Abstract
In this article, we are concerned with a diffusive SVEIR epidemic model with nonlinear incidences. We first obtain the well-posedness of solutions for the model. Then, the basic reproduction number \(R_{0}\) and the local basic reproduction number \({\overline{R}}_{0}(x)\) are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between \(R_{0}\) and \({\overline{R}}_{0}(x)\) as well as the asymptotic properties of \(R_{0}\) when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators \(L_{1}\) and \(L_{2}\). Using the theory of monotone dynamical systems and the persistence theory of dynamical systems, we show that the disease-free equilibrium is globally asymptotically stable when \(R_{0}<1\), while the disease is uniformly persistent when \(R_{0}>1\). Furthermore, in the spatially homogeneous case, by using the Lyapunov functions method and LaSalle’s invariance principle, we completely obtain that the disease-free equilibrium is globally asymptotically stable if \(R_{0}\le 1\), and the endemic equilibrium is globally asymptotically stable if \(R_{0}>1\) and an additional condition is satisfied.
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1 Introduction
It is common knowledge that infectious diseases remain to threaten the survival and development of mankind. Therefore, one of the most significant and frequently discussed subjects nowadays is the prevention and control of disease transmission. In recent years, the convenient transportation provides the possibility of spatiotemporal transmission of diseases, and dynamic modeling has played a key role in describing spatiotemporal feature on diseases. Review these studies, the important and valuable results have been obtained and applied in realistic prevention and control of disease [1,2,3,4].
Vaccination is important to control measures in preventing the transmission of diseases. Thus, many countries provide routine vaccination against some the diseases, such as the avian influenza vaccine, the spondylitis vaccine, and so on. Over time, however, the vaccine loses its protection. Hence, developing mathematical models describing and understanding this phenomenon are particularly important [2, 5,6,7,8,9]. Kribs-Zaleta and Velasco-Hernández [5] investigated a model of an SIS epidemic with vaccination. Li et al. [6] demonstrated that vaccine efficacy is crucial in preventing and controlling diseases. In particular, Liu et al. [2] established and studied the following model to characterize vaccination strategy:
Here, S(t), V(t), I(t) and R(t) denote the densities of susceptible, vaccinated, infected and recovered individuals, respectively. For model (1.1), the authors showed that there exhibits rigorous threshold dynamics and vaccination contributes to disease control by reducing the basic reproduction number.
Noting that model (1.1) does not take into account incubation periods. But before the host becomes contagious, many diseases have an incubation period, and the duration of this time varies from disease to disease [10]. Additionally, we are aware that disease spread is not only time-dependent but also space-dependent. Webby [11] pointed out that infectious cases can be discovered in one place and then spread to other locations. As a result, studying epidemic models with spatial diffusion is intriguing. It has been confirmed by numerous studies in various models [12,13,14,15,16].
On the other hand, incidence rate is also an important substance in modeling the dynamics of epidemic systems. However, incidence functions in many infectious disease models are bilinear or standard incidence. From the perspective of the development mechanism of infectious diseases, the bilinear incidence is generally used for small-scale susceptible group and exposure. As the population size increases, the bilinear incidence will become infinite and lose its practical significance. At this time, the standard incidence is adopted, which is applicable to a large number of people. More and more nonlinear incidence has been mentioned many times (See [17,18,19,20]), such as nonlinear incidences \(\beta Sg(I)\), \(\beta f(S)g(I)\) and \(\beta f(S,I)\). At this time, the nonlinear incidence is more realistic and achieves more exact results. Therefore, it is very interesting and natural to investigate epidemic model with nonlinear incidences.
Let \(\Omega \subset {\mathbb {R}}^{n}\) be a bounded domain with smooth boundary \(\partial \Omega \), and n be the outward unit normal vector on \(\partial \Omega \). Based on the above discussion, we take into account latency, temporal and spatial factors, and nonlinear incidences in model (1.1). As a result, we propose the diffusive SVEIR model with nonlinear incidences shown below:
with the homogeneous Neumann boundary conditions
and the initial conditions
Here, the densities of susceptible, vaccinated, latent, infected and recovered individuals at time t and spatial location x are denoted by S(t, x), V(t, x), E(t, x), I(t, x) and R(t, x), respectively. \(\Lambda (x)\) is the input rate of population; \(\xi (x)\) is the vaccination rate of susceptible individuals; \(\alpha (x)\) is the rate of vaccinees obtain immunity; \(\sigma (x)\) represents the transition rate from E to I; \(\delta (x)\) is the disease-induced death rate; \(\gamma (x)\) is the recovery rate after infection; \(\beta _{1}(x)\) and \(\beta _{2}(x)\) are the infection rate of S and V infected by I in spatial location x, respectively; \(\mu _{k}(x)\;(1\le k\le 5)\) denote the natural death rates of susceptible, vaccinated, latent, infected and recovered individuals, respectively; \(d_{k}(x)\;(1\le k\le 5)\) are the diffusion rate of susceptible, vaccinated, latent, infected and recovered individuals, respectively; and \(f_{1}(S)\), \(f_{2}(V)\) and g(I) denote the nonlinear incidence rates. For the purposes of this article, we assume that all parameters in model (1.2) are continuous, nonnegative and bounded defined on \({{\overline{\Omega }}}\).
The main purpose in this paper is to investigate the basic reproduction number and the dynamics of model (2.1) in the spatially heterogeneous and homogeneous cases, respectively. The model proposed in this paper can more reasonably characterize the spread of epidemics. The main contribution and innovations are summarized as follows:
(1) The well-posedness of solutions for model (2.1) is established. It is proved that for any nonnegative initial value, model (2.1) has a unique nonnegative and ultimately bounded solution by using the comparison principle, \(C_0\)-semigroup and the integral expression of model (2.1).
(2) The expression of the basic reproduction number \(R_{0}\) for model (2.1) is calculated, which depends diffusion rates \(d_{i}(x)(1\le i\le 4)\). Furthermore, we investigate the asymptotic properties of \(R_{0}\) when the diffusive rates tend to infinity or zero.
(3) We establish the threshold criteria on the extinction and persistence of solutions for model (2.1) in the spatially heterogeneous case by using the theory of monotone dynamical systems and the persistence theory of dynamical systems.
(4) We establish the complete results for the global stability of model (2.1) in the spatially homogeneous case. Specially, the disease-free equilibrium \(P_{0}\) of model (5.1) is also globally asymptotically stable when \(R_{0}=1\).
The organization of this paper is as follows. In Sect. 2, we prove the existence and ultimate boundedness of solutions for model (2.1). In Sect. 3, we calculate the basic reproduction number \(R_{0}\) of model (2.1) and local basic reproduction number \({\overline{R}}_{0}(x)\) by using the next-generation operator method. And then, two compact linear operators \(L_{1}\) and \(L_{2}\) are introduced to investigate the relationship between \(R_{0}\) and \({\overline{R}}_{0}(x)\) and the asymptotic properties of \(R_{0}\) when the diffusive rates are constants and tend to infinity or zero. In Sect. 4, the threshold dynamics of model (2.1) are discussed. It is demonstrated that disease-free equilibrium \(P_{0}(x)\) is globally stable and that the solutions persist uniformly. In Sect. 5, we discuss the global asymptotic stability of the model in the homogeneous space.
2 Well-posedness of solutions
In this section, we focus on the well-posedness of solutions of model (2.1). Because R(t, x) does not appear in the first four equations of model (1.2), we only discuss the following subsystem of (1.2)
Denote \({\bar{k}}=\max _{x\in {\overline{\Omega }}}k(x)\) and \({\underline{k}} =\min _{x\in {\overline{\Omega }}}k(x)\) for any bounded function k(x) defined on \({\overline{\Omega }}\). Let \({\mathbb {Y}}:=C({\overline{\Omega }},{\mathbb {R}})\) with the supremum norm \(\Vert \phi \Vert _{{\mathbb {Y}}}=\sup _{x\in {\overline{\Omega }}}|\phi (x)|\). Let \({\mathbb {X}}:=C({\overline{\Omega }},{\mathbb {R}}^{4})\) be the Banach space of continuous functions \(\phi : {\overline{\Omega }}\rightarrow {\mathbb {R}}^{4}\) with the norm \(\Vert \phi \Vert _{{\mathbb {X}}}=\max \{\Vert \phi _{1}\Vert _{{\mathbb {Y}}}, \Vert \phi _{2}\Vert _{{\mathbb {Y}}}, \Vert \phi _{3}\Vert _{{\mathbb {Y}}},\Vert \phi _{4}\Vert _{{\mathbb {Y}}}\}\), where \(\phi =(\phi _{1},\phi _{2},\phi _{3},\phi _{4})\in {\mathbb {X}}\), and \(\phi _{i}\in {\mathbb {Y}}\; (1\le i\le 4)\). Let \(\mathbb {X^{+}}:=C({\overline{\Omega }}, {\mathbb {R}}^{4}_{+})\) be the positive cone of \({\mathbb {X}}\).
In model (2.1), functions \(f_{1}(S)\), \(f_{2}(V)\) and g(I) satisfy the following assumptions.
\(({\textbf{A}}_{1})\) Functions \(f_{1}\), \(f_{2}\) and \(g: {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}\) are continuously differentiable, \(f_{1}(0)=f_{2}(0)=g(0)=0\), and \(g'(0)>0\).
