Stability in a Ross epidemic model with road diffusion

1.   Introduction

Mosquito was firstly discovered a transmission vector of Malaria by Ronald Ross in 1898. Ross was also the first to use differential equation to study epidemiology. In order to build an intra-host epidemic model, the mosquito population is divided into two compartments: healthy mosquitoes (susceptible mosquitoes) and infected mosquitoes, while the human population is divided into two compartments: susceptible humans and infected humans. He constructed a two-component system composed with the infected humans $u$ and the infectious mosquitoes $v$ as the following form:

He proposed out the threshold (the predecessor of basic reproduction number) to determine the asymptotic stability of the disease-free equilibrium and endemic equilibrium.

It is assumed that the densities of mosquitoes and humans are homogeneous in the above model. The heterogeneous spatial distribution has formed a reasonable basis for studying insect dispersal (Lewis ^[1]). For example, in 1986 Aedes albopictus was found for the first time in northern counties in Florida (Peacock et al. ^[2]). Then it kept spreading southward, slowly but steadily, and had spread all over the 67 counties (O'Meara et al. ^[3]) in six years' time. By 2008, Aedes albopictus had spread over to 36 states and was still continuing its expansion (Hawley et al. ^[4], Enserink ^[5], and Hahnet et al. ^[6]). In 2013, Rochlin et al.^[7] predicted that, especially in urban areas, in the next 20 years, Aedes albopictus population, will be over three times more. A recent survey on the current distribution of Aedes albopictus (Parker et al. ^[8]) shows that Aedes albopictus have been tracked in 56 of all 67 Florida counties. The mosquito of Aedes albopictus had also been reported to spread along roads. Bennett et al. ^[9] found that the infestation rate of Aedes albopictus was high in garages trading used tires along the highways, and this road provided a channel for fast dispersal across Panama.

Considering the effect of highways on the spreading of mosquito, Berestycki, Roquejoffre and Rossi ^[10] introduced the road diffusion into a reaction-diffusion system. They proposed the model where a two-dimensional environment contains a "line" inside which fast diffusion occurs, while reproduction and the usual diffusion occur only outside. Once "plane" is a field and "line" is a road, this system is a combination of the density of the field with that of the road, between which exists a population exchange satisfied Fickian conservation law. Berestycki et al. have studied the qualitative properties of road-field system (see ^[11,12]). Since the fact the motion of mosquitoes and humans obey the Gaussian distribution, we take the effect of the dispersal into account by using a Laplacian diffusion. Moreover, we study the spread of mosquitoes along the highways with the help of the road diffusion. Thus we improve Ross epidemic model as the form:

where $u(x, y, t)$ represents the density of the infected humans, $v(x, y, t)$ represents the density of the infected mosquitoes, and $w(x, t)$ represents the density of the infected humans living on the road. Meanwhile, $\mathbb{R}\times \mathbb{R}^+$ and $\mathbb{R}$ represent field and road respectively. The first equation accounts for the infected humans dynamic on the field, the second and third for the infected mosquitoes dynamic in the field and road respectively, and the fourth for the exchanges between the field and the road. $d_1$, $d_2$ and $D$ denote the diffusion rates. $\nu$ represents the ratio moving from field to road, $\mu$ for the ratio moving from road to field. $m$ is the rate of the bitten mosquitoes over the whole mosquito community. $a$ is the number of the humans bitted by a mosquito per unit of time. $b_1$ is the probability of a susceptible hunman becoming infectious after a bite of an infected mosquito. $b_2$ is the probability of a susceptible mosquito becoming infectious after it bites a infectd human. $\gamma_1$ and $\gamma_2$ are the infection cycles of the mosquitos and humans respectively. Mosquito and human infectious individual remains infectious on a mean time $\frac{1}{\gamma_1}$ and $\frac{1}{\gamma_2}$ respectively. Epidemic models without road diffusion have been applied to study the spreading of infectious disease through the investigation of asymptotic behavior, seeing for example Allen et al. ^[13,14], Yang et al. ^[15], Lin and Zhu ^[16], Li et al. ^[17], Lei et al. ^[18], and the references therein. Besides the classical Laplacian diffusion in (1.2), the new generalized fractional derivative has been presented in ^[19,20,21] to study the memory effect of epidemic model dynamics. Recently, some important real examples of epidemic model dynamics have been proposed in ^[22,23,24].

