Threshold Dynamics and Competitive Exclusion in a Virus Infection Model with General Incidence Function and Density-Dependent Diffusion

In this paper, we investigate single-strain and multistrain viral infection models with general incidence function and densitydependent diffusion subject to the homogeneous Neumann boundary conditions. For the single-strain viral infection model, by using the linearization method and constructing appropriate Lyapunov functionals, we obtain that the global threshold dynamics of the model is determined by the reproductive numbers for viral infectionR0. For the multistrain viral infection model, we have discussed the competitive exclusion problem. If the reproduction number Ri for strain i is maximal and larger than one, the steady state Ei corresponding to the strain i is globally stable. *us, competitive exclusion happens and all other strains die out except strain i. Meanwhile, we can prove that the single-strain and multistrain viral infection models are well posed. Furthermore, numerical simulations are also carried out to illustrate the theoretical results, which is seldom seen in the relevant known literatures.


Introduction
Nowadays, more and more people in the world are dying of various diseases such as AIDS, avian influenza, cholera, Ebola, and Zika virus. To explore the mechanisms of these diseases, scientists have proposed plenty of mathematical models describing the transmission of disease such as the infectious disease model (SI, SIR, SEI, and SIS) and the within-host virus model (HBV, HCV, HIV, Ebola, and Zika). Since Nowak et al. [1,2] has established the basic within-host virus dynamics model, more and more scholars have devoted to investigate the within-host virus dynamics model and obtained so many significant dynamical results [3][4][5][6][7][8][9][10][11][12][13][14][15][16].
Assuming that the motion of the virus belongs to the Fickian diffusion [17], Wang et al. [5] have firstly investigated a delayed diffusive hepatitis B virus model. Here, we have to mention that the equation with Fickian diffusion implies that random walkers in the system have no interactions. In fact, random walkers in the system may have interactions and the density-dependent diffusion happens [17] and we may establish the equation with density-dependent diffusion. So, parabolic or elliptic systems with density-dependent diffusion have been widely studied by many researchers [18][19][20][21]. More recently, Wang et al. have investigated a virus infection model with density-dependent diffusion and Holling II type [6] or Beddington-DeAngelis type incidence function [7] subjected to the homogeneous Neumann boundary conditions. And, they have obtained the significant results of threshold dynamics and competitive exclusion.
It is well known that there are many factors which affect the dynamics of within-host models. One crucial component is incidence function, which can be classified into many different types, such as bilinear type [2], Holling II type [22], Beddington-DeAngelis type [23], and Crowley-Martin type [24]. Many researchers have concentrated on these withinhost models or epidemic models and have obtained some interesting results such as global threshold dynamics [8,9], delay-induced Hopf bifurcation [10,11], competitive exclusion [6,7], traveling wave [12,13], and pattern formation [14][15][16]. Recently, Korobeinikov [25] studied an ordinary differential equation model with general incidence function and dose-dependent parasite reproduction and virulence. Huang et al. [26] investigated a viral infection model involving time delay and general incidence function. McCluskey and Yang [27] considered a diffusive viral infection model with general incidence function and time delay. Hattaf and Yousfi [28] studied a diffusive HBV model with two delays and general incidence function. Miao et al. [29] considered the global stability of a diffusive humoral immunity virus infection model with time delay and general incidence function and so on.
Based on the aforementioned works, we firstly propose a single-strain viral infection model with density-dependent diffusion and general incidence function as follows: where D u (u) � u m 1 , D w (w) � w m 2 , and D v (v) � v m 3 are the density-dependent diffusion coefficients for m i > 0, i � 1, 2, 3; u(x, t) represents the densities of uninfected cells; w(x, t) represents the densities of infected cells; v(x, t) represents the densities of free virus at position x and at time t, respectively; constant λ represents the production rate of the uninfected cells; the death rates of uninfected cells, infected cells, and free virus are represented by the parameters d, δ, and c, respectively; the infected cells produce the free virions at a rate pw; further, we assume that the incidence function in this paper is a general incidence function given by g (u, v), which represents the contacts between target cells and viruses; and the general incidence function g(u, v) satisfies the following conditions [27].
Model (1) we considered in this paper satisfies the following Neumann boundary conditions: and initial conditions Here, we give out a bounded domain Ω in R n and its boundary zΩ is smooth, and the outward normal derivative on zΩ is denoted by (z/z]). e nonnegative functions ϕ i (x, 0) (i � 1, 2, 3) are Hölder continuous and satisfy (zϕ i /z]) � 0 (i � 1, 2, 3) on zΩ.
For (1), we can obtain its steady-state system, which satisfies boundary conditions (2). We denote and let f i (u, v, w)(i � 1, 2, 3) satisfy the following conditions. (H 2 ) e following relations 2 Complexity hold for constants c i ≥ 0 (i � 1, 2, 3). Since Gause has discussed the principles that govern species competitive exclusion [30], competitive exclusion has been considered in such settings such as within-host models, ecological models, and epidemiological models. Here, we have to point out that Bremermann and ieme [31] have given out the first rigorous proof of the competitive exclusion principle in the epidemiological model setting. For the multistrain SIR model, they have proved that the strain with the largest reproduction number persists in the population, while the remaining strains die out. is principle was also used in within-host models and has been studied by many scholars [32][33][34][35][36][37]. For instance, De Leenheer and Pilyugin [32] proved the global competitive exclusion in an ODE version with a multistrain virus. Feng and Velasco-Hernández [33] discussed competitive exclusion in a vectorhost model for the dengue fever. Iggidr et al. [34] analyzed globally a new malaria within-host model through a competitive exclusion principle. Wang et al. investigated the competitive exclusion in the virus infection model with density-dependent diffusion and two types of incidence functions [6,7]. Duan et al. [36] considered the competitive exclusion in the diffusive virus infection model with age of infection and general incidence function, and so on. So, in this paper, we further investigate the competitive exclusion problem for the multistrain virus infection model. Setting i � 1, 2, . . . , n, we can obtain the following multistrain viral infection model with density-dependent diffusion and general incidence function: Here, we have to stress that the Neumann boundary conditions, initial conditions, and steady-state system of Model (7) are described in Section 3. However, to the best of our knowledge, there are no results on the above models (1) and (7) such as global threshold dynamics and competitive exclusion. So, in this paper, we shall focus on the global threshold dynamics for the single-strain viral infection model (1) and competitive exclusion for the multi-strain viral infection model (7). It is worthwhile to mention that the incidence function discussed in [6,7] is a special incidence function. But, the opposite case is more difficult and complicated. rough constructing an appropriate Lyapunov functional, Lemma 2, and mathematic analysis skills, we overcome the difficulties successfully and obtain the global threshold dynamics for the single-strain viral infection model (1) and competitive exclusion for the multi-strain viral infection model (7). e rest of this paper is organized as follows: in Section 2, for the single-strain viral infection model (1), we shall prove the existence, uniqueness, and boundedness of solutions to problems (1)-(4) and consider the global threshold dynamics of the model through constructing appropriate Lyapunov functionals, which is determined by the reproductive numbers for viral infection R 0 . In Section 3, for the multistrain viral infection model (7), we also prove the existence, uniqueness, and boundedness of solutions to problems (7) and (42) and discuss the competitive exclusion problem. If the reproduction number R i for strain i is maximal and larger than one, the steady state E i corresponding to the strain i is globally stable. us, competitive exclusion happens and all other strains die out except strain i. In Section 4, we illustrate our results with numerical simulations, which support and extend the theoretical results. e paper ends with a conclusion.

