Startseite Mathematik Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
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Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment

  • Jinliang Wang EMAIL logo , Wenjing Wu und Chunyang Li
Veröffentlicht/Copyright: 1. März 2023
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 12 Heft 1

Abstract

In this article, we formulate and perform a strict analysis of a reaction–diffusion mosquito-borne disease model with total human populations stabilizing at H(x) in a spatially heterogeneous environment. By utilizing some fundamental theories of the dynamical system, we establish the threshold-type results of the model relying on the basic reproduction number. Specifically, we explore the mutual impacts of the spatial heterogeneity and diffusion coefficients on the basic reproduction number and investigate the existence, uniqueness, and global attractivity of the nontrivial steady state by utilizing the arguments of asymptotically autonomous semiflows. For the case that all parameters are independent of space, the global attractivity of the nontrivial steady state is achieved by the Lyapunov function.

Keywords: mosquito-borne disease model; basic reproduction number; spatial heterogeneity; uniform persistence; Lyapunov function
MSC 2010: 35K57; 35J57; 35B40; 92D25

1 Introduction

Malaria, dengue fever, zika, and chikungunya belong to mosquito-borne diseases that the pathogens are transmitted to humans by biting from infectious mosquitoes, which brings the global burden of disease and threatens public health [4]. Since the pioneering work of Ross, reaction–diffusion mosquito-borne disease models with climate change [12], repeated age-structure [7], exposure [13,25,37], vector-bias [3,6,10], growth domain [42], and seasonality [3] have received much attentions, which have been proved to be a powerful tool to study the spread and control of vector-borne disease.

In epidemiology, spatial effects have also been introduced to better understand the spatial geographical spread of mosquito-borne diseases. It is believed that the spatiotemporal diffusion and spatial heterogeneity remarkably affect the dynamics of infectious diseases. In [36], the authors considered the asymptotic profiles of the endemic equilibrium for a reaction–diffusion susceptible-infective-susceptible (SIS) model with a bilinear infection mechanism. Later, Li et al. [14] found that the disease becomes harder to control by adopting the varying total population and frequency-dependent infection mechanism for an SIS model. Thereafter, Li et al. [15] showed that infection mechanism, as well as variation in the total population, can influence the transmission dynamics of diseases by analyzing four diffusive SIS epidemic models. There are some contributions that have been made to explore the impacts of the dispersal of human and mosquito populations on the disease spread, see, for example, but are not limited to, [3,5,12,13,19,24,30,32,39,40]. In [13], Lou and Zhao introduced a time-delayed reaction–diffusion model to study the spatial movement of mosquitoes during the incubation period. Later, Bai et al. [3] extended the model in [13] to a model with seasonality and “vector bias” effect and analytically studied the threshold-type results relying on the basic reproduction number. Here, the so-called “vector bias” was proposed in [10] to describe the preference that mosquitoes prefer to bite infected humans. Very recently, Zhu et al. [42] formulated a reaction–diffusion dengue model involving the impact of regional evolution on dengue transmission with a time-dependent domain Ω t ⊂ R n , where Ω t grows uniformly and isotropically to a bounded domain with a growing boundary ∂ Ω t . The authors found that the growth of domain brings a negative impact on dengue control.

It is commonly known that it is challenging to understand comprehensively the spatial effects on the geographical spread of mosquito-borne diseases both theoretically and empirically [3,13,19,24]. Therefore, constructing epidemic models in spatially heterogeneous environments may be more realistic and applicative. Let Ω ∈ R n ( n ≥ 1 ) be the spatial bounded domain with smooth boundary ∂ Ω . The model contains space-dependent parameters accounting for the interactions of female mosquitoes and humans in Ω . Throughout the article, we let x ∈ Ω and t ∈ R be the location and time variables, respectively. Let Δ = ∂ 2 ∂ x 1 2 + ∂ 2 ∂ x 2 2 + ⋯ + ∂ 2 ∂ x n 2 be the Laplacian operator and ν be the unit normal vector on ∂ Ω . Denote by S m ≔ S m ( x , t ) and I m ≔ I m ( x , t ) (respectively S h ≔ S h ( x , t ) and I h ≔ I h ( x , t ) ), respectively, the densities of susceptible and infectious mosquitoes (respectively humans). Denote by N h ≔ N h ( x , t ) = S h + I h the total human population. We assume that N h fulfills

(1.1) ∂ N h ∂ t = D h Δ N h + b h N h 1 − N h K ( x ) , x ∈ Ω , t > 0 ; ∂ N h ∂ ν = 0 , x ∈ ∂ Ω , t > 0 .

Here, the positive constant D h represents diffusion coefficient; the positive constant b h is the growth rate; and the positive function K ( x ) accounts for the carrying capacity, allowing for diversity in habitats. In model (1.1), we consider the homogeneous Neumann boundary conditions, which means zero population flux across the boundary ∂ Ω , i.e., the populations are confined in Ω . According to [3, Theorem 3.1.5], system (1.1) possesses a unique positive steady state and satisfies lim t → ∞ N h ( ⋅ , t ) = H ( x ) for any positive initial values. Based on this, it allows us to biologically assume that, for any x ∈ Ω , N h ( ⋅ , t ) = H ( x ) , t ≥ 0 , i.e., the total human density stabilizes at H ( x ) , x ∈ Ω . Thus, by [2,3,13], we suppose that the forces of infection for humans and mosquito population are given as follows:

c β ( x ) H ( x ) − I h H ( x ) I m and b β ( x ) H ( x ) S m I h ,

respectively, where the positive function β ( x ) accounts for the biting rate and the positive constants c and b stand for the transmission probability. Let μ ( x ) and d m ( x ) be the recruitment and death rate of mosquito populations. The parameter functions β , μ , and d m are assumed to be positive Hölder continuous functions with respect to x on Ω ¯ .

In this article, we shall investigate the following reaction–diffusion system in a spatially heterogeneous environment:

(1.2) ∂ I h ∂ t − D h Δ I h = c β ( x ) H ( x ) ( H ( x ) − I h ) I m − ( d h + ρ h ) I h , x ∈ Ω , t > 0 , ∂ S m ∂ t − D m Δ S m = μ ( x ) − b β ( x ) H ( x ) S m I h − d m ( x ) S m , x ∈ Ω , t > 0 , ∂ I m ∂ t − D m Δ I m = b β ( x ) H ( x ) S m I h − d m ( x ) I m , x ∈ Ω , t > 0 , ∂ W ∂ ν = 0 , W = I h , S m , I m , x ∈ ∂ Ω , t > 0 , ( I h ( x , 0 ) , S m ( x , 0 ) , I m ( x , 0 ) ) = ( I h 0 ( x ) , S m 0 ( x ) , I m 0 ( x ) ) , x ∈ Ω ,

where D m is the diffusion coefficient for mosquitoes. As stated in [3,13], the longevity of a human is longer than the life span of a mosquito, we also assume that spatial heterogeneity or climate factor has little impact on human activities. Thus, we take the parameters related to humans, d h and ρ h , as constants, where d h and ρ h represent, respectively, the natural death and recovery rate of humans, i.e., 1 ∕ ρ is the human infectious period. The initial values I h 0 , S m 0 , and I m 0 are nonnegative continuous functions on Ω ¯ . We note here that model (1.2) is the one in [13] without considering the mobility of the latent mosquitoes in the extrinsic incubation period confined in a spatial domain.

Here, we provide a comprehensive analysis of the dynamics of the model (1.2). We shall solve the following questions on (1.2): (Q1) Is model (2) well-posed? (Q2) How to characterize the mutual effects of diffusion rates on the basic reproduction number with two infective components? and (Q3) What are the threshold dynamics (including disease extinction and persistence) relying on the basic reproduction number? To answer the question (Q1), we will show the existence, uniqueness, nonnegativity, and ultimate boundedness of the solution. Model (1.2) contains two infective components, i.e., I h and I m . Inspired by a recent work [19] that the authors addressed the relationship between the basic reproduction number and the local basic reproduction number, we shall pay much attention to the mutual impacts of spatial heterogeneity and diffusion coefficients on the basic reproduction number. Specifically, we write the next-generation operator as the product of the local basic reproduction number and strongly positive compact linear operators and then investigate the linkage between the basic reproduction number and local basic reproduction number to answer the question (Q2). To answer the question (Q3), we appeal to the comparison principle and the theory of uniform persistence. For the case that all parameters are independent of space, the global attractivity of the nontrivial steady state is achieved by the Lyapunov function. We would like to mention that the analysis carried out here is valid for other epidemic models with random diffusion and two infective components as will be seen in later sections.

We organize the article as follows: the existence, uniqueness, nonnegativity, and ultimate boundedness of the solution of (1.2) are explored in Section 2. In Section 3, we define the basic reproduction number and local basic reproduction number and pay much attention to the mutual impacts of the spatial heterogeneity and diffusion coefficients on the basic reproduction number. In Section 4, we analyze the threshold-type result through the basic reproduction number. By constructing the Lyapunov function, we verify the global attractivity of the positive constant steady state when all parameters are independent of space variables. We end up the article with a summary and discussion in Section 5.

2 The well-posedness of the model

Let X ≔ C ( Ω ¯ , R 3 ) be the Banach space of continuous functions, and let X + ≔ C ( Ω ¯ , R + 3 ) be its positive cone. Denote Y ≔ C ( Ω ¯ , R ) and Y + ≔ C ( Ω ¯ , R + ) . Let

(2.1) X H ≔ { ϕ ∈ X + : 0 ≤ ϕ 1 ( x ) ≤ H ( x ) , ∀ x ∈ Ω ¯ } .

Under the Neumann boundary condition, we let T 1 ( t ) , T 2 ( t ) : Y → Y , t ≥ 0 , be, respectively, the strongly continuous semigroups with D h △ − ( d h + ρ h ) ≔ A 1 and D m △ − d m ( x ) ≔ A 2 . An application of [29, Section 7.1 and Corollary 7.2.3] yields that T i ( t ) , i = 1 , 2 , t ≥ 0 , is compact and strongly positive. Moreover, T ( t ) = ( T 1 ( t ) , T 2 ( t ) , T 2 ( t ) ) : X → X , t ≥ 0 , is a strongly continuous semigroup generated by A = diag { A 1 , A 2 , A 2 } on D ( A ) = D ( A 1 ) × D ( A 2 ) × D ( A 2 ) , where

D ( A j ) ≔ s ∈ C ( Ω ¯ ) : lim t → 0 + ( T j ( t ) − I d ) s t exists , j = 1 , 2 ,

I d is the identity operator. Our main result in this section reads as follows.

Theorem 2.1

For any ϕ ∈ X H , (1.2) admits a unique global classical solution u ( x , t , ϕ ) , t ≥ 0 , with u ( x , 0 ) = ϕ . Moreover, system (1.2) admits a global attractor in X H .

Proof

For ∀ x ∈ Ω ¯ , ϕ = ( ϕ 1 , ϕ 2 , ϕ 3 ) T ∈ X H , we let ℱ = ( F 1 , F 2 , F 3 ) : X H → X be the nonlinear part defined by F 1 ( ϕ ) ( x ) = c β ( x ) H ( x ) ( H ( x ) − ϕ 1 ) ϕ 3 , F 2 ( ϕ ) ( x ) = μ ( x ) − b β ( x ) H ( x ) ϕ 2 ϕ 1 , and F 3 ( ϕ ) ( x ) = b β ( x ) H ( x ) ϕ 2 ϕ 1 . It follows that system (1.2) can be rewritten as d u ( t ) d t = A u ( t ) + ℱ ( u ( t ) ) , u ( 0 ) = ϕ ∈ X H , where u ( t ) = ( I h ( ⋅ , t ) , S m ( ⋅ , t ) , I m ( ⋅ , t ) ) ∈ X H , t > 0 , ϕ ≔ ( ϕ 1 , ϕ 2 , ϕ 3 ) = ( I h 0 , S m 0 , I m 0 ) ∈ X H .

Let β ∗ = max x ∈ Ω ¯ β ( x ) and H ∗ = min x ∈ Ω ¯ H ( x ) . For ∀ ϕ ∈ X H , h ≥ 0 , we then check that

ϕ + h ℱ = ϕ 1 + h c β ( x ) H ( x ) ( H ( x ) − ϕ 1 ) ϕ 3 ϕ 2 + h μ ( x ) − b β ( x ) H ( x ) ϕ 2 ϕ 1 ϕ 3 + h b β ( x ) H ( x ) ϕ 2 ϕ 1 ≥ ϕ 1 1 − h c β ∗ H ∗ ϕ 3 ϕ 2 1 − h b β ∗ H ∗ ϕ 1 ϕ 3

and

H ( x ) − ( ϕ 1 + h F 1 ( ϕ ) ( x ) ) = ( H ( x ) − ϕ 1 ) 1 − h c β ( x ) H ( x ) ϕ 3 .