This paper is mainly aimed to study the existence and stability of the solutions to system (1.2). Section 2 proves the global existence and uniqueness of solutions to the road diffusion problem (1.2). Section 3 investigates the asymptotic stabilities of the disease-free equilibrium and endemic equilibrium based on the theory of the basic reproduction number. Section 4 gives out some discussions and conclusions.

2.   Existence and uniqueness

We use the approach of upper and lower solutions to study the existence and uniqueness of solutions.

Definition 2.1. ${\rm{Suppose}}\; {\rm{that}}\; \tilde{\bf{u}}: = (\tilde u, \tilde v, \tilde w)$ and $\hat{\bf{u}}: = (\hat u, \hat v, \hat w)$ is continuous for $t\in [0, \infty)$. $\hat{\bf{u}}$ is called a lower solution of system (1.2) if it satisfies (1.2) with the $=$ signs replaced by $\leq$ signs. Similarly, $\tilde{\bf{u}}$ is called a upper solution $satisfies\; (1.2) \;with \;the\;$ = $signs \;replaced \;by \;\geq \;signs$ .

In order to show the uniqueness, we need the following comparison principle.

Theorem 2.1. (Comparison principle) If nonnegative $(\tilde u, \tilde v, \tilde w)$ and $(\hat u, \hat v, \hat w)$ are respectively a upper solution and lower solution of system (1.2) satisfying $\hat {u}\leq \tilde {u}\leq 1, \hat {v}\leq\tilde{v}\leq 1, \hat{w}\leq\tilde {w}\leq 1$ at $t = 0$. Then $\hat {u}\leq \tilde {u}, \hat {v}\leq\tilde{v}, \hat{w}\leq\tilde {w}$ for all $t>0$.

Proof. For $K>0$, we define functions $(\underline {u}, \underline {v}, \underline {w}) = (\hat{u}, \hat{v}, \hat{w})e^{-K t}, (\overline {u}, \overline {v}, \overline {w}) = (\tilde {u}, \tilde {v}, \tilde {w})e^{-K t}$. Let $\chi:\; \mathbb{R}\rightarrow \mathbb{R}$ be a nonnegative smooth function satisfying:

where $\chi''$ is the second derivative of $\chi$, $x_0 = \max\{x_1, x_2\}$, here $x_1$ and $x_2$ are determined in the following (2.8) and (2.12).

For $\varepsilon > 0$, set

It is easy to verify that $(\check {u}, \check {v}, \check {w})$ is strictly above $(\underline{u}, \underline{v}, \underline{w})$ at $t = 0$. In order to show $\hat {u}\leq \tilde {u}, \hat {v}\leq\tilde{v}, \hat{w}\leq\tilde {w}$ for all $t > 0$, we need to show that $\underline{u} < \check {u}, \underline{v} < \check {v}, \underline{w} < \check {w}$ for all $t > 0$ owing to the arbitrariness of $\varepsilon$. Assume by contradiction that this property is not true for all $t > 0$. Then,

According to the continuity of the functions, $\check{u}$, $\check{v}$ and $\check{w}$ tend to $+\infty$ as the space variable goes to infinity, uniformly in time, implies that $T > 0$ and $\check{u}-\underline{u} = 0$ or $\check{v}-\underline{v} = 0$ or $\check{w}-\underline{w} = 0$ at time $T$. By choosing

we argue differently depending on $\check{u}-\underline{u} = 0$ or $\check{v}-\underline{v} = 0$ or $\check{w}-\underline{w} = 0$ at time $T$.

Case 1. In the case $\min_{\mathbb{R}}(\check{w}-\underline{w})(\cdot, T) = 0$. In view of the definition of $\overline{w}$, we have

Applying (2.2) to the above inequality, we have

where the last inequality comes from (2.1).

