key: cord-0997295-pqrc5w4d
authors: Dong, Lingmin; Li, Bo; Zhang, Guanghui
title: Analysis on a Diffusive SI Epidemic Model with Logistic Source and Saturation Infection Mechanism
date: 2022-02-23
journal: Bull Malays Math Sci Soc
DOI: 10.1007/s40840-022-01255-7
sha: 59009945aca2e1e71550b0b0a516dfed67d31dab
doc_id: 997295
cord_uid: pqrc5w4d

In this paper, we consider an SI epidemic reaction–diffusion model with logistic source and saturation infection mechanism. We first establish the uniform boundedness and the extinction and persistence of the infectious disease in terms of the basic reproductive number. We also discuss the global stability of the unique endemic equilibrium when the spatial environment is homogeneous. Then we investigate the asymptotic behavior of the endemic equilibria in the heterogeneous environment when the movement rate of the susceptible and infected populations is small. Our results, together with the other two related epidemic models , not only show that the logistic growth, the infection mechanism, and the population movement can play an important role in the transmission dynamics of disease, but also suggest that increasing the inhibitory effect of the susceptible individuals instead of reducing the mobility of the populations can control the epidemic disease modeled by the SI system under consideration.

where the habitat Ω is a bounded domain in R n with the smooth boundary ∂Ω; S(x, t) and I (x, t) represent, respectively, the densities of susceptible and infected individuals at location x ∈ Ω and time t > 0; the positive constants d S and d I represent the diffusion coefficients of the susceptible individuals and infected individuals, respectively; the positive Hölder continuous function β(x) and γ (x) stand for, respectively, the rates of disease transmission and recovery at x. The homogeneous Neumann boundary conditions are imposed which means no flux can cross the boundary ∂Ω. Various kinds of SIS (susceptible-infected-susceptible) epidemic reaction-diffusion systems with standard incidence rates have been extensively studied; one may refer to [7-9, 16, 18, 21, 26, 27, 29, 31-37, 44, 46, 48, 49, 51, 53-55, 58] and the references therein. By direct calculation, we can find that in (1) that the total population Ω (S(x, t) + I (x, t))dx = Ω (S 0 (x) + I 0 (x))dx = N is fixed for all t > 0. In general, the total population number can not always keep a constant in the real world. On the other hand, the birth date of susceptible population and the death rate induced by disease are important factors in the evolution of disease transmission; see [19, 20] . Then Li, Peng and Wang [35] used the recruitment term to describe the growth of the susceptible population. Moreover, most mathematical models imply that the logistic source seems to be a suitable choice of describing the intrinsic growth of the susceptible individuals; see, for instance, [12, 15] . Therefore, Li et al. [32] introduced the logistic source a(x)S − b(x)S 2 in the first equation of (1) to model the susceptible population growth. More precisely, the model proposed in [32] reads as follows.

S(x, 0) = S 0 (x), I (x, 0) = I 0 (x),

where a(x) and b(x) are positive Hölder continuous functions, a(x) and a(x) b(x) represent, respectively, the birth of susceptible populations and the intrinsic carrying capacity.

On the other hand, Huo et al. [22] discussed the following SIS epidemic model with saturated incidence rate and the logistic sources

The models with the above-mentioned saturated incidence rate have been investigated by many mathematicians, see [24, 52, 56, [60] [61] [62] . Motivated by the above works, in this paper, we consider the following SI epidemic model:

where the parameters μ(x) and η(x) are the positive Hölder continuous functions on Ω, μ(x) accounts for the rate of natural death and η(x) is the death rate caused by the disease; α is a positive constant and measures the inhibitory effect. It is worth mentioning that Zhang et. al [62] studied a time-delay SIR epidemic model with the same incidence rate and the logistic source. One may further refer to [41, Chapter 15] for the derivation of the ODE version of model (4) . As in [4, 28, 64] , we introduce the following favorable set:

in order to reflect the feature of the heterogeneous environment. We always assume that the favorable set F + is nonempty through this paper. Then one can apply the standard theory of semilinear parabolic systems to conclude that (4) has a classical solution provided that the initial functions S 0 (x) and I 0 (x) are nonnegative continuous functions. If additionally Ω I 0 dx > 0, it follows from the strong maximum principle and the Hopf boundary Lemma of parabolic equations that S(x, t) > 0 and I (x, t) > 0 for x ∈ Ω and t > 0. The steady-state problem corresponding to (4) satisfies the following elliptic system:

Biologically, only nonnegative solutions of (5) are of our interest. The solution (S(x), I (x)) of (5) is called a disease-free equilibrium (DFE) if I (x) ≡ 0 for all x ∈ Ω; and the solution (S(x), I (x)) of (5) is called an endemic equilibrium (EE) if I (x) > 0 for some x ∈ Ω. Then we obtain that any EE satisfies S(x) > 0 and I (x) > 0 for x ∈ Ω by the strong maximum principle and the Hopf boundary lemma of elliptic equations.

