Identification of propagated defects to reduce software testing cost via mutation testing

1.   Introduction

It is now widely believed that the actual spread of many infectious diseases occurs in a diverse or dispersed population ^[1]. Subpopulations (or compartments) can be determined not only on the basis of disease-related factors such as mode of transmission, latent period, infectious period, and genetic susceptibility or resistance, but also on the basis of social, cultural, economic, demographic, and geographic factors ^[2]. In the viewpoint of epidemiology above, let $S(t)$, $I(t)$ and $R(t)$ be the density of susceptible, infectious and recover individuals at time $t$, respectively, and we suppose that the dynamics of the disease transmission is governed by the following equations:

where $\mu, \delta, \gamma$ are all positive constants, $\mu$ the birth and death rate, $\delta$ the recovery rate of infectious individuals, and $\gamma$ the rate of removed individuals who lose immunity and return to susceptible individuals class. Here, we assume that the disease does not have vertical transmission and is generally non–pathogenic and ignores the death induced by the disease. The infectious individuals force $g(S, I)$ plays a key role in determining disease dynamics ^[3,4]. Traditionally, the density-dependent transmission (or the bilinear incidence rate, $g(S, I) = \beta SI$, and $\beta$ is the proportionality constant) and the frequency-dependent transmission (or the standard incidence rate, $g(S, I) = \frac{\beta SI}{S+I}$) are two extreme forms of disease transmission, which have been frequently used in well-known epidemic models ^[5,6,7]. There are several different nonlinear transmission functions proposed by researchers, to see more details, we refer to ^{[8,9,10,11,12]} and the references therein. Especially, Yuan and Li ^[13] studied a ratio-dependent nonlinear incident rate which takes the following form (1.2):

where the parameters $l$ and $h$ are positive constant, $\alpha$ is the parameter which measures the psychological or inhibitory effect. It is worthy to note that in the special case of $\alpha = 1$ and $h = l = 1$, (1.2) becomes the well–known frequency-dependent transmission rate $\frac{\beta SI}{S+I}$. In this case, the nonlinear incidence rate (1.2) can be seen as an extension form of the frequency-dependent transmission rate.

Before 1970s, ecological population modelers (involving epidemic models) typically used ordinary differential equations (ODE, e.g. model (1.1)), seeking equilibria and analyzing stability. These models provided important insights, such as when species can stably coexist and when susceptible and infectious densities oscillate over time ^[14]. The ODE models that have been described so far assume that the populations experience the same homogeneous environment. In reality, individual organisms are distributed in space and typically interact with the physical environment and other organisms in their spatial neighborhood ^[15]. More recently, many studies have shown that the spatial epidemic model is an appropriate tool for investigating the fundamental mechanism of complex spatiotemporal epidemic dynamics. In these studies, reaction-diffusion equations have been intensively used to describe spatiotemporal dynamics ^{[16,17,18,19,20,21,22,23,24,25]}.

Many studies show that spatial heterogeneity generated by species dynamics is mathematically more interesting and also biologically more important ^[14]. In fact, relationships between individual-level processes and ecological dynamics often depend on population spatial structure, and epidemic dynamics can be governed by localized spatial processes of contact between susceptible and infectious individuals ^[27]. It has been suggested that spatial heterogeneity may address many of the deficiencies of epidemic models and play an important role in the spread of an epidemic ^{[16,28,29,30,31,32,34,33]}. Of them, Grenfell, Bjornstad and Kappey ^[31] showed that measles waves spreading from large cities to small towns in England and Wales are determined by the spatial hierarchy of the host population structure; Keeling et al. ^[32] showed that the spatial distribution of farms influences the regional variability of foot-and-mouth outbreaks in UK; Hufnagel et al. ^[34], Colizza et al. ^[33] showed that the high degree of predictability of the worldwide spread of infectious diseases is caused by the strong heterogeneity of the transport network; Merler and Ajelli ^[30] showed that spatial heterogeneity in population density results in a relevant delay in epidemic onset between urban and rural areas. Hence, understanding the role of the spatial heterogeneity in epidemic dynamics is challenging both theoretically and empirically.