By (2.2), it follows from the above inequality that

On the other hand, we have

Combining (2.4) and (2.5), for $x\in \mathbb{R}$ and $t\in (0, T]$, we have

Using the parabolic strong maximum principle, we can yield that $\min_{\mathbb{R}}(\check{w}-\underline{w}) = 0$ in $\mathbb{R}\times[0, T]$, which is contrary to $\check{w}(x, 0)-\underline{w}(x, 0) > 0$. Thus $\min_{\mathbb{R}}(\check{w}-\underline{w})(\cdot, T) = 0$ is not true.

Case 2. In the case $\min_{\mathbb{R}\times \mathbb{R}^+}(\check{u}-\overline{u})(\cdot, T) = 0$. By the definition of $\underline {u}$ and $\overline {u}$, we have

By (2.2), it follows from (2.6) that

where the second inequality is induced from (2.1).

Thus we have

where the last inequality is due to the definition of $K$ from (2.3).

Owing to $\hat u\leq1$, we have

for $(x, y)\in \mathbb{R}\times \mathbb{R}^+$ and $t\in(0, T]$. By applying the parabolic strong maximum principle, $\min_{\mathbb{R}\times \mathbb{R}^+}(\check{u}-\underline u)(\cdot, T) = 0$ must be attained on the boundary of $\mathbb{R}\times \mathbb{R}^+$. Thus we have

Moreover, we have

where the last inequality is by using the Hopf lemma.

This contradiction of (2.8) and (2.9) implies that $\min_{\mathbb{R}\times \mathbb{R}^+}(\check{u}-\underline u)(\cdot, T) = 0$ is not true.

Case 3. In the case $\min_{\mathbb{R}\times \mathbb{R}^+}(\check{v}-\underline v)(\cdot, T) = 0$. Directly calculations show that

By (2.2), it follows from (2.10) that

where the second inequality is induced from (2.1). Inserting (2.10) into the above inequality, we have

where the last inequality is because of the definition of $K$ from (2.3).

Owing to $\hat v\leq1$, we have

for $(x, y)\in \mathbb{R}\times \mathbb{R}^+$ and $t\in(0, T]$. By applying the parabolic strong maximum principle, $\min_{\mathbb{R}\times \mathbb{R}^+}(\check v-\underline v)(\cdot, T) = 0$ must be attained on the boundary of $\mathbb{R}\times \mathbb{R}^+$. Thus we have

Using the Hopf lemma yields that

In view of (2.2) and (2.12), we have

In view of (2.2), we have

where the last inequality is because of the definition of $\chi$ in (2.1). The conclusion (2.15) contradicts to $\partial_y\tilde v|_{y = 0}\leq 0\leq \partial_y\hat v|_{y = 0}$. Thus $\min_{\mathbb{R}\times \mathbb{R}^+}(\check v-\underline v)(\cdot, T) = 0$ is not true.

In the above three cases, we all reached a contradiction. This completes the proof.

As for a given pair of coupled upper and lower solutions $\tilde{\bf{u}}$ and $\hat{\bf u},$ we denote

where

We denote the functions of system (1.2) by

We denote the linear operators of system (1.2) by

By using ${\bf{\underline u}}^{(0)} = \hat{\bf u}$ and ${\bf{\overline u}}^{(0)} = \tilde{\bf{u}}$ as the initial iterations we can construct sequences $\{{\bf{\underline u}}^{(m)}\}_{m = 1}^\infty$ and $\{{\bf{\overline u}}^{(m)}\}_{m = 1}^\infty$ satisfying the same initial functions from the iteration process of scalar equations

Here we choose

$\{{\bf{\underline u}}^{(m)}\}_{m = 1}^\infty$ satisfies the above equation with the superscripts replaced by subscripts. Indeed, (2.20) is reduced to a linear parabolic equation with half-space homogeneous Neumamm condition with respect to $u$ and $v$, and a Cauchy problem of linear parabolic equation with respect to $w$. Thus $\{{\bf{\overline u}}^{(m)}\}_{m = 1}^\infty$ and $\{{\bf{\underline u}}^{(m)}\}_{m = 1}^\infty$ are well-defined. By using the monotone dynamical system method (Smith ^[25]), we have the following lemma.