\(({\textbf{A}}_{2})\) \(f_{1}(S)\), \(f_{2}(V)\) and g(I) are nondecreasing for all \(S>0\), \(V>0\) and \(I>0\), respectively, and \(\frac{g(I)}{I}\) is nonincreasing for all \(I>0\).
Consider the following scalar reaction–diffusion equation:
where d(x), \(\mu (x)\) and \(\beta (x)\) are positive, continuous and bounded functions defined on \({\overline{\Omega }}\). The following conclusions are held in light of Lemma 1 in [21].
Lemma 2.1
(1) Model (2.2) has a unique positive steady state \(u_{0}(x)\) that satisfies the equation
(2) The steady state \(u_{0}(x)\) is globally asymptotically stable in \(C({\overline{\Omega }},{\mathbb {R}}_{+})\).
(3) If \(\mu (x)\equiv \mu \) and \(\beta (x)\equiv \beta \) are positive constants, then \(u_{0}(x)\equiv \frac{\mu }{\beta }\).
(4) If \(d(x)\equiv d\) is a positive constant, then \(\lim _{d\rightarrow 0}u_{0}(x)=\frac{\mu (x)}{\beta (x)}\) and \(\lim _{d\rightarrow \infty }u_{0}(x)=\frac{\int _{\Omega }\mu (x)dx}{\int _{\Omega }\beta (x)dx}.\)
Based on the results in [22,23,24] and [25], we first obtain that operator \(\nabla \cdot (d_{i}(\cdot )\nabla )-\rho _{i}(\cdot )\) subjects to Neumann boundary condition (1.3) generates a \(C_{0}\)-semigroup \(T_{i}(t): C({\overline{\Omega }},{\mathbb {R}})\rightarrow C({\overline{\Omega }},{\mathbb {R}})\), where \(\rho _{1}(x)=\mu _{1}(x)+\xi (x),\rho _{2}(x)=\mu _{2}(x)+\alpha (x),\rho _{3}(x)=\mu _{3}(x)+\sigma (x)\) and \(\rho _{4}(x)=\mu _{4}(x)+\delta (x)+\gamma (x)\), respectively. Then, we have the expression
where \(\Gamma _{i}(t,x,y)\) is the Green function of operator \(\nabla \cdot (d_{i}(\cdot )\nabla )-\rho _{i}(\cdot )\) subjects to the Neumann boundary condition. Furthermore, we can obtain that \(T_{i}(t)\;(i=1,2,3,4)\) are strongly positive and compact for every \(t>0\), and there exist constants \(M_i>0\;(i=1,2,3,4)\) such that \(\Vert T_{i}(t)\Vert \le M_ie^{\omega _{i}t}\) for \(t\ge 0\), where \(\omega _{i}<0\) is the principle eigenvalue of operator \(\nabla \cdot (d_{i}(\cdot )\nabla )-\rho _{i}(\cdot )\) subjects to the Neumann boundary condition.
Define \(F=(F_{1},F_{2},F_{3},F_{4})\) as follows:
where \(x\in \Omega \) and \(\psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})\in {\mathbb {X}}^{+}\). Let \(u(t,\cdot ,\psi )=(S(t,\cdot ,\psi ),V(t,\cdot ,\psi ),E(t,\cdot ,\psi ), I(t,\cdot ,\psi ))\) be the solution of model (2.1) with initial value \(\psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})\in {\mathbb {X}}^{+}\) at start time \(t=0\). Then, model (2.1) can be rewritten as the following integral equations:
With the use of basic calculation and the assumptions \((A1)-(A2)\), we can prove that the following subtangential conditions hold:
Hence, from Corollary 4 in [26] we directly obtain
Lemma 2.2
For any initial value \(\psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})\in {\mathbb {X}}^{+}\), model (2.1) has a unique noncontinuable mild solution \(u(t,\cdot ,\psi )=(S(t,\cdot ,\psi ),V(t,\cdot ,\psi ),E(t,\cdot ,\psi ),I(t,\cdot ,\psi ))\in {\mathbb {X}}^{+}\) on the existence interval \([0,\tau _{\psi })\) with \(\tau _{\psi }\le \infty \). Furthermore, this solution also is a classical solution.
Next, the following results are established regarding the existence and ultimate boundedness of the global solution of model (2.1).
Theorem 2.1
For any initial value \(\psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})\in {\mathbb {X}}^{+}\), model (2.1) has a unique nonnegative solution \(u(t,\cdot ,\psi )=(S(t,\cdot ,\psi ),V(t,\cdot ,\psi ),E(t,\cdot ,\psi ), I(t,\cdot ,\psi ))\) defined on \([0,\infty )\times {\overline{\Omega }}\) and is ultimately bounded.
Proof
According to Lemma 2.2, model (2.1) has a unique classical solution \(u(t,x)=(S(t,x),V(t,x),E(t,x), I(t,x))\) defined for \(t\in [0,\tau _{\psi })\) and \(x\in {\overline{\Omega }}\), where \(\tau _{\psi }\le +\infty \). Now, we prove \(\tau _{\psi }=+\infty \). On the contrary, suppose that \(\tau _{\psi }<\infty \), then we have \(\Vert u(t,x)\Vert _{{\mathbb {X}}}\rightarrow \infty \) as \(t\rightarrow \tau _{\psi }\) from Theorem 2 in [26]. In fact, from the first equation of model (2.1), we get
where \(t\in [0,\tau _{\psi })\) and \(x\in \Omega \). By the comparison principle of reaction–diffusion equation and Lemma 2.1, there is a constant \(M_{1}>0\) such that \(S(t,x)\le M_{1}\) for all \(t\in [0,\tau _{\psi })\) and \(x\in {\overline{\Omega }}\). Furthermore, from the second equation of model (2.1), we get
Similarly, there exists a constant \(M_2>0\) such that \(V(t,x)\le M_2\) for all \(t\in [0,\tau _{\psi })\) and \(x\in {\overline{\Omega }}\).
Define
Calculating the derivative of W(t), and using the divergence theorem [27] and the homogeneous Neumann boundary conditions, we can obtain
where \(a=\min \{ {\underline{\mu }}_{1},{\underline{\mu }}_{2}+{\underline{\alpha }},{\underline{\mu }}_{3},{\underline{\mu }}_{4} +{\underline{\delta }}+{\underline{\gamma }}\}\) and \(\mid \Omega \mid \) is the measure of \(\Omega \). The comparison principle implies
for all \(t\in [0,\tau _{\psi })\). This shows that W(t) is bounded on \([0,\tau _{\psi })\). Consequently, there is a constant \(M_3>0\) such that \(\int \limits _{\Omega }E(t,x)dx \le M_{3}\) and \(\int \limits _{\Omega }I(t,x)dx \le M_{3}\) for all \(t\in [0,\tau _{\psi })\).
It is clear from [28] that
where \(\pi _{n}\) is the eigenvalue of \(\nabla \cdot (d_{3}(x)\nabla )-(\mu _{3}(x)+\sigma (x))\) subject to the Neumann boundary condition corresponding to the eigenfunction \(\varphi _{n}(x)\) and satisfies \(\pi _{1}>\pi _{2}> \pi _{3}> \cdots> \pi _{n}> \cdots \). Since \(\{\varphi _{n}(x)\}\) is uniformly bounded on \({\overline{\Omega }}\), there exists constant \(\kappa _{1}>0\) such that
Furthermore, let \(\tau _{n}\;(n=1,2,\cdots )\) be the eigenvalue of \(\nabla \cdot (d_3(x)\nabla )-({\underline{\mu }}_{3}+{\underline{\sigma }} )\) subject to the Neumann boundary condition satisfying \(\tau _{1}>\tau _{2}> \tau _{3}> \cdots> \tau _{n}> \cdots \). We have \(\tau _{1}=-({\underline{\mu }}_{3}+{\underline{\sigma }})\), and by Theorem 2.4.7 in [29], \(\pi _{i}\le \tau _{i}\) for all \(i\in {\mathbb {N}}_{+}\). Due to the fact that \(\tau _{i}\) decreases like \(-i^{2}\), then there is a constant \(\kappa _{2}>0\) such that
From (2.4) and (2.5), for any \(t\in [0,\tau _{\psi })\) and \(x\in \Omega \), we have that
For a similar argument as in above, from the fourth equation of (2.1), we also have that for any \(t\in [0,\tau _{\psi })\) and \(x\in \Omega \)
The above discussions imply that solution (S(t, x), V(t, x), E(t, x), I(t, x)) of system (2.1) is bounded for \(t\in [0,\tau _{\psi })\) and \(x\in {\overline{\Omega }}\). This leads to a contradiction. Therefore, \(\tau _{\psi }=\infty \). That is, solution \((S(t,x),V(t,x),E(t,x),I(t,x))\) is defined for all \(t\ge 0\) and \(x\in {\overline{\Omega }}\).
