Abstract
In this paper, we investigate an SVEIR epidemic model with reaction–diffusion and nonlinear incidence. We establish the well-posedness of the solutions and the basic reproduction number \(\mathfrak{R}_{0}\). Moreover, we show that the disease-free steady state is globally asymptotically stable when \(\mathfrak{R}_{0}<1\), whereas the disease will be persistent when \(\mathfrak{R}_{0}>1\). Furthermore, using the method of Lyapunov functional, we prove the global stability of the positive steady state for the spatial homogeneous model.
Similar content being viewed by others
1 Introduction
In the recent years, much effort has been paid on epidemic models by many researchers due to their important role in describing the dynamical evolution of infectious diseases. For better understanding of epidemiological scheme and intervening spreading of infectious diseases, see [1–8] and references therein.
Vaccination is one of the effective control measures to prevent and weaken the transmission of infectious diseases. Currently, various modeling studies have been made to explain the effect of vaccination on the spread of diseases [3, 9–11]. In particular, Liu et al. [3] proposed and studied the following model:
where S, I, R, and V denote the numbers of compartments of susceptible, infected, recovered, and vaccinated individuals, respectively. The global stability of the equilibrium of model (1.1) has been studied by constructing Lyapunov functions. For more biological background of model (1.1), we refer to [3].
Note that the latent period was not considered in model (1.1). However, many diseases have a latent period before the hosts become infectious, and the length of latent period differs from disease to disease [12]. Besides, it well known that the spatial structure has also been considered as an important factor that affects the spatial spreading of disease due to the carrier hosts of infectious sources randomly moving in space. There are many mathematical studies of the influence of the spatial aspect and mobility of host populations on the dynamics of diseases; see [13–18] and the references therein. Taking into consideration the latent period and individuals movements, we consider the diffusive version of model (1.1) with latency, which is a more realistic biological model. Thus we consider the diffusive SVEIR model
with e homogeneous Neumann boundary conditions
and the initial conditions
where Ω is a bounded domain in \(\mathbb{R}^{n}\) with smooth boundary ∂Ω, and ν is the outward normal vector to ∂Ω; \(S=S(t,x)\), \(V=V(t,x)\), \(E=E(t,x)\), \(I=I(t,x)\), and \(R=R(t,x)\) stand for the densities of the susceptible, vaccinated, latent, infective, and recovered individuals at time t and spatial location x, respectively; \(\Lambda (x)\) is the input rate of S in spatial location x; \(\mu _{k}(x) \) (\(k=1,2,3,4,5\)) denote the natural death rates of S, V, E, I, and R in spatial location x, respectively; \(\delta (x)\) is the death rate induced by the disease in spatial location x; \(\xi (x)\) is the vaccination rate of S in spatial location x; \(\gamma (x)\) is the rate of recovery from infection in spatial location x; \(\sigma (x)\) represents the transition rate from E to I; \(\alpha (x)\) is the rate of obtaining immunity by vaccinees; \(\beta _{1}(x)\) and \(\beta _{2}(x)\) are the infection rates of S and V infected by I in spatial location x, respectively; \(d_{k}(x)\) (\(k=1,2,3,4,5\)) are the diffusion rates of S, V, E, I, and R in spatial location x, respectively. All location-dependent parameters are continuous and strictly positive d on Ω̅; \(f_{j}(u,I)\) (\(j=1,2\); \(u\in \{S,V\}\)) denotes the force of infection. We assume that the function \(f_{j}(u, I)\) satisfies the following properties:
Obviously, \(f_{j}(u, I)=uI\), \(f_{j}(u, I)=\frac{uI}{1+I}\), and \(f_{j}(u, I)=\frac{uI}{(1+u)(1+I)}\) satisfy these assumptions. For convenience, let \(f_{jI}(u,I)=\frac{\partial{f_{j}(u,I)}}{\partial{I}}\).
Since the last equation of model (1.1) is decoupled from other equations, we indeed need to study the following subsystem of model (1.2):
The rest of this paper is organized as follows. In Sect. 2, we introduce some preliminaries for the well-posedness of the model. In Sect. 3, we define the basic reproduction number \(\mathfrak{R}_{0}\) and establish the threshold dynamics in terms of \(\mathfrak{R}_{0}\). A particular case is performed as a supplementary to the theoretical results in Sect. 4. A brief conclusion ends the paper.
2 Well-posedness
For convenience, denote \(\overline{k}=\max_{x\in \overline{\Omega}}{k(x)}\) and \(\underline{k}=\min_{x\in \overline{\Omega}}{k(x)}\). Let \(\mathbb{X}=C(\overline{\Omega},\mathbb{R}^{4})\) with norm \(\|\cdot \|_{\mathbb{X}}\), and let \(\mathbb{X}^{+}=C(\overline{\Omega},\mathbb{R}_{+}^{4})\). Denote by
the \(C_{0}\) semigroups associated with \(\triangledown \cdot (d_{i}(\cdot )\triangledown )-\rho _{i}(\cdot )\) subject to (1.3), 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 \(\rho _{4}(x)=\mu _{4}(x)+\delta (x)+\gamma (x)\). Thus we have
where \(\Gamma _{i}(t,x,y)\) are the Green functions associated with \(\triangledown \cdot (d_{i}(\cdot )\triangledown )-\rho _{i}(\cdot )\) (\(i=1,2,3,4\)) subject to the Neumann boundary condition. It follows from [19] that \(T_{i}(t)\) is strongly positive and compact for all \(t>0\). Thus there exists \(M>0\) such that \(\|T_{i}(t)\|\leq M e^{\omega _{i} t}\) for all \(t\geq 0\), where \(\omega _{i}<0\) denotes the principal eigenvalue of \(\triangledown \cdot (d_{i}(\cdot )\triangledown )-\rho _{i}(\cdot )\) (\(i=1,2,3,4\)) subject to Neumann boundary condition (1.3).
Define \(F=(F_{1}, F_{2}, F_{3}, F_{4}):\mathbb{X}\rightarrow \mathbb{X}\) by
for \(t\geq 0\), \(x\in \overline{\Omega}\), and \(\psi =(\psi _{1}, \psi _{2}, \psi _{3},\psi _{4})\in \mathbb{X}^{+}\). Then model (1.6) can be written as
where \(u(t,x)=(S(t,x), V(t,x), E(t,x), I(t,x))\) and \(T(t)=\mathit{diag}(T_{1}(t),T_{2}(t),T_{3}(t), T_{4}(t))\).
It is easy to check that
For any \(\psi \in \mathbb{X}^{+}\), it follows from expressions (2.2) that
Firstly, if \(\psi _{i}>0\), then \(\psi +\rho F(\psi )>0\) for all \(\rho >0\) sufficiently small, so that (2.4) holds. Secondly, if \(\psi _{i}=0\), then as \(F_{i}(\psi )|_{\psi _{i}=0}\geq 0\) by (2.2), we have \(\psi _{i}+\rho F(\psi )\geq 0\) for all \(\rho >0\). Thus \(\psi +\rho F(\psi )\in \mathbb{X}^{+}\) when ρ is sufficiently small. Then it follows from [20] that (2.4) holds.
By Corollary 4 in [20] we have have the following:
Lemma 2.1
For \(\psi =(\psi _{1}, \psi _{2}, \psi _{3},\psi _{4})\in \mathbb{X}^{+}\), model (1.6) has a unique nonnegative 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 its maximal existence interval \([0, \tau _{\psi})\), where \(\tau _{\psi}\leq \infty \). Moreover, this solution is a classical solution.
Next, we will show the existence of solutions of model (1.6).
Theorem 2.1
Model (1.6) has a unique solution \(u(t, x, \psi )\) on \([0, \infty )\) with \(\psi =(\psi _{1}, \psi _{2}, \psi _{3}, \psi _{4})\in \mathbb{X}^{+}\). Furthermore, the solution semiflow \(\Phi (t)=u(t, \cdot ): \mathbb{X}^{+}\rightarrow \mathbb{X}^{+}\) of model (1.6) is defined by
admits a global compact attractor.
