Stability analysis of a schistosomiasis model with delays

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 [1]. 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 [11], 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.

In this section, we derive a mathematical model for the spread of schistosomiasis. Here, we consider human and snail populations. We assume that all newborns are susceptible, and that the infection does not result in death of human and snail populations. Further, it is assumed that a susceptible host became infected only by contact with water, in which there exist cercariae from infected snails, and a susceptible snail became infected by contact with miracidia coming from parasite eggs released in feces and urine of infected hosts (see Figure 1 below).

We denote by

We also denote by

Note that the support of $k_1$ and $k_2$ 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 ${\int }_0^h{k}_1({\tau }_1)d{\tau }_1=1$ and ${\int }_0^h{k}_2({\tau }_2)d{\tau }_2=1$.

As general as possible, the incidence functions f and g must satisfy technical conditions. Thus, we assume that

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 ${\nu }_1,{\nu }_2\in C([-h,0],{\mathbf{R}}^{+})$. We also define the sup norm on $C([-h,0],{\mathbf{R}}^{+})$ as $\parallel {\nu }_i\parallel ={sup}_{\theta \in [-h,0]}{\nu }_i(\theta )$, $i=1,2$. Standard theory of functional differential equations (see [14]) can be used to show that solutions of system (2.1) exist and are differentiable for all $t>0$.

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 [11]).

is positively invariant for system (2.1).

To prove Theorem 2.2, we need the following result.

Theorem 2.3 [15]

Let $L:{\mathbf{R}}^n⟶\mathbf{R}$ be a differentiable function, and let $a\in \mathbf{R}$. Let $X(x)$ be the vector field, and let G be the closed set $G=\{x\in {\mathbf{R}}^n:L(x)\le a\}$ such that $\mathrm{\nabla }L(x)\ne 0$ for all $x\in L^{-1}(a)=\{x\in {\mathbf{R}}^n:L(x)=a\}$. If $〈X(x)|\mathrm{\nabla }L(x)〉\le 0$ for all $x\in L^{-1}(a)$, then the set G is positively invariant.

Then $〈X(x)|\mathrm{\nabla }L(x)〉=-{\mathrm{\Lambda }}_h<0$. This proves that $\{H_s\ge 0\}$ is positively invariant. Similarly, we prove that $\{H_i\ge 0\}$, $\{S_s\ge 0\}$, $\{S_i\ge 0\}$ are positively invariant. Then $\{(H_s,H_i,S_s,S_i)\in {\mathbf{R}}^4:H_s\ge 0,H_i\ge 0,S_s\ge 0,S_i\ge 0\}$ 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 $t>0$.

Theorem 2.4 Assume that $H(t)=H_s(t)+H_i(t)$ and $S(t)=S_s(t)+S_i(t)$.

where $H(0)=H_s(0)+H_i(0)$. Thus, as $t\to \mathrm{\infty }$, $H(t)\le \frac{{\mathrm{\Lambda }}_h}{d_h}+\frac{ϵ}{d_h}$.

Similarly, we prove that $S(t)\le \frac{{\mathrm{\Lambda }}_s}{d_s}+\frac{ϵ}{d_s}$. □

where

According to [17], we conclude the proof. □

The basic reproduction number ${\mathcal{R}}_0$ represents the average number of new case generated by a single infected individual in a completely susceptible population (see [18]).

Theorem 3.2 If ${\mathcal{R}}_0<1$, then ${\mathcal{E}}_0$ 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 ${\mathcal{E}}_0$ is locally asymptotically stable. □

Now, we will study the behaviour of system (2.1) when ${\mathcal{R}}_0>1$.

Theorem 3.3 If ${\mathcal{R}}_0>1$, then there exists an endemic equilibrium.

Thus, ${\mathcal{R}}_0>1$. □

In this section, we study the global behaviour of the disease-free equilibrium. For that, we assume that

We have the following result.

Theorem 4.1 Let $\delta =max(d_s,d_h)$. Assume that H3 and H4 hold, and $\delta \le 1$, then the disease-free equilibrium is globally asymptotically stable if ${\mathcal{R}}_0\le 1$.

with equality only if $S_i=0$ and $H_i=0$. According to LaSalle’s extension to Lyapunov’s method [19], the limit set of each solution is contained in the largest invariant set, for which $S_i=0$ and $H_i=0$, which is the singleton $\{{\mathcal{E}}_0\}$. This means that the disease-free equilibrium ${\mathcal{E}}_0$ is globally asymptotically stable on ${\mathrm{\Gamma }}_{ϵ}$. □

In this section, we assume that f and g satisfies the conditions

Theorem 5.1 Assume that H5 and H6 hold, then if ${\mathcal{R}}_0>1$ the endemic equilibrium ${\mathcal{E}}_{\ast }$ is globally asymptotically stable.

where $A({\tau }_1)=2-\frac{f(H_s^{\ast },S_i^{\ast })}{f(H_s,S_i^{\ast })}+\frac{f(H_s,S_{{\tau }_1}^i)}{f(H_s,S_i^{\ast })}-\frac{H_i^{\ast }}{H_i}\frac{f(H_s,S_{{\tau }_1}^i)}{f(H_s^{\ast },S_i^{\ast })}-\frac{H_i}{H_i^{\ast }}$.