Now, we prove that the solution is ultimately bounded. In fact, from (2.6) we can get \(\limsup _{t \rightarrow \infty }S(t,x) \le \frac{{\overline{\Lambda }}}{{\underline{\mu }}_{1}+{\underline{\xi }}}:=N_1\) uniformly for \(x\in {\overline{\Omega }}\). This implies that S(t, x) is ultimately bounded. Namely, for an enough small constant \(\varepsilon _0>0\), there is a \(t_{1}>0\) such that \(S(t,x)\le N_{1}+\varepsilon _0\) for all \(t>t_{1}\) and \(x\in {\overline{\Omega }}\). Furthermore, from (2.7) we can get \(\limsup _{t\rightarrow \infty }V(t,x)\le \frac{{\overline{\xi }}(N_1+\varepsilon _0)}{{\underline{\mu }}_2+{\underline{\alpha }}}:=N_2\) uniformly for \(x\in {\overline{\Omega }}\). Hence, V(t, x) is ultimately bounded. Namely, there is a \(t_{2}>t_1\) such that \(V(t,x)\le N_{2}+\varepsilon _0\) for all \(t>t_{2}\) and \(x\in {\overline{\Omega }}\). From (2.8), we can get \(\limsup _{t\rightarrow \infty }W(t)\le \frac{{\overline{\Lambda }}|\Omega |}{a}:=N_3\). Thus, there is a \(t_3>t_2\) such that \(\int \limits _{\Omega }E(t,x)dx\le N_3+\varepsilon _0\) and \(\int \limits _{\Omega }I(t,x)dx\le N_3+\varepsilon _0\) for all \(t>t_3\). Furthermore, similar to (2.9) and (2.10), we also can obtain that
uniformly for \(x\in {\overline{\Omega }}\). This shows that E(t, x) and I(t, x) are also ultimately bounded. This completes the proof. \(\square \)
Base on Theorem 2.2.6 in [30], Theorem 3.4.8 in [31] and Theorem 2.1, we can get the following result.
Corollary 2.1
All nonnegative solutions \(u(t,\cdot ,\psi )\) of model (2.1) generate a solution semiflow \(\Phi (t): {\mathbb {X}}^{+}\rightarrow {\mathbb {X}}^{+}\) defined by \(\Phi (t)\psi =u(t,\cdot ,\psi )\) for any \(t\ge 0\) and initial value \(\psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})\in {\mathbb {X}}^{+}\). Moreover, \(\Phi (t)\) has a global compact attractor.
3 Basic reproduction number
In this section, we want to calculate the basic reproduction number of model (2.1) and find its display expression and explore its some properties. When \((E(t,x),I(t,x))\equiv (0,0)\) in model (2.1), we get the reaction–diffusion equation shown below.
It follows from Lemma 2.1 that the following system
admits a unique positive steady state \(S_{0}(x)\), satisfying the equation
with \(\frac{\partial S_{0}(x)}{\partial n}=0\) for \(x\in \partial \Omega \), which is globally asymptotically stable in \(C({\overline{\Omega }},{\mathbb {R}}_{+})\). Then, from the second equation of system (3.1), we have the limit system
By Lemma 2.1 and Corollary 4.3 in [32], system (3.3) exists a unique positive steady state \(V_{0}(x)\) satisfying the equation
with \(\frac{\partial V_{0}(x)}{\partial n}=0\) for \(x\in \partial \Omega \), which is globally asymptotically stable in \(C({\overline{\Omega }},{\mathbb {R}}_{+})\). Thus, we obtain that model (2.1) has disease-free equilibrium \(P_{0}(x)=(S_{0}(x),V_{0}(x),0,0)\).
At equilibrium \(P_{0}(x)\), we linearize the last two equations of model (2.1), and then, we obtain the following linearized subsystem
We define the matrix operators
Define \(T(x): C({\overline{\Omega }},{\mathbb {R}}^{2})\rightarrow C({\overline{\Omega }},{\mathbb {R}}^{2})\) be the \(C_0\)-semigroup produced by the following system
where \(w=(w_{1},w_{2})^{T}\). Assume that the illness is introduced at time \(t = 0\) and the initial infection\('\) distribution is represented by the expression \(\psi (x)=(\psi _{3}(x),\psi _{4}(x))^{T}\). As a result, the distribution of new infection becomes \(F(x)T(t)\psi (x)\) at time t as time evolves. Therefore, \(\int \limits _{0}^{+\infty }F(x)T(t)\psi (x)\) represents the distribution of total new infective numbers. Define the operator \(L: C({\overline{\Omega }},{\mathbb {R}}^{2})\rightarrow C({\overline{\Omega }},{\mathbb {R}}^{2})\) as follows
We call operator L by the next-generation operator. Obviously, L is a continuous and positive operator which maps the initial infection distribution \(\psi (x)\) to the distribution of the total infective people produced during the infection period. According to [33], the spectral radius of L is defined as the basic reproduction number \(R_{0}\) of model (2.1). Namely, \(R_{0}=r(L)\).
We can obtain by calculating
where \(b_{21}(x)=(\nabla \cdot d_{3}(x)\nabla -a_{1}(x))^{-1}\sigma (x)(\nabla \cdot d_{4}(x)\nabla -a_{2}(x))^{-1}\), \(a_{1}(x)=\mu _{3}(x)+\sigma (x), a_{2}(x)=\mu _{4}(x)+\delta (x)+\gamma (x)\). Further, we get
where \(f_{12}(x)=\beta _{1}(x)f_{1}(S_{0}(x))g'(0)+\beta _{2}(x)f_{2}(V_{0}(x))g'(0)\). Thus, we can obtain
Namely, \(R_{0}\) is defined as the spectral radius of operator \(b_{21}(x)f_{12}(x)\). This tells that \(R_{0}\) is the principal eigenvalue of the following eigenvalue problem:
Thus, there exists a strictly positive eigenfunction \(\phi =\phi _{*}\) satisfying
That is
and \(\frac{\partial \phi _{*}(x)}{\partial n}=0\) for \(x\in \partial \Omega \). Multiplying by \(\phi _{*}\) in both sides of (3.5) and then integrating on \(\Omega \), then we have
On the other hand, we consider the following eigenvalue problem:
From Krein–Rutman theorem, there is a principal eigenvalue \(\lambda =\lambda ^{*}\) and a corresponding strictly positive eigenfunction \(\psi =\psi _{*}\) such that
That is
Then, we get
Multiplying by \(\psi _{*}\) in both sides of (3.5), we have
Then, from (3.6) we also obtain
Integrating on \(\Omega \), we further have
This shows that
Let \((U_{3}(t,x),U_{4}(t,x)) =e^{\lambda t}(\psi _{3}(x),\psi _{4}(x))\), from model (3.4), we can obtain
We can derive the following conclusion from Theorem 7.6.1 in [32].
Lemma 3.1
The eigenvalue problem (3.8) admits a principal eigenvalue \(\lambda _{0}(d_{3},d_{4}, P_{0}(x))\) with a strictly positive eigenfunction \(\psi (x)=(\psi _{3}(x),\psi _{4}(x)).\)
Furthermore, we can draw the following conclusions from Theorem 3.1 in [33].
Lemma 3.2
\((\textrm{i})\) \(\lambda _{0}\) and \(R_{0}-1\) have the same sign.
\((\textrm{ii})\) If \(R_{0}<1\), then disease-free equilibrium \(P_{0}(x)\) of model (2.1) is locally asymptotically stable.
\((\textrm{iii})\) If \(R_{0}>1\), then equilibrium \(P_{0}(x)\) for model (2.1) is unstable.
When diffusion coefficients \(d_{i}(x)=0~(i=1,2,3,4)\), then model (2.1) is transformed into the corresponding ordinary differential equation model as follows:
Theorem 3.1
For model (3.9), in each position \(x\in \Omega \), the local basic reproduction number \({\overline{R}}_{0}(x)\) is expressed as
where
Theorem 3.1 can be easily proved by using the next-generation matrix method. We here omit it.
Based on the above analysis, for the relation between \(R_{0}\) and \({\overline{R}}_{0}(x)\), we can have the following obvious observation.
Remark 3.1
The following facts hold.
\((\textrm{i})\) If \(R_{i}(x)(i=1,2,3)\) are constants, then \({\overline{R}}_{0}(x)\equiv {\overline{R}}_{0}\) also is a constant, and \(R_{0}={\overline{R}}_{0}\).
\((\textrm{ii})\) If \(\min _{1\le i \le 3}\{R_{i}(x)\}>1\), then \({\overline{R}}_{0}(x)>1\) always holds.
Now, we define two linear operators \(L_{1}\) and \(L_{2}: C({\overline{\Omega }},{\mathbb {R}})\rightarrow C({\overline{\Omega }}, {\mathbb {R}})\) as follows
and let the operators \(L_{i}\;(i=1,2)\) subject to the Neumann boundary condition. Then, it is easy to see that \(r(L_{1}L_{2})=r(L_{2}L_{1})\le \Vert L_{1}\Vert \Vert L_{2}\Vert ,\) where \(r(L_{i})\;(i=1,2)\) denote the spectral radius of \(L_{i}\).
Next we can have the following result.
Theorem 3.2
\(L_{1}\) and \(L_{2}\) in (3.12) are strongly positive compact linear operators on \(C({\overline{\Omega }},{\mathbb {R}})\), and the basic reproduction number \(R_{0}\) of model (2.1) is expressed by
where \(R_{i}(x)\;(i=1,2,3)\) defined in (3.11) are additive operators on \(C({\overline{\Omega }},{\mathbb {R}})\).