Proof
Suppose to the contrary that \(\tau _{\psi}<\infty \). Then \(\|u(t, x, \psi )\|\rightarrow +\infty \) as \(t\rightarrow \tau _{\psi}\) by Theorem 2 in [20]. Recalling the first equation of model (1.6), we have
By the comparison principle and Lemma 2.2 in [21] there exists a constant \(\mathcal{Q}_{1}>0\) such that \(S(t, x)\leq \mathcal{Q}_{1}\) for \(t\in [0, \tau _{\psi})\), \(x\in \overline{\Omega}\). Furthermore, a similar procedure can be applied to the second equation of model (1.6). Then there exists a constant \(\mathcal{Q}_{2}>0\) such that \(V(t, x)\leq \mathcal{Q}_{2}\) for \(t\in [0, \tau _{\psi})\), \(x\in \overline{\Omega}\). Then from the last two equations of model (1.6) we have
where \(f_{1I}(\mathcal{Q}_{1},0)= \frac{\partial{f_{1}(\mathcal{Q}_{1},0)}}{\partial{I}}\) and \(f_{2I}(\mathcal{Q}_{2},0)= \frac{\partial{f_{2}(\mathcal{Q}_{2},0)}}{\partial{I}}\).
Consider the linear system
The standard Krein–Rutman theorem (see [22]) implies that the eigenvalue problem of model (2.6) admits a principal eigenvalue λ with strongly positive eigenfunction \(\varphi =(\varphi _{1},\varphi _{2})\). Thus model (2.6) has a solution \(\varsigma e^{\lambda t}\varphi (x)\) for \(t\geq 0\), where ς is a positive constant such that \(\varsigma \varphi =(u_{1}(0,x),u_{2}(0,x))\geq (E(0, x), I(0, x))\) for \(x\in \overline{\Omega}\). Then from the comparison principle it follows that
Therefore there exists a constant κ such that \(E(t,x)\leq \kappa \), \(I(t,x)\leq \kappa \), \(x\in \overline{\Omega}\), which leads to a contradiction. Hence the global existence of \(u(t, \cdot , \psi )\) is derived.
Furthermore, it follows from the comparison principle and Lemma 2.2 in [21] that there exist \(t_{1}>0\) and \(\mathcal{L}_{1}>0\), \(\mathcal{L}_{2}>0\) such that \(S(t, x)\leq \mathcal{L}_{1}\) and \(V(t, x)\leq \mathcal{L}_{2}\) for all \(t\geq t_{1}\) and \(x\in \overline{\Omega}\). Set
Then we have
Thus there exist \(t_{2}>0\) and \(\mathcal{L}_{3}>0\) such that \(P(t)\leq \mathcal{L}_{3}\) for all \(t\geq t_{2}\). It follows from [23] that
where \(\pi _{n}\) are the eigenvalues of \(\triangledown \cdot (d_{3}(x)\triangledown )-(\mu _{3}(x)+\sigma (x))\) subject to the Neumann boundary condition with eigenfunction \(\varphi _{n}(x)\) and satisfy \(\pi _{1}>\pi _{2}\geq \pi _{2}\geq \cdots \geq \pi _{n}\geq \cdots \) . Then for some \(\kappa _{1}>0\), we have \(\Gamma _{3}(t,x,y)\leq \kappa _{1}\sum_{n\geq 1}e^{\pi _{n}t}\) for \(t>0\). Moreover, we assume that \(\tau _{n}\) are the eigenvalues of \(\triangledown \cdot (\underline{d}_{3}\triangledown )- ( \underline{\mu}_{3}+\underline{\sigma})\) subject to the Neumann boundary condition, which satisfy \(\tau _{1}=-(\underline{\mu}_{3}+\underline{\sigma}) >\tau _{2}\geq \tau _{3}\geq \cdots \geq \tau _{n}\geq \cdots \) . Following Theorem 2.4.7 in Wang [24], we get \(\tau _{i}\geq \pi _{i}\) for all \(i\in \mathbb{N}_{+}\). Then for some \(\kappa _{2}>0\), we have
