A Mathematical Model for Schistosomiasis Japonicum with Harmless Delay ()
1. Introduction
Mathematical models ([1] -[7] , etc.) have been used to study the transmission and control of schistosomiasis since the first model that has been given by MacDonald in [8] . MacDonald’s model consists of two differential equations in two state variables that correspond to average parasite burden in the definitive hosts and the prevalence of infection in snails. DAS et al. [5] added a layer of biological realism to these early models to study the delay effect on schistosomiasis transmission with control measures. The model is given by
(1)
where
is the current number of egg laying schistosomes in the human host population and
is the current number of infected snails in the environment. Here,
is the human population density per unit accessible water area;
is the multiplication rate due to the infected snail population;
and
are the intrinsic death rates of two populations
and
respectively;
is the simple contact rate;
is the constant decay rate due to chemotherapy; and
is the constant decay rate by predation or harvesting. Further
is the incubation period for becoming
to be infectious. For simplicity, it is assumed that
is the constant total population of snails.
In [5] , for the sake of mathematical simplicity, they assumed the development of schistosoma is instantaneous. In fact, the developmental time of schistosome is not short. Under normal circumstances, the transit time from parasite eggs to miracidia to infect snail is about 21 days, cercariae are produced about 44 - 159 days after the miracidium penetration in snail hosts. In this paper, we also assume that
and
are the proportions of chemotherapy and predation or harvesting, respectively. Based on the above description, a schistosomiasis model with two time delays is proposed:
(2)
where
is the incubation period for becoming infected human host population and
is the transit time from parasite eggs to miracidia to infect snail. We assume that all parameters are positive.
From biological view, we assume that system (2) holds for the time
with given nonnegative initial conditions:
(3)
where
, the Banach space of continuous functions mapping the interval ![]()
into
, where
.
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 [9] that system (2) has a unique solution
satisfying initial conditions (3). It is easy to show that all solutions of system (2) corresponding to initial conditions (3) are defined on
and remain positive for all
.
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.
2. Stability Analysis of the Disease-Free Equilibrium
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
always exists.
Define the basic reproductive number by
![]()
Then for system (2), it is easy to obtain the following result:
(i) If
, system (2) has a unique disease-free equilibrium
;
(ii) If
, system (1.2) has two equilibria, the disease-free equilibrium
and the unique endemic equilibrium
, where
![]()
In the following, we study the global stability of the disease-free equilibrium
of system (2).
Theorem 2.1. If
, the disease-free equilibrium
of system (2) is locally asymptotically stable.
Proof. First, according to [9] , the Jacobian matrix at
of system (2) can be written as
(4)
Then the characteristic equation of system (2) at ![]()
(5)
where
,
,
,
.
When
, (5) becomes into
(6)
If
, the roots of the equation (6) have negative real parts. Note that
is equivalent to
. Therefore, if
and
,
is locally asymptotically stable.
Assume that there exists a
such that (5) has pure imaginary roots
. Then we have from (5) that
![]()
Separating real and image parts:
![]()
Adding up the squares of both equations, we obtain that
(7)
Note that
![]()
and
Thus,
, which implies that (7) has no positive roots, i.e.,
does not exist. This yields that all roots of (5) have negative real parts if
.
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
, the disease-free equilibrium
is globally asymptotically stable in
for all
.
Proof. Let
be any positive solution of system (2) with initial conditions (3).
Define
,
where
.
Calculating the derivative of
along positive solutions of system (2), it follows that
(8)
Define
(9)
We derive from (8) and (9) that
(10)
Define
. (11)
It follows from (10) and (11) that
(12)
On substituting
and
into (12), we obtain that
(13)
If
, that is
, it then follows from (13) that
. By Theorem
5.
3.1
in [9] , solutions limit to
, the largest invariant subset of
. Clearly, we see from (13) that
if and only if
. Noting that
is invariant, for each element in
, we have
. It therefore follows from the second equation of system (2) that
![]()
which yields
. Hence,
if and only if
. Accordingly, the global asymptotic stability of
follows from LaSalle’s invariance principle.
3. Stability Analysis of the Endemic Equilibrium
It is obtained that the endemic equilibrium
of system (2) is local stable in this section. Further, the global stability of
is shown if
.
Similar to the proof of Theorem 2.1, the following result is obtained.
Theorem 3.1. If
, the endemic equilibrium
of system (2) is locally asymptotically stable.
Proof. First, according to [9] , the Jacobian matrix at
can be written as
(14)
Then the characteristic equation of system (2) at
:
, (15)
where
.
When
, (15) becomes into
, (16)
If
, then
. It is shown that all the roots of the Equation (16) have negative real parts, suggesting
is locally asymptotically stable.
Assume that there exists a
such that (15) has pure imaginary roots
Then we have from (15) that
![]()
Separating real and image parts:
![]()
Adding up the squares of both equations, we obtain that
(17)
We know that
if
, so (17) has no positive roots, i.e.,
does not exist. This yields that all roots of (17) have negative real parts if
.
Now, we are interested in the global stability of
. Then its global stability is investigated by means of Bendixson theorem.
Theorem 3.2. If
, the endemic equilibrium
is globally asymptotically stable in
when
.
Proof. It is easy to check that equilibrium
of system (2) is unstable if
. By the above discussion, we know that equilibrium
is locally stable if
and all solutions of system (2) are ultimately bounded in
. To prove the second assertion, we only prove that system (2) has not periodic orbits in the interior of
if
.
When
,
![]()
It follows that
![]()
which leads to the nonexistence of periodic orbits by Bendixson theorem, therefore,
is globally asymptotically stable.
4. Numerical Simulations
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
and
in this paper. Further, in this section, we perform some numerical simulations and sensitivity analysis using the following value of parameters:
,
,
,
,
,
,
,
.
Thus, we can obtain
, the disease-free equilibrium
is asymptotically stable (Figure 1(a)). When
, the value of other parameters is fixed, we can obtain
and the unique endemic equilibrium is asymptotically stable (Figure 1(b)). In addition, fixing
in simulations, we find that the number of parasite eggs and infectious snails increases as
decreases, respectively (Figure 2).
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
, we can see that when
, the endemic equilibrium exists and is stable, when
, the endemic equilibrium doesn’t exist. But the disease-free equilibrium is stable (Figure 3(a)). Analogously, fixing
, from Figure 3(b), it is obvious that the disease-free equilibrium is stable when
.
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
on
and
. The derivation is respectively
![]()
From Figure 4(a), we can see that when
,
decreases rapidly with the increase of
, the decline of
is not obvious. Similarly,
decreases rapidly with the increase of
when
(Figure 4(b)).
![]()
![]()
Figure 3. Forward bifurcation diagrams for the parasite eggs population.
In brief, the basic reproductive number
is more sensitive when
and
are small.
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.
5. Conclusions
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
, the disease-free equilibrium is globally asymptotically stable (Figure 5(a)); when
, the endemic-free equilibrium is locally asymptotically stable and globally asymptotically stable if
. Thus,
plays an important part in controlling schistosomiasis.
Finally, we guess that the endemic equilibrium should be global asymptotic stable when
. And this guess is verified by numerical simulations (Figure 5(b)). This issue will be addressed in future studies.
Acknowledgements
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).
NOTES
*Corresponding author.