Proof
Recall the calculation of \(R_{0}\), we know
According to the elliptic estimates and maximum principles, we know that \(L_{1}\) and \(L_{2}\) are strongly positive compact linear operators on \(C({\overline{\Omega }},{\mathbb {R}})\). This completes the proof. \(\square \)
Lemma 3.3
\(\Vert L_{1}\Vert =1\) and \(\Vert L_{2}\Vert =1.\)
Proof
Let \(L_{1}(1)=u_{1}\), then we have \(((\mu _{3}(x)+\sigma (x))-\nabla \cdot d_{3}(x)\nabla )^{-1}(\mu _{3}(x)+\sigma (x))(1)=u_{1}(x).\) Further, we have \((\mu _{3}(x)+\sigma (x))(1)=((\mu _{3}(x)+\sigma (x))-\nabla \cdot d_{3}(x)\nabla )u_{1}(x).\) That is, \(\nabla \cdot d_{3}(x)\nabla u_{1}(x)-(\mu _{3}(x)+\sigma (x))(1-u_{1}(x))=0.\) Therefore, we have \(u_{1}(x)=1\). This shows that \(L_{1}(1)=1\). Similarly, we also have \(L_{1}(-1)=-1\) and \(L_{2}(\pm 1)=\pm 1\).
For any \(u\in C({\overline{\Omega }},{\mathbb {R}})\) with \(\Vert u\Vert _{\infty }\le 1\), we have \(-1\le u(x)\le 1\). By the comparison principle, we have \(-1=L_{i}(-1)\le L_{i}(u)\le L_{i}(1)=1\) for \(i=1,2.\) Therefore, \(\Vert L_{i}u\Vert \le 1 =\Vert u\Vert _{\infty }\), which implies \(\Vert L_{i}\Vert \le 1\) for \(i=1,2\). Further, from \(L_{1}(1)=1\) and \(L_{2}(1)=1\), we have \(\Vert L_{1}\Vert =\Vert L_{2}\Vert =1\). This completes the proof. \(\square \)
Lemma 3.4
\(r(L_{1})=r(L_{2})=r(L_{1}L_{2})=1\).
Proof
Because \(L_{1}\) and \(L_{2}\) are strongly positive compact linear operators on \(C({\overline{\Omega }},{\mathbb {R}})\), so is \(L_{1}L_{2}\). By the general result on the Krein–Rutman theorem (Theorem 2.5 [34]), we know that \(r(L_{1}),r(L_{2})\) and \(r(L_{1}L_{2})\) are simple positive eigenvalues of operators \(L_{1},L_{2}\) and \(L_{1}L_{2}\) associated with positive eigenvectors, respectively. Moreover, there is no other such eigenvalue for \(L_{1},L_{2}\) and \(L_{1}L_{2}\), respectively. Owing to \(L_{1}(1)=L_{2}(1)=L_{1}L_{2}(1)=1\), we have \(r(L_{1})=r(L_{2})=r(L_{1}L_{2})=1\). This completes the proof. \(\square \)
By (3) of Theorem 3.6 in [34], we can have the following result.
Lemma 3.5
The following inequalities hold:
where \({\overline{R}}_{0*}=\min _{x\in \Omega }{\overline{R}}_{0}(x)\) and \({\overline{R}}_{0}^{*}=\max _{x\in \Omega }{\overline{R}}_{0}(x)\).
Proof
Since \(R_{i}(x)\le R_{i}^{*}\) for \(i=1,2,3\), we have \(L_{1}R_{1}L_{2}R_{2}+L_{1}R_{1}L_{2}R_{3}\le L_{1}R_{1}^{*}L_{2}R_{2}^{*}+L_{1}R_{1}^{*}L_{2}R_{3}^{*}.\) Therefore,
Similar proof for the opposite part can take. This completes the proof. \(\square \)
3.1 The large diffusion rates
In this subsection, we investigate \(R_{0}\) quantitatively when the diffusion rates increase to infinity. For this purpose, assume \(d_{3}(x)\equiv d_3\) and \(d_{4}(x)\equiv d_4\) are constants. We firstly give a useful lemma before we will do this.
Lemma 3.6
(see [34]) Let W be an ordered Banach space with positive cone \(W_{+}\) such that \(W_{+}\) has nonempty interior. Let \(T_{n}(n\ge 1)\), and T be strongly positive compact linear operators on W. Suppose \(T_{n} \xrightarrow {SOT} T\) (strong operator topology) which means \(T_{n}(u)\rightarrow T(u)\) for any \(u\in W\). If \(\mathop {\cup }_{n\ge 1}T_{n}(B)\) is precompact, where B is the closed unit ball of W, and \(r(T_{n})\ge r_{0}\) for some \(r_{0}>0\), then \(r(T_{n})\rightarrow r(T)\).
The proof of this lemma can be found in Theorem 4.1 in [34].
Lemma 3.7
Let \(R_{0}\) be defined in (3.7), and \(d_{i}(x)\equiv d_i\;(i=1,2,3,4)\) be constants. For any given \(d_{1}>0\) and \(d_{2}>0\), assume \(\sigma (x)=\beta _{1}(x)f(S_{0}(x))g'(0)+\beta _{2}(x)f(V_{0}(x))g'(0)\) or both \(\sigma (x)\) and \(\beta _{1}(x)f(S_{0}(x))g'(0)+\beta _{2}(x)f(V_{0}(x))g'(0)\) are constants. Then, \(R_{0}\) is decreasing function with respect to \(d_{3}\) and \(d_{4}\).
Proof
Firstly, we prove \(R_{0}\) is decreasing in \(d_{3}\). Let \(k=\frac{1}{R_{0}}\). According to the Krein–Rutman theory, there is an eigenvalue k and a corresponding eigenfunction \(\psi \) with \(\Vert \psi \Vert _{2}=1\), which is strictly positive and satisfies
Therefore, we have
Differentiating both sides of (3.13) with respect to \(d_{3}\), we get
Multiplying (3.14) by \(\psi \) and (3.13) by \(\psi _{d_{3}}\), and integrating their difference over \(\Omega \), we have
and
By divergence theorem and conditions of Lemma 3.7, we have \(\int \limits _{\Omega }\psi \Delta \psi dx= -\int \limits _{\Omega }|\nabla \psi |^{2}dx,\) and
Combine (3.15) and (3.16), we obtain
Since \((\mu _{3}(x)+\sigma (x))(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})\) is strongly positive, \((\mu _{3}(x)+\sigma (x))(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})\psi >0\). Therefore, \(k_{d_{3}}\ge 0\); that is, k is increasing in \(d_{3}\). Hence, \(R_{0}\) is decreasing in \(d_{3}\). Using a similar approach, we can also prove that \(R_{0}\) is decreasing in \(d_{4}\). This completes the proof. \(\square \)
Define two integral operators \(L_{1,\infty }\), \(L_{2,\infty }: C({\overline{\Omega }},{\mathbb {R}})\rightarrow C({\overline{\Omega }},{\mathbb {R}})\) by
Lemma 3.8
\(L_{1}\xrightarrow {SOT} L_{1,\infty }\) in \(C({\overline{\Omega }},{\mathbb {R}})\) as \(d_{3}\rightarrow \infty \).
Proof
For any given \(u\in C({\overline{\Omega }},{\mathbb {R}})\), we just need to show that \(L_{1}(u)\rightarrow L_{1,\infty }(u)\) in \(C({\overline{\Omega }},{\mathbb {R}})\) as \(d_{3}\rightarrow \infty \). For any \(d_{3}>0\), let \(Q_{d_{3}}=L_{1}(u)\), that is \(Q_{d_{3}}=((\mu _{3}(x)+\sigma (x))-d_{3}\Delta )^{-1}(\mu _{3}(x)+\sigma (x))u\). Further, we have \((\mu _{3}(x)+\sigma (x))Q_{d_{3}}-d_{3}\Delta Q_{d_{3}}= (\mu _{3}(x)+\sigma (x))u\). Since the operator \(L_{1}\) subjects to the Neumann boundary condition, we have that \(Q_{d_{3}}\) is the solution of the following problem
Under the comparison principle, we know \(-\Vert u\Vert _{\infty }\le Q_{d_{3}}\le \Vert u\Vert _{\infty }\) for all \(d_{3}>1\). Hence by the \(L^{p}\) estimate, \(\{Q_{d_{3}}\}_{d_{3}>1}\) is uniformly bounded in \(W^{2,p}(\Omega )\) for any \(p>1\). By the embedding theorems, we see that \(W^{2,p}(\Omega )\subseteq C({\overline{\Omega }},{\mathbb {R}})\) is compact for \(p>n\), up to a subsequence, \(Q_{d_{3}}\rightarrow Q\) weakly in \(W^{2,p}(\Omega )\) and strongly in \(C({\overline{\Omega }},{\mathbb {R}})\) for some \(Q\in W^{2,p}(\Omega )\) as \(d_{3}\rightarrow \infty \). Moreover, Q satisfies
Under the maximum principle, we know that Q must be constant. Then integrating both sides of the first equation of (3.17) and taking \(d_{3}\rightarrow \infty \), we have \(Q=\frac{\int \limits _{\Omega }(\mu _{3}(x)+\sigma (x))u(x)dx}{\int \limits _{\Omega }(\mu _{3}(x)+\sigma (x))dx}.\) This completes the proof. \(\square \)
Lemma 3.9
\(L_{2}\xrightarrow {SOT} L_{2,\infty }\) in \(C({\overline{\Omega }},{\mathbb {R}})\) as \(d_{4}\rightarrow \infty \).