Let \(t_{3}=\max \{t_{1},t_{2}\}\). For all \(t\geq t_{3}\), we obtain
which yields that
Similarly, we can also obtain that there exists a positive constant \(\mathcal{L}_{4}>0\) such that \(\limsup_{t\rightarrow \infty}\|I(t,x)\| \leq \mathcal{L}_{4}\). Thus the above discussions imply that the system is point dissipative. Furthermore, by Theorem 2.2.6 in [25] the solution semiflow \(\Phi (t)\) is compact for all \(t>0\). Therefore it follows from Theorem 3.4.8 in [26] that \(\Phi (t)\) has a global compact attractor. □
3 Threshold dynamics
3.1 Basic reproduction number
Considering the following subsystem of model (1.6):
It follows from the Lemma 2.2 in [21] that the system
admits a unique positive steady state \(S^{0}(x)\) that satisfies the equation
with \(\frac{\partial{S^{0}(x)}}{\partial{\nu}}=0\) for \(x\in \partial{\Omega}\), which is globally asymptotically stable in \(C(\overline{\Omega}, \mathbb{R}_{+})\). Then the second equation of model (3.1) is asymptotic to
By Lemma 2.2 in [21] and Corollary 4.3 in [27] we easily obtain that model (1.6) admits a unique disease-free steady state \(P_{0}(x)= (S^{0}(x), V^{0}(x), 0, 0 )\). Furthermore, if all the parameters of model (1.6) are positive constants, then we have \(S^{0}(x)=\frac{\Lambda}{\mu _{1}+\xi}\) and \(V^{0}(x)=\frac{\Lambda \xi}{(\mu _{2}+\alpha )(\mu _{1}+\xi )}\). Linearizing model (1.6) at \(P_{0}(x)\), we obtain the linearized subsystem
Letting \((E(t,x),I(t,x))=e^{\lambda t}(\psi _{3}(x),\psi _{4}(x))\), it follows from model (3.4) that
It follows from the Krein–Rutman theorem that system (3.5) admits a unique principal eigenvalue \(\lambda _{0}(S^{0}(x),V^{0}(x))\) with strongly positive eigenfunction \(\psi (x)=(\psi _{3}(x), \psi _{4}(x))\).
Let \(\Psi (t): C(\bar{\Omega},\mathbb{R}^{2})\rightarrow C(\bar{\Omega}, \mathbb{R}^{2})\) be the semigroup associated with the following system:
Define
Assuming that the distribution of initial infection is \(\psi (x)=(\psi _{3}(x), \psi _{4}(x))\), the distribution of totally new infective numbers is given by
According to [28], the basic reproduction number is defined by \(\mathfrak{R}_{0}=r(\mathcal{L})\), where \(r(\mathcal{L})\) is the spectral radius of the operator \(\mathcal{L}\). Furthermore, by Theorem 3.1 in [28] we have
Lemma 3.2
The principal eigenvalue \(\lambda _{0}\) and \(\mathfrak{R}_{0}-1\) have the same sign, and the disease-free steady state \(P_{0}(x)\) is locally asymptotically stable if \(\mathfrak{R}_{0}<1\) and unstable if \(\mathfrak{R}_{0}>1\).
3.2 Persistence analysis
In this section, we investigate the extinction and persistence of the disease in terms of \(\mathfrak{R}_{0}\).
Theorem 3.2
If \(\mathfrak{R}_{0}<1\), then the disease-free steady state \(P_{0}(x)\) is globally asymptotically stable.