Lemma 2.1. The sequences $\{{\bf{\overline u}}^{(m)}\}_{m = 1}^\infty$ and $\{{\bf{\underline u}}^{(m)}\}_{m = 1}^\infty$ governed by (2.20) possess the monotonicity property

Moreover, for each $m = 1, 2, \cdots$, $\bf{\overline u^{(m)}}$ and $\bf{\underline u^{(m)}}$ are coupled upper and lower solutions of (1.2).

Proof. Let $\underline s^{(1)} = \underline u^{(1)}-\underline u^{(0)}$. We necessarily apply the comparison principle of standard parabolic equation to $\underline s^{(1)}$. By (2.21), $\underline s^{(1)}$ satisfies

Using comparison principle of standard parabolic equation yields $\underline s^{(1)}\geq0$. It follows that $\underline u^{(0)}\leq \underline u^{(1)}$. On the other hand, we set $\underline q^{(1)} = \underline v^{(1)}-\underline v^{(0)}$. By (2.20), $\underline q^{(1)}$ satisfies

the parabolic comparison yields $\underline v^{(0)}\leq \underline v^{(1)}$. Meanwhile, using the similar process yields $\underline w^{(0)}\leq \underline w^{(1)}$. Thus, we have $\bf{\underline u^{(0)}}\leq \bf{\underline u^{(1)}}$. Likewise, we have $\bf{\overline u^{(0)}}\geq \bf{\overline u^{(1)}}$.

Letting $z^{(1)} = \overline u^{(1)}-\underline u^{(1)}$, it follows from (2.21) that

It follows again from comparison principle that ${\overline u}^{(1)}\geq{\underline u}^{(1)}$, and thus we have ${\underline v}^{(1)}\leq {\overline v}^{(1)}$. The above conclusions show that

Now we show that ${\bf{\overline u}}^{(1)}$ and ${\bf{\underline u}}^{(1)}$ are upper and lower solutions of (1.2).

Next we use an induction method. By choosing ${\bf{\overline u}}^{(1)}$ and ${\bf{\underline u}}^{(1)}$ as the coupled upper and lower solutions $\bf{\tilde u}$ and $\bf{\hat u}$, after a similar argument as above, we have

so ${\bf{\overline u}}^{(2)}$ and ${\bf{\underline u}}^{(2)}$ are coupled upper and lower solutions of (1.2). The conclusion of the lemma follows from the induction principle.

In view of Lemma 2.1, the pointwise limits

exist. In the following theorem we show that $(\overline u, \overline v, \overline w)$ and $(\underline u, \underline v, \underline w)$ are respectively the maximal and minimal solutions of system (1.2).

Theorem 2.2. Assume that $\tilde{\bf{u}}$ and $\hat{\bf u}$ be upper and lower solutions of system (1.2). Let $(\overline u, \overline v, \overline w)$ and $(\underline u, \underline v, \underline w)$ be given by (2.25). Then $(\overline u, \overline v, \overline w)$ and $(\underline u, \underline v, \underline w)$ are the solutions of system (1.2). Moreover,

Proof. (ⅰ) To show that the limit $(\overline u, \overline v, \overline w)$ in (2.25) is the solution of (1.2), we use an integral representation for the solution of the Cauchy problem and the linear parabolic Eq (2.20) under Neumann boundary condition. Let $\mathcal{G}(t, x, y; \tau, \xi_1, \xi_2)$ be the Green function given by

We denote

By the integral representation of linear parabolic boundary value problems ^[26], the solution of (2.20) is given by

The integrands in the above equation are integrable because the definition of Green function $\mathcal{G}$ satisfies the admissible growth in the time variable. Using the dominated convergence theorem yields that the limits $(\overline u, \overline v, \overline w)$ satisfy the relation

Then $(\overline u, \overline v, \overline w)$ is a solution of (1.2). A similar argument shows that $(\underline u, \underline v, \underline w)$ is also a solution of (1.2).