In the present paper, our first goal is to study the extinction and persistence of the disease via the basic reproduction number R 0 . Indeed, Theorem 2 tells us that the disease vanishes if R 0 < 1, whereas if R 0 > 1 and I 0 (x) ≡ 0, the solutions of system (4) is uniformly persistent and so the EE exists. Furthermore, we establish the global stability of EE by constructing a suitable Lyapunov function; see Theorem 3. As our second goal, we study the asymptotic profile of EE as the immigration rate of susceptible or infected individuals tends to zero (see . Our results show that the infectious disease always persists though the movement rate of susceptible or infected populations is controlled to be sufficiently small. Similar conclusions still hold with systems (2) and (3). These theoretical results imply that the controlling of the mobility of susceptible or infected populations in the epidemic model with logistic sources is not an effective strategy to eradicate the disease infection. In the discussion section, we will compare our results with those for the other two related models (2) and (3), in order to understand the effect of the incidence rate, the logistic sources, and the mobility of the population; see the last section for more details.

The rest of this paper is organized as follows. In Sect. 2, the dynamics of the epidemic model (4) are analyzed in terms of the basic reproduction number. First, the uniform boundedness of solutions to (4) is established; then, the definition and properties of the basic reproduction number are studied, and finally, the long-time behavior of system (4) by R 0 is obtained. Section 3 is devoted to studying the global stability of EE and exploring the spatial distribution of the disease if the movement of the susceptible or infected populations is small. In the last section, the discussion of our paper and the comparisons of the results between our problem (4) and the related systems (2) and (3) are given.

In this section, we aim to establish the dynamical behaviors of (4) in terms of the basic reproduction number R 0 . First of all, we study the auxiliary parabolic problem

The associated steady-state problem satisfies

Denote byS(x) the unique positive solution of (7) if it exists. Then, we use [4, Theorem A.1] to derive the following conclusion Lemma 1 Suppose that F + is nonempty. Consider the eigenvalue problem with indefinite weight:

If Ω (a(x) − μ(x)) dx < 0, let Λ 1 (a(x) − μ(x)) be the principal positive eigenvalue of (8) , and if Ω (a(x) − μ(x)) dx ≥ 0, set Λ 1 (a(x) − μ(x)) = 0. Then the problem has a unique positive steady-stateS, which is a global attractor for nonnegative solutions when 0 < d S < 1/Λ 1 (a(x) − μ(x)). When d S ≥ 1/Λ 1 (a(x) − μ(x)), there is no positive steady state for (6) , and all nonnegative solutions to (6) decay to 0 as t → ∞.

From now on, for the sake of simplicity, let us denote

The uniform bounds of solutions of (4) are given as follows.

Theorem 1 There exists a positive constant C independent of initial data such that

for some large time T > 0.

Proof It follows from the first equation of (4) that

Let W be the solution of the following problem

Applying the standard comparison principal, we get

Moreover, it is well known that

In what follows, we use C to represent a positive constant, which is independent of d S but allows to vary from line to line. Thus, we can find a large time T 1 > 0 such that

Denote

Then, it follows from (4) that

Due to (10), we obtain

where k = Ca * |Ω| and m = μ * . This yields

That is,

Therefore, we have lim sup

By setting

Now, we use [14, Lemma 2.1] (due to [30] ) with σ = p 0 = 1 to the system (4) conclude that there exists a constant C > 0 independent of initial data such that

for some T 2 ≥ T 1 . This completes our proof.

In this subsection, we will define the basic reproduction number R 0 and show the properties of R 0 . Linearizing the equation of I of system (4) at DFE (S(x), 0), we have the following parabolic problem

As in [1] , we define the basic reproduction number R 0 :

It should be noticed that the basic reproduction number R 0 defined by (13) implicitly depends not only on the diffusion rate d I of the infected population but also on the diffusion rate d S of the susceptible population and saturation rate α. To stress the dependence of R 0 on these parameters, we denote by R 0 (d I , d S , α) as the basic reproduction number of system (4). Let (λ 1 , ψ 1 ) be the principal eigenpair of the eigenvalue problem:

Then, we have the following properties.

(a) For fixed d S , α > 0, then

sign for x ∈ Ω. Then there exists a threshold value d I ∈ (0, ∞) so that

The proof of Proposition 1 is similar to that of [1, Lemma 2.3], and hence, the details are omitted.

, the unique positive solution of (7) satisfies

Then the principal eigenvalue λ 1 of the eigenvalue problem (14) converges to λ * 1 as

Furthermore, R * 0 > 1 when λ * 1 < 0, R * 0 = 1 when λ * 1 = 0 and R * 0 < 1 when λ * 1 > 0.

It turns out that the long-time dynamics of solutions of (4) is completely determined by R 0 . More precisely, we have Theorem 2 Let (S, I ) be the unique solution of (4). Then the following statements hold.

uniformly for x ∈ Ω, whereS(x) is the unique positive solution of (6) . (4) is uniformly persistent in the sense that for I (·, 0) ≡ 0, there exists a constant 0 > 0 independent of the initial data, such that any solution (S, I ) satisfies

uniformly for x ∈ Ω. Furthermore, (4) admits at least one EE.

Proof By the first equation of (4), we see that

Then, using the standard comparison principle for parabolic equation, we obtain

where S is the unique solution of (6). Moreover, it follows from Lemma 1 that

For any given small ε ≥ 0, there exists a large time T > 0 such that

Note that R 0 < 1. Then, we use Proposition 1 to conclude that λ 1 > 0. Let λ 1 (ε) be the principal eigenvalue of (14) withS(x) replaced by (S(x) +ε) and let ψ 1 (x) > 0 be the corresponding eigenfunction. Thus, we can choose small ε > 0 such that λ 1 (ε) > 0 by the continuous dependence the principal eigenvalue on the parameters. For such ε, we apply (18) and the second equation of (4) to see that I satisfies

It is easily seen that u(x, t) satisfies the following auxiliary system

It follows from the parabolic comparison principle that

Using this fact, for any small ε > 0, we can findT such that

It is easily seen from the first equation in (4) that

Denote byW the unique positive solution of the following problem

Then, we use the standard comparison principle for parabolic equations to infer that

Here we used the fact that a(x) − εβ * − μ(x) > 0 for some x ∈ Ω, which ensures the existence of the positive solutionS ε (x). By the arbitrariness of ε, letting ε → 0, we easily obtain lim inf t→∞ S(x, t) ≥S(x) uniformly for all x ∈ Ω.

Therefore, in light of (17), one can observe that

This completes the proof of (i).

Next, we will claim that (ii) holds. Assume that R 0 > 1. We make use of the arguments of [40, Theorem 2.1]. Let X = C(Ω, R 2 + ),

and X = X 0 ∪ ∂ X 0 . For a given (S 0 , I 0 ) ∈ X , the system (4) has a semiflow, denoted by Ψ (t), and

where the (S(·, t), I (·, t)) is the unique solution of (4). In light of Theorem 1, Ψ (t) is point dissipative. It also follows from the standard parabolic L p -theory and embedding theorems that Ψ (t) is compact from X to X for any fixed t > 0. By the uniqueness of solutions, we observe that I (x, t) ≡ 0 for all t ≥ 0. Then, by the similar process as in the proof of assertion (i), we can get S(x, t) →S(x) uniformly on Ω as t → ∞.

This proves that (S(x), 0) attracts (S 0 , I 0 ) ∈ ∂ X 0 .

Set M 0 = (S(x), 0). For any given >, we are going to show that

Suppose that

for some (S 0 , I 0 ) ∈ X 0 . Without loss of generality, there exists T 0 > 0 such that d(Ψ (t)(S 0 , I 0 ), M 0 ) < . Then, it is clearly that

Due to R 0 > 1, it follows from Proposition 1(d) that λ 1 < 0. We can choose a positive constant small enough such that λ 1 ( ) < 0, and (λ 1 ( ), Φ 1 ) is the eigenpair of the eigenvalue problem

where the positive constant δ will be chosen below.

Then ω satisfies

It follows from (20) that the second equation of (4) satisfies

x ∈ Ω, we obtain that I is an upper solution to the problem (21) , that is, ω(x, t) ≤ I (x, t + T 0 ) for x ∈ Ω and t > 0. It follows from λ 1 ( ) < 0 that I (x, t) → ∞ uniformly on Ω as t → ∞, which contradicts Theorem 1. Finally, we can use the argument of [40, 50] to derive the desired conclusion of (ii). The proof is complete.