Proof
For any given \(u\in C({\overline{\Omega }},{\mathbb {R}})\), we just need to show that \(L_{2}(u)\rightarrow L_{2,\infty }(u)\) in \(C({\overline{\Omega }},{\mathbb {R}})\) as \(d_{4}\rightarrow \infty \). For any \(d_{4}>0\), let \(Q_{d_{4}}=L_{2}(u)\). The \(Q_{d_{4}}\) is the solution of the following problem
For similar discussions as in Lemma 3.8, we can conclude that \(L_{2}\rightarrow L_{2,\infty }\) in \(C({\overline{\Omega }},{\mathbb {R}})\) holds as \(d_{4}\rightarrow \infty \). This completes the proof. \(\square \)
Theorem 3.3
The following conclusions hold.
\((\textrm{1})\) For fixed \(d_{4}>0\), then \(R_{0}\rightarrow r[L_{1,\infty }(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})]\) as \(d_{3}\rightarrow +\infty \).
\((\textrm{2})\) For fixed \(d_{3}>0\), then \(R_{0}\rightarrow r[L_{2,\infty }(R_{2}L_{1}R_{1}+R_{3}L_{1}R_{1})]\) as \(d_{4}\rightarrow +\infty \).
Proof
Two bounded linear operators \(H_{1,\infty },H_{2,\infty }: C({\overline{\Omega }},{\mathbb {R}})\rightarrow C({\overline{\Omega }},{\mathbb {R}})\) are defined as follows
and
Then, \(H_{1,\infty }=L_{1,\infty }(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})\) and \(H_{2,\infty }=L_{2,\infty }(R_{2}L_{1}R_{1}+R_{3}L_{1}R_{1}).\) By Lemmas 3.8-3.9, we have that \(L_{1}R_{1}L_{2}R_{2}+L_{1}R_{1}L_{2}R_{3}\xrightarrow {SOT} H_{1,\infty }\) when \(d_{3}\rightarrow +\infty \) and \(L_{2}R_{2}L_{1}R_{1}+L_{2}R_{3}L_{1}R_{1}\xrightarrow {SOT} H_{2,\infty }\) when \(d_{4}\rightarrow +\infty .\)
We obverse that the operators \(L_{1}R_{1}L_{2}R_{2}+L_{1}R_{1}L_{2}R_{3}\), \(L_{2}R_{2}L_{1}R_{1}+L_{2}R_{3}L_{1}R_{1}\), \(H_{1,\infty }\) and \(H_{2,\infty }\) are strongly positive compact operators on \(C({\overline{\Omega }},{\mathbb {R}})\). For similar discussion of Theorem 4.5 in [34] and Theorem 3.2, we have that \(R_{0}=r(L_{1}R_{1}L_{2}R_{2}+L_{1}R_{1}L_{2}R_{3})\rightarrow r(H_{1,\infty })\) when \(d_{3}\rightarrow +\infty \) and \(R_{0}=r(L_{2}R_{2}L_{1}R_{1}+L_{2}R_{3}L_{1}R_{1})\rightarrow r(H_{2,\infty })\) when \(d_{4}\rightarrow +\infty \). Finally, we can know that the eigenfunctions of \(H_{1,\infty }\) and \(H_{2,\infty }\) must be constants, and
This completes the proof. \(\square \)
Remark 3.2
The following conclusions also hold.
\((\textrm{i})\) If \(R_{2}(x)\equiv \) constant and \(R_{3}(x) \equiv \) constant, then we have \(L_{2}R_{2}=R_{2}\), \(L_{2}R_{3}=R_{3}\) and \(R_{0}\rightarrow \frac{\int \limits _{\Omega }(\mu _{3}(x)+\sigma (x)){\overline{R}}_{0}dx}{\int \limits _{\Omega }(\mu _{3}(x)+\sigma (x))dx}\) as \(d_{3}\rightarrow +\infty ,\) which is independent of \(d_{4}\).
\((\textrm{ii})\) If \(R_{1}(x)\equiv \) is constant, then we have \(L_{1}R_{1}=R_{1}\) and \(R_{0}\rightarrow \frac{\int \limits _{\Omega }(\mu _{4}(x)+\delta (x) +\gamma (x)){\overline{R}}_{0}dx}{\int \limits _{\Omega }(\mu _{4}(x)+\delta (x)+\gamma (x))dx}\) as \(d_{4}\rightarrow +\infty ,\) which is independent of \(d_{3}\).
Define
Theorem 3.4
The following conclusions always hold.
\((\textrm{i})\) When \(d_{4}\rightarrow \infty \), then \(r[L_{1,\infty }(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})]\rightarrow {\widetilde{R}}_{1}{\widetilde{R}}_{2}+{\widetilde{R}}_{1}{\widetilde{R}}_{3}.\)
\((\textrm{ii})\) When \(d_{3}\rightarrow \infty \), then \(r[L_{2,\infty }(R_{2}L_{1}R_{1}+R_{3}L_{1}R_{1})]\rightarrow {\widetilde{R}}_{1}{\widetilde{R}}_{2}+{\widetilde{R}}_{1}{\widetilde{R}}_{3}.\)
Proof
By Theorem 3.3, we have
The proof of (ii) is similar, and we omit it. This completes the proof. \(\square \)
Remark 3.3
The following conclusions are valid.
\((\textrm{i})\) By Theorems 3.3 and 3.4, we have \(\mathop {\lim }_{d_{3}\rightarrow \infty }\mathop {\lim }_{d_{4}\rightarrow \infty } R_{0}=\mathop {\lim }_{d_{4}\rightarrow \infty }\mathop {\lim }_{d_{3}\rightarrow \infty } R_{0}={\widetilde{R}}_{1}{\widetilde{R}}_{2}+{\widetilde{R}}_{1}{\widetilde{R}}_{3}.\)
\((\textrm{ii})\) Since \(L_{1}R_{1}L_{2}R_{2}+L_{1}R_{1}L_{2}R_{3}\xrightarrow {SOT} L_{1,\infty }R_{1}L_{2,\infty }R_{2}+L_{1,\infty }R_{1}L_{2,\infty }R_{3} \) as \((d_{3},d_{4})\rightarrow (\infty ,\infty )\), then from Lemma 3.6, we have \(\mathop {\lim }_{(d_{3},d_{4})\rightarrow (\infty ,\infty )} R_{0}={\widetilde{R}}_{1}{\widetilde{R}}_{2}+{\widetilde{R}}_{1}{\widetilde{R}}_{3}.\)
3.2 The small diffusion rates
Now, we investigate the asymptotic properties of basic reproduction number \(R_{0}\) when the diffusion rates decrease to zero. We first have the result as follows.
Theorem 3.5
The following statements hold.
\((\textrm{i})\) For fixed \(d_{4}>0\), when \(d_{3}\rightarrow 0\), then \(R_{0}\rightarrow r({\overline{R}}_{0}L_{2}).\)
\((\textrm{ii})\) For fixed \(d_{3}>0\), when \(d_{4}\rightarrow 0\), then \(R_{0}\rightarrow r({\overline{R}}_{0}L_{1}).\)
Proof
For any \(\phi \in C({\overline{\Omega }}), L_{1}\phi \rightarrow \phi \) in \(C({\overline{\Omega }})\) as \(d_{3}\rightarrow 0\). Hence, we have \((R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})L_{1}\xrightarrow {SOT} R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3}\) as \(d_{3}\rightarrow 0\). Let \({\mathcal {B}}\) be the closed unit ball in \(C({\overline{\Omega }})\). Since \(L_{1}({\mathcal {B}}) \subseteq {\mathcal {B}}\), we have \(\cup _{d_{3}< 1}(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})L_{1}({\mathcal {B}}) \subseteq (R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})({\mathcal {B}})\). By the compactness of \(L_{2}\), \(\cup _{d_{3}< 1}(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})L_{1}({\mathcal {B}})\) is precompact in \(C({\overline{\Omega }})\). By Lemma 3.5, we have \(r[(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})L_{1}]\ge {\overline{R}}_{0*}>0\). Owing to \((R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3})L_{1}\) and \(R_{1}L_{2}R_{2}+R_{1}L_{2}R_{3}\) that are strongly positive compact operators in \(C({\overline{\Omega }})\), by Lemma 3.6, we have
The proof of (ii) is similar to (i), and we omit it. This completes the proof. \(\square \)
Theorem 3.6
The following statements hold.
\((\textrm{i})\) When \(d_{4}\rightarrow 0\), then \(r({\overline{R}}_{0}L_{2})\rightarrow {\overline{R}}_{0}^{*}.\)
\((\textrm{ii})\) When \(d_{3}\rightarrow 0\), then \(r({\overline{R}}_{0}L_{1})\rightarrow {\overline{R}}_{0}^{*}.\)
Proof
Let \(r_{d_{4}}:=r({\overline{R}}_{0}L_{2})=r(L_{2}{\overline{R}}_{0})\) and \(k_{d_{4}}=\frac{1}{r_{d_{4}}}\), where \(k_{d_{4}}\) is the principal eigenvalue of the following system
By (3.21), we have
It concludes that \(\liminf _{d_{4}\rightarrow 0}k_{d_{4}}\ge \frac{1}{{\overline{R}}_{0}^{*}}\).