Hence, we have

lim h → 0 + 1 h dist ( ϕ + h ℱ ( ϕ ) , X H ) = 0 , ∀ ϕ ∈ X H .

An application of [21, Corollary 4] and [35, Corollary 8.1.3] ensures that for each ϕ ∈ X H , (1.2) admits a mild solution ω ( ⋅ , t , ϕ ) , on [ 0 , T max ) with ω ( x , 0 ) = ϕ , where T max < ∞ . It remains to prove T max = ∞ for any ϕ ∈ X H . For this purpose, we follow a standard argument from [27] obtaining that ( I h , S m , I m ) remains nonnegative for t ∈ [ 0 , T max ) . We next confirm the boundedness of the solution in Ω × [ 0 , T max ) .

Note that the S m -equation of system (1.2) is governed by

(2.2) ∂ S ˜ m ∂ t − D m Δ S ˜ m = μ ( x ) − d m ( x ) S ˜ m , x ∈ Ω , t > 0 , ∂ S ˜ m ∂ ν = 0 , x ∈ ∂ Ω , t > 0 .

By [13, Lemma 1], (2.2) possesses a unique and globally attractive positive steady state S ˆ ( x ) . This combined with the comparison theorem [29, Theorem 7.3.4] yields that

(2.3) limsup t → ∞ S m ≤ lim t → ∞ S ˜ m = S ˆ ( x ) , uniformly for x ∈ Ω ¯ .

Due to N h = S h + I h and (1.1), we know that I h is bounded on [ 0 , T max ) . Hence, the I m -equation of (1.2) is dominated by

(2.4) ∂ I m ∂ t − D m Δ I m = M − d m ( x ) I m , x ∈ Ω , t > 0 , ∂ I m ∂ ν = 0 , x ∈ ∂ Ω , t > 0 ,

for M > 0 . Again from the comparison principle and [13, Lemma 1], one knows that I m is bounded on [ 0 , T max ) . Hence, u ( x , t , ϕ ) is bounded on [ 0 , T max ) . Therefore, the solution exists globally on [ 0 , ∞ ) , and system (1.2) forms a semiflow Φ ( t ) : X H → X H , i.e., ( Φ ( t ) ϕ ) ( x ) = u ( x , t , ϕ ) , ∀ x ∈ Ω ¯ .

Furthermore, from (2.3), one can obtain

limsup t → ∞ ‖ S m ( ⋅ , t ) ‖ ≤ ‖ S ˆ ‖ .

In addition, we have limsup t → ∞ ‖ I h ( ⋅ , t ) ‖ ≤ ‖ H ‖ and limsup t → ∞ ‖ I m ( ⋅ , t ) ‖ ≤ ‖ M d m ( x ) ‖ . That is to say, Φ ( t ) : X H → X H is point dissipative. Furthermore, Φ ( t ) : X H → X H is compact. Thus, the existence of a global compact attractor of Φ ( t ) is a consequence of [20, Theorem 2.9]. For the local existence and uniqueness of the solution by using the contraction mapping principle, the existence of global solutions and blow up in a finite time of solution, we refer the readers to [17,18]. This proves Theorem 2.1.□

3 Basic reproduction number

As will be seen in later discussions, the basic reproduction number, ℜ 0 , is a threshold value of model (1.2) in the sense that if ℜ 0 > 1 , the disease persists, while the disease vanishes if ℜ 0 < 1 . In general, for one infectious compartment in reaction–diffusion models, ℜ 0 has a relationship with the principal eigenvalue of an elliptic system. The explicit expression of ℜ 0 can be obtained by taking advantage of the variational characterization. It should be mentioned that the classical methods for calculating basic reproduction numbers for reaction–diffusion epidemic models, stream population models, and nonlocal and time-delayed reaction–diffusion epidemic models with or without periodic environments have attracted the interest from many researchers, see, for example, [9,13,23,26,34,38] and the references therein. Here we define the basic reproduction number with spatially dependent coefficients according to [24,38].

Obviously, (1.2) admits a disease-free equilibrium, ( 0 , S ˆ , 0 ) , where S ˆ fulfills

(3.1) − D m Δ S ˆ = μ ( x ) − d m ( x ) S ˆ , x ∈ Ω , ∂ S ˆ ∂ ν = 0 , x ∈ ∂ Ω .

Furthermore, from [16, Theorem 1.1], we directly obtain that

lim D m → 0 S ˆ = μ ( x ) d m ( x ) and lim D m → ∞ S ˆ = ∫ Ω μ ( x ) d x ∫ Ω d m ( x ) d x .

To find the next-generation operator, we linearize (1.2) at ( 0 , S ˆ , 0 ) to obtain

(3.2) ∂ I h ∂ t − D h Δ I h = c β ( x ) I m − ( d h + ρ h ) I h , x ∈ Ω , t > 0 , ∂ I m ∂ t − D m Δ I m = b β ( x ) H ( x ) S ˆ I h − d m ( x ) I m , x ∈ Ω , t > 0 , ∂ I h ∂ v = ∂ I m ∂ v = 0 , x ∈ ∂ Ω .

Let Π ( t ) be the solution semiflow of (3.2) on C ( Ω ¯ , R 2 ) with the generator

(3.3) ℬ = D h Δ − ( d h + ρ h ) c β ( x ) b β ( x ) H ( x ) S ˆ D m Δ − d m ( x ) ≔ B + F ,

where

B = D h Δ − ( d h + ρ h ) c β ( x ) 0 D m Δ − d m ( x ) , F = 0 0 b β ( x ) H ( x ) S ˆ 0 ,

Note that both ℬ and B are resolvent-positive operators, and ℬ is a positive perturbation of B . Since B is cooperative, we know that the C 0 -semigroup generated by B , T ˜ ( t ) : C ( Ω ¯ , R 2 ) → C ( Ω ¯ , R 2 ) , satisfies T ˜ ( t ) C ( Ω ¯ , R + 2 ) ⊆ C ( Ω ¯ , R + 2 ) . Hence, the next-generation operator ℒ is defined as follows:

ℒ ϕ ( x ) = ∫ 0 ∞ F ( x ) T ˜ ( t ) ϕ ( x ) d t = F ( x ) ∫ 0 ∞ T ˜ ( t ) ϕ ( x ) d t ϕ ∈ C ( Ω ¯ , R 2 ) , x ∈ Ω ¯ .

We then follow the procedure in [38] to define the basic reproduction number as

ℜ 0 ≔ r ( ℒ ) = sup { ∣ λ ∣ , λ ∈ σ ( L ) } ,

where σ ( ℒ ) is the spectrum of ℒ . Note that (3.2) is cooperative and irreducible. The following result comes from [34, Theorem 3.5] and [38, Lemma 2.2].

Lemma 3.1

Let s ( ℬ ) be the spectral bound of ℬ . Then ℜ 0 − 1 (or, r ( − F B − 1 ) − 1 ) has the same sign as λ 0 = s ( ℬ ) , where λ 0 is the principal eigenvalue of

(3.4) λ φ − D h Δ φ = c β ( x ) ϕ − ( d h + ρ h ) φ , x ∈ Ω , λ ϕ − D m Δ ϕ = b β ( x ) H ( x ) S ˆ φ − d m ( x ) ϕ , x ∈ Ω , ∂ φ ∂ v = ∂ ϕ ∂ v = 0 , x ∈ ∂ Ω ,

associated with a positive eigenvector ( φ ∗ , ϕ ∗ ) .

The following result is devoted to characterizing ℜ 0 .

Lemma 3.2

Let F : C ( Ω ¯ , R 2 ) → C ( Ω ¯ , R 2 ) be defined in (3.3), D = diag ( D h , D m ) and

− W 1 ( ⋅ ) = − ( d h + ρ h ) c β ( ⋅ ) 0 − d m ( ⋅ ) .

Then, the following linear elliptic eigenvalue problem

(3.5) − D Δ φ + W 1 φ = μ F φ , x ∈ Ω , ∂ φ i ∂ v = 0 , i = 1 , 2 , x ∈ ∂ Ω ,

has a unique principal eigenvalue μ 0 > 0 , associated with a strictly positive eigenvector ( φ 1 ∗ , φ 2 ∗ ) . Furthermore, ℜ 0 = 1 μ 0 .

Proof

Let φ = ( φ 1 ( x ) , φ 2 ( x ) ) T . Note that solution ( μ , φ ) of (3.5) satisfies

(3.6) − D h △ φ 1 + ( d h + ρ h ) φ 1 = c β ( x ) φ 2 , x ∈ Ω , − D m △ φ 2 + d m ( x ) φ 2 = μ b β ( x ) H ( x ) S ˆ φ 1 , x ∈ ∂ Ω .

Recall that for each t ≥ 0 , T 1 , T 2 : Y → Y is strongly positive and compact. From [29, Theorem 7.6.1], one knows that there exists a μ 0 ∈ R , such that resolvent-positive ( μ I d − A i ) − 1 is compact and strongly positive for all μ > μ 0 , where A i , i = 1 , 2 , and I d are defined in Section 2. Furthermore, from [34, Theorem 3.12], one can obtain

(3.7) ( μ I d − A i ) − 1 φ = ∫ 0 ∞ e − μ t T i ( t ) φ d t , ∀ μ > s ( A i ) , φ ∈ Y , i = 1 , 2 .

Note that the spectral bound s ( A 1 ) = − ( d h + ρ h ) < 0 and s ( A 2 ) = − min x ∈ Ω ¯ d m ( x ) < 0 . Letting μ = 0 in equation (3.7), we have

− A i − 1 φ = ∫ 0 ∞ T i ( t ) φ d t , ∀ φ ∈ Y .

An application of [29, Theorem 7.6.1] yields that − A i − 1 ( i = 1 , 2 ) is strongly positive and compact. We then rewrite (3.5) as follows:

− A 1 φ 1 = c β ( ⋅ ) φ 2 , x ∈ Ω , − A 2 φ 2 = μ b β ( ⋅ ) H ( ⋅ ) S ˆ φ 1 , x ∈ Ω ,

which implies that φ 1 = − c β ( ⋅ ) A 1 − 1 φ 2 and ( μ , φ 2 ) fulfills

(3.8) φ 2 μ = b β ( ⋅ ) c β ( ⋅ ) H ( ⋅ ) S ˆ A 2 − 1 A 1 − 1 φ 2 .

Since b β ( x ) c β ( x ) H ( x ) S ˆ > 0 ∀ x ∈ Ω ¯ , it follows that b β ( x ) c β ( x ) H ( x ) S ˆ A 2 − 1 A 1 − 1 is strongly positive and compact on Y . By [1, Theorem 3.2] and [29, Theorem 7.6.1], we know that (3.8) has a unique principal eigenvalue μ 0 > 0 with φ 2 ∗ ≫ 0 in Y . Let φ 1 ∗ = − c β ( ⋅ ) A 1 − 1 φ 2 ∗ , then φ 1 ∗ ≫ 0 . From [38, Theorem 3.2], we directly obtain ℜ 0 = 1 μ 0 . This proves Lemma 3.2.□

When there are no diffusion terms in system (1.2), we directly have

(3.9) d I h d t = c β ( x ) H ( x ) ( H ( x ) − I h ) I m − ( d h + ρ h ) I h , x ∈ Ω , t > 0 , d S m d t = μ ( x ) − b β ( x ) H ( x ) S m I h − d m ( x ) S m , x ∈ Ω , t > 0 , d I m d t = b β ( x ) H ( x ) S m I h − d m ( x ) I m , x ∈ Ω , t > 0 .

The basic reproduction number of (3.9) at a specific location x , called as local basic reproduction number, is calculated as follows:

(3.10) ℜ ( x ) = ℜ 1 ( x ) ℜ 2 ( x ) , where ℜ 1 = c β ( x ) d h + ρ h and ℜ 2 = b β ( x ) S ˆ H ( x ) d m ( x ) ,

where both ℜ 1 and ℜ 2 are multiplication operators defined on C ( Ω ¯ ) , and ℜ 1 ( x ) (resp. ℜ 2 ( x ) ) describes the effect of one infectious mosquito (resp. human) on susceptible human (resp. mosquitoes).