In this subsection, we study the global stability of the EE of problem (4) in the spatially homogeneous environment. That is, we consider

where γ, α, β, μ, η are positive constants and a > μ. Clearly, (22) [b+α(a−μ)](μ+η) > 1.

By elementary calculation, we have from the two equations of (22) that

Now, we consider the global stability of the EE under certain conditions as follows.

Proof We choose the following Lyapunov functional

For convenience, let us denote 

By (23), we have Then it is easy to see that the maximal invariant subset of E is (Ŝ,Î ). By some standard arguments, we see that

Moreover, since we have the L ∞ estimates of S and I in Theorem 1, by some standard arguments, we know

for some positive constant C 0 . Therefore, the Sobolev embedding theorem allows one to assert

that is, (Ŝ,Î ) attracts all solutions of (4). Furthermore, using a similar process as in [43, Lemma 3.1], we see that the EE is globally asymptotically stable. This completes the proof.

If α = 0, condition (23) always holds, and thus (Ŝ,Î ) is globally asymptotically stable as long as it exists. However, due to the appearance of the saturated incidence rate (α > 0), (Ŝ,Î ) may be unstable. Let us denote f (S, I ), g(S, I ) as in the proof of Theorem 3. Then the Jacobian of the system (22) evaluated at (Ŝ,Î ) can be obtained easily as

By checking the conditions of Turing instability [41] , (Ŝ,Î ) is unstable if

However, the above conditions fail when α = 0, because we have

This section is devoted to the investigation of the asymptotic behavior of the EE of (4) for the small mobility of susceptible or infected individuals in the spatially heterogeneous environment.

In this subsection, we aim to establish the asymptotic profiles of any positive solution of (4) as d S → 0 while d I > 0 is fixed. Our main result can be stated as follows. 

where I * * is a positive solution to

Proof We divide our proof into three steps as follows.

Step 1. The estimates of (S(x), I (x)) of (5). Let S( 

Thus, it holds that

Here and in what follows, the positive constant C does not depend on d S > 0 which varies from place to place. We rewrite the second equation of (5) as follows

As

we use the Harnack-type inequality (see, e.g., [38] or [43, Lemma 2.2] ) to conclude that

In view of (5), it follows that

Now, we use (27) and integrate the second equation of system (5) over Ω to get

In view of (26) and (27), it follows that

Suppose that I has no positive lower bound, we can find a subsequence of d S → 0, say d n := d S,n , satisfying d n → 0 as n → ∞, and a corresponding positive solution (S n , I n ) := (S d S,n , I d S,n ) of (5) with d S = d n , such that min Ω I n → 0. Then, we apply (26) to obtain that I n → 0 uniformly on Ω, as n → 0.

We may choose arbitrarily small > 0 such that

for all large n.

This fact, together with the first equation of (5), implies that for all large n, (S n , I n ) satisfies

Given any large n, consider the following two auxiliary systems:

and

Denote by w n and v n the unique positive solution of (30) and (31), respectively. Using a simple comparison argument, we deduce that w n ≤ S n ≤ u n on Ω, for all large n.

By the similar argument to those in [13, Lemma 2.2], it is not hard to show that

Letting n → ∞ in (32) gives

Due to the arbitrary choice of small , one immediately gets

From the second equation of (5), I n satisfies

DefineĨ n := I n I n L ∞ (Ω) .

Then Ĩ n L ∞ (Ω) = 1 for all n ≥ 1, andĨ n solves

By a standard compactness argument for elliptic equations and passing to a further subsequence (if necessary), we may assume that

whereĨ ∈ C 1 (Ω) withĨ ≥ 0 on Ω and Ĩ L ∞ (Ω) = 1. In view of (34), (36) and (37), it follows thatĨ satisfies

From the Harnack-type inequality (see, [38] or [43, Lemma 2.2]), it follows thatĨ > 0 on Ω. For the uniqueness of the principal eigenvalue, it follows that λ * 1 = 0. This is a contradiction with our assumption that λ * 1 < 0. Thus, I has a positive lower bound C, which is independent of 0 < d S ≤ 1.

Step 2. Convergence of I . Obviously, I satisfies

In view of (25) and (29), we have

By the standard L p -estimate for elliptic equations (see, e.g., [17] ), we see that I W 2, p (Ω) ≤ C for any given p > 1.