(ⅱ) We denote by

By plugging (2.18) into (2.27), we obtain

In view of $\mathcal{G}_3$, we have

We also obtain

Since the definition of $K_1$ in (2.16), we have $|f_1(\overline u, \underline v)+K_1\overline u-f_1(\underline u, \overline v)-K_1\underline u| \leq 2K_1(|\overline u-\underline u|+|\overline v-\underline v|)$. Recalling $\mathcal{G}_2$, we have

By (2.27), we have

Since the definition of $K_2$ in (2.21), we have $|f_2(\overline u, \overline v)+K_2\overline v-f_2(\underline u, \underline v)-K_2\underline v| \leq 2K_2(|\overline u-\underline u|+|\overline v-\underline v|)$. We also have $|g({\overline u}, {\overline w})-g({\underline u}, {\underline w})| \leq (\mu+\nu)(|\overline u-\underline u|+|\overline w-\underline w|)$. Recalling $\mathcal{G}_2$, we have

Combining (2.28), (2.29) and (2.30), by choosing $K: = \max\{\frac{\nu}{\mu}, \frac{\mu+\nu}{d_2K_2}+2\}$ and $\gamma = \max\{K_1, K_2, \mu\}$, we obtain

Thus there exist a constant $T: = ln\frac{K}{K-1}$ such that $||\overline u-\underline u||_{L^\infty (E_t^*)}+||\overline v-\underline v||_{L^\infty (E_t^*)}+ ||\overline w-\underline w||_{L^\infty (E_t^{**})} = 0$, that is $\overline u\equiv \underline u$, $\overline v\equiv \underline v$, and $\overline w\equiv \underline w$ for $t\in (0, T]$. Owing to the fact the above $\gamma$ and $K$ do not depend on the initial value, $T$ can be extended to the time $\infty$.

Notice that the above theorem is also valid when the upper and the lower solutions are constant vectors. Moreover we can induce the global asymptotic convergence of system (1.2). Suppose that there are constant vectors ${\bf{\tilde c}}: = (\tilde c_1, \tilde c_2, \tilde c_3)$ and ${\bf{\hat c}} = (\hat c_1, \hat c_2, \hat c_3)$ such that $f_i(\tilde c_1, \tilde c_2)\leq0$ and $f_i(\hat c_1, \hat c_2)\geq0$ for $i = 1, 2$. Here $\tilde c_3 = \frac{\nu}{\mu} \tilde c_2$, $\hat c_3 = \frac{\nu}{\mu}\hat c_2$, and $\hat c_i\leq\tilde c_i$ for $i = 1, 2, 3$. In view of Definition 2.1, ${\bf{\tilde c}}$ and ${\bf{\hat c}}$ are upper and lower solutions of system (1.2). Thus using ${\bf{\tilde c}}$ and ${\bf{\hat c}}$ as initial iteration, after the procedure (2.20), we can obtain the maximal solution ${\bf{\overline c}}: = (\overline c_1, \overline c_2, \overline c_3)$ and minimal solution ${\bf{\underline c}}: = (\underline c_1, \underline c_2, \underline c_3)$. Since the solution of system (2.20) is unique, the sequences $\{\overline c_i^{(m)}\}$ and $\{\underline c_i^{(m)}\}$ are constants. The limits of $\{\overline c_i^{(m)}\}$ and $\{\underline c_i^{(m)}\}$ satisfy

We give the convergence result in the next theorem.

Theorem 2.3. Suppose that the initial functions of system (1.2) satisfy $\hat c_1\leq u_0(x, y)\leq \tilde c_1$, $\hat c_2\leq v_0(x, y)\leq \tilde c_2$, and $\hat c_3\leq w_0(x)\leq \tilde c_3$. If $\overline c_i = \underline c_i: = c_i$ for $i = 1, 2, 3$, then for $(x, y)\in \mathbb{R}\times \mathbb{R}^+$,

Proof. We denote $(\tilde u_1, \tilde u_2, \tilde u_3)$ by the solution of the following system

It is easy to see that ${\bf{\overline c}}$ is a lower solution of (2.33). Then we have $\overline c_i\leq \tilde u_i$ for $i = 1, 2, 3$. By using Theorem 2.1, $\tilde u_i$ is time-nonincreasing. The limit of $\tilde u_i$ exists as $t\rightarrow \infty$. We denote by $\lim_{t\rightarrow \infty}\tilde u_i(t, \cdot) = \overline u_i(\cdot)$. Thanks to the Schauder estimates, $\overline u_i$ is the stationary solution of system (1.2). That is to say, $(\overline u_1, \overline u_2, \overline u_3)$ satisfies