In the following, we shall explore the relationship between ℜ ( x ) and ℜ 0 . We first define L 1 and L 2 as follows:

(3.11) L 1 ≔ ( ( d h + ρ h ) − D h Δ ) − 1 ( d h + ρ h ) and L 2 ≔ ( d m ( x ) − D m Δ ) − 1 d m ( x ) .

Obviously, both L 1 and L 2 are strongly positive compact linear operators on C ( Ω ¯ ) .

3.1 General diffusion rates

Theorem 3.1

Let ℜ i ( x ) , i = 1 , 2 , and L i , i = 1 , 2 , be defined in (3.10) and (3.11), respectively. Then, we have the following results:

  1. r ( L 1 L 2 ) = r ( L 1 ) = r ( L 2 ) = 1 ;

  2. ℜ 0 = r ( L 1 ℜ 1 L 2 ℜ 2 ) ;

  3. If ℜ i ( x ) < 1 , i = 1 , 2 , ∀ x ∈ Ω ¯ , then ℜ 0 < 1 .

Proof

Proof of (i). L 1 L 2 is strongly positive compact linear operator on C ( Ω ¯ ) . By the general results in [24, Theorem 2.5], r ( L 1 ) , r ( L 2 ) , and r ( L 1 L 2 ) are principal eigenvalues of L 1 , L 2 , and L 1 L 2 , respectively. Since L 1 ( 1 ) = L 2 ( 1 ) = L 1 L 2 ( 1 ) = 1 , we directly have r ( L 1 ) = r ( L 2 ) = r ( L 1 L 2 ) = 1 . This proves (i).

Proof of (ii). By elementary computation, we directly have

B − 1 = ( D h △ − ( d h + ρ h ) ) − 1 − ( D h △ − ( d h + ρ h ) ) − 1 c β ( x ) ( D m △ − d m ( x ) ) − 1 0 ( D m △ − d m ( x ) ) − 1 .

Hence, − F B − 1 can be calculated as follows:

− F B − 1 = 0 0 b β ( x ) H ( x ) S ˆ ( D h △ − ( d h + ρ h ) ) − 1 b β ( x ) H ( x ) S ˆ ( D h △ − ( d h + ρ h ) ) − 1 c β ( x ) ( D m △ − d m ( x ) ) − 1 .

Then, the spectral radius of − F B − 1 takes the following form:

ℜ 0 = r ( − F B − 1 ) = r b β ( x ) H ( x ) S ˆ ( ( d h + ρ h ) − D h △ ) − 1 c β ( x ) ( d m ( x ) − D m △ ) − 1 .

Since L i , i = 1 , 2 , is strongly positive compact linear operator, the spectral radius of L i , r ( L i ) , i = 1 , 2 , can be derived by

r ( L i ) = lim n → ∞ ‖ L i n ‖ 1 ∕ n ( Gelfand’s formula ) .

It can be easily checked that

r ( L 2 L 1 ) = lim n → ∞ ‖ L 2 L 1 ⋯ L 2 L 1 ‖ 1 ∕ n = lim n → ∞ ‖ L 2 ⋅ L 1 L 2 ⋯ L 1 L 2 ⋅ L 1 ‖ 1 ∕ n ≤ lim n → ∞ ‖ L 2 ‖ 1 ∕ n ‖ ( L 1 L 2 ) n − 1 ‖ 1 ∕ n ‖ L 1 ‖ 1 ∕ n = lim n → ∞ ‖ ( L 1 L 2 ) n − 1 ‖ 1 n − 1 ⋅ n − 1 n = r ( L 1 L 2 ) ,

and vice versus, i.e., r ( L 1 L 2 ) ≤ r ( L 2 L 1 ) . Hence, one can obtain

(3.12) r ( L 1 L 2 ) = r ( L 2 L 1 ) .

It then follows that

ℜ 0 = r b β ( x ) H ( x ) S ˆ ( ( d h + ρ h ) − D h △ ) − 1 c β ( x ) ( d m ( x ) − D m △ ) − 1 = r ( ( d h + ρ h ) − D h △ ) − 1 ( d h + ρ h ) c β ( x ) d h + ρ h ( d m ( x ) − D m △ ) − 1 d m ( x ) b β ( x ) S ˆ H ( x ) d m ( x ) = r ( L 1 ℜ 1 L 2 ℜ 2 ) .

This proves (ii).

Proof of (iii). Obviously, L i ( ± 1 ) = ± 1 for i = 1 , 2 . Due to the strong positivity and linear property, we know that for i = 1 , 2 , L i ( 1 ) = L i ( u + 1 − u ) = L i ( u ) + L i ( 1 − u ) , which implies that L i ( 1 ) ≥ L i ( u ) . On the other hand, L i ( − 1 ) = L i ( u − 1 − u ) = L i ( u ) + L i ( − 1 − u ) , which implies that L i ( − 1 ) ≤ L i ( u ) . Hence, for any u ∈ C ( Ω ¯ ) with ‖ u ‖ ∞ ≤ 1 , we have

− 1 = L i ( − 1 ) ≤ L i u ≤ L i 1 = 1 for i = 1 , 2 .

Therefore, ‖ L i ‖ ≤ 1 for i = 1 , 2 . Furthermore, from L i ( 1 ) = 1 , we deduce that ‖ L 1 ‖ = ‖ L 2 ‖ = 1 .

Due to Gelfand’s formula, we obtain

r ( L 1 L 2 ) = lim n → ∞ ‖ ( L 1 L 2 ) n ‖ 1 ∕ n ≤ lim n → ∞ ( ‖ L 1 ‖ n ‖ L 2 ‖ n ) 1 ∕ n = ‖ L 1 ‖ ‖ L 2 ‖ .

It follows that

(3.13) ℜ 0 = r ( L 1 ℜ 1 L 2 ℜ 2 ) = lim n → ∞ ‖ ( L 1 ℜ 1 L 2 ℜ 2 ) n ‖ 1 ∕ n ≤ lim n → ∞ ( ‖ L 1 ‖ n ‖ ℜ 1 ‖ n ‖ L 2 ‖ n ‖ ℜ 2 ‖ n ) 1 ∕ n = ‖ L 1 ‖ ‖ ℜ 1 ‖ ‖ L 2 ‖ ‖ ℜ 2 ‖ = ‖ ℜ 1 ‖ ‖ ℜ 2 ‖ < 1 .

This proves (iii).□

Theorem 3.2

Let ℜ 1 ( x ) and ℜ 2 ( x ) be defined by (3.10)and let ℜ i m = min { ℜ i ( x ) : x ∈ Ω ¯ } , i = 1 , 2 , and ℜ i M = max { ℜ i ( x ) : x ∈ Ω ¯ } , i = 1 , 2 . We then have the following statements:

  1. If ℜ i ( x ) ≥ 1 , i = 1 , 2 , ∀ x ∈ Ω ¯ , then ℜ 0 ≥ 1 . If, in addition, ℜ 1 ( x ) ≢ 1 or ℜ 2 ( x ) ≢ 1 , then ℜ 0 > 1 .

  2. If ℜ i ( x ) ≤ 1 , i = 1 , 2 , ∀ x ∈ Ω ¯ , then ℜ 0 ≤ 1 . If, in addition, ℜ 1 ( x ) ≢ 1 or ℜ 2 ( x ) ≢ 1 , then ℜ 0 < 1 .

  3. ℜ 1 m ℜ 2 m ≤ ℜ 0 ≤ ℜ 1 M ℜ 2 M .

Proof

We first prove (i). If ℜ i ( x ) ≥ 1 for all x ∈ Ω ¯ , by the strong positivity of L 2 , we know that L 2 ( ℜ 2 − 1 ) ≥ 0 , i.e., L 2 ℜ 2 ≥ L 2 . We further have ℜ 1 L 2 ℜ 2 ≥ L 2 . This together with the strong positivity of L 1 yields that L 1 ( ℜ 1 L 2 ℜ 2 − L 2 ) ≥ 0 , i.e.,

L 1 ℜ 1 L 2 ℜ 2 ≥ L 1 L 2 .

By [24, Theorem 2.5] and (i) of Theorem 3.1, on can obtain that ℜ 0 = r ( L 1 ℜ 1 L 2 ℜ 2 ) ≥ r ( L 1 L 2 ) = 1 .

Let ϕ be the positive eigenfunction of ℜ 0 . If, in addition, ℜ 1 ( x ) ≢ 1 or ℜ 2 ( x ) ≢ 1 , combined with the strong positivity of L i , i = 1 , 2 , one knows that

ℜ 0 ϕ = L 1 ℜ 1 L 2 ℜ 2 ϕ ≫ L 1 L 2 ϕ .

Hence, there is ε > 0 such that ℜ 0 ϕ ≥ ( 1 + ε ) L 1 L 2 ϕ . Let ϕ m ≡ min x ∈ Ω ¯ ϕ ( x ) > 0 . Then, by the positivity of L 1 L 2 and L 1 L 2 1 = 1 , we have

ℜ 0 ϕ ≥ ( 1 + ε ) L 1 L 2 ϕ ≥ ( 1 + ε ) L 1 L 2 ϕ m = ( 1 + ε ) ϕ m .

Therefore, ℜ 0 ϕ ≥ ( 1 + ε ) ϕ m , which implies ℜ 0 ≥ 1 + ε > 1 .

Analogous to the proof of (i), the assertions (ii) and (iii) directly follow. This proves Theorem 3.2.□

3.2 Large diffusion rates

This subsection is spent on investigating ℜ 0 quantitatively when the diffusion rates D m and D h approach ∞ .

Lemma 3.3

Let L 1 and L 2 be defined by (3.11) and let L 1 , ∞ , L 2 , ∞ : C ( Ω ¯ ) → C ( Ω ¯ ) be defined by

L 1 , ∞ ( ϕ ) = ∫ Ω ϕ ( x ) d x ∣ Ω ∣ and L 2 , ∞ ( ϕ ) = ∫ Ω d m ( x ) ϕ ( x ) d x ∫ Ω d m ( x ) d x , ∀ ϕ ∈ C ( Ω ¯ ) .

We then have

  1. L 1 ⟶ SOT L 1 , ∞ in C ( Ω ¯ ) as D h → ∞ ;

  2. L 2 ⟶ SOT L 2 , ∞ in C ( Ω ¯ ) as D m → ∞ ,

where the symbol “ ⟶ SOT ” means that for ∀ ϕ ∈ C ( Ω ¯ ) , L i ( ϕ ) → L i , ∞ ( ϕ ) , i = 1 , 2 .

Proof

We first prove (i). Given ϕ ∈ C ( Ω ¯ ) and D h > 0 , we let L 1 ( ϕ ) = ψ . Note that ψ satisfies

(3.14) ( d h + ρ h ) ψ − D h Δ ψ = ( d h + ρ h ) ϕ , x ∈ Ω , ∂ ψ ∂ v = 0 , x ∈ ∂ Ω .

For all D h > 1 , by the comparison principle, one can obtain − ‖ ϕ ‖ ∞ ≤ ψ ≤ ‖ ϕ ‖ ∞ . From the L p -estimate for elliptic equations, one knows that for any given p > 1 ,

‖ ψ ‖ W 2 , p ( Ω ) ≤ C .

Letting p be sufficiently large and utilizing the Sobolev embedding theorem directly yield

‖ ψ ‖ C 1 + α ( Ω ) ≤ C with α = 1 − n p ,

where 0 < α < 1 . Since the embedding W 2 , p ( Ω ) ⊆ C ( Ω ¯ ) is compact for p > n , there is a subsequence of D h → 0 , written D n ≔ D h , n , fulfilling D n → 0 as n → ∞ , such that for some ψ ˜ ∈ W 2 , p ( Ω ) , ψ → ψ ˜ . Here, the convergence is weak in W 2 , p ( Ω ) and strong in C ( Ω ¯ ) . Furthermore, ψ ˜ satisfies − Δ ψ ˜ = 0 , x ∈ Ω , with ∂ ψ ˜ ∂ v = 0 , x ∈ ∂ Ω . With the help of the maximum principle, one can obtain that ψ ˜ is a constant. Taking D h → ∞ and integrating (3.14), one directly obtains that ψ = L 1 , ∞ . This proves (i).