We just need to prove \(\limsup _{d_{4}\rightarrow 0}k_{d_{4}}\le \frac{1}{{\overline{R}}_{0}^{*}}\). Assume to the contrary, i.e., \(\limsup _{d_{4}\rightarrow 0}k_{d_{4}}> \frac{1}{{\overline{R}}_{0}^{*}}\). Then, there exists \(\varepsilon _{0}>0\) and a sequence \(\{d_{4,n}\}\) with \(d_{4,n}\rightarrow 0\) such that \(k_{d_{4,n}}>\frac{1}{{\overline{R}}_{0}^{*}-\varepsilon _{0}}\). Suppose that \({\overline{R}}_{0}(x)>{\overline{R}}_{0}^{*}-\frac{\varepsilon _{0}}{2}\) in \(B(x_{0},\delta )\) for \(x_{0}\in \Omega \) and \(\delta >0\). Assume that system (3.21) has a principal eigenvalue \(k_{d_{4,n}}\) and a corresponding eigenfunction \(v_{d_{4,n}}\), which is positive. Then in \(B(x_{0},\delta )\), we obtain
It leads to in \(B(x_{0},\delta )\)
This implies
Let \(k'\) be the principal eigenvalue of \(-\Delta \) in domain \(B(x_{0},\delta )\) with a Dirichlet boundary condition. According to a minimax formulation of \(k'\) [35], we get
Noting that the right-hand side of (3.22) tends to \(\infty \) as \(d_{4,n}\rightarrow 0\). This is a contradiction. It shows that \(\limsup _{d_{4}\rightarrow 0}k_{d_{4}}\le \frac{1}{{\overline{R}}_{0}^{*}}\) holds. Hence, \(k_{d_{4}}\rightarrow \frac{1}{{\overline{R}}_{0}^{*}}\) and \(r_{d_{4}}={\overline{R}}_{0}^{*}\) as \(d_{4}\rightarrow 0\). This completes the proof. \(\square \)
Combining Theorems 3.5-3.6, we actually have the following repeated limit formula
Lemma 3.10
Let \(R_{0}\) be defined in (3.7), then we have
Proof
We suppose \({\overline{R}}_{0}^{*}=1\), next, we need to show that \(R_{0}\rightarrow 1\) as \((d_{3},d_{4})\rightarrow (0,0)\). Let \(k=\frac{1}{R_{0}}\) and view it as a function of \((d_{3},d_{4})\). Since \(R_{0}\) is the principal eigenvalue of \(L_{1}R_{1}L_{2}R_{2}+L_{1}R_{1}L_{2}R_{3}\), there is a positive \(\Phi _{0}=(\varphi _{0},\psi _{0})^{T}\) satisfying homogeneous Neumann boundary conditions such that k
where
and
For any positive \(a,d_{3}\) and \(d_{4}\), let \(b=b(a,d_{3},d_{4})\) be the principal eigenvalue of the following eigenvalue problem:
Then, we have \(b(k,d_{3},d_{4})=0\).
According to Theorem 1.4 in [36], we have \(\lim _{(d_{3},d_{4})\rightarrow (0,0)}b=\max _{x\in {\overline{\Omega }}}{\hat{b}}(D_{a}(x)),\) where \({\hat{b}}(D_{a}(x))\) is an eigenvalue of the matrix \(D_{a}(x)\) with a greater real part for each \(x\in {\overline{\Omega }}\) and
Therefore, for each \(a,b=b(a,d_{3},d_{4})\) can be expanded to be a continuous function of \((d_{3},d_{4})\) on \((0,\infty )\times (0,\infty )\bigcup \{(0,0)\}\) by \(b(a,0,0):=\max _{x\in {\overline{\Omega }}}{\hat{b}}(D_{a}(x))\).
We now claim that b is increasing in a for each \((d_{3},d_{4})\in (0,\infty )\times (0,\infty )\). Let \(\Phi =(\varphi ,\psi )\) be a positive eigenvector with \(\Vert \varphi \Vert _{2}^{2}+\Vert \psi \Vert _{2}^{2}=1\) of (3.24). Then, differentiate both sides of (3.24) with respect to a, and we get
Multiplying (3.25) by \(\Phi ^{T}\) and (3.24) by \(\Phi _{a}^{T}\), and integrating their difference over \(\Omega \), we get \(\Phi ^{T}C\Phi =b_{a}\Phi ^{T}\Phi \). Therefore, \(b_{a}=\int \limits _{\Omega }(\mu _{3}(x)+\sigma (x))R_{1}\varphi \psi dx>0\); that is, b is strictly increasing in a.
Since \({\overline{R}}_{0}^{*}=1\), it is simple to verify that \(b(a,0,0)=\max _{x\in \Omega }{\hat{b}}(D_{a}(x))\) if and only if \(a=1\). Furthermore, b(a, 0, 0) is strictly increasing in a. On the contrary, we assume that \(k(d_{3},d_{4})\nrightarrow 1\) as \((d_{3},d_{4})\rightarrow (0,0)\). Then, there is a sequence \(\{(d_{3,n},d_{4,n})\}_{n=1}^{\infty }\) and \(a_{0}\ne 1\) such that \(k_{n}:= k(d_{3,n},d_{4,n})\rightarrow a_{0}\) as \(n\rightarrow \infty \). Without loss of generality, we can assume \(a_{0}>1\). Choose \(\epsilon _{0}>0\) such that \(a_{0}-\epsilon _{0}>1\), which implies \(b(a_{0}-\epsilon _{0},0,0)>b(1,0,0)=0 \). Then, there exists \(N>0\) such that \(k_{n}>a_{0}-\epsilon _{0}\) for all \(n\ge N\). By the monotonicity and continuity of b, let \(n\rightarrow \infty \), we have
which is a contradiction. Therefore, \(k(d_{3},d_{4})\rightarrow 1\) as \((d_{3},d_{4})\rightarrow (0,0)\).
Then, we drop the assumption \({\overline{R}}_{0}^{*}=1\). We have that when \((d_{3},d_{4})\rightarrow (0,0)\)
This means \(R_{0}\rightarrow {\overline{R}}_{0}^{*}\) as \((d_{3},d_{4})\rightarrow (0,0)\). This completes the proof. \(\square \)
4 Threshold dynamics
In this section, we discuss the global stability of disease-free equilibrium \(P_{0}(x)\) and the uniform persistence of model (2.1). Firstly, the following conclusions are established.
Theorem 4.1
Assume \(R_{0}<1\), then disease-free equilibrium \(P_{0}(x)\) is globally asymptotically stable.
Proof
Let \(u(t,x)=(S(t,x),V(t,x),E(t,x),I(t,x))\) be any nonnegative solution of model (2.1) defined on \([0,\infty )\times {\overline{\Omega }}\). It follows from the first equation of (2.1) that
According to the comparison principle and Lemma 2.1, we directly acquire \(\limsup _{t\rightarrow \infty }S(t,x)\le S_{0}(x)\) uniformly for \(x\in {\overline{\Omega }}\). Without loss of generality, we can assume \(S(t,x)\le S_{0}(x)\) for all \(t\ge 0\) and \(x\in {\overline{\Omega }}\). Furthermore, by the second equation of model (2.1) we obtain
Similarly, we directly acquire \(\limsup _{t\rightarrow \infty }V(t,x)\le V_0(x)\) uniformly for \(x\in {\overline{\Omega }}\). Without loss of generality, we also can assume \(V(t,x)\le V_{0}(x)\) for all \(t\ge 0\) and \(x\in {\overline{\Omega }}\). Therefore, from assumptions \((A_1)-(A_2)\) and the last two equations of (2.1) we have
The corresponding comparison system is
By Lemma 3.2, if \(R_{0}<1\), then \(\lambda _{0}<0\). Hence, system (4.2) has the solution \((e(t,x),i(t,x))=e^{\lambda _{0}t}(\psi _{3}(x), \psi _{4}(x))\) tend to (0, 0) uniformly for \(x\in {\overline{\Omega }}\) as \(t\rightarrow \infty \). We choose the constant \(\zeta >0\) such that \((E(0,x),I(0,x))\le \zeta (\psi _{3}(x),\psi _{4}(x))\) for all \(x\in {\overline{\Omega }}\). Owing to \(\zeta e^{\lambda _{0}t}(\psi _{3}(x),\psi _{4}(x))\) also is the solution of system (4.2), by the comparison principle and (4.1) we have \((E(t,x),I(t,x))\le \zeta e^{\lambda _{0}t}(\psi _{3}(x),\psi _{4}(x))\) for all \(x\in \Omega \) and \(t\ge 0\). Thus, \((E(t,x),I(t,x))\rightarrow (0,0)\) uniformly for \(x\in {\overline{\Omega }}\) as \(t\rightarrow \infty \).