Meanwhile, ${\bf{\overline c}}$ is also a stationary solution of system (1.2). Moreover, in view of the iterative process of ${\bf{\overline c}}$, $(\overline c_1^{(1)}, \overline c_2^{(1)}, \overline c_3^{(1)})$ satisfies

where the righthand inequality sign is due to $\overline u_i\leq\tilde c_i$ for $i = 1, 2, 3$. By using comparison principle (Theorem 2.1), we induce that $\overline c_i^{(1)}\geq \overline u_i$ for $i = 1, 2, 3$. By using an induction method, $(\overline c_1^{(2)}, \overline c_2^{(2)}, \overline c_3^{(2)})$ satisfies

By using comparison principle (Theorem 2.1), we induce that $\overline c_i^{(2)}\geq \overline u_i$ for $i = 1, 2, 3$. After an induction argument, we have $\overline c_i^{(m)}\geq \overline u_i$ for $i = 1, 2, 3$. Here $m = 1, 2, \cdots$. By letting $m\rightarrow \infty$, we have

On the other hand, We denote $(\hat u_1, \hat u_2, \hat u_3)$ by the solution of the following system

In a similar way, we can induce

Since $\overline c_i = \underline c_i: = c_i$ for $i = 1, 2, 3$, it follows from (2.38) and (2.39) that

For any initial functions satisfying $\hat c_1\leq u_0(x, y)\leq \tilde c_1$, $\hat c_2\leq v_0(x, y)\leq \tilde c_2$, and $\hat c_3\leq w_0(x)\leq \tilde c_3$, the solution $(u, v, w)$ is a lower solution of system (2.33) and a upper solution of system (2.38). By using comparison principle (Theorem 2.1), we have $(\hat u_1, \hat u_2, \hat u_3)\leq(u, v, w)\leq (\tilde u_1, \tilde u_2, \tilde u_3)$. Thus we induce that

Theorem 2.4. If the initial functions of system (1.2) satisfy $0\leq u_0(x, y)\leq 1$, $0\leq v_0(x, y)\leq 1$, and $0\leq w_0(x)\leq \frac{\nu}{\mu}$. then system (1.2) possesses a unique solution $(u, v, w)$ for $t\in (0, \infty)$. Moreover, for $(x, y)\in\mathbb{R}\times \mathbb{R}^+$,

Proof. In order to utilize Theorem 2.2, we need to construct upper and lower solutions of system (1.2). We set

It is easy to verify that $(M_1, M_2, M_3)$ and $(0, 0, 0)$ are upper and lower solutions of system (1.2). Using Theorem 2.2, system (1.2) has a unique solution. Moreover we can obtain (2.41).

3.   Stabilities of the equilibria

Based on the basic reproduction $R_0: = \frac{ma^2b_1b_2}{\gamma_1\gamma_2}$, we examine the asymptotic stability of the disease-free equilibrium $(0, 0, 0)$ and endemic equilibrium.

Theorem 3.1. Suppose that the initial functions of system (1.2) satisfy $0\leq u_0(x, y)\leq 1$, $0\leq v_0(x, y)\leq 1$ and $0\leq w_0(x)\leq \frac{\nu}{\mu}$, if $R_0 < 1$, then the solution to system (1.2) satisfies

Proof. We use the method of upper and lower solutions. we set

It is easy to verify that $(M_1, M_2, M_3)$ and $(0, 0, 0)$ are upper and lower solutions of system (1.2). By using Theorem $\ref{theob2}$, we can construct the maximal solution ${\bf{\overline c}}$ and minimal solution ${\bf{\underline c}}$, which satisfy

By a directly computation, we have

Since $R_0 < 1$, we induce that $\overline c_1\overline c_2 = 0$. Combing with $mab_1\overline c_2(1-\overline c_1)-\gamma_1 \overline c_1 = 0$, we have

On the other hand, a similar argument yields that

It follows from (3.2) and (3.3) that

By applying Theorem 2.3, we have

Theorem 3.2. Suppose that $\delta$ is an arbitrary small positive constant such that $\delta\leq u_0(x, y)\leq 1$, $\delta\leq v_0(x, y)\leq 1$ and $\delta\leq w_0(x, y)\leq \frac{\nu}{\mu}$. If $R_0 > 1$, then the solution to system (1.2) satisfies

where $(u^*, v^*)$ is a unique positive solution of

and $w^* = \frac{\nu}{\mu} v^*$.