Analogous to (i), the assertion (ii) directly follows. This proves Lemma 3.3.□

Denote by

(3.15) H 1 , ∞ = L 1 , ∞ ℜ 1 L 2 ℜ 2 and H 2 , ∞ = L 2 , ∞ ℜ ˆ 2 L 1 ℜ 1 ,

where ℜ ˆ 2 = b β ( x ) H ( x ) d m ( x ) ∫ Ω μ ( x ) d x ∫ Ω d m ( x ) d x , i.e., for ∀ ϕ ∈ C ( Ω ¯ ) ,

H 1 , ∞ ( ϕ ) = ∫ Ω ℜ 1 L 2 ℜ 2 ϕ d x ∣ Ω ∣ and H 2 , ∞ ( ϕ ) = ∫ Ω d m ( x ) ℜ ˆ 2 L 1 ℜ 1 ϕ d x ∫ Ω d m ( x ) d x .

It follows that H i , ∞ : C ( Ω ¯ ) → C ( Ω ¯ ) , i = 1 , 2 , is bounded linear operator.

Note that L 1 ⟶ SOT L 1 , ∞ and L 2 ⟶ SOT L 2 , ∞ in C ( Ω ¯ ) as D h → ∞ and D m → ∞ , respectively. With the aim to explore the relation between the spectral radius of L 1 ℜ 1 L 2 ℜ 2 (resp. L 2 ℜ 2 L 1 ℜ 1 ) and H 1 , ∞ (resp. H 2 , ∞ ), according to [24, Theorem 4.1], we only need to verify the the following conditions:

  1. Both L 1 ℜ 1 L 2 ℜ 2 (resp. L 2 ℜ 2 L 1 ℜ 1 ) and H 1 , ∞ (resp. H 2 , ∞ ) are strongly positive compact linear operators on C ( Ω ¯ ) .

  2. L 1 ℜ 1 L 2 ℜ 2 ⟶ SOT H 1 , ∞ as D h → ∞ and L 2 ℜ 2 L 1 ℜ 1 ⟶ SOT H 2 , ∞ as D m → ∞ .

  3. For the closed unit ball B of C ( Ω ¯ ) , ∪ D h > 1 L 1 ℜ 1 L 2 ℜ 2 ( B ) (resp. ∪ D m > 1 L 2 ℜ 2 L 1 ℜ 1 ( B ) ) is precompact.

  4. r ( L 1 ℜ 1 L 2 ℜ 2 ) > r 0 (resp. r ( L 2 ℜ 2 L 1 ℜ 1 ) > r 0 ) for some r 0 > 0 .

Lemma 3.4

Let H 1 , ∞ and H 2 , ∞ be defined in (3.15). The following statements hold:

  1. Fix D m > 0 . Let D h → ∞ , then

    ℜ 0 = r ( L 1 ℜ 1 L 2 ℜ 2 ) → r ( H 1 , ∞ ) = ∫ Ω ℜ 1 L 2 ℜ 2 d x ∣ Ω ∣ .

  2. Fix D h > 0 . Let D m → ∞ , then

    ℜ 0 = r ( L 2 ℜ 2 L 1 ℜ 1 ) → r ( H 2 , ∞ ) = ∫ Ω d m ( x ) ℜ ˆ 2 L 1 ℜ 1 d x ∫ Ω d m ( x ) d x .

Proof

Our argument comes from [24, Theorem 4.5]. By Lemma 3.3, we know that conditions (C1) and (C2) hold. In what follows, we prove that (C3) and (C4) valid. From (iii) of Theorem 3.1, for the closed unit ball B of C ( Ω ¯ ) , L i ( B ) ⊂ B , i = 1 , 2 . This combined with the proof of (iii) of Theorem 3.1 indicates that ∪ D h > 1 L 1 ℜ 1 L 2 ℜ 2 ( B ) ⊂ L 1 ℜ 1 ( ℜ 2 M B ) and ∪ D m > 1 L 2 ℜ 2 L 1 ℜ 1 ( B ) ⊂ L 2 ℜ 2 ( ℜ 1 M B ) are precompact in C ( Ω ¯ ) , where ℜ 1 M and ℜ 2 M are defined in (iii) of Theorem 3.2. Thus, (C3) holds. Furthermore, from (iii) of Theorem 3.2, r ( L 1 ℜ 1 L 2 ℜ 2 ) = r ( L 2 ℜ 2 L 1 ℜ 1 ) ≥ ℜ 1 m ℜ 2 m > 0 . Hence, (C4) holds. Consequently, by [24, Theorem 4.5], one can obtain that

ℜ 0 = r ( L 1 ℜ 1 L 2 ℜ 2 ) → r ( H 1 , ∞ ) as D h → ∞ and ℜ 0 = r ( L 2 ℜ 2 L 1 ℜ 1 ) → r ( H 2 , ∞ ) as D m → ∞ .

It is noted that the eigenfunctions of H 1 , ∞ and H 2 , ∞ are constants. We finally obtain that

r ( H 1 , ∞ ) = ∫ Ω ℜ 1 L 2 ℜ 2 d x ∣ Ω ∣ and r ( H 2 , ∞ ) = ∫ Ω d m ( x ) ℜ ˆ 2 L 1 ℜ 1 d x ∫ Ω d m ( x ) d x .

This proves Theorem 3.4.□

We now further pay attention to r ( H 1 , ∞ ) = r ( L 1 , ∞ ℜ 1 L 2 ℜ 2 ) and r ( H 2 , ∞ ) = r ( L 2 , ∞ ℜ ˆ 2 L 1 ℜ 1 ) . Note that L 1 ⟶ SOT L 1 , ∞ in C ( Ω ¯ ) as D h → ∞ and L 2 ⟶ SOT L 2 , ∞ in C ( Ω ¯ ) as D m → ∞ (stated in Theorem 3.3). We next investigate what will occur for r ( L 1 , ∞ ℜ 1 L 2 ℜ 2 ) (resp. r ( L 2 , ∞ ℜ ˆ 2 L 1 ℜ 1 ) ) as D m → ∞ (resp. D h → ∞ ).

The main result of this subsection reads as follows.

Theorem 3.3

Let ℜ 1 and ℜ 2 be defined in (3.10). We further let

ℜ ˘ 1 ≔ ∫ Ω c β ( x ) d x ( d h + ρ h ) ∣ Ω ∣ a n d ℜ ˘ 2 ≔ ∫ Ω μ ( x ) d x ∫ Ω d m ( x ) d x 2 ∫ Ω b β ( x ) H ( x ) d x .

We then have the following statements:

  1. When D m → ∞ , we have r ( H 1 , ∞ ) = r ( L 1 , ∞ ℜ 1 L 2 ℜ 2 ) → ℜ ˘ 1 ℜ ˘ 2 , i.e., lim D m → ∞ lim D h → ∞ ℜ 0 = ℜ ˘ 1 ℜ ˘ 2 ;

  2. When D h → ∞ , we have r ( H 2 , ∞ ) = r ( L 2 , ∞ ℜ ˆ 2 L 1 ℜ 1 ) → ℜ ˘ 1 ℜ ˘ 2 , i.e., lim D h → ∞ lim D m → ∞ ℜ 0 = ℜ ˘ 1 ℜ ˘ 2 ;

  3. lim ( D h , D m ) → ( ∞ , ∞ ) ℜ 0 = ℜ ˘ 1 ℜ ˘ 2 .

Proof

Recall from Lemma 3.3 that

L 1 , ∞ ( ϕ ) = ∫ Ω ϕ ( x ) d x ∣ Ω ∣ and L 2 , ∞ ( ϕ ) = ∫ Ω d m ( x ) ϕ ( x ) d x ∫ Ω d m ( x ) d x , for any ϕ ∈ C ( Ω ¯ ) .

Direct calculation yields that

L 2 ℜ 2 → ∫ Ω d m ( x ) ℜ ˆ 2 d x ∫ Ω d m ( x ) d x = ∫ Ω μ ( x ) d x ∫ Ω d m ( x ) d x 2 ∫ Ω b β ( x ) H ( x ) d x , in C ( Ω ¯ ) , as D m → ∞ ,

and

L 1 ℜ 1 → ∫ Ω ℜ 1 d x ∣ Ω ∣ = ∫ Ω c β ( x ) d x ( d h + ρ h ) ∣ Ω ∣ , in C ( Ω ¯ ) , as D m → ∞ .

Hence, assertions (i) and (ii) directly follow.

Again from [24, Theorem 4.1], we know that (C1)–(C4) conditions still valid for the L 1 ℜ 1 L 2 ℜ 2 (resp. L 2 ℜ 2 L 1 ℜ 1 ) and L 1 , ∞ ℜ 1 L 2 , ∞ ℜ ˆ 2 (resp. L 2 , ∞ ℜ ˆ 2 L 1 , ∞ ℜ 1 ). Hence,

ℜ 0 = r ( L 1 ℜ 1 L 2 ℜ 2 ) = r ( L 2 ℜ 2 L 1 ℜ 1 ) → r ( L 1 , ∞ ℜ 1 L 2 , ∞ ℜ ˆ 2 ) = r ( L 2 , ∞ ℜ ˆ 2 L 1 , ∞ ℜ 1 ) = ℜ ˘ 1 ℜ ˘ 2

as ( D h , D m ) → ( ∞ , ∞ ) . This proves Theorem 3.3.□

3.3 Small diffusion rates

This subsection is devoted to investigating ℜ 0 quantitatively when the diffusion rates D m and D h approach zero.

Lemma 3.5

Let L 1 , L 2 , and ℜ ( x ) be defined in (3.10).

  1. Fix D m > 0 . Let D h → 0 , then ℜ 0 → r ( ℜ L 2 ) ;

  2. Fix D h > 0 . Let D m → 0 , then ℜ 0 → r ( ℜ 1 ℜ 2 ∗ L 1 ) , where ℜ 2 ∗ = b β ( x ) μ ( x ) H ( x ) d m 2 ( x ) .

Proof

Our argument comes from [24, Theorem 4.9]. By (3.12), it follows that r ( L 1 ℜ 1 L 2 ℜ 2 ) = r ( ℜ 1 L 2 ℜ 2 L 1 ) . We next pay attention to the operator ℜ 1 L 2 ℜ 2 L 1 . Note that ℜ 1 L 2 ℜ 2 L 1 and ℜ 1 L 2 ℜ 2 are strongly positive compactor operators on C ( Ω ¯ ) . When D h → 0 , we know that L 1 ϕ → ϕ in C ( Ω ¯ ) . It then follows that ℜ 1 L 2 ℜ 2 L 1 ⟶ SOT ℜ 1 L 2 ℜ 2 as D h → 0 . Hence, the conditions (C1) and (C2) hold for ℜ 1 L 2 ℜ 2 L 1 and ℜ 1 L 2 ℜ 2 . In what follows, we prove that (C3) and (C4) valid for ℜ 1 L 2 ℜ 2 L 1 and ℜ 1 L 2 ℜ 2 .

For a given closed unit ball in C ( Ω ¯ ) , by L 1 ( B ) ⊂ B , one can obtain that ∪ D h > 1 ℜ 1 L 2 ℜ 2 L 1 ( B ) ⊂ ℜ 1 L 2 ℜ 2 ( B ) . Furthermore, from the compactness of L 2 , ∪ D h > 1 ℜ 1 L 2 ℜ 2 L 1 ( B ) is precompact in C ( Ω ¯ ) . Thus, the condition (C3) holds.

Furthermore, from (iii) of Theorem 3.2, r ( ℜ 1 L 2 ℜ 2 L 1 ) = r ( L 2 ℜ 2 L 1 ℜ 1 ) ≥ ℜ 1 m ℜ 2 m > 0 . Hence, (C4) holds. Consequently,

ℜ 0 = r ( L 1 ℜ 1 L 2 ℜ 2 ) = r ( ℜ 1 L 2 ℜ 2 L 1 ) → r ( ℜ 1 L 2 ℜ 2 ) = r ( ℜ 2 ℜ 1 L 2 ) = r ( ℜ L 2 ) , as D h → 0 .

Analogously, the assertion (ii) can be proved by verifying that ℜ 2 L 1 ℜ 1 L 2 ⟶ SOT ℜ 2 ∗ L 1 ℜ 1 as D m → 0 . We omit the details here, as it follows from (i). This proves Theorem 3.5.□

We next pay attention to r ( ℜ L 2 ) and r ( ℜ 1 ℜ 2 ∗ L 1 ) . We list the main results on what will occur for r ( ℜ L 2 ) (resp. r ( ℜ 1 ℜ 2 ∗ L 1 ) ) as D m → 0 (resp. D h → 0 ). Furthermore, the result on the specific expression of ℜ 0 on ( D m , D h ) → ( 0 , 0 ) is also addressed. We omit the proof here, as it only takes slight modifications to [24, Theorem 4.10 and 4.11].