Proof
It follows from Lemma 3.1 that \(\lambda _{0}(S^{0}(x),V^{0}(x))<0\) for \(\mathfrak{R}_{0}<1\). By the continuity there is \(\varepsilon >0\) such that \(\lambda _{0}(S^{0}(x)+\varepsilon ,V^{0}(x)+\varepsilon )<0\). Furthermore, according to the first two equation of model (1.6), there exists \(t_{1}>0\) such that \(S(t,x)\leq S^{0}(x)+\varepsilon \) and \(V(t,x)\leq V^{0}(x)+\varepsilon \) for \(t\geq t_{1}\) and \(x\in \Omega \). Then we have
Assume that \(\zeta (\psi _{3}(x),\psi _{4}(x))\geq (E(t_{1},x), I(t_{1},x))\) with \(\zeta >0\) and the eigenfunction \(\psi =(\psi _{3}(x),\psi _{4}(x))\) corresponding to the principal eigenvalue \(\lambda _{0}\). Since \(\lambda _{0}(S^{0}(x)+\varepsilon ,V^{0}(x)+\varepsilon )<0\), according to the comparison principle, we obtain
which yields \(\lim_{t\rightarrow \infty}(E(t,x,) I(t,x))=0\) uniformly for \(x\in \overline{\Omega}\). Thus model (1.6) is asymptotic to (3.1). Consequently, we have \(\lim_{t\rightarrow \infty}S(t,x)=S^{0}(x)\) and \(\lim_{t\rightarrow \infty}V(t,x)=V^{0}(x)\). Therefore the disease-free steady state \(P_{0}(x)\) is globally asymptotically stable. □
Theorem 3.3
If \(\mathfrak{R}_{0}>1\), then there exists a constant \(\rho >0\) such that for any \(\psi \in \mathbb{X}^{+}\) with \(\psi _{3}\not \equiv 0\) and \(\psi _{4}\not \equiv 0\),
uniformly for all \(x\in \overline{\Omega}\).
Proof
Let
Then we have
Thus \(\mathbb{X}_{0}\) is positively invariant for \(\Phi (t)\). Set
and let \(\omega (\psi )\) be the omega limit set of the orbit \(\mathcal{O}^{+}(\psi )=\{\Phi (t)\psi : t\geq 0\}\). We first prove the following claim.
Claim 1. \(\bigcup_{\psi \in \mathcal{M}_{\partial}}\omega (\psi )=P_{0}(x)\).
For \(\psi \in \mathcal{M}_{\partial}\), \(u_{t}(\psi )\in \partial{X}_{0}\), since \(u_{t}(\psi )=u(t,\cdot ,\psi )\), for \(t\geq 0\), there are two possible cases, either \(E(t,\cdot ,\psi )\equiv 0\) or \(I(t,\cdot ,\psi )\equiv 0\). If \(E(t,\cdot ,\psi )\equiv 0\) for \(t\geq 0\), then it follows from the last equation of model (1.6) that \(\lim_{t\rightarrow +\infty}I(t,\cdot ,\psi )=0\) uniformly for \(x\in \Omega \). Consequently, we have \(\lim_{t\rightarrow +\infty}S(t,\cdot ,\psi )=S^{0}(x)\) and \(\lim_{t\rightarrow +\infty}V(t,\cdot ,\psi )=V^{0}(x)\) for \(x\in \Omega \). If \(E(t,\cdot ,\psi )\not \equiv 0\) for some \(t_{0}>0\), then \(E(t,\cdot ,\psi )>0\) for \(t>t_{0}\), \(x\in \Omega \), and thus \(I(t,\cdot ,\psi )\equiv 0\) for \(t\geq t_{0}\), which implies that \(E(t,\cdot ,\psi )=0\), a contradiction. Hence the proof of the claim is completed.
Claim 2. \(P_{0}(x)\) is a uniform weak repeller for \(\mathbb{X}_{0}\) in the sense that
Suppose this is not true. Then there exists \(\psi _{0}\in \mathbb{X}_{0}\) such that
Then there exists \(t_{1}>0\) such that
Therefore from model (1.6) we have
Let \((\psi _{3}(x), \psi _{4}(x))\) be the eigenfunction associated with principle eigenvalue \(\lambda _{0}(S^{0}(x)-\eta ,V^{0}(x)-\eta )>0\). Suppose that \((E(t_{1},x),I(t_{1},x))\geq \xi (\psi _{3}(x), \psi _{4}(x))\) for some \(\xi >0\). Then we obtain that
which implies that \(\limsup_{t\rightarrow \infty}E(t,x)=\infty \) and \(\limsup_{t\rightarrow \infty}I(t,x)=\infty \). This gives a contradiction.