In this paper, we mainly focus on the impact of spatial heterogeneity of the disease dynamics of an SIRS epidemic model corresponding to model (1.1). To incorporate the random diffusion, described by Laplacian operator ^[26], coming from the random wandering of susceptible, infectious and recover individuals, we add the diffusion terms to model (1.1) for the susceptible, infectious and recover individuals, respectively. To incorporate the spatial heterogeneity, we consider $\beta: = \beta(x)$, and $\beta(x)$ is a positive Hölder continuous function on $\bar{\Omega}$, which is the space–dependent rate of disease transmission by infectious individuals at position $x\in\Omega$, $\beta(x) I^l$ measures the infection force of the disease, and $1/(1+\alpha I^h)$ describes the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals is very large. For the sake of convenient analysis, we adopt $l = 1$. Then the reaction-diffusion model corresponding to model (1.1) is the following model system which governs the spatial heterogeneity and population mobility:

where the habitat $\Omega\subset \mathbb{R}^m (m\geq 1)$ is a bounded domain with smooth boundary $\partial \Omega$ (when $m > 1$), and $\bf{n}$ is the outward unit normal vector on $\partial \Omega$. Moreover $N = S+I+R$ is the total population, $d_S$, $d_I$ and $d_R$ are diffusion coefficients for the susceptible, infectious and recover individuals, respectively.

Assume that the initial values satisfy

$(\bf{H1})$ $S(x, 0), I(x, 0)$ and $R(x, 0)$ are nonnegative continuous functions in $\bar{\Omega}$, $\int_\Omega I(x, 0)dx > 0$ and

Mathematical models later confirm that spatial subdivision is important for the persistence of populations.

The rest of the paper is organized as follows. In Section 2, we give the global existence and uniform boundedness of solution. In Section 3, we investigate the threshold dynamics in terms of the basic reproduction number and study the asymptotic behavior of endemic equilibrium with respect to small diffusion rate of susceptible individuals. Finally, in Section 4, we provide the summary of the main results.

2.   Global existence and uniform boundedness

The first goal of this paper is to concern with the global existence of classical solutions to model (1.3).

Theorem 2.1. Model (1.3) has a unique global classical solution $(S(x, t), I(x, t), R(x, t)) \in [C([0, \infty)\times\bar{\Omega}) \cap C^{2, 1}((0, \infty)\times\bar{\Omega})]^3$ satisfying $S(x, t), I(x, t), R(x, t)\geq 0$ for all $t > 0$ and

where $C(N_0) > 0$ is a constant dependent of $N_0$.

Proof. The local existence and uniqueness of the solutions of model (1.3) follow from a classical result ([35, Theorem 3.3.3]). It follows from the strong maximum principle ^[36] that $S(x, t), I(x, t)$ and $R(x, t)$ are nonnegative for $x\in \bar{\Omega}$ and $t\in(0, T_{\max})$, here $T_{\max}$ is the maximal existence time for solutions of model (1.3). In what follows, we prove that the local solution can be extended to a global one, that is $T_{\max} = \infty$. The method of the proof is similar to ([37,Theorem 2.2]).

Let

be the total population size at time $t$. We can obtain that

The population size $V$ is a constant, i.e.,

which shows that $\|S(\cdot, t)\|_{L^1(\Omega)}, \|I(\cdot, t)\|_{L^1(\Omega)}$ and $\|R(\cdot, t)\|_{L^1(\Omega)}$ are bounded in $[0, T_{max})$.

From model (1.3) we easily deduce that

It follows from [38, Lemma 2.1] with $q = p_0 = 1$ that $\|S(\cdot, t)\|_{L^\infty(\Omega)}, \|I(\cdot, t)\|_{L^\infty(\Omega)}$ and $\|R(\cdot, t)\|_{L^\infty(\Omega)}$ are also bounded in $[0, \infty)$. Thus, we obtain (2.1) and complete the proof.