Theorem 3.4

Let ℜ M = max { ℜ ( x ) : x ∈ Ω ¯ } and let ℜ ( x ) and L 1 , L 2 be defined in (3.10) and (3.11), respectively. We then have

  1. Let D m → 0 , then r ( ℜ L 2 ) → ℜ M , i.e., lim D m → 0 lim D h → 0 ℜ 0 = ℜ M ;

  2. Let D h → 0 , then r ( ℜ 1 ℜ 2 ∗ L 1 ) → ℜ M , i.e., lim D h → 0 lim D m → 0 ℜ 0 = ℜ M ;

  3. lim ( D h , D m ) → ( 0 , 0 ) ℜ 0 = ℜ M .

The result stated below can be viewed as a consequent of Theorem 3.4.

Corollary 3.1

Let ℜ ( x ) be defined in (3.10).

  1. Provided that ℜ ( x ) < 1 , ∀ x ∈ Ω ¯ , then for a small enough number D ˆ > 0 , ℜ 0 < 1 for all ( D h , D m ) with D h , D m ≤ D ˆ ;

  2. Provided that ℜ ( x ) > 1 , for some x ∈ Ω ¯ , then for a small enough number D ˜ > 0 , ℜ 0 > 1 for all ( D h , D m ) with D h , D m ≤ D ˜ .

4 Threshold dynamics

This section aims to explore the threshold-type results of system (1.2) relying on ℜ 0 .

4.1 Global stability of E 0

A steady state of (1.2) is a nonnegative solution of

(4.1) − D h Δ I h = c β ( x ) H ( x ) ( H ( x ) − I h ) I m − ( d h + ρ h ) I h , x ∈ Ω , − D m Δ S m = μ ( x ) − b β ( x ) H ( x ) S m I h − d m ( x ) S m , x ∈ Ω , − D m Δ I m = b β ( x ) H ( x ) S m I h − d m ( x ) I m , x ∈ Ω , ∂ I h ∂ v = ∂ I m ∂ v = ∂ S m ∂ v = 0 , x ∈ ∂ Ω .

We first show that the disease-free equilibrium E 0 is globally asymptotically stable.

Theorem 4.1

Provided that ℜ 0 < 1 , then the disease-free equilibrium E 0 is globally asymptotically stable.

Proof

Substituting ( I h ( ⋅ , t ) , I m ( ⋅ , t ) ) = e λ t ( φ ( ⋅ ) , ϕ ( ⋅ ) ) into (3.2), we then obtain (3.4). By Lemma 3.1, one can obtain that λ 0 < 0 when ℜ 0 < 1 . Hence, E 0 is stable.

We next aim to show that ∀ ( I h 0 ( x ) , S m 0 ( x ) , I m 0 ( x ) ) ∈ X H ,

(4.2) lim t → ∞ ‖ ( I h ( ⋅ , t ) , S m ( ⋅ , t ) , I m ( ⋅ , t ) ) − E 0 ‖ X H = 0 .

When ℜ 0 < 1 , Lemma 3.1 implies that λ 0 ≔ λ 0 ( S ˆ ) < 0 , where λ 0 ( S ˆ ) is the principle eigenvalue of (3.4) having a positive eigenfunction. Due to lim ε → 0 λ 0 ( S ˆ + ε ) = λ 0 ( S ˆ ) < 0 , there exists a small ε 2 > 0 such that λ 0 ( S ˆ + ε 2 ) < 0 . For fixed ε 2 > 0 , by (2.2) and (2.3), there is t 2 such that S m ( ⋅ , t ) ≤ S ˆ + ε 2 ∀ x ∈ Ω ¯ , t ≥ t 2 . Therefore,

∂ I h ∂ t − D h Δ I h ≤ c β ( x ) I m − ( d h + ρ h ) I h , x ∈ Ω , t ≥ t 2 , ∂ I m ∂ t − D m Δ I m ≤ b β ( x ) H ( x ) ( S ˆ + ε 2 ) I h − d m ( x ) I m , x ∈ Ω , t ≥ t 2 , ∂ I h ∂ v = ∂ I m ∂ v = 0 x ∈ ∂ Ω

holds for ( I h ( ⋅ , t ) , S m ( ⋅ , t ) , I m ( ⋅ , t ) ) ∈ X H . Denote by ψ ¯ = ( ψ 1 ε 2 , ψ 2 ε 2 ) the strongly positive eigenfunction related to λ 0 ( S ˆ + ε 2 ) ≔ λ 0 ε 2 < 0 . Selecting ξ 1 > 0 such that ( I h 0 ( x ) , I m 0 ( x ) ) ≤ ξ 1 ( ψ 1 ε 2 , ψ 2 ε 2 ) ∀ t ≥ t 2 . Obviously, e λ 0 ε 2 ( t − t 2 ) ψ ¯ is a solution of

∂ I h ∂ t − D h Δ I h = c β ( x ) I m − ( d h + ρ h ) I h , x ∈ Ω , t ≥ t 2 , ∂ I m ∂ t − D m Δ I m = b β ( x ) H ( x ) ( S ˆ + ε 2 ) I h − d m ( x ) I m , x ∈ Ω , t ≥ t 2 , ∂ I h ∂ v = ∂ I m ∂ v = 0 , x ∈ ∂ Ω .

Following the comparison principle, one can obtain

( I h ( x , t ) , I m ( x , t ) ) ≤ ξ 1 e λ 0 ε 2 ( t − t 2 ) ( ψ 1 ε 2 , ψ 2 ε 2 ) ∀ x ∈ Ω ¯ and t ≥ t 2 .

Therefore,

lim t → ∞ ( I h ( x , t ) , I m ( x , t ) ) = ( 0 , 0 ) uniformly for x ∈ Ω ¯ .

It follows that the S m -equation is asymptotic to (2.2), lim t → ∞ S m ( x , t ) = S ˆ ( x ) uniformly for x ∈ Ω ¯ (see [33, Corollary 4.3]). This proves Theorem 2.1.□

4.2 Uniform persistence

Let Φ ( t ) : X H → X H be the semiflow generated by the solution u ( x , t ; ϕ ) = ( I h , S m , I m ) of (1.2) with initial value ϕ ∈ X H , i.e., Φ ( t ) ϕ = u ( ⋅ , t ) for t ≥ 0 . According to [8] and Theorem 2.1, Φ ( t ) is point dissipative in X H .

Before going into details, let

(4.3) X H 0 ≔ { ϕ ∈ X H : ϕ 1 ( ⋅ , 0 ) ≢ 0 and ϕ 3 ( ⋅ , 0 ) ≢ 0 }

and

∂ X H 0 ≔ { ϕ ∈ X H : ϕ 1 ( ⋅ , 0 ) ≡ 0 or ϕ 3 ( ⋅ , 0 ) ≡ 0 } .

Then, X H 0 is an open subset of X H with X H = X ¯ H 0 . Furthermore, X H = X H 0 ∪ ∂ X H 0 with the boundary ∂ X H 0 = X H − X H 0 being closed in X H . If there is ς > 0 small enough such that liminf t → ∞ d ( Φ ( t ) u , ∂ X H 0 ) ≥ ς for all u ∈ X H 0 , where d is the distance function X H , we say that Φ ( t ) is uniformly persistent corresponding to ( X H 0 , ∂ X H 0 ) .

Theorem 4.2

Suppose that ℜ 0 > 1 , then there exists ς > 0 such that ∀ ϕ = ( ϕ 1 , ϕ 2 , ϕ 3 ) ∈ X H 0 with ϕ i ≢ 0 , i = 1 , 3 ,

liminf t → ∞ u ( ⋅ , t ; ϕ ) ≥ ( ς , ς , ς ) , uniformly f o r x ∈ Ω ¯ ,

i.e., (1.2) is uniformly persistent. Furthermore, Φ ( t ) admits at least one positive steady state in X H 0 .

We will prove Theorem 4.2 by the following lemmas.

Lemma 4.1

Considering system (1.2), we have the following statements:

  1. If there exists some t 1 ≥ 0 such that I h ( ⋅ , t 1 , ϕ ) ≢ 0 or I m ( ⋅ , t 1 , ϕ ) ≢ 0 , then

    I h ( ⋅ , t , ϕ ) , I m ( ⋅ , t , ϕ ) > 0 ∀ t ≥ t 1 x ∈ Ω ¯ .

  2. For any ϕ ∈ X H and some ε 1 > 0 , S m ( ⋅ , t , ϕ ) , ∀ t > 0 , x ∈ Ω ¯ . Furthermore,

    liminf t → ∞ S m ( x , t , ϕ ) ≥ ε 1

    uniformly for x ∈ Ω ¯ .

Proof

From the I h and I m -equations of (1.2), we know that I h ( x , t , ϕ ) and I m ( x , t , ϕ ) satisfy

∂ I h ∂ t − D h Δ I h ≥ − ( d h + ρ h ) I h , x ∈ Ω , t > 0 , ∂ I h ∂ v = 0 , x ∈ ∂ Ω ,

and

∂ I m ∂ t − D m Δ I m ≥ − d m ( x ) I m , x ∈ Ω , t > 0 , ∂ I m ∂ v = 0 , x ∈ ∂ Ω ,

respectively. If I h ( t 1 , ⋅ , ϕ ) ≢ 0 or I m ( t 1 , ⋅ , ϕ ) ≢ 0 , for some t 1 ≥ 0 , combined with the strong maximum principle and Hopf boundary lemma, the results in (i) directly follows.

By letting w ( x , t , ϕ ) be the solution of

∂ ϖ ∂ t − D m Δ ϖ = μ ( x ) − ( b β ( x ) + d m ( x ) ) ϖ , x ∈ Ω , t > 0 , ∂ ϖ ∂ v = 0 , x ∈ ∂ Ω , t > 0 , ϖ ( x , 0 ) = ϕ 1 ( x ) , x ∈ Ω ,

we know that S m ( x , t , ϕ ) ≥ ϖ ( x , t , ϕ ) > 0 ∀ t > 0 , x ∈ Ω ¯ . This together with [13, Lemma 1] and the comparison principle yields that liminf t → ∞ S m ( x , t , ϕ ) ≥ μ ∗ b β ∗ + d m ∗ uniformly for x ∈ Ω ¯ , where μ ∗ = min x ∈ Ω ¯ μ ( x ) and P ∗ = max x ∈ Ω ¯ P ( x ) with P = β , d m , respectively. This proves Lemma 4.1.□

Lemma 4.2

Let ω ( ϕ ) be the omega limit set of the orbit γ + ( ϕ ) ≔ { Φ ( t ) ϕ : ∀ t ≥ 0 } . Let us define

M ∂ ≔ { ϕ ∈ ∂ X H 0 : Φ ( t ) ϕ ∈ ∂ X H 0 , ∀ t ≥ 0 } .

For any ψ ∈ M ∂ , we have ω ( ψ ) = { E 0 } .

Proof

Given ψ ∈ M ∂ , one knows that Φ ( t ) ψ ∈ M ∂ , ∀ t ≥ 0 . Hence, for t ≥ 0 , either I h ( ⋅ , t , ψ ) ≡ 0 or I m ( ⋅ , t , ψ ) ≡ 0 . For I h ( ⋅ , t , ϕ ) ≡ 0 for all t ≥ 0 , we see that the I m -equation of (1.2) satisfies ∂ I m ∂ t − D m Δ I m ≤ − d m ( x ) I m , for x ∈ Ω ¯ , t > 0 . With the help of the comparison principle, one can obtain lim t → ∞ I m ( x , t ) = 0 uniformly for x ∈ Ω ¯ . From the S m -equation in (1.2), we directly obtain that lim t → ∞ S m ( x , t ) = S ˆ uniformly for x ∈ Ω , see [33, Corollary 4.3]. If I h ( ⋅ , t 4 , ϕ ) ≢ 0 for some t 4 ≥ 0 , then I h ( ⋅ , t , ϕ ) > 0 ∀ t > t 4 , x ∈ Ω ¯ , see Lemma 4.1. So I m ( ⋅ , t , ϕ ) ≡ 0 ∀ t ≥ t 4 . By the I h -equation in (1.2), we see that lim t → ∞ I h ( x , t ) = 0 uniformly for x ∈ Ω ¯ . Again from [33, Corollary 4.3]), we directly obtain that lim t → ∞ S m ( x , t ) = S ˆ uniformly for x ∈ Ω . Therefore, the assertion in Lemma 4.2 directly follows. Let Φ ∂ : Φ ( t ) ∣ ∂ X H 0 be the semiflow of Φ ( t ) restricted on ∂ X H 0 . Based on the aforementioned statements, one knows that Φ ∂ has a compact global attractor A ∂ , and A ˜ ∂ ≔ ∪ ϕ ∈ A ∂ ω ( ϕ ) = { E 0 } .□

Lemma 4.3

Suppose that ℜ 0 > 1 . The disease-free equilibrium E 0 is a uniform weak repeller with respect to X H 0 , i.e., there exists ς > 0 such that ∀ ϕ ∈ X H 0 with ϕ i ( ⋅ ) ≢ 0 , i = 1 , 3 , u ( x , t ; ϕ ) of (1.2) satisfies the following statements:

lim t → ∞ sup ‖ u ( ⋅ , t ; ϕ ) − E 0 ‖ X H ≥ ς .