Define the continuous function \(p:\mathbb{X}^{+}\rightarrow [0, +\infty )\) by
Clearly, \(p^{-1}(0,+\infty )\subseteq \mathbb{X}_{0}\), and p has the property that if either \(p(\psi )>0\) or \(p(\psi )=0\) and \(\psi \in \mathbb{X}_{0}\), then \(p(\Phi (t)\psi )>0\). Moreover, we can show that \(P_{0}(x)\) is isolated in \(\mathbb{X}^{+}\) and \(W^{s}(P_{0}(x))\cap \mathbb{X}_{0}=\emptyset \), where \(W^{s}(P_{0}(x))\) is the stable set of \(P_{0}(x)\). It then follows from Theorem 3 in [29] that there exists a constant \(\varrho >0\) such that \(\liminf_{t\rightarrow \infty}p(\Phi (t)\psi )\geq \varrho \) for all \(\psi \in \mathbb{X}_{0}\). The proof is complete. □
4 A particular case
In this section, we consider a particular case of model (1.6) to study global stabilities of steady states. Select \(f_{1}(S,I)=Sf_{1}(I)\) and \(f_{2}(V,I)=Vf_{2}(I)\) and consider the model
It follows that
where \(S_{0}=\frac{\Lambda}{\mu _{1}+\xi}\) and \(V_{0}=\frac{\xi \Lambda}{(\mu _{2}+\alpha )(\mu _{1}+\xi )}\). If \(\mathfrak{R}_{0}>1\), then the positive steady state \(P_{*}=(S_{*},V_{*},E_{*},I_{*})\) of model (4.1) satisfies the following equations:
Then we have
and \(I_{*}\) is the positive root of the equation
It is easy to show that \(\lim_{I\rightarrow 0^{+}}F(I) = \frac{(\mu _{3}+\sigma )(\mu _{4}+\delta +\gamma )}{\sigma}( \mathfrak{R}_{0}-1)>0\), \(\lim_{I\rightarrow +\infty}F(I)<0\), and
where
It follows from the assumption on the functions \(f_{j}(u,I)\) (\(j=1,2\)) that \(\Theta _{1}<0\), \(\Theta _{2}<0\). Then we have \(F'(I)<0\). Thus there exists a positive steady state \(P_{*}\).
Theorem 4.4
If \(\mathfrak{R}_{0}>1\), then the unique positive steady state \(P_{*}\) of model (4.1) is globally asymptotically stable.
Proof
Define
Using (4.2) and the function \(\varphi (z)=1+\ln z-z\) with global maximum \(\varphi (1)=0\). Then we have
Thus by the LaSalle invariance principle it is clear that \(P_{*}\) is globally asymptotically stable. □
5 Conclusions
In this paper, we investigated an SEVIR model with diffusion and nonlinear incidence rate. The basic reproduction number \(\mathfrak{R}_{0}\), which serves as a threshold index, is defined. By applying the comparison principle we prove that the disease-free steady state \(P_{0}(x)\) is globally asymptotically stable when \(\mathfrak{R}_{0}<1\). If \(\mathfrak{R}_{0}>1\), then the disease will persist. Furthermore, for the spatially homogeneous model, we establish the global stability of the positive steady state in terms of the corresponding basic reproduction number. Some existing global dynamical results can be covered and improved (see [18]). Though the well-posedness, the basic reproduction number and persistence of model (1.6) are established. The uniqueness and stability of the positive steady state remain an open problem. In addition, the global stability of the positive steady state of model (4.1) of general forms \(f_{1}(S,I)\) and \(f_{2}(V,I)\) is still an open problem. We leave these problems for future investigation.
Availability of data and materials
The analysis in this paper did not generate data.