As expected, the steady state will play a central role in the dynamics of model (1.3). A steady state solution of model (1.3) is a time–independent (classical) solution and therefore can be viewed as a function $(\tilde{S}, \tilde{I}, \tilde{R})\in [C^2(\Omega)\cap C^1(\bar{\Omega})]^3$ satisfying

where $\tilde{S}(x), \tilde{I}(x)$ and $\tilde{R}(x)$ denote the density of the steady state solution at location $x\in \Omega$. In the view of (2.2), the steady state solutions also satisfy

In epidemiological model, there are two typical constant steady state solutions, namely, disease–free equilibrium and endemic equilibrium. A disease–free equilibrium (DFE) is a steady state solution of (2.4) in which $\tilde{S}(x) > 0$, both $\tilde{I}(x)$ and $\tilde{R}(x)$ vanish at $x\in \Omega$, i.e., $\tilde{I}(x) = \tilde{R}(x) = 0$. An endemic equilibrium (EE) is a steady state solution in which $\tilde{S}(x), \tilde{I}(x), \tilde{R}(x) > 0$ for some $x\in \Omega$.

3.   Threshold dynamics

This section aims to establish the threshold dynamics of (1.3) in terms of the basic reproduction number.

3.1.   Basic reproduction number

The basic reproductive number, denoted by $\mathcal{R}_0$, which is defined as the average number of secondary infections generated by a single infected individual introduced into a completely susceptible population, is one of the important quantities in epidemiology ^[41,44].

The DFE of model (1.3) is $E_0 = \left(\dfrac{N_0}{|\Omega|}, 0, 0\right)$, where $|\Omega|$ is the Lebesgue measure of $\Omega$. Linearizing model (1.3) at $E_0 = \left(\dfrac{N_0}{|\Omega|}, 0, 0\right)$, we get the following system for the infection related variable $I$:

Using the next generation approach for spatial heterogenous populations ^[44], we characterize the basic reproduction number $\mathcal{R}_0$ for model (1.3) is

where $\lambda_0$ is a unique positive eigenvalue with a positive eigenfunction $\Psi(x)$ on $\Omega$ for the elliptic eigenvalues problem

Let $\lambda^*$ be the principal eigenvaluewith a positive eigenfunction ${\psi^*}(x)$ on $\Omega$ for the following eigenvalue problem :

We have the following result.

Lemma 3.1. $\mathrm{sign}(1-\mathcal{R}_0) = \mathrm{sign}\, \lambda^*$.

Proof. Following [42,Lemma 2.3], we consider (3.1) and (3.2),

We multiply the first equation in (3.3) by $\psi^*$ and the second equation in (3.3) by $\Psi$, integrate by parts on $\Omega$, and subtract the two resulting equations,

Since $\beta, \Psi, \psi^*$ are positive, we have $\mathrm{sign}(1-\mathcal{R}_0) = \mathrm{sign}\lambda^*$.

3.2.   Stability of the DFE

In this subsection, we show that the stability of the DFE is determined entirely by the magnitude of $\mathcal{R}_0$.

Theorem 3.2. For model (1.3), if $\mathcal{R}_0 < 1$, then $(S(x, t), I(x, t), R(x, t))\rightarrow \left(\dfrac{N_0}{|\Omega|}, 0, 0\right)$ as $t\rightarrow \infty$. That is, the DFE is globally asymptotically stable.

Proof. Suppose that $\mathcal{R}_0 < 1$. By Lemma 3.1, it implies that $\lambda^* > 0$. Observe from the second equality of model (1.3) that

Note that the following linear system

admits a solution $a e^{-\lambda^*t}{\psi^*}(x)$, and $a$ is chosen so large that $I(x, 0)\leq Z(x, 0)$ for every $x\in \Omega$. The comparison principle implies that $I(x, t)\leq a e^{-\lambda^*t}{{\psi^*}(x)}$, and it then follows that $I(x, t)\rightarrow 0$ as $t\rightarrow \infty$ for $x\in\bar{\Omega}$.