Proof

Assume to contrary that for some ς > 0 ,

(4.4) lim t → ∞ sup ‖ u ( ⋅ , t ; ϕ ) − E 0 ‖ X < ς .

Hence, there is a t 3 > 0 such that

S m ( ⋅ , t ) > S ˆ − ς , 0 < I h ( ⋅ , t ) < ς and 0 < I m ( ⋅ , t ) < ς , t ≥ t 3 .

It leads to

∂ I h ∂ t − D h Δ I h ≥ c β ( x ) H ( x ) ( H ( x ) − ς ) I m − ( d h + ρ h ) I h , x ∈ Ω , t ≥ t 3 , ∂ I m ∂ t − D m Δ I m ≥ b β ( x ) H ( x ) ( S ˆ − ς ) I h − d m ( x ) I m , x ∈ Ω , t ≥ t 3 , ∂ I h ∂ v = ∂ I m ∂ v = 0 , x ∈ ∂ Ω .

From Lemma 3.1, one knows that λ 0 ( S ˆ ) > 0 when ℜ 0 > 1 . Let φ ¯ ς = ( φ 1 ς , φ 2 ς ) be the positive eigenfunction related to λ 0 ( S ˆ − ς ) ≔ λ ¯ 0 ς > 0 . By choosing C > 0 such that ( I h ( ⋅ , t 3 , ϕ ) , I m ( ⋅ , t 3 , ϕ ) ) ≥ C φ ¯ ς , ∀ x ∈ Ω ¯ . Since e λ ¯ 0 ς ( t − t 3 ) φ ¯ ς is a solution of

∂ I h ∂ t − D h Δ I h = c β ( x ) H ( x ) ( H ( x ) − ς ) I m − ( d h + ρ h ) I h , x ∈ Ω , ∂ I m ∂ t − D m Δ I m = b β ( x ) H ( x ) ( S ˆ − ς ) I h − d m ( x ) I m , x ∈ Ω , ∂ I h ∂ v = ∂ I m ∂ v = 0 , x ∈ ∂ Ω ,

∀ t > t 3 , combined with the comparison principle, one directly has

( I h ( x , t , ϕ ) , I m ( x , t , ϕ ) ) ≥ C e λ ¯ 0 ς ( t − t 3 ) ( φ 1 ς , φ 2 ς ) ∀ x ∈ Ω ¯ and ∀ t > t 3 .

Thanks to λ ¯ 0 ς > 0 , we directly obtain that I h ( ⋅ , t , ϕ ) and I m ( ⋅ , t , ϕ ) are unbounded, which results in a contradiction. This proves 4.3.□

Proof of Theorem 4.2

Lemma 4.1 ensures that X H 0 is invariant under Φ ( t ) , i.e., for ∀ ϕ ∈ X H 0 , Φ ( t ) ϕ ∈ X H 0 , ∀ t > 0 . Lemma 4.2 indicates that E 0 is isolated invariant with respect to Φ ( t ) on ∂ X H 0 . Lemma 4.3 confirms that the disease-free equilibrium E 0 is a uniform weak repeller when ℜ 0 > 1 . Lemma 4.2 together with Lemma 4.3 yields that W s ( E 0 ) ∩ X H 0 = ∅ , where W s ( E 0 ) denotes the stable set of E 0 . By [28, Theorem 3], we can define

p ( ϕ ) ≔ min { min x ∈ Ω ¯ ϕ 1 ( x , 0 ) , min x ∈ Ω ¯ ϕ 3 ( x , 0 ) } ∀ ϕ ∈ X H 0 ,

as the distance function for Φ ( t ) . In addition, there exists ς such that min p ( Φ ( t ) ϕ ) > ς if p ( ϕ ) = 0 and ϕ ∈ X H 0 or p ( ϕ ) > 0 . By choosing ς small enough and Lemma 4.1, Φ ( t ) is uniformly persistent corresponding to ( X H 0 , ∂ X H 0 ) . Furthermore, Φ ( t ) is point dissipative. Hence, the existence of the global attractor A ˆ of Φ ( t ) is a consequence of [20, Theorem 3.7 and Remark 3.10]. Furthermore, from Lemma 4.1 and [20, Theorem 4.7], Φ ( t ) has at least a positive steady state in X H 0 . This proves Theorem 4.2.□

4.3 The stability analysis of positive steady state

This section is spent on proving the global attractivity of the positive steady state for system (1.2) by using the theory of asymptotically autonomous semiflows [19,33,41]. By letting S = S m + I m . Then, S satisfies (2.2). We shall study the forthcoming limiting system:

(4.5) ∂ I h ∂ t − D h Δ I h = c β ( x ) H ( x ) ( H ( x ) − I h ) I m − ( d h + ρ h ) I h , x ∈ Ω , t > 0 , ∂ I m ∂ t − D m Δ I m = b β ( x ) H ( x ) ( S ˆ ( x ) − I m ) + I h − d m ( x ) I m , x ∈ Ω , t > 0 , ∂ I h ∂ v = ∂ I m ∂ v = 0 , x ∈ ∂ Ω , I h ( x , 0 ) = I h 0 ( x ) , I m ( x , 0 ) = I m 0 ( x ) , x ∈ Ω .

The following result reveals the well-posedness of system (4.5).

Lemma 4.4

For any nonnegative initial data ( I h 0 , I m 0 ) ∈ X H ∗ ≔ { ( I h 0 , I m 0 ) : 0 ≤ I h 0 ≤ H ( x ) , 0 ≤ I m 0 ≤ S ˆ ( x ) , ∀ x ∈ Ω ¯ } , the solution of (4.5) satisfies that for all x ∈ Ω ¯ , t > 0 ,

  1. If I h 0 , I m 0 is nontrivial, then I h > 0 and I m > 0 ;

  2. I h , I m ≤ M ˆ , for some M ˆ > 0 .

Proof

We first prove (i). Let ℱ 1 ( I h , I m ) = c β ( x ) H ( x ) ( H ( x ) − I h ) I m − ( d h + ρ h ) I h and ℱ 2 ( I h , I m ) = b β ( x ) H ( x ) ( S ˆ − I m ) + I h − d m ( x ) I m . By simple calculation, we have ∂ ℱ 1 / ∂ I m ≥ 0 and ∂ ℱ 2 / ∂ I h ≥ 0 . It tells us that system (4.5) is cooperative. By the standard comparison principle, one knows that I h ( ⋅ , t ) , I m ( ⋅ , t ) ≥ 0 ∀ x ∈ Ω ¯ , and t ≥ 0 . If I m 0 ≠ 0 , then I m -equation of (4.5) satisfies

(4.6) ∂ I m ∂ t − D m Δ I m ≥ − d m ( x ) I m , x ∈ Ω ¯ , t > 0 ; ∂ I m ∂ v = 0 , x ∈ ∂ Ω , t > 0 .

Thanks to the comparison principle, I m ( x , t ) > 0 ∀ x ∈ Ω ¯ and t > 0 . Furthermore, since ( H ( x ) − I h ( x , t ) ) is nontrivial, I h -equation of (4.5) satisfies

∂ I h ∂ t − D h Δ I h > − ( d h + ρ h ) I h , x ∈ Ω ¯ , t > 0 ; ∂ I h ∂ v = 0 , x ∈ ∂ Ω , t > 0 ,

for some x ∈ Ω ¯ . Again from the comparison principle, I h ( x , t ) > 0 ∀ x ∈ Ω ¯ and t > 0 .

If I m 0 = 0 , it follows that I h 0 ≠ 0 as ( I h 0 , I m 0 ) is nontrivial. Hence, I h -equation of (4.5) satisfies

∂ I h ∂ t − D h Δ I h ≥ − ( d h + ρ h ) I h , x ∈ Ω ¯ , t > 0 ; ∂ I h ∂ v = 0 , x ∈ ∂ Ω , t > 0 .

Due to the comparison principle, one can obtain I h > 0 ∀ x ∈ Ω ¯ and t > 0 . This together with the continuity of I m yields that there is t 5 > 0 such that ( S ˆ − I m ) + > 0 ∀ x ∈ Ω ¯ and t ∈ ( 0 , t 5 ] . Hence, I m -equation of (4.5) satisfies

∂ I m ∂ t − D m Δ I m > − d m ( x ) I m , x ∈ Ω ¯ , t ∈ ( 0 , t 5 ] ; ∂ I m ∂ v = 0 , x ∈ ∂ Ω , t ∈ ( 0 , t 5 ] .

Again from the comparison principle, we have I m > 0 ∀ ( x , t ) ∈ Ω ¯ × ( 0 , t 5 ] . Therefore, with the help of (4.6), I m > 0 ∀ x ∈ Ω ¯ and t > 0 . This proves (i).

We next prove (ii). By (2.2), we know that S m + I m → S ˆ ( x ) in C ( Ω ¯ , R ) as t → ∞ if S m 0 + I m 0 ≠ 0 . Let M 1 = max { ‖ S ˆ ‖ ∞ , ‖ I m 0 ‖ ∞ } . From the I m -equation of (4.5), combined with the comparison principle, one can obtain that I m ≤ M 1 ∀ x ∈ Ω ¯ and t > 0 . Hence, the I h -equation of (4.5) fulfills

∂ I h ∂ t − D h Δ I h ≤ c β ( x ) M 1 − ( d h + ρ h ) I h , x ∈ Ω , t > 0 ; ∂ I h ∂ v = 0 , x ∈ ∂ Ω , t > 0 ,

i.e., I h is the lower solution of

∂ ϖ ∂ t − D h Δ ϖ = c β ( x ) M 1 − ( d h + ρ h ) ϖ , x ∈ Ω , t > 0 , ∂ ϖ ∂ v = 0 , x ∈ ∂ Ω , t > 0 , ϖ ( x , 0 ) = I h 0 ( x ) , x ∈ Ω .

This tells us that I h ( x , t ) ≤ ϖ ( x , t ) < M 2 , where M 2 = max { c ‖ β ( x ) ‖ ∞ M 1 ( d h + ρ h ) , ‖ I h 0 ‖ ∞ } . Hence, (ii) directly follows from M ˆ = max { M 1 , M 2 } . This proves Lemma 4.4.□

We next pay attention to the positive steady state of (4.5), denoted by ( I ˆ h ( x ) , I ˆ m ( x ) ) , which is the positive solution of

(4.7) − D h Δ I h = c β ( x ) H ( x ) ( H ( x ) − I h ) I m − ( d h + ρ h ) I h , x ∈ Ω , − D m Δ I m = b β ( x ) H ( x ) ( S ˆ ( x ) − I m ) + I h − d m ( x ) I m , x ∈ Ω , ∂ I h ∂ v = ∂ I m ∂ v = 0 , x ∈ ∂ Ω .

We next prove that if a positive steady state of (4.5) exists, then it is globally stable in { ( I h 0 ( x ) , I m 0 ( x ) ) ∈ X H ∗ : I h 0 ( x ) ≠ 0 , I m 0 ( x ) ≠ 0 } . The assertion below indicates that the positive steady state of (4.5) is unique if it exists.

Lemma 4.5

If the positive steady state of (4.5), ( I ˆ h ( x ) , I ˆ m ( x ) ) , exists, then it is unique.

Proof

We will prove Lemma 4.5 by the following claims.

Claim I. I ˆ h ( x ) , I ˆ m ( x ) > 0 ∀ x ∈ Ω ¯ .