Taking p to be sufficiently large, we see from the embedding theorem (see, e.g., [17] ) that

Therefore, there exists a subsequence of d S → 0, say d n := d S,n , satisfying d n → 0 as n → ∞, and a corresponding positive solution (S n , I n ) := (S d S,n , I d S,n ) of (5) with d S = d n , such that I n → I * * uniformly on Ω, as n → ∞,

where I * * ∈ C 1 (Ω) and I * * > 0.

Step 3. Convergence of S. From the first equation of (5), S n satisfies

Due to (39) , given any small > 0, we have for all large n that

Here, g 1, + (x, I * * (x)) and g 1, − (x, I * * (x)) are the root of the following equation with respect to the unknown function g:

For large enough n, we consider the following auxiliary problem

In view of the bounds of S and I in the proof of step 1, we can further assume that g 1, + > g 1, − ≥ 0 on Ω. In addition, S n is a subsolution to (40) and a sufficiently large positive constant C is a supersolution. Hence, (40) has at least one positive solution denoted by W n which satisfies S n ≤ W n ≤ C.

By the similar argument as in proof of [ 

On the other hand, by (39) , for all large n we have

Here, g 2, + (x, I * * (x)) and g 2, − (x, I * * (x)) are the root of the following equation with respect to the unknown function g:

Consider the following auxiliary problem:

In the same fashion, we also get g 2, + > g 2, − ≥ 0. Observe that S n and 0 is a pair of upper and lower solution of (42) . Hence, one can assert that (42) admits at least one positive solution, and further get lim inf n→∞ S n (x) ≥ g 2, + (x, I * * (x)) uniformly on Ω.

As

by the arbitrariness of , it immediately follows from (41) and (43) that S n (x) → G(x, I * * (x)) uniformly for x ∈ Ω, as n → ∞.

Furthermore, it is easily seen that I * * satisfies (24) by (35) .

We now fix d S > 0 and analyze the asymptotic behavior of positive solution of (5) as d I → 0. Due to mathematical difficulty, we will consider one space dimension case by taking Ω = (0, 1). By Proposition 1 (a1) and Theorem 2 (ii), to ensure the existence of positive solutions of (5) for all small d I , it is necessary to assume that Proof It is easy to check that (25) , (27) and (28) remain true for all d I > 0. Note that

One can use the well-known elliptic L 1 -theory [3] (or see [42, Lemma 2.2] ) to (44) to find that S W 1, p (0,1) ≤ C for any given p > 1.

By taking a properly large p and using the Sobolev embedding theorem, we have

This tells us that there exists a subsequence of d I → 0, say d n := d I ,n , satisfying d n → 0 as n → ∞, and a corresponding positive solution (S n , I n ) := (S d I ,n , I d I ,n ) of (5) with d I = d n , such that

On the other hand, by (27) , up to a further subsequence of d n if necessary, it follows that 1 0 I n dx → I 0 as n → ∞. In what follows, we are going to show I 0 , S 0 > 0. We first prove I 0 > 0. To this end, we use a contradiction argument and suppose that I 0 = 0. By integrating (44) from 0 to x, we have By means of S n (x) − S n (0) = x 0 S n (y)dy for any n ≥ 1, it is easily seen that S 0 satisfies

Again, one can integrate (44) from x to 1 and apply a similar process as before to deduce that (S 0 ) (1) = 0. Hence, by virtue of (45), we can conclude that S 0 =S, which means that S n →S uniformly on [0, 1] as n → ∞.

One can easily observe that λ n 1 = 0, ∀n ≥ 1, where λ n 1 is the principal eigenvalue of the following eigenvalue problem

Applying the same analysis as in [ 

Clearly, this leads to a contradiction because {β(x)S/(1 + αS) > (η(x) + μ(x)) :

x ∈ (0, 1)} is non-empty by our assumption. Thus, we must have I 0 > 0. To show S 0 > 0 on [0, 1], we proceed indirectly again and suppose that S 0 (x) = 0 for some x ∈ [0, 1]. Then, applying the Harnack inequality to the S-equation, one will see immediately that S 0 (x) = 0 for all x ∈ [0, 1]. As a result, we have 1 0 S n dx → 0 as n → ∞. Integrating the second equation of (5) from 0 to 1, one gets In this paper, we have studied the SI epidemic model (4) with logistic source and saturation infection mechanism. For the parabolic problem (4), we have established the uniform boundedness and the extinction and persistence of the infectious disease in terms of the basic reproductive number R 0 . We also obtained the global stability of the unique endemic equilibrium when the spatial environment is homogeneous. For the steady-state solution problem (5), we have investigated the asymptotic behavior of the endemic equilibria in the heterogeneous environment when the movement rate of the susceptible and infected populations is small.