Single-Strain Viral Infection Model
In this section, we shall prove firstly that the single-strain viral infection model (1) is well posed by the method used in [18]. en, we continue to discuss the existence of equilibria for model (1). Finally, through constructing appropriate Lyapunov functionals, we investigate the global threshold dynamics of model (1), which is determined by the reproductive numbers for viral infection R 0 .

Well Posedness.
In this section, we give out some definitions and results first, which can help us to prove that the solutions to problems (1)-(4) exist and are unique and bounded. Similar to the definition method in [6,7], we can definite that two pairs of functions U � (u, w, v), are called coupled upper and lower solutions to system (1)-(3) and to system (4), respectively. en, we can easily conclude that each pair of coupled upper and lower solutions U s , U s of problems (1)-(3) is ones of (4). In what follows, we shall search a pair of constant solutions of (4), which is called upper and lower solutions of (4).
Proof. Obviously, the constant pair L � (L 1 , L 2 , L 3 ) and l � (l 1 , l 2 , l 3 ) satisfy the homogeneous Neumann boundary conditions (2). Further, by means of condition (H 1 ), we can confirm that the differential inequalities in (4) can also be satisfied by the constant pair L and l if and then, the above inequalities can be fulfilled for constants l 1 and L 1 , sufficiently large L j , and small l j 3 , To obtain the upper solution of (4), we only need to take L 1 � (λ/d) and sufficiently large L 2 and L 3 satisfying (ληL 3 /dδ) ≤ L 2 ≤ (cL 3 /p). Meanwhile, choosing sufficiently small l 3 , l 2 � (cl 3 /p), and l 1 satisfies (cδ/p) ≤ g v (l 1 , l 3 ) and l 1 ≤ (λ/(d + ηL 3 )), and we can also obtain the lower solution of (4). is means that the coupled upper and lower solutions of (4) are a constant pair L and l. is completes the proof of Lemma 1.
According to the results of Lemma 1 and method in [6,7], we can conclude that problems (1)-(4) is well-posed similarly. at is, the solutions to problems (1)-(4) exist and are unique and bounded. □ 2.2. Equilibria. To describe that a single infected cell generates the average number of newly infected cells when the disease has just entered the body, we define the following basic reproduction number: which is independent of spatial variety. Obviously, is always an infection-free equilibrium of model (1). Similar to the analysis and discussion in [27], together with the basic reproduction number R 0 , it is not hard to see that model (1) has only a positive equilibrium So, we have the results on the equilibria of model (1) as follows. (1). In this section, we shall investigate the local stability of model (1). Under the homogeneous Neumann boundary conditions, assume that 0 � ] 0 < ] 1 < · · · < ] q < · · · be the eigenvalues of − Δ on Ω and for q � 0, 1, 2, . . ., and E(] q ) be the eigenfunctions space corresponding to ] q in C 1 (Ω). Suppose that E(] q ) has an orthonormal basis 3 , and X qm � cφ qm : c ∈ R 3 . us, we obtain

e Local Stability of Model
Linearizing system (1) at the equilibrium (u, w, v) is as follows:

Complexity
Note that, under the linearization, X q is an invariant for each q � 0, 1, 2, . . .. We let the expression of solution be us, the characteristic equation of (12) is When the equilibrium of model (1) is E 0 � ((λ/d), 0, 0), the characