Dynamics analysis of a nonlocal diffusion dengue model

Due to the unrestricted movement of humans over a wide area, it is important to understand how individuals move between non-adjacent locations in space. In this research, we introduce a nonlocal diffusion introduce for dengue, which is driven by integral operators. First, we use the semigroup theory and continuously Fréchet differentiable to demonstrate the existence, uniqueness, positivity and boundedness of the solution. Next, the global stability and uniform persistence of the system are proved by analyzing the eigenvalue problem of the nonlocal diffusion term. To achieve this, the Lyapunov function is derived and the comparison principle is applied. Finally, numerical simulations are carried out to validate the results of the theorem, and it is revealed that controlling the disease’s spread can be achieved by implementing measures to reduce the transmission of the virus through infected humans and mosquitoes.


Model and preliminaries
To assess the impact of nonlocal diffusion on the dengue model, we begin by introducing the SIR-SI model detailed in 21 , the parameters are defined in Table 1.
It's worth noting that mosquitoes generally have a limited, activity range, typically flying only tens to hundreds of meters.The furthest recorded flight distance is one to two kilometers.Given this, the nonlocal spread of mosquitoes was disregarded.Also, since the third equation doesn't feature in the other equations of system (1), we focus on the subsequent dengue model: with Neumann boundary condition (the derivative is zero when x is at the boundary) and initial condition where Eq. ( 4) represents the value in the individual at the initial time (namely, t=0).d 1 and d 2 represent the dif- fusion coefficients, and d 1 > 0 , d 2 > 0 .µ h (x) , µ(x) , β H (x) , b(x), γ H (x) , β v (x) and ν(x) are positive continuous functions on .The dispersal kernel function J is continuous and satisfies the following properties Let us consider the following function spaces and positive cones.
Well-posedness of the solution.In this section, we will prove the existence and uniqueness of the solution for system (2).
] be a linear operator on Y defined as follow: By calculating, we have due to the coefficients are positive and bounded, we have that the last term in the right-hand of this equation is (a 1 , a 2 , a 3 , a 4 ) ∈ Y.
(6) where Due to A be the infinitesimal generator of e tA t≥0 and F is continuously Fréchet differentiable on Y .From 25 , Proposition 4.16, the result holds.
Proof By calculation, we have and For all t ∈ [0, t 0 ) and x ∈ ¯ .Due to (S H,0 , I H,0 , S V ,0

Lemma 2.2
For any initial data (S H,0 , I H,0 , S V ,0 , I V ,0 ) and t ∈ [0, t 0 ) , the solution (S H (x, t), I H (x, t), S V (x, t), I V (x, t)) of system (2) satisfy that Proof By (2) and ( 5), we have where | | denotes the volume of .By virtue of the variation of constants formula and take limit as t → ∞ , we can obtain that Basic reproduction number.For a more abstract representation of the basic reproduction number, we utilize the next-generation matrix method 26 and evaluate the linearized equations surrounding the disease-free equilibrium E 0 = (S 0 H (x), 0, S 0 V (x), 0): System (10) be equivalent to where and By virtue of 27 , Chapter 11, we obtain that the following linear equation Vol:.(1234567890) www.nature.com/scientificreports/Let T(t) be the solution semigroup with respect to the linear Eq. (11).Define

Scientific
In terms of the next infection operator, the spectral radius of K can be defined as the basic reproduction number We consider the following eigenvalue problem with respect to system (10).
Meanwhile, by virtue of 28 , for system (12), there exists a principal eigenvalue 0 with respect to a pair positive continuous eigenfunction (� 0 (x), � 0 (x)) satisfy that the following lemma.
Proof The proof procedure can be referred to reference 14 , Theorem 2.10.

Global stability and uniform persistence
Global stability of the disease-free equilibrium.Global stability of the disease-free equilibrium is to be demonstrated.Before proving its global asymptotic stability, certain lemmas are presented.Additionally, we investigate an eigenvalue problem previously examined Garc ía-Meliá n and Rossi 13 .