In what follows, we first want to compare the influence of immigration rate, logistic sources, and incidence rate on the basic reproduction number of models (1)-(4).

Allen et al. [1] introduced the epidemic model (1) with standard incidence rate S I S+I and defined the basic reproduction number

It is clear that the R 0 defined here depends on the immigration rate of infected individuals d I , the transmission rate β(x), and the recovery rate γ (x). Then, Li et al. [32] added the logistic source a(x)S − b(x)S 2 to system (1); but the basic reproduction number is the same as the one that without logistic source. In other words, the logistic sources have no influence on the definition of the basic number of the epidemic model with standard incidence rate.

As the infection mechanism changes to the saturated incidence rate S I 1+m I , Huo and Cui in [22] defined a basic reproduction number of system (3) as

whereŜ is the unique solution of (7) with μ(x) ≡ 0. It is easily seen that such R 0 depends onŜ, which is continuously dependent on the logistic sources and parameter d S , except the coefficients d I , β(x) and γ (x). However, the basic reproduction number R 0 of our model (4) not only depends on the parameters d S , d I , a(x), b(x), β(x) and γ (x) but also depends on the death rate μ(x) and the saturated coefficient α.

In the spatially heterogeneous environment, we have obtained that threshold dynamics in terms of the basic reproduction number, that is, the uniform persistence property holds if R 0 > 1, and the disease extinction occurs if R 0 < 1. Based on the ultimately uniform boundedness in Theorem 1, we established the uniform persistence property of (4), that is, there existed at least one EE when R 0 > 1. Moreover, the disease will die out if the saturation factor α is large enough. Biologically, the epidemic will be extinct in the long run provided that the effective prevention measures (α) are taken. For example, people disinfected, washed, and blocked the market of infected food during the outbreak of the COVID-19. Therefore, we know how important the effective prevention and control strategy is in the absence of sufficient medical treatment and vaccines.

Finally, we discussed that the global stability and asymptotic profiles of the endemic equilibrium. When the environment is spatially homogeneous, that is, all the parameters in (4) are positive constants, the global stability of endemic equilibrium has been shown by establishing suitable Lyapunov function for the basic reproduction number R 0 > 1; see Theorem 3. By Remark 2, when α = 0, condition (23) always holds. Hence, (Ŝ,Î ) is globally asymptotically stable as long as it exists. However, if the saturated incidence rate α > 0, (Ŝ,Î ) may be unstable.

Furthermore, in the case of d S → 0, Theorem 4 shows that the disease exists in the entire habitat. On the other hand, Theorem 5 suggests that the susceptible population is positive while the total infected population tends to a positive constant as d I → 0 in the one-dimensional interval. The results have suggested that the density of the infected population will not vanish when the mobility of the susceptible or infected population goes to zero. The above discussion reveals that more effective measures α should be taken to control the sources of infection and cut off the channels of transmission so as to eradicate the disease.

Indeed, [22, Theorems 4.2 and 4.3] , [32, Theorems 4.1-4.2] and Theorems 4-5 here have shown that the infectious disease does not die out for the low diffusion rate of susceptible or infected individuals, and thus the epidemic disease cannot be eliminated by controlling the mobility of individuals. Combined with the discussion in [22, 32] , we can conclude that the logistic source enhances the persistence of the disease, and the infectious disease will become threatening and harder to control. Our results here, together with the other two related epidemic models [22, 32] , show that the logistic growth, the infection mechanism, and the population movement play an important role in the transmission dynamics of disease.

In summary, our discussion above shows that, in order to eradicate the disease modeled of the susceptible individuals, instead of reducing the mobility of the populations.

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An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment

On the global stability of SIS, SIR and SIRS epidemic models with standard incidence

Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate

A delayed epidemic model with pulse vaccination

Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism

Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism

Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment

Asymptotic profiles of the endemic equilibrium of a diffusive SIS epidemic system with saturated incidence rate and spontaneous infection

Analysis of a delayed SIR model with nonlinear incidence rate

Pulse vaccination delayed SEIRS epidemic model with saturation incidence

The diffusive logistic model with a free boundary in heterogeneous environment

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