As a result, the equation for $R(x, t)$ is asymptotic to

From the comparison principle, we can get that $R(x, t)\rightarrow 0$ as $t\rightarrow\infty$ for $x\in\bar{\Omega}$. Similarly, we can get that $S(x, t)\rightarrow \dfrac{N_0}{|\Omega|}$ as $t\rightarrow\infty$ for $x\in\bar{\Omega}$. This yields the desired result.

3.3.   The existence of the endemic equilibrium

Suppose that $\mathbb{X} = C(\overline{\Omega}; \mathbb{R}^3)$ have a supremum norm $\|\cdot\|$, then $\mathbb{X}$ is an ordered Banach space with the cone $\mathbb{P}$ consisting of all nonnegative functions in $\mathbb{X}$, and $\mathbb{X}$ has nonempty interior, denoted by $\mathrm{int}(\mathbb{P})$. Set

and $\mathbb{U} = \mathbb{P}\cap \mathbb{X}_0$. It is easy to verify that model (1.3) defines a dynamic system on $\mathbb{U}$. Denote the unique solution of model (1.3) with initial value $(S_0, I_0, R_0) \in \mathbb{U}$ by $\Phi_t(S_0, I_0, R_0) = (S(\cdot, t), I(\cdot, t), R(\cdot, t))$ for any $t > 0$. $\Phi_t$ is continuous and compact for $t > 0$. $\Phi_t$ is pointwisely dissipative. Therefore, $\Phi_t$ has a global attractor ^[46].

Theorem 3.3. If $\mathcal{R}_0 > 1$, model (1.3) admits at least one endemic equilibrium.

Proof. We appeal to the uniform persistence theory developed in ^[46,47]. Denote

Note that $\mathbb{U} = \mathbb{U}_0\cup \partial \mathbb{U}_0$. Moreover, $\mathbb{U}_0$ and $\partial \mathbb{U}_0$ are relatively open and closed subsets of $\mathbb{U}$, respectively, and $\mathbb{U}_0$ is convex. We divide the proof into three steps.

Step 1. A direct result of the strong maximum principle for parabolic equations is $\Phi_t\mathbb{U}_0\subset \mathbb{U}_0$ for all $t > 0$.

Step 2. Let $A_\partial$ be the maximal positively invariant set for $\Phi_t$ in $\partial \mathbb{U}_0$, i.e.

It is easy to verify that $A_\partial: = \left\{W_0\in \mathbb{U} \Big|I_0 = 0\right\}$.

Denote $\omega(W_0)$ as the $\omega$–limit set of $W_0$ in $U$ and

We now prove $\widehat{A}_\partial = \{E_0\}$. For any $W_0\in A_\partial$, i.e. $I_0 = 0$, then $I(x, t) = 0$ for all $x\in\Omega, t\geq0$, and model (1.3) becomes

which implies $R(\cdot, t)\rightarrow0, S(\cdot, t) \rightarrow \dfrac{N_0}{|\Omega|}$ uniformly as $t\rightarrow \infty$. Hence, $\widehat{A}_\partial = \{E_0\}$. Therefore, $\{E_0\}$ is a compact and isolated invariant set for $\Phi_t$ restricted in ${A}_\partial$.

Step 3. We prove that there exists some constant $\epsilon_0$ independent of initial values such that

Suppose, on the contrary, that for any $\epsilon_1 > 0$, there exists some initial value $W_0^*$ such that

For any given small $\epsilon_2 > 0$, let $\lambda^*(\epsilon_2)$ be the unique principal eigenvalue of the following eigenvalue problem with a positive eigenfunction $\phi_I$

Note that $\lim\limits_{\epsilon_2\rightarrow0}\lambda^*(\epsilon_2) = \lambda^* < 0$, where $\lambda^*$ is the principal eigenvalue of eigenvalue problem (3.2). Therefore, we can choose $\epsilon_2$ such that $\lambda^*(\epsilon_2) < 0$. Since $\epsilon_1$ is arbitrary, we choose $\epsilon_1 = 2\epsilon_2$. In view of (3.6), there exists $T > 0$ such that

By the strong maximum principal of parabolic equations, $(S^*(\cdot, x), I^*(\cdot, x), R^*(\cdot, x))\in\mathrm{ int}(\mathbb{P})$ for all $t > 0$. Then we can choose a sufficiently small number $c_* > 0$ such that $I^*(T, x)\geq c_* \phi_I$. Note that $c_* e^{-\lambda^*(\epsilon_2)(t-T)}\phi_I$ is a solution of the following linear system

It follows from the comparison principal that

and, hence, $I^*(x, t)\rightarrow\infty$ uniformly in $\overline{\Omega}$ as $t\rightarrow\infty$, which contradicts (3.6).

The result of Step 3 implies that $\{E_0\}$ is an isolated invariant set for $\Phi_t$ in $\mathbb{U}$, and $W^S(\{E_0\}) \cap \mathbb{U}_0$ is an empty set, where $W^S(\{E_0\})$ is the stable set of $\{E_0\}$ for $\Phi_t$.

Finally, Combining Steps 1–3 and [46,Theorem 1.3.1], we have that $\Phi_t$ is uniformly persistent with respect to $(\mathbb{U}, \partial \mathbb{U}_0)$. Moreover, by [46,Theorem 1.3.7], model (1.3) admits at least one endemic equilibrium.

3.4.   Asymptotic properties of the endemic equilibrium

In this section, we are concerned with the asymptotic behavior of the EE of model (1.3). Our aim is to investigate the effect of the slow movement of susceptible individuals on the spatial distribution of the infectious disease. From now on, unless otherwise specified, we always assume $\mathcal{R}_0 > 1$ and

(H2) $\beta(x) > \mu$ for all $x\in \bar{\Omega}$.

Consider the linear eigenvalue problem

and denote the smallest eigenvalue of (3.8) by $\lambda_1: = \lambda_1\left(-d_R\Delta+(\mu+\gamma)(1-\frac{\delta}{\beta-\mu})\right)$.

Denote $\xi: = d_S \tilde{S}(x)+d_I \tilde{I}(x)+d_R \tilde{R}(x)$ and set

Model (2.4) is equivalent to

The following results hold:

Lemma 3.4. $(\tilde{S}(x), \tilde{I}(x), \tilde{R}(x))$ is a solution of model (2.4) if and only if $(S(x), I(x), R(x))$ is a solution of model (3.9). Moreover

Lemma 3.5. Assume that $\mathcal{R}_0 > 1$. For model (3.9), $I\rightarrow I^*, R\rightarrow R^*$ in $C^1(\bar{\Omega})$ as $d_S\rightarrow0$ for some $I^*, R^*\in C^1(\bar{\Omega})$ with $I^*\geq0, R^* > 0$ on $\bar{\Omega}$.

Proof. In the view of $d_SS+d_II+d_RR = 1$, $\dfrac{\beta(x)S^hI}{S^h+\alpha I^h}$ is uniformly bounded for any $d_S > 0$. It follows from $L^p$-estimate that $\|I\|_{W^{2, p}}$ is bounded for any $p > 1$. Thus, $\|I\|_{C^{1, \tau}}$ is bounded for any $\tau\in(0, 1)$ by Sobolev embedding theorem. Passing to a subsequence if necessary, $I\rightarrow I^*$ in $C^1(\Omega)$ as $d_S\rightarrow0$ where $I^*(x) \geq0$ for $x\in\Omega$ and $\dfrac{\partial I}{\partial \bf{n}} = 0$ for $x\in\Omega$. By similar arguments, $R\rightarrow R^*$ in $C^1(\Omega)$ as $d_S\rightarrow0$ where $R^*(x) \geq0$ for $x\in\Omega$, which satisfies