Because (E(t, x), I(t, x)) tend to (0, 0) uniformly for \(x\in {\overline{\Omega }}\) as \(t\rightarrow \infty \), from the first equation of model (2.1), we get the following limit equation
From the theory of asymptotically autonomous semiflows (see [32]) and Lemma 2.1, we further acquire that \(S(t,x)\rightarrow S_{0}(x)\) uniformly for \(x\in {\overline{\Omega }}\) as \(t\rightarrow \infty \). Similarly, we also obtain that \(V(t,x)\rightarrow V_{0}(x)\) uniformly for \(x\in {\overline{\Omega }}\) as \(t\rightarrow \infty \). Thus, by Lemma 3.2, we know that disease-free equilibrium \(P_{0}(x)\) is globally asymptotically stable. This completes the proof. \(\square \)
Theorem 4.2
Assume \(R_{0}>1\), then there exists a constant \(\rho >0\) such that for any initial function \(\psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})\in {\mathbb {X}}^{+}\) with \(\psi _{3}(x)\ne 0\) and \(\psi _{4}\ne 0\), and the solution \(u(t,x)=(S(t,x),V(t,x),E(t,x),I(t,x))\) of model (2.1) with initial value \(\psi \) satisfies
uniformly for \(x\in {\overline{\Omega }}.\)
Proof
Set
Clearly, we get
and \({\mathbb {X}}_{0}\) is the positive invariant set for solution semiflow \(\Phi (t)\) of model (2.1). Furthermore, we define the set
and \(\omega (\psi )\) is the omega limit set of the orbit \({\mathcal {N}}^{+}(\psi )=\{\Phi (t)\psi : t\ge 0\}\). We first prove the following two claim.
Claim 1
\(\bigcup _{\psi \in {\mathcal {M}}_{\partial }}{\omega (\psi )}=\{P_{0}(x)\}.\)
Since \(\Phi (t)P_{0}(x)=P_{0}(x)\) for all \(t\ge 0\), we have \(\{P_{0}(x)\}\subset \bigcup _{\psi \in {\mathcal {M}}_{\partial }}{\omega (\psi )}.\) Now, we will prove \(\bigcup _{\psi \in {\mathcal {M}}_{\partial }}{\omega (\psi )}\subset \{P_{0}(x)\}\). For any given \(\psi \in \mathcal {M}_{\partial },\) because \(\Phi (t)\psi \in \partial {\mathbb {X}}_{0}\) for all \(t\ge 0\), then for \(t\ge 0\) we have \(E(t,x)\equiv 0\) or \(I(t,x)\equiv 0\).
Assume \(E(t,x)\equiv 0\) for \(t\ge 0\), by the third equation of model (2.1), we have \(I(t,x)\equiv 0\). Thus, by equations of S(t, x) and V(t, x) in model (2.1), we further acquire
From Lemma 2.1, we get \(\lim _{t\rightarrow \infty }S(t,x)= S_{0}(x)\) and \(\lim _{t\rightarrow \infty }V(t,x)= V_{0}(x)\) for \(x\in \Omega \). This implies that \(\omega (\psi )= P_{0}(x).\)
If \(I(t,x)\equiv 0\) for \(t\ge 0\), then by the fourth equation of model (2.1), we have \(E(t,x)\equiv 0\). Similarly, we also get \(\lim _{t\rightarrow \infty }S(t,x)= S_{0}(x)\) and \(\lim _{t\rightarrow \infty }V(t,x)= V_{0}(x)\) for \(x\in \Omega \). This also shows \(\omega (\psi )= P_{0}(x)\).
Based on the above discussion, it follows that \(\bigcup _{\psi \in {\mathcal {M}}_{\partial }}\omega (\psi )\subset \{P_{0}(x)\}\). Therefore, \(\bigcup _{\psi \in {\mathcal {M}}_{\partial }}\omega (\psi )= \{P_{0}(x)\}.\)
Claim 2
\(P_{0}(x)\) is uniform weak repeller for \({\mathbb {X}}_{0}\); that is, there is a constant \(\eta >0\) such that for any \(\psi \in {\mathbb {X}}_0\) the solution \(u(t,\cdot )\) with initial value \(u(0,\cdot )=\psi \) satisfies
Supposing that \(\textbf{Claim}\) 2 is not true, namely for any constant \(\eta >0\) there exists a \(\psi \in {\mathbb {X}}_0\) such that \(\limsup _{t\rightarrow \infty }\Vert u(t,\cdot )-P_{0}(x)\Vert <\eta \). This implies that there exists an enough large \(t_{4}>0\) such that
Therefore, from model (2.1) and assumption \((A1)-(A2)\), we obtain
We have the comparison system
For any initial value \(\psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})\in {\mathbb {X}}^{+}\) with \(\psi _{3}(x)\ne 0\) and \(\psi _{4}\ne 0\), from the parabolic maximum principle (see [29]), we can obtain \(E(t,x)> 0\) and \(I(t,x)>0\) for all \(t>0\) and \(x\in {\overline{\Omega }}\). By Lemma 3.2, we have \(\lambda _{0}>0\) when \(R_{0}>1\).
We consider the following eigenvalue problem associated with system (4.5)
Let \(\lambda _{0}(\eta )\) be the principle eigenvalue of problem (4.6). Since \(\lim _{\eta \rightarrow 0}\lambda _0(\eta )=\lambda _0\), there exists an enough small constant \(\eta \in (0,1)\) such that \(\lambda _{0}(\eta )>0\), \(g'(0)-\eta >0\), \(S_{0}(x)-\eta >0\) and \(V_{0}(x)-\eta >0\) for all \(x\in {\overline{\Omega }}\). By Lemma 3.1, \((\xi _{1}(x),\xi _{2}(x))\) is also strictly positive eigenvector corresponding to \(\lambda _{0}(\eta )\) for \(x\in {\overline{\Omega }}.\)
It is obvious that system (4.5) has the solution \((y_{1}(t,x),y_{2}(x))=e^{\lambda _{0}(\eta )(t-t_{4})}(\xi _{1}(x),\xi _{2}(x))\). Due to \((E(t,x),I(t,x))>0\) for \(t>0\) and \(x\in {\overline{\Omega }}\), there is a constant \(\varrho >0\) such that \((E(t_4,x),I(t_4,x))\ge \varrho (\xi _{1}(x),\xi _{2}(x))\) for \(x\in {\overline{\Omega }}\). Note that \((\varrho y_{1}(t,x),\varrho y_{2}(t,x))\) is also the solution of system (4.5). According to the comparison principle and equation (4.4), we can obtain \((E(t,x),I(t,x))\ge (\varrho y_{1}(t,x),\varrho y_{2}(t,x))\) for all \(t>t_{4}\) and \(x\in {\overline{\Omega }}.\) Owing to \(\lambda _{0}(\eta )>0\), we get \(\lim _{t\rightarrow \infty } y_{i}(t,x)=\infty \;(i=1,2)\). This implies \(\lim _{t\rightarrow \infty } E(t,x)=\infty \) and \(\lim _{t\rightarrow \infty } I(t,x)=\infty \), which is a contradiction with the boundedness of (E(t, x), I(t, x)) by Theorem 2.1. Therefore, \(\textbf{Claim}\) 2 holds.
Theorem 2.1 implies that there exist constants \(M>0\) and \(t_{0}>0\) such that \(S(t,x)\le M\), \(V(t,x)\le M\), \(E(t,x)\le M\) and \(I(t,x)\le M\) for all \(t\ge t_{0}\) and \(x\in {\overline{\Omega }}.\) Hence, it follows from the first equation of model (2.1) and assumptions \((A_1)\) and \((A_2)\) that
where \(M_{6}=\sup \{\frac{f_{1}(S)}{S}g(I): 0\le S,I\le M\}\). According to Lemma 2.1 and comparison principle, we can see that S(t, x) has a positive lower bound \(\varrho _{1}:=\frac{{\overline{\Lambda }}}{{\underline{\mu }}_{1}+{\underline{\xi }}+{\underline{\beta }}_{1}M_{6}}\), indicating that S(t, x) in model (2.1) is uniformly persistent.
By the second equation of model (2.1) and assumption \((A_2)\), we can obtain
where \(M_{7}=\sup \{\frac{f_{2}(V)}{V}g(I): 0\le V, I\le M\}\). By Lemma 2.1 and comparison principle, we can get that V(t, x) has a positive lower bound \(\varrho _{2}:=\frac{{\overline{\xi }}{\overline{\Lambda }}}{({\underline{\mu }}_{2} +{\underline{\alpha }}+{\underline{\beta }}_{2}M_{7})({\underline{\mu }}_{1} +{\underline{\xi }}+{\underline{\beta }}_{1}M_{6})}\), which implies that V(t, x) in model (2.1) is uniformly persistent.
Define a continuous function \(l: {\mathbb {X}}^{+}\rightarrow {\mathbb {R}}^{+}\) as follows:
Obviously, \(l^{-1}(0,+\infty )\subseteq {\mathbb {X}}_{0}\). By the parabolic maximum principle [29], we can obtain that \(E(t,x)>0\) and \(I(t,x)>0\) for all \(t\ge 0\) and \(x\in {\overline{\Omega }}\) when \(\psi _{3}(x)\ne 0\) and \(\psi _{4}(x)\ne 0\). Thus, if \(l(\psi )>0\) or \(l(\psi )=0\) and \(\psi \in {\mathbb {X}}_{0}\), then \(l(\Phi (t)\psi )>0\). By the definition of the generalized distance function (See Theorem 3 in [37]), we know that l is a generalized distance function for semiflow \(\Phi (t):{\mathbb {X}}^{+} \rightarrow {\mathbb {X}}^{+}\).