From the I h -equation of (4.7), we have ( ( d h + ρ h ) − D h Δ ) I ˆ h = c β ( x ) H ( x ) ( H ( x ) − I ˆ h ) I ˆ m , which indicates that I ˆ h ≠ 0 and I ˆ m ≠ 0 . By the maximum principle yields, one directly has I ˆ h ( x ) , I ˆ m ( x ) > 0 ∀ x ∈ Ω ¯ . This proves Claim I.

Claim II. I ˆ h is a fixed point of F on a bounded set, where F : B ⊂ C ( Ω ¯ ) → C ( Ω ¯ ) takes the following form

F ( Θ ) = ( C 2 − D h Δ ) − 1 c β ( x ) H ( x ) ( H ( x ) − Θ ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I ˆ m ) + + [ C 2 − ( d h + ρ h ) ] Θ ,

and

B = { Θ ∈ C ( Ω ¯ ; R + ) : ‖ Θ ‖ ∞ ≪ C 1 } ,

for any C 1 , C 2 > 0 .

Solving the I ˆ m from (4.7) yields that I ˆ m = ( d m − D m △ ) − 1 b β ( x ) H ( x ) ( S ˆ − I ˆ m ) + I ˆ h . It then follows that

− D h Δ I ˆ h = c β ( x ) H ( x ) ( H ( x ) − I ˆ h ) ( d m − D m △ ) − 1 b β ( x ) H ( x ) ( S ˆ − I ˆ m ) + I ˆ h − ( d h + ρ h ) I ˆ h .

By adding the term C 2 I ˆ h on both sides of the above equation, we obtain the fixed point problem of F ( Θ ) for I ˆ h as long as C 1 is sufficiently large. This proves Claim II.

Claim III. The fixed point problem of F ( Θ ) is monotone on a bounded set.

For any I h , I h + ℏ ∈ B and ℏ ≥ 0 , we next show that F ( I h ) ≤ F ( I h + ℏ ) . Let us denote

F ˜ ( I h ) = c β ( x ) H ( x ) ( H ( x ) − I h ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) + + [ C 2 − ( d h + ρ h ) ] I h .

It then follows that

F ˜ ( I h + ℏ ) − F ˜ ( I h ) = − c β ( x ) H ( x ) ℏ ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) + I h + c β ( x ) H ( x ) ( H ( x ) − I h − ℏ ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) + + [ C 2 − ( d h + ρ h ) ] ℏ = ℏ c β ( x ) H ( x ) ( H ( x ) − 2 I h − ℏ ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) + + C 2 − ( d h + ρ h ) ≥ ℏ c β ( x ) H ( x ) ( − 2 I h − ℏ ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) + + C 2 − ( d h + ρ h ) .

By appealing to the elliptic estimate, one knows that

c β ( x ) H ( x ) ( − 2 I h − ℏ ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) +

is bounded for I h , I h + ℏ ∈ B and ℏ ≥ 0 . Therefore, F ˜ ( I h + ℏ ) − F ˜ ( I h ) ≥ 0 as long as C 2 is sufficiently large. This proves Claim III.

Claim IV. The fixed point problem of F ( Θ ) is sublinear on a bounded set in the sense that for any κ ∈ ( 0 , 1 ) and I h ∈ B with I h ≫ 0 , κ F ( I h ) ≪ F ( κ I h ) for all x ∈ Ω ¯ .

Simple calculations imply that

κ F ˜ ( I h ) = c β ( x ) H ( x ) ( H ( x ) − I h ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) + + [ C 2 − ( d h + ρ h ) ] κ I h < c β ( x ) H ( x ) ( H ( x ) − κ I h ) ( d m − D m Δ ) − 1 b β ( x ) H ( x ) ( S ˆ − I m ) + + [ C 2 − ( d h + ρ h ) ] κ I h = F ˜ ( κ I m ) .

Furthermore, due to the strong positivity of the operator ( C 2 − D h Δ ) − 1 , Claim IV directly follows.

Based on above preparations, we are now in a position to show the uniqueness of ( I ˆ h ( x ) , I ˆ m ( x ) ) whenever it exists. If there are two distinct positive steady states, denoted by ( I h 1 , I m 1 ) and ( I h 2 , I m 2 ) , by the I h -equation of (4.7), we know that I h 1 ≠ I h 2 . In the following, we assume that I h 1 ≰ I h 2 . Set

κ = max { κ ˜ ≥ 0 : κ ˜ I h 1 ≤ I h 2 } .

It follows that κ ∈ ( 0 , 1 ) . According to the definition of κ , one can obtain κ I h 1 ≤ I h 2 and κ I h 1 ( x 0 ) = I h 2 ( x 0 ) for some x 0 ∈ Ω ¯ . By Claim I–IV and selecting suitable C 1 and C 2 such that F ( I h 1 ) = I h 1 and F ( I h 2 ) = I h 2 , we consequently obtain that

κ I h 1 = κ F ( I h 1 ) ≪ F ( κ I h 1 ) ≤ F ( I h 2 ) = I h 2 , ∀ x ∈ Ω ¯ ,

i.e., κ I h 1 ≪ I h 2 ∀ x ∈ Ω ¯ , which results in a contradiction with κ I h 1 ( x 0 ) = I h 2 ( x 0 ) . This proves Lemma 4.5.□

The following result demonstrates that the positive steady state, ( I ˆ h ( x ) , I ˆ m ( x ) ) , of (4.5) is globally asymptotically stable whenever it exists and ( I h 0 , I m 0 ) ∈ X H ∗ .

Theorem 4.3

Provided that the positive steady state ( I ˆ h ( x ) , I ˆ m ( x ) ) of (4.5) exists. For any nonnegative nontrivial initial value ( I h 0 , I m 0 ) ∈ X H ∗ , the solution of (4.5) satisfies

(4.8) lim t → ∞ ( I h ( ⋅ , t ) , I m ( ⋅ , t ) ) = ( I ˆ h ( x ) , I ˆ m ( x ) ) in X H ∗ .

Proof

Recall that system (4.5) is cooperative. Let Φ ˜ ( t ) : X H ∗ → X H ∗ be the semiflow generated by the solution of (4.5), i.e., Φ ˜ ( t ) ( I h 0 , I m 0 ) = ( I h ( ⋅ , t ) , I m ( ⋅ , t ) ) for all t ≥ 0 . Following the arguments as those in [11,29], Φ ˜ ( t ) is monotone.

Based on Lemma 4.4 and 4.5, we let ( I ˆ h ( x ) , I ˆ m ( x ) ) be the unique positive steady state of (4.5). Choosing sufficiently small ζ 1 > 0 such that ( I ̲ h , I ̲ m ) = ζ 1 ( I ˆ h , I ˆ m ) satisfies

(4.9) − D h Δ I ̲ h ≤ c β ( x ) H ( x ) ( H ( x ) − I ̲ h ) I ̲ m − ( d h + ρ h ) I ̲ h ( x , t ) , x ∈ Ω , − D m Δ I ̲ m ≤ b β ( x ) H ( x ) ( S ˆ − I ̲ m ) + I ̲ h − d m ( x ) I ̲ m , x ∈ Ω , ∂ I ̲ h ∂ t = ∂ I ̲ m ∂ t = 0 , x ∈ ∂ Ω , I ̲ h ( x ) ≤ I h 0 ( x ) , I ̲ m ( x ) ≤ I m 0 ( x ) , x ∈ Ω .

This together with [29, Corollary 7.3.6] and Lemma 4.5 ensures that Φ ˜ ( t ) ( I ̲ h , I ̲ m ) is monotone increasing in t , and Φ ˜ ( t ) ( I ̲ h , I ̲ m ) → ( I ˆ h , I ˆ m ) in X H ∗ as t → ∞ .

Similarly, choosing sufficiently large ζ 2 > 0 such that ( I ¯ h , I ¯ m ) = ζ 2 ( I ˆ h , I ˆ m ) satisfies the inverse inequalities of (4.9). Again from [29, Corollary 7.3.6] and Lemma 4.5, Φ ˜ ( t ) ( I ¯ h , I ¯ m ) is monotone decreasing in t , and Φ ˜ ( t ) ( I ¯ h , I ¯ m ) → ( I ˆ h , I ˆ m ) in X H ∗ as t → ∞ .

Thanks to ( I ̲ h , I ̲ m ) ≤ ( I h 0 , I m 0 ) ≤ ( I ¯ h , I ¯ m ) and the monotonicity of Φ ˜ ( t ) , we directly obtain that

Φ ˜ ( t ) ( I ̲ h , I ̲ m ) ≤ Φ ˜ ( t ) ( I h 0 , I m 0 ) ≤ Φ ˜ ( t ) ( I ¯ h , I ¯ m ) ∀ t ≥ 0 .

The assertion (4.8) directly follows.

By letting ζ ¯ > 0 be any sufficiently small number with ( 1 − ζ ¯ ) ( I ˆ h , I ˆ m ) ≤ ( I h 0 , I m 0 ) ≤ ( 1 + ζ ¯ ) ( I ˆ h , I ˆ m ) . Again from [29, Corollary 7.3.6], one knows that

( 1 − ζ ¯ ) ( I ˆ h , I ˆ m ) ≤ ( I h ( ⋅ , t ) , I m ( ⋅ , t ) ) ≤ ( 1 + ζ ¯ ) ( I ˆ h , I ˆ m ) , t ≥ 0 .

Thus, the local stability of ( I ˆ h , I ˆ m ) directly follows. Consequently, we arrive at the conclusion that the unique positive steady state, ( I ˆ h , I ˆ m ) , is globally asymptotically stable.□

In the following, by appealing to the arguments of asymptotically autonomous semiflows [33, Theorem 4.1], we show the main result on the unique positive steady state of (1.2).

Theorem 4.4

Let ( I ˆ h ( ⋅ ) , S ˆ m ( ⋅ ) , I ˆ m ( ⋅ ) ) be the unique positive steady state of (1.2) and X H 0 be defined in (4.3). Then, for any ( I h 0 ( ⋅ ) , S m 0 ( ⋅ ) , I m 0 ( ⋅ ) ) ∈ X H 0 , ( I ˆ h ( ⋅ ) , S ˆ m ( ⋅ ) , I ˆ m ( ⋅ ) ) is globally asymptotically stable provided that ℜ 0 > 1 , i.e., the solution of (1.2) fulfills

lim t → ∞ ( I h ( ⋅ , t ) , S m ( ⋅ , t ) , I m ( ⋅ , t ) ) = ( I ˆ h ( ⋅ ) , S ˆ m ( ⋅ ) , I ˆ m ( ⋅ ) ) uniformly o n Ω ¯ .

Proof

Combined with ℜ 0 > 1 , Theorem 4.2 and the previous results, the existence and uniqueness of the positive steady state of (1.2) are ensured. We are now in a position to rewrite the H i and V i equations of (1.2) as follows:

(4.10) ∂ I h ∂ t − D h Δ I h = c β ( x ) H ( x ) ( H ( x ) − I h ) I m − ( d h + ρ h ) I h , x ∈ Ω , t > 0 , ∂ I m ∂ t − D m Δ I m = b β ( x ) H ( x ) ( S ˆ − I m ) + I h + Q − d m ( x ) I m , x ∈ Ω , t > 0 , ∂ I h ∂ t = ∂ I m ∂ t = 0 , x ∈ ∂ Ω , t > 0 , I h ( 0 , x ) = I h 0 ( x ) , I m ( 0 , x ) = I m 0 ( x ) , x ∈ Ω ,

with Q ( ⋅ , t ) = b β ( x ) H ( x ) ( S m − ( S ˆ − I m ) + ) I h . Thanks to ∣ S m − ( S ˆ − I m ) + ∣ ≤ ∣ S m − S ˆ + I m ∣ , one knows that Q ( ⋅ , t ) → 0 as t → ∞ . According to [22, Proposition 1.1], one knows that (4.10) is asymptotic to (4.5). By Theorems 4.2 and 4.3, the ω -limit set of system (4.10) is contained in a stable set of ( I ˆ h , I ˆ m ) of (4.5). Furthermore, from the generalized results in [33, Theorem 4.1], one directly obtain that ( I h , I m ) → ( I ˆ h ( ⋅ ) , I ˆ m ( ⋅ ) ) as t → ∞ in C ( Ω ¯ ; R 2 ) . On the other hand, as ( S m + I m ) → S ˆ and S ˆ m ( ⋅ ) + I ˆ m ( ⋅ ) = S ˆ , it gives S m → S ˆ m ( ⋅ ) in C ( Ω ¯ , R ) as t → ∞ . This proves Theorem 4.4.□

4.4 Global stability of positive steady state: a homogeneous case

By Lyapunov functions, we next explore the stability of equilibria of the following system:

(4.11) ∂ I h ∂ t − D h Δ I h = c β H ( H − I h ) I m − ( d h + ρ h ) I h , x ∈ Ω , t > 0 , ∂ S m ∂ t − D m Δ S m = μ − b β H S m I h − d m S m , x ∈ Ω , t > 0 , ∂ I m ∂ t − D m Δ I m = b β H S m I h − d m I m , x ∈ Ω , t > 0 , ∂ I h ∂ ν = ∂ S m ∂ ν = ∂ I m ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , ( I h ( x , 0 ) , S m ( x , 0 ) , I m ( x , 0 ) ) = ( I h 0 ( x ) , S m 0 ( x ) , I m 0 ( x ) ) ∈ C ( Ω , R + 2 ) , x ∈ Ω .

Here, (4.11) is the homogeneous case of (1.2), i.e., all parameter functions are supposed to be positive constants. Obviously, (4.11) admits a disease-free equilibrium, written by [ E 0 ] = ( 0 , S ˘ m , 0 ) , where S ˘ m = μ d m . The basic reproduction number of (1.2) will reduce to

(4.12) [ ℜ 0 ] = c b β 2 μ H ( d h + ρ h ) d m 2 .

It should be mentioned here that the main Theorems 4.1 and 4.2 are still valid for system (4.11).

A constant positive steady state of (4.11), denoted by E ∗ = ( I h ∗ , S m ∗ , I m ∗ ) , satisfies the following algebraic equations:

(4.13) 0 = c β H ( H − I h ∗ ) I m ∗ − ( d h + ρ h ) I h ∗ , 0 = μ − b β H S m ∗ I h ∗ − d m S m ∗ , 0 = b β H S m ∗ I h ∗ − d m I m ∗ .

After elementary calculations, we can obtain

I h ∗ = d m H ( [ ℜ 0 ] − 1 ) d m [ ℜ 0 ] + b β , S m ∗ = μ H b β I h ∗ + 1 d m , and I m ∗ = ( d h + ρ h ) I h ∗ H c β ( H − I h ∗ ) .

Hence, (4.11) admits a unique constant positive steady state E ∗ when [ ℜ 0 ] > 1 .

The main result is as follows.

Theorem 4.5

Let [ ℜ 0 ] be defined in (4.12) and any initial value ( I h 0 ( x ) , S m 0 ( x ) , I m 0 ( x ) ) ∈ X H . We have the following statements:

  1. If [ ℜ 0 ] < 1 , then [ E 0 ] is globally asymptotically stable.

  2. If [ ℜ 0 ] > 1 , then system (4.11) is uniformly persistent, and the unique positive steady state of (4.11), E ∗ , is globally attractive.

Proof

We first prove (i). Let g ( α ) = α − 1 − ln α , α > 0 . It is well-known that g ( α ) ≥ 0 and g ( α ) = 0 iff α = 1 . Constructing a Lyapunov functional for [ E 0 ] ,

L E 0 ( t ) = ∫ Ω ( L 1 ( x , t ) + L 2 ( x , t ) + L 3 ( x , t ) ) d x ,

where L 1 = 1 H I h , L 2 = c β d m H S ˘ m g ( S m S ˘ m ) , and L 3 = c β d m H I m . Differentiating L i , i = 1 , 2 , 3 , with respect to the solution for system (4.11), one has

∂ L 1 ∂ t = 1 H D h Δ I h + c β H ( H − I h ) I m − ( d h + ρ h ) I h , ∂ L 2 ∂ t = c β d m H 1 − S ˘ m S m D m Δ S m + d m S ˘ m − d m S m − b β H S m I h , ∂ L 3 ∂ t = c β d m H D m Δ I m + b β H S m I h − d m I m .

Therefore,

d L d t = ∫ Ω 1 H D h Δ I h d x + ∫ Ω c β d m H 1 − S ˘ m S m D m Δ S m d x + ∫ Ω c β d m H D m Δ I m d x + ∫ Ω 1 H c β H ( H − I h ) I m − ( d h + ρ h ) I h d x + ∫ Ω c β d m H 1 − S ˘ m S m ( d m S ˘ m − d m S m ) d x + ∫ Ω c b β 2 H 2 d m S ˘ m I h − c β H I m d x .

Since all equations of system (4.11) subject to Neumann boundary condition, we have ∫ Ω Δ u i u i d x = ∫ Ω ‖ ∇ u i ‖ 2 u i 2 d x and ∫ Ω Δ u i d x = 0 . Rearranging the terms yields that

d L d t = − c β d m H D m S ˘ m ∫ Ω ‖ ∇ S m ‖ 2 S m 2 d x − c β H S m ∫ Ω ( S ˘ m − S m ) 2 d x + ( d h + ρ h ) H ∫ Ω I h ( ℜ 0 − 1 ) d x − c β H 2 ∫ Ω I m I h d x .

Consequently, the global attractivity of [ E 0 ] is obtained if ℜ 0 < 1 . Combined with Theorem 4.1, we conclude that the [ E 0 ] is globally attractive.

We next prove (ii). The uniform persistence of (4.11) directly follows from Theorem 4.2. Construct a Lyapunov function for E ∗ ,

L E ∗ ( t ) = ∫ Ω ( V 1 ( x , t ) + V 2 ( x , t ) + V 3 ( x , t ) ) d x ,

where V 1 = c β H ( d h + ρ h ) I m ∗ I h ∗ I h ∗ g I h I h ∗ , V 2 = c β H d m S m ∗ g S m S m ∗ , and V 3 = c β H d m I m ∗ g I m I m ∗ .

Differentiating V i , i = 1 , 2 , 3 with respect to the solution for system (4.11) and with the help of (4.13), one obtains

∂ V 1 ∂ t = c β H ( d h + ρ h ) I m ∗ I h ∗ 1 − I h ∗ I h D h Δ I h + c β H ( H − I h ∗ ) I m + c β H ( I h ∗ − I h ) I m − c β H ( H − I h ∗ ) I m ∗ I h I h ∗ = 1 − I h ∗ I h c β H ( d h + ρ h ) I m ∗ I h ∗ D h Δ I h + c β H I m − c β H I m ∗ I h I h ∗ + c β H ( I h ∗ − I h ) I m c β H ( d h + ρ h ) I m ∗ I h ∗ = c β H I m ∗ I m I m ∗ − I h I h ∗ − I m I m ∗ I h ∗ I h + 1 + c β ( I h ∗ − I h ) I m H ( d h + ρ h ) I h ∗ 1 − I h ∗ I h + 1 − I h ∗ I h c β H ( d h + ρ h ) I m ∗ I h ∗ D h Δ I h ,

∂ V 2 ∂ t = c β d m H 1 − S m ∗ S m D m Δ S m + d m S m ∗ − d m S m − b β H S m I h + b β H S m ∗ I h ∗ = c β d m H 1 − S m ∗ S m D m Δ S m − d m S m ( S m − S m ∗ ) 2 c β d m H + c β d m H 1 − S m ∗ S m − b β H S m I h + b β H S m ∗ I h ∗

and

∂ V 3 ∂ t = c β d m H 1 − I m ∗ I m D m Δ I m ( x , t ) + b β H S m I h − b β S m ∗ I h ∗ I m H I m ∗ .

Therefore,

d L E ∗ d t = ⊕ + ∫ Ω c β H I m ∗ I m I m ∗ − I h I h ∗ − I m I m ∗ I h ∗ I h + 1 + c β ( I h ∗ − I h ) I m H ( d h + ρ h ) I h ∗ 1 − I h ∗ I h d x + c b β 2 H 2 d m S m ∗ I h ∗ ∫ Ω − S m I h S m ∗ I h ∗ + 1 + I h I h ∗ − S m ∗ S m + S m I h S m ∗ I h ∗ − I m I m ∗ − S m I h I m ∗ S m ∗ I h ∗ I m + 1 d x = ⊕ + ∫ Ω c β H I m ∗ 3 − S m ∗ S m − I m I m ∗ I h ∗ I h − I m ∗ S m I h I m S m ∗ I h ∗ d x − ∫ Ω c 2 β 2 ( I h ∗ − I h ) 2 I m ∗ I m H 2 ( d h + ρ h ) I h ∗ I h d x ,

where

⊕ = − c β H ( d h + ρ h ) I m ∗ D h ∫ Ω ‖ ∇ I h ‖ 2 I h 2 d x − c β d m H D m S m ∗ ∫ Ω ‖ ∇ S m ‖ 2 S m 2 d x − c β d m H ∫ Ω d m S m ( S m − S m ∗ ) 2 d x − c β d m H D m I m ∗ ∫ Ω ‖ ∇ I m ‖ 2 I m 2 d x .

It follows from the nonnegativity of H ( x ) and [31, Section 9.9], we have that E ∗ is globally attractive. This proves Theorem 4.5.□

5 Summary and discussion

This article performs a comprehensive analysis of the global dynamics of system (1.2). System (1.2) is formulated with the assumption that the total populations stabilizing at H ( x ) in a spatially heterogeneous environment and all model parameters being dependent on the spatial variable x . Considering such general settings are meaningful and important, although model (1.2) is the one in [13] without considering the mobility of the latent mosquitoes in extrinsic incubation period confined in a spatial domain. We aim to establish a framework for studying such problems from a technical perspective as be seen in the whole article.

We shall briefly summarize the main results in this article. The first result that we want to obtain is the well-posedness of model (1.2). Theorem 2.1 confirms that the existence, uniqueness, nonnegativity, and ultimate boundedness of the solution, i.e., the solution exists globally for any initial data ϕ ∈ X H . Furthermore, system (1.2) generates a semiflow admitting a connected global attractor in X H . In a general theoretical approach, the basic reproduction number is defined by the spectral radius of the next-generation operator, lacking visualization. For the case a model containing only one infective component, the basic reproduction number can be demonstrated by a variational characterization, or the sign of the principal eigenvalue of an eigenvalue problem. Note that model (1.2) contains two infective components, i.e., I h and I m , which makes the analysis more difficult. Inspired by a recent work [19], we address the relation between the basic reproduction number and the local basic reproduction number, where local basic reproduction number is a function of x ∈ Ω . Specifically, we write the next-generation operator as the product of local basic reproduction number and strongly positive compact linear operators with spectral radii one, see Theorem 3.1. Lemmas 3.3 and 3.4 and Theorem 3.3 investigate ℜ 0 quantitatively when the diffusion rates D m and D h approach ∞ . Lemma 3.5 and Theorem 3.4 investigate ℜ 0 quantitatively when the diffusion rates D m and D h approach zero. Indeed, as can be seen from the above results, the basic reproduction number depends on the spatial heterogeneity and the diffusion coefficients, although such dependence is difficult to analyze.

By appealing to the comparison principle and the associated eigenvalue problem, we show the threshold-type results relying on the basic reproduction number, see Theorems 4.1 and 4.2. Biologically, the asymptotic behaviors of the solution of (1.2) are dominated by the basic reproduction number. Specifically, although the disease spreads in a spatially heterogeneous environment, the disease will be extinct if ℜ 0 < 1 , and while the disease will spread if ℜ 0 > 1 . Furthermore, we investigate the existence and uniqueness of the positive steady state. The global attractivity of the positive steady state for system (1.2) is established by using the theory of asymptotically autonomous semiflows [19,33,41] (see Theorem 4.4). For the case that all parameters are independent of space, the global attractivity of the positive steady state is achieved by Lyapunov function (see Theorem 4.5). We would like to mention that the analysis carried out here is also valid for the other epidemic model with random diffusion and two infective components.

Acknowledgments

The authors would like to thank the referees and editor for many helpful comments. J. Wang was supported by the National Natural Science Foundation of China (nos. 12071115 and 11871179) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems.

  1. Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this article.

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Received: 2022-02-26
Revised: 2023-01-25
Accepted: 2023-01-27
Published Online: 2023-03-01

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2022-0295
Schlagwörter für diesen Artikel
mosquito-borne disease model; basic reproduction number; spatial heterogeneity; uniform persistence; Lyapunov function
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Artikel in diesem Heft

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
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  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
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Heruntergeladen am 7.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/anona-2022-0295/html
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