Now we show that $I^*(x) \not\equiv0$ on $\Omega$ by contradiction argument. If $I^*(x) = 0$, then we obtain by (3.10) that $R^* = 0$, which implies that $S\rightarrow\infty$ is almost everywhere (abbreviate a.e.) as $d_S\rightarrow0$. Thus

Define

Note that $\hat{I}, \hat{R} > 0$ and $\|\hat{I}\|_{L^\infty(\Omega)}+\|\hat{R}\|_{L^\infty(\Omega)} = 1$. Then by a standard compactness argument for elliptic equations, after passing to a further subsequence if necessary,

where $\hat{I}^*, \hat{R}^*\geq0$ for $x\in\Omega$ and

It follows from (3.11) that $\hat{I}^*$ is a weak solution of

By elliptic regularity, we have $\hat{I}^*\in C^2({\bar{\Omega}})$, which gives

It follows from maximum principle together with (3.12) that $\hat{I}^*(x), \hat{R}^*(x) > 0$. We conclude that $(\lambda, \psi) = (0, \hat{I}^*(x))$ is a solution of (3.2). Since $\hat{I}^*(x) > 0$ on $\Omega$, it must be that $\lambda^* = 0$, which implies that $\mathcal{R}_0 = 1$. This contradiction yields $I^*(x)\not\equiv0$. Therefore, again by maximum principle together with (3.10), we obtain $R^* > 0$.

Note that $I(x), R(x) > 0$ for any $x\in\Omega$, $d_S > 0$. Denote

Let

where $M^*(x): = d_II^*+d_RR^*$. Observe that $J^-\bigcup J^+ = \bar{\Omega}.$

Lemma 3.6. Assume that $\mathcal{R}_0 > 1$.

(i) The set $J^+$ has positive Lebesgue measure.

(ii) If further assume that $\lambda_1 < 0$, then the set $J^-$ has positive Lebesgue measure.

Proof. We prove $|J^+| > 0$ by contradiction. If $|J^+| = 0$, i.e., $0 < M^*(x) < 1$ on $\Omega$ a.e., then it follows from $d_SS = 1-d_II -d_RR$ that $S\rightarrow\infty$ a.e. as $d_S\rightarrow0$ and thus $\dfrac{\beta(x)S^hI}{S^h+\alpha I^h}\rightarrow \beta(x)I^*$ a.e. as $d_S\rightarrow0$. Therefore, $I^*$ is a weak solution of

By elliptic regularity, we have $I^*\in C^2(\bar{\Omega})$, which yields

In light of (3.14) and $R^* > 0$, we have $I^* > 0$. We conclude that $(\lambda, \psi) = (0, I^*(x))$ is a solution of (3.2). Since $I^*(x) > 0$ on $\Omega$, it must be that $\lambda^* = 0$, which implies that $\mathcal{R}_0 = 1$. This contradiction implies $|J^+| > 0$.

We next prove part (ⅱ) by contradiction. Now assume that $|J^-| = 0$, i.e., $M^*(x) = 1$ on $\Omega$ a.e.. Denote

and choose $\varphi\in C^1(\bar{\Omega})$ such that $\varphi\geq0$ on $\Omega$. Multiplying the first two equations in (3.9) by $\varphi$, adding them together and integrating on $\Omega$, we have

As $d_S\rightarrow 0$, $M(x)\rightarrow M^*(x) = 1$ a.e on $\Omega$. Thus, we obtain

for any $\varphi\in C(\bar{\Omega})$ such that $\varphi\geq0$ on $\Omega$.

Let $\psi_0$ be a positive eigenfunction of $\lambda_1\left(-d_R\Delta +(\mu+\gamma)\left(1-\dfrac{\delta}{\beta-\mu}\right)\right)$, i.e.