From the discussion in \(\textbf{Claim}\) 1 and \(\textbf{Claim}\) 2, we can see that \(P_0(x)\) is isolated invariant set in \({\mathbb {X}}^{+}\) and \(W^{s}(P_{0}(x))\bigcap {\mathbb {X}}_{0}=\varnothing \), where \(W^{s}(P_{0}(x))\) is the stable set of \(P_{0}(x)\). Thus, \(W^{s}(P_{0}(x))\bigcap p^{-1}(0,\infty )=\varnothing \). It follows from Theorem 3 in [37] that there is a constant \(\varrho _{3}>0\) such that \(\liminf _{t\rightarrow \infty }p(\Phi (t)\psi )\ge \varrho _{3}\) for all \(\psi \in {\mathbb {X}}_{0}\). This completes the proof. \(\square \)
As a result of Theorem 4.2, we can obtain
Corollary 4.1
When \(R_{0}>1\), model (2.1) has at least one endemic equilibrium \(P_{*}(x)=(S_{*}(x),V_{*}(x),E_{*}(x), I_{*}(x))\).
Remark 4.1
We here have proved the existence of endemic equilibrium, but its uniqueness and stability are still an open question. In the following section, we will prove the global asymptotic stability of endemic equilibrium in the spatial homogeneous environment.
5 Spatial homogeneous case
In this section, we consider a special case of model (2.1) to establish the complete results for the global stability of model (2.1). This special case exactly is the spatial homogeneous environment. That is, all parameters of model (2.1) are positive constants. As a result, model (2.1) becomes into
Obviously, model (5.1) has always disease-free equilibrium \(P_{0}=\big (S_{0},V_{0},0,0\big )\) with \(S_{0}=\frac{\Lambda }{\mu _{1}+\xi }\) and \(V_{0}=\frac{\Lambda \xi }{(\mu _{1}+\xi )(\mu _{2}+\alpha )}\). From (3.7), we can get that model (5.1) has the basic reproduction number
From Corollary 4.1, we can obtain that when \(R_{0}>1\), model (5.1) has an endemic equilibrium \(P_{*}=(S_{*},V_{*},E_{*},I_{*})\) satisfying the following equations
Theorem 5.1
If \(R_{0}\le 1\), then disease-free equilibrium \(P_{0}=(S_{0},V_{0},0,0)\) of model (5.1) is globally asymptotically stable.
Proof
If \(R_0<1\), we can directly obtain that disease-free equilibrium \(P_{0}=(S_{0},V_{0},0,0)\) of model (5.1) is globally asymptotically stable from Theorem 4.1.
Now, we consider the case of \(R_0=1\). Choose the Lyapunov function
Then along any positive solution (S(t, x), V(t, x), E(t, x), I(t, x)) of model (5.1), combining the divergence theorem and the Neumann boundary conditions, we can get
If \(R_0=1\), then we have \(\sigma (\beta _1f_1(S_0)g'(0)+\beta _2f_2(V_0)g'(0))=(\mu _3+\sigma )(\mu _4+\delta +\gamma )\). Hence, from (5.3) we can obtain
Obviously, \(\frac{dL(t)}{dt}\le 0\) from assumption \((A_2)\), and \(\frac{dL(t)}{dt}\equiv 0\) implies that \(S(t,x)\equiv S_0\) and \(V(t,x)\equiv V_0\). Directly from model (5.1), it follows that \(E(t,x)\equiv 0\) and \(I(t,x)\equiv 0\).
Thus, by LaSalle’s invariable principle, we finally get that the equilibrium \(P_0\) is globally asymptotically stable when \(R_0\le 1\). This completes the proof. \(\square \)
Theorem 5.2
Assume that \(R_{0}>1\), and the following inequalities hold
Then, endemic equilibrium \(P_{*}=(S_{*},V_{*},E_{*},I_{*})\) of model (5.1) is globally asymptotically stable.
Proof
Define a Lyapunov function as follows:
Calculating the time derivative of \(L_{1}\), we have
Due to the divergence theorem, the Neumann boundary conditions and (5.2), we can obtain
Thus, we know that \(\frac{dL_{1}}{dt}\le 0\). Furthermore, we know that \(\frac{dL_{1}}{dt}= 0\) if and only if \(S=S_{*}\), \(V=V_{*}\), \(E=E_{*}\) and \(I=I_{*}\). Thus, by LaSalle’s invariable principle, it clear that endemic equilibrium \(P_{*}\) is globally asymptotically stable. This completes the proof. \(\square \)
Remark 5.1
In Theorem 5.2, an additional condition (5.4) is added. We easily see that when \(f_1(S)=S\) and \(f_2(V)=V\), then condition (5.4) holds. However, if we take \(f_1(S)=S\), \(f_2(V)=V\) and \(g(I)=\frac{I}{1+\omega _3I^2}\) with constant \(\omega _3>0\), then condition (5.4) does not hold. In fact, when \(f_1(S)=S\) and \(f_2(V)=V\), then condition (5.4) is reduced to
Clearly, assumption \((A_2)\) implies that this inequality holds. When \(g(I)=\frac{I}{1+\omega _3I^2}\), then g(I) is not increasing for all \(I>0\), and by calculating we obtain
Clearly, when \(I>0\) is enough large, then \((1-\frac{g(I_{*})}{g(I)})(\frac{g(I)}{g(I_*)}-\frac{I}{I_{*}})>0\).
Remark 5.2
In Theorem 5.1, we see that only condition \(R_0\le 1\) is required. In addition, from Remark 5.1 we obtain that when \(f_1(S)=S\) and \(f_2(V)=V\) condition (5.4) holds. Therefore, in Theorem 5.2 whether we also only need condition \(R_0>1\) to exactly prove the global asymptotic stability of endemic equilibrium \(P_*\). This will be an interesting open problem.
6 Conclusions
In this paper, we propose a diffusive SVEIR epidemic model with nonlinear incidences. We mainly concern to the dynamical behaviors and some properties of the basic reproduction number. On nonlinear incidence rate function \(f_{1}(S), f_{2}(V)\) and g(I), we have introduced assumptions \((A_{1})-(A_{2}).\) Firstly, the well-posedness of solutions is established. Secondly, we calculate the expression of the basic reproduction number \(R_{0}\) and the local basic reproduction number \({\overline{R}}_{0}(x)\), which are characterized as the spectral radius of the next-generation operator. Following, we investigate the relation of \(R_{0}\) and \({\overline{R}}_{0}(x)\) by defined compact linear operators \(L_{1}\) and \(L_{2}\) with spectral radius one and additive operators \(R_{i}(x)\;(i=1,2,3)\). Furthermore, when the diffusion coefficients \(d_{i}(x)\equiv d_{i}\;(i=3,4)\) are constant, we establish quantitative connection of \(R_{0}\) and \({\overline{R}}_{0}(x)\), that is,
and
Thirdly, we show the disease-free equilibrium \(P_{0}(x)\) is globally asymptotically stable if \(R_{0}<1\), while the disease is persistent if \(R_{0}>1\), and model (2.1) has at least one endemic equilibrium \(P_{*}(x)=(S_{*}(x),V_{*}(x),E_{*}(x),I_{*}(x))\). Furthermore, applying Lyapunov functions, we established the global stability of the equilibria in the spatially homogeneous model. That is, when \(R_{0}\le 1\), then disease-free equilibrium \(P_{0}\) of model (5.1) is globally asymptotically stable; when \(R_{0}>1\) and the conditions of (5.4) are satisfied, then endemic equilibrium \(P_{*}\) of model (5.1) is globally asymptotically stable. It is covered and improved some existing global dynamical results (see [38]).
There are still some open problems for model (2.1) that we need to look into in the future. For example, one open problem is when the basic reproduction number \(R_{0}>1\) is the uniqueness and stability of the endemic equilibrium in the spatially heterogeneous environments for model (2.1). The other open problem is that when \(R_{0}=1\), the threshold dynamics of model (2.1) are not establish in this paper. In addition, in the spatial homogeneous environment whether we also only need condition \(R_{0}>1\) to exactly prove the global asymptotic stability of endemic equilibrium \(P_{*}\) as in Theorem 5.2. As well as, the asymptotic profiles of endemic equilibrium for model (2.1) as the diffusive coefficients tend to infinity or zero also are not investigated.
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Acknowledgements
This work was supported the National Natural Science Foundation of Xinjiang Province of China (Grant Nos. 2021D01C281) and by the National Natural Science Foundation of China (Grant Nos. 11961071, 12061079, 12101529 and 72064036).
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Zhou, P., Wang, J., Teng, Z. et al. Dynamical analysis for a diffusive SVEIR epidemic model with nonlinear incidences. Z. Angew. Math. Phys. 74, 173 (2023). https://doi.org/10.1007/s00033-023-02057-y
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DOI: https://doi.org/10.1007/s00033-023-02057-y