Stability analysis of a schistosomiasis model with delays
© Guiro et al.; licensee Springer. 2013
Received: 12 May 2013
Accepted: 23 September 2013
Published: 8 November 2013
In this work, a nonlinear deterministic model for schistosomiasis transmission including delays with two general incidence functions is considered. Rigourous mathematical analysis is done. We show that the stability of the disease-free equilibrium and the existence of an endemic equilibrium for the model are stated in terms of key thresholds parameters known as basic reproduction number . This study of the dynamic of the model is globally asymptotically stable if , and the unique endemic equilibrium is globally asymptotically stable when . Some numerical simulations are provided to support the theoretical result with respect to in this paper.
Schistosomiasis is a serious health problem in developing countries. Indeed, despite the remarkable achievements in schistosomiasis control over the past five decades, there are about 240 million people infected worldwide, and more than 700 million people live in endemic areas . There are two patterns of schistosomiasis. We note the urinary schistosomiasis and the intestinal schistosomiasis. The first one is caused by Schistosoma haematobium, when the second is caused by any of the organisms Schistosoma intercalatum, Schistosoma mansoni, Schistosoma japonicum and Schistosoma mekongi. Mathematical modeling of schistosomiasis transmission can help in the development of the strategies for control. Thus, several mathematical models for this disease have been done (see [2–13] and the references therein). In , a discrete delay model for the transmission is studied. The delay appears in the incidence term including masse action SI (S: susceptible, I: infectious). It appears that the incidence function form is determinative in the study of the model system. Then, changing the form of the incidence can potentially change the behaviour of the system. In this paper, a mathematical model is derived with a bounded delay distributed and two general incidence functions term f and g. The model described here considers two population hosts, humans and snails, and is structured as follows: Susceptible (uninfected) and infectious humans and susceptible (uninfected) and infected snails. The paper is organized as follows. In Section 2, we present the mathematical model, and we study the mathematical properties of the model system. In Section 3, we derive some results about the basic reproduction number, the disease-free equilibrium and the endemic equilibrium. Section 4 is devoted to the global stability of the disease-free equilibrium. In Section 5, we study the global stability of the endemic equilibrium. Section 6 is devoted to numerical simulation. Finally, in Section 7, we end by a conclusion.
2 The mathematical model
We denote by
the susceptible (uninfected) human population size;
the infected human population size (infectious humans);
the susceptible (uninfected) snail population size;
the infected and shedding snail population size (shedding snail population size).
We also denote by
the recruitment rate of susceptible humans;
the recruitment rate of susceptible snails;
the per capita natural death rate of humans;
the per capita natural death rate of snails;
the transit time from cercaria in water to schistosomule in a human host;
the transit time from parasite eggs to miracidia to infect a snail;
and the Lebesgue integrable functions, which give the relative infectivity of snails and humans (respectively) of different infection ages.
Note that the support of and has a positive measure in any open interval having supremum h, so that the interval of integration is not artificially extended by concluding with an interval, for which the integral is automatically zero. On the other hand, we choose in the model two real numbers α and γ, so that and .
As general as possible, the incidence functions f and g must satisfy technical conditions. Thus, we assume that
H1 f and g are non-negative functions on the non-negative quadrant,
H2 for all , and .
Remark 2.1 f and g are two incidence functions, which explain the contact between two species. Therefore, f and g are non-negative. Note also that when there is no one infected in the human and snail populations, then the incidence functions are equal to zero. The incidence functions are also equal to zero, when there is no one susceptible in the human and snail populations.
where . We also define the sup norm on as , . Standard theory of functional differential equations (see ) can be used to show that solutions of system (2.1) exist and are differentiable for all .
The delay is inspired by the life history of the schistosomiasis. Indeed, it is possible that some hosts or intermediate hosts (snails) die due to natural death during the incubation period, respectively (see ).
is positively invariant for system (2.1).
To prove Theorem 2.2, we need the following result.
Theorem 2.3 
Let be a differentiable function, and let . Let be the vector field, and let G be the closed set such that for all . If for all , then the set G is positively invariant.
Then . This proves that is positively invariant. Similarly, we prove that , , are positively invariant. Then is positively invariant for system (2.1). □
Therefore, the model is mathematically well posed and epidemiologically reasonable since all the variables remain non-negatives for all .
Theorem 2.4 Assume that and .
where . Thus, as , .
Similarly, we prove that . □
3 Basic reproduction number, disease-free equilibrium and endemic equilibrium
denote the rate of appearance of new infections in each class j,
denote the rate of transfer into each class j by all other means,
denote the rate of transfer out of each class j.
According to , we conclude the proof. □
The basic reproduction number represents the average number of new case generated by a single infected individual in a completely susceptible population (see ).
Theorem 3.2 If , then is locally asymptotically stable.
This implies that λ cannot be a solution of the characteristic equation. Hence, all eigenvalues have negative real part, and then is locally asymptotically stable. □
Now, we will study the behaviour of system (2.1) when .
Theorem 3.3 If , then there exists an endemic equilibrium.
Thus, . □
4 Global stability of the disease-free equilibrium
In this section, we study the global behaviour of the disease-free equilibrium. For that, we assume that
H3 for all , and ,
H4 and .
We have the following result.
Theorem 4.1 Let . Assume that H3 and H4 hold, and , then the disease-free equilibrium is globally asymptotically stable if .
with equality only if and . According to LaSalle’s extension to Lyapunov’s method , the limit set of each solution is contained in the largest invariant set, for which and , which is the singleton . This means that the disease-free equilibrium is globally asymptotically stable on . □
5 Global stability of the endemic equilibrium
In this section, we assume that f and g satisfies the conditions
H5 for all , and ,
H6 for all , and .
Theorem 5.1 Assume that H5 and H6 hold, then if the endemic equilibrium is globally asymptotically stable.
for all with equality only for , , and . Hence, the endemic equilibrium is the only positively invariant set of system (2.1) contained in . Then it follows that is globally asymptotically stable on (see ). □
6 Numerical simulation
In this paper, a deterministic model of transmission of schistosomiasis with two general nonlinear incidence functions including distributed delay is derived. The global behaviour of the model system was studied. We proved that, if holds, then the disease-free equilibrium is globally asymptotically stable, which implies that the disease fades out from the population. If , then there exists a unique endemic equilibrium which is globally asymptotically stable, and this implies that the disease will persist in the population. This result suggests that the latent period in infection affects the prevalence of schistosomiasis, and it is an effective strategy on schistosomiasis control to lengthen in prepatent period on infected definitive hosts by drug treatment, for example.
Threshold analysis of the basic reproduction number shows that the use of public health education campaign could have positive, more determinant impact on the control of the schistosomiasis. Overall, an effective education campaign which focuses on drug treatment with reasonable coverage level could be helpful for countries concerned with the disease.
The authors express their deepest thanks to the editor and an anonymous referee for their comments and suggestions on the article.
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