Since $-d_R\Delta R+(\mu+\gamma)R-\delta I = 0$ and $S, I, R > 0$ on $\Omega$, we have

Multiplying (3.18) by $\psi_0$, integrating by parts over $\Omega$ and applying (3.17), we obtain

Let $d_S\rightarrow0$. It immediately follows from (3.16) that $\lambda_1 \int_\Omega\psi_0R^*dx\geq0$. Since $\psi_0, R^* > 0$ on $\Omega$, we see that $\lambda_1\geq0$. This contradiction yields (ⅲ).

Theorem 3.7. Let (H2) hold. Assume that $\mathcal{R}_0 > 1$ and $\lambda_1\left(-d_R\Delta+(\mu+\gamma)(1-\frac{\delta}{\beta-\mu})\right) < 0$, then the following assertions hold:

(i) As $d_S\rightarrow0$, $\tilde{S}$ subject to a sequence,

for some $M^*(x)$ satisfying $0 < M^*(x)\leq1$ in $\Omega$. Moreover, $S^* = 0$ on $J^+\subset \bar{ \Omega}$, $S^* > 0$ on $J^-\subset \bar{ \Omega}$ and $\int_\Omega \tilde{S}^* = N_0$.

(ii) There exist positive constants $C_1, C_2$, independent of $d_S$, such that for sufficiently small $d_S$

That is $\tilde{I}, \tilde{R}\rightarrow0$ uniformly in $\Omega$ as $d_S\rightarrow0$.

Proof. By (3.9), we have

It follows from $S, I, R > 0$ and $d_SS+d_II+d_RR = 1$ that $I, R$ are uniformly bounded with respect to $d_S$. Thus,

In view of Lemmas 3.5 and 3.6, we have

Therefore,

This limit is well-defined because $J^-$ has positive measure. It follows from $\tilde{S} = \dfrac{\xi}{d_S}(1-M(x))$ that

and $\int_\Omega \tilde{S}^* = N_0$.

Now we verify (ii). It follows from $d_SS+d_II+d_RR = 1$ and $\tilde{I} = \dfrac{\xi}{d_S}d_SI, \tilde{R} = \dfrac{\xi}{d_S}d_SR$ that

Hence (3.19) implies that

Next, by contradiction, we prove

Assume that $\min\left\{ \inf_{\Omega}\tilde{I}, \, \inf_{\Omega}\tilde{R}\right\}/d_S = o(d_S)$. By [48,Lemma 2.3] and (2.4), there exists a positive constant $\theta$ such that

Hence

which implies

Noting that

and combining (3.19) and (3.21), we can obtain

which contradicts Lemma 3.6 (ⅱ) (i.e., $|J^-| > 0)$. We complete the proof of part (ⅱ).

4.   Concluding remarks

It is now widely believed that the mathematical models have been revealed as a powerful tool to understand the mechanism that underlies the spread of the disease. In this paper, we proposed an epidemic model with ratio-dependent incidence rate incorporating both mobility of population and spatial heterogeneity, and focus on how spatial diffusion and environmental heterogeneity affect the basic reproductive number and disease dynamics of the model.

The value of our study lies in two aspects. Mathematically, we prove that the basic reproductive number $\mathcal{R}_0$ can be used to govern the threshold dynamics of the model: if $\mathcal{R}_0 < 1$, the unique DFE is globally asymptotically stable (see Theorem 3.2), while if $\mathcal{R}_0 > 1$, there is at least one endemic equilibrium (see Theorem 3.3). Epidemiologically, we find that restricting the movement of susceptible population can effectively control the number of infectious individuals (see Theorem 3.7 (ⅱ)). Simply speaking, our results may provide some potential applications in disease control.

Acknowledgements

The authors would like to thank the anonymous referees for very helpful suggestions and comments which led to improvement of our original manuscript. This research was supported by the National Natural Science Foundation of China (Grant numbers 61672013, 11601179 and 61772017), and the Huaian Key Laboratory for Infectious Diseases Control and Prevention (HAP201704).

Conflict of interest

The authors declare that they have no competing interests.