Proof. We first consider the nullclines of system (1.2), which is written by

By taking the derivatives of (3.6), we have

Hence the two nullclines are monotone increasing. In the domain $(u, v)\in [0, 1]\times [0, \infty)$, they have the same start point $(0, 0)$, but the different end points $(1, \infty)$ and $(1, \frac{ab_2}{ab_2+\gamma_2})$. The nullclines has a unique positive cross points $(u^*, v^*)$.

Next we use the method of upper and lower solutions. we set

It is easy to verify that $(M_1, M_2, M_3)$ and $(\delta, \delta, \delta)$ are upper and lower solutions of system (1.2). By using Theorem 2.3, we can construct the maximal solution ${\bf{\overline c}}$ and minimal solution ${\bf{\underline c}}$, which satisfy $\overline c\geq\underline c_i > 0$ for $i = 1, 2, 3$. Moreover, $(\overline c_1, \overline c_2)$ and $(\underline c_1, \underline c_2)$ are the positive solutions of the following nullclines

Since the nullclines has a unique positive cross points $(u^*, v^*)$, we obtain that $\overline c_1 = \underline c_1 = u^*$, $\overline c_2 = \underline c_2 = v^*$. Then $\overline c_3 = \underline c_3 = w^*$.

By applying Theorem 2.3, we have

4.   Discussion

Since mosquito is a prominent vector of Malaria, it is of crucial importance to study the qualitative spreading behavior of mosquitoes in implementing vector control strategies and preventing mosquito-borne diseases. A large proportion of the current studies on epidemic transmission dynamics, using ordinary differential systems, focuses on the temporal development and control of infectious diseases, thus uncertainty still exists concerning how population mobility affects on epidemic outbreaks. The spatial factor should be taken into consideration in the modeling processes in order to study the geographic spread of infectious diseases. Laplacian diffusion systems with spatially homogeneous parameters (Murray ^[27], Ruan ^[28]) and spatially heterogeneous parameters (Allen et al. ^[14], Lei et al. ^[18], Li et al. ^[17], Song et al. ^[1]) have been proposed to study the spatio-temporal dynamics epidemic models. These studies exhibit an equal probability of mosquitoes' movement to any direction. Yet our road diffusion model is different since mosquitoes move at a faster speed along the highway. We studied the long-time dynamical behaviors of Ross epidemic model. To the best of our knowledge, the road diffusion has never been applied to describe the two compartments spread of infectious diseases in any epidemic model in the literature.

Our results indicate that mosquitoes has an impact on long-time dynamics of infectious diseases in road diffusion models. According to Theorem 3.1, when the basic reproduction number $R_0 < 1$, system (1.2) admits a globally asymptotically stable disease-free equilibrium, and an asymptotic stable endemic equilibrium when the basic reproduction number $R_0 > 1$ in view of Theorem 3.2. So we have generalized the threshold dynamics of the classical Ross model to that in a road diffusion model. Although many factors related to a real disease have been simplified in our model, we capture the dynamics of mosquito-borne diseases, which helps us to control and prevent the spread of diseases.

5.   Conclusions

Since Ronald Ross discovers the transmission of Malaria by mosquitoes, the differential equations have been used to study the spread of infectious disease (see ^[29,30,31]). A natural question is how to describe the dynamical system when the infectious disease spreads along the directed diffusion. We introduce the road-field diffusion into a Ross epidemic model which describes the dynamical behaviors of infected Mosquitoes and humans. Our result reveals that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 while the endemic equilibrium asymptotically stable if the basic reproduction number is greater than 1.

Acknowledgments

The authors would like to thank the anonymous referees for the helpful comments and suggestions. Canrong Tian was was partially supported by NSFC grant (NSFC-61877052), the Jiangsu Province 333 Talent Project, and Jiangsu Province Qinglan Project.

Conflict of interest

The authors declare that they have no conflict of interest.