References
McKendrick, A.G.: Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 125(44), 98–130 (1925)
Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases. Wiley, Chichester (2000)
Liu, X., Takeuchi, Y., Iwami, S.: SVIR epidemic models with vaccination strategies. J. Theor. Biol. 253(1), 1–11 (2008)
Wang, L., Wang, J., Zhao, H., et al.: Modelling and assessing the effects of medical resources on transmission of novel coronavirus (COVID-19) in Wuhan, China. Math. Biosci. Eng. 17(4), 2936–2949 (2020)
Jiao, J., Liu, Z., Cai, S.: Dynamics of an SEIR model with infectivity in incubation period and homestead-isolation on the susceptible. Appl. Math. Lett. 107, 106442 (2020)
Zhou, W., Wang, A., Xia, F., et al.: Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak. Math. Biosci. Eng. 17(3), 2683–2707 (2020)
Tang, B., Xia, F., Tang, S., et al.: The effectiveness of quarantine and isolation determine the trend of the COVID-19 epidemics in the final phase of the current outbreak in China. Int. J. Infect. Dis. 95, 288–293 (2020)
Tang, B., Wang, X., Li, Q., et al.: Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions. J. Clin. Med. 9(2), 462 (2020)
Li, J., Ma, Z.: Global analysis of SIS epidemic models with variable total population size. Math. Comput. Model. 39, 1231–1242 (2004)
Pei, Y.Z., Liu, S.Y., Chen, L.S., et al.: Two different vaccination strategies in an SIR epidemic model with saturated infectious force. Int. J. Biomath. 1, 147–160 (2008)
Gumel, A.B., Moghadas, S.M.: A qualitative study of a vaccination model with non-linear incidence. Appl. Math. Comput. 143, 409–419 (2003)
Anderson, R.M., May, R.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1991)
Ruan, S.: Spatiotemporal epidemic models for rabies among animals. Infect. Dis. Model. 2, 277–287 (2017)
Yamazaki, K., Wang, X.: Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete Contin. Dyn. Syst., Ser. B 21, 1297–1316 (2016)
Yamazaki, K., Wang, X.: Global stability and uniform persistence of the reaction–convection–diffusion cholera epidemic model. Math. Biosci. Eng. 14, 559–579 (2017)
Yu, X., Zhao, X.Q.: A nonlocal spatial model for Lyme disease. J. Differ. Equ. 261, 340–372 (2016)
Wang, W., Wang, X., Guo, K., Ma, W.: Global analysis of a diffusive viral model with cell-to-cell infection and incubation period. Math. Methods Appl. Sci. 43(9), 5963–5978 (2020)
Xu, Z., Xu, Y., Huang, Y.: Stability and traveling waves of a vaccination model with nonlinear incidence. Comput. Math. Appl. 75(2), 561–581 (2017)
Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Math. Surveys Monger. Am. Math. Soc., Providence (1995)
Martin, R.H., Smith, H.L.: Abstract functional-differential equations and reaction–diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990)
Guo, Z., Wang, F.B., Zou, X.: Threshold dynamics of an infective disease model with a fixed latent period and non-local infections. J. Math. Biol. 65, 1387–1410 (2012)
Du, Y.: Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Maximum Principles and Applications. World Scientific, Singapore (2006)
Guenther, R.B., Lee, J.W.: Partial Differential Equations of Mathematical Physics and Integral Equations. Dover, Mineola (1996)
Wang, M.: Nonlinear Elliptic Equations. Science Public, Beijing (2010)
Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)
Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Am. Math. Soc., Providence (1988)
Thieme, H.R.: Convergence results and a Poincaré–Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30, 755–763 (1992)
Wang, W., Zhao, X.: Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11, 1652–1673 (2012)
Smith, H.L., Zhao, X.: Robust persistence for semidynamical systems. Nonlinear Anal., Real World Appl. 47, 6169–6179 (2001)
Acknowledgements
The authors wish to thank the editor and the anonymous reviewers for their helpful feedback to improve the quality of this paper.
Funding
This work was supported by National Natural Science Foundation of China (#11701445, #11971379, #11801439), by Natural Science Basic Research Plan in Shaanxi Province of China (2022JM-042, 2021JM-320), and Scientific Research Program Funded by Shaanxi Provincial Education Department (20JK0642).
Author information
Authors and Affiliations
Contributions
The author Jinhu Xu wrote the whole manuscript text.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Xu, J. Global dynamics for an SVEIR epidemic model with diffusion and nonlinear incidence rate. Bound Value Probl 2022, 80 (2022). https://doi.org/10.1186/s13661-022-01660-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-022-01660-8