Proof
We first prove that S H (x, t) → S 0 H (x) on x as t → +∞ , let h 1 (x, t) = S H (x, t) − S 0 H (x) .Furthermore, we have Let H(t) = � h 2 1 (x, t)dx , we can obtain By calculation yields that Vol.:(0123456789) www.nature.com/scientificreports/Hence, there exists constant c 0 , we have By virtue of Eq. ( 14), we can obtain Applying the h ölder inequality to the following equation, there exists some positive constant satisfy that Combine ( 16) and ( 17), there exists some positive constants c i (i = 1, 2) we have Hence, as t → ∞ , h 1 (x, t) → 0 uniformly on x ∈ .Furthermore, we obtain that S H (x, t) → S 0 H (x). Next, we prove , by virtue of the above argument, there exists some positive constant c 0 > 0 satisfy that Hence, equation ( 18) be equivalent to

By calculation yields that
Hence, for some positive constants k i (i = 1, 2, 3, 4) , we have

By virtue of system (2), we can obtain
Applying the h ölder inequality to the following equation, there exists some positive constant satisfy that Combine ( 19) and ( 20), there exists some positive constants ki (i = 1, 2, 3, 4) we have www.nature.com/scientificreports/Since R 0 < 1 , we know that 0 < 0 , hence, as t → ∞ , I H (x, t) → 0 uniformly on x ∈ .Moreover, we prove that S V (x, t) → S 0 V (x) on x as t → +∞ , let h 2 (x, t) = S V (x, t) − S 0 V (x) , then, we have Due to I H (x, t) → 0 as t → ∞ , by virtue of the above argument, we know that h 2 (x, t) → 0 as t → ∞ .using the the constant variation method with respect to the last equation of (2), we can obtain that I V (x, t) → 0 as t → ∞ .
Uniform persistence.In this section, we consider the uniform persistence of system (2).To get these goals, we first consider the following problem.Theorem 3.2 For R 0 > 1 , then there exists a function Ŵ(x) , such that hence, the disease uniform persistence.
rep- resents that the endemic equilibrium ).It means that there exists a t 1 > 0 satisfy that S H (x, t) > S H,0 − κ and S V (x, t) > S V ,0 − κ for t ≥ t 1 and x ∈ .For x ∈ �, t > t 1 , according to the comparison principle, we can obtain Define ( I H (x, t), I V (x, t), ) = (Me t ̺ 1 (x), Me t ̺ 2 (x)) , ( I H (x, t), V (x, t), ) satisfy that the following equation where ( ̺ 1 (x), ̺ 2 (x)) is the eigenfunction with respect to < 0 .According to the comparison principle, we know On the basis of the Lemma (2.2), we know that there exists a constants K > 0 and t 2 such that Then, S H and S V satisfy that the following equation Hence The disease uniform persistence is obtained.

Numerical simulations
This section presents the theoretical results supported by numerical simulations are presented in this section.The parameter values and initial value are chosen as follows: initial value: Moreover, the nonlocal kernel function 23 is selected as follows: Here 1 for the evolution path of kernel function J(x).

Global dynamics of system (2).
In this section, we choose to change β H to illustrate the result of the theorem.Let β H = 0.015(1 − 0.65cosx) and see Table 2 for other parameters, then R 0 = 0.949319338848686 < 1 .Figure 2 illustrates the long-term dynamic behavior of the system (2).As time t approaches infinity, the density of infected humans and mosquitoes both converge to 0, indicating the extinction of the disease.If the human transmission rate β H increases to 10β H , we can obtain R 0 = 3.002011337607015 > 1 .At this point, Fig. 3 shows that the solution of system (2) eventually stabilizes, implying disease persistence.

Conclusions
We conducted research on the threshold dynamics of a nonlocal diffusion dengue model with spatial heterogeneity.To establish the existence, uniqueness, positivity, and boundedness of the solution, we utilized the semigroup theory and the variation of constants formula.The expression of the basic reproduction number was abstractly determined using the next-generation matrix method.By constructing a Lyapunov function and applying the comparison principle, we proved the system's global stability and uniform persistence.Numerical simulations were performed to verify the theorem.This study explored the evolution of disease extinction and persistence by adjusting the human transmission rate β H .We also considered the impact of diffusion on infected humans and mosquitoes.The simulation results indicate that an increase in the diffusion coefficient leads to greater persistence of the disease in both humans and mosquitoes.This finding highlights the importance of controlling the spread of humans and mosquitoes during disease outbreaks.To achieve better disease control, we recommend implementing appropriate measures to reduce their transmission.

Figure 3 .Figure 5 .
Figure 3.The evolution path of S H , I H , S V , I V for system (2) with R 0 = 3.002011337607015 > 1.

Figure 6 .
Figure 6.Numerical simulation of I H , I V for system (2) with d 1 = d 2 = 0.060 (where R 0 = 2.956695436468467 > 1 ).Left: The evolution path of I H , I V .Right: The distribution of I H , I V in time and space.

Table 1 .
Definitions of all parameters.
H The recovery rate of humanAThe recruitment rate of mosquitoes ν The nature death rate of mosquitoesβ VThe transmission rate of dengue to the mosquito from human m The densities of alternative hosts www.nature.com/scientificreports/Next, we define the linear operators on X.

Table 2 .
The parameter values.