furthermore, to reduce the prevalence of schistosomiasis infection, to some extent, increasing the rate of predation or harvesting by some measures could achieve better results than increasing the rate of chemotherapy.

In this paper, we propose a system of delayed differential equations for schistosomiasis japonicum transmission and obtain sufficient conditions for the existence and local stability of equilibria. Further, global asymptotic stability of the disease-free equilibrium is also studied by constructing suitable Lyapunov functions. When

Finally, we guess that the endemic equilibrium should be global asymptotic stable when

The research has been partially supported by The Natural Science Foundation of China (No. 11261004), China Postdoctoral Science Foundation funded project (No. 2012M 510039), the National Key Technologies R & D Program of China (2009BAI78B01, 2009BAI78B02) and the Natural Science Foundation of Jiangxi (20114 BAB201013, 20122BAB211010).

HuahuaCao,ShujingGao,XiangyuZhang,YouquanLuo, (2014) A Mathematical Model for Schistosomiasis Japonicum with Harmless Delay. Applied Mathematics,05,2807-2814. doi: 10.4236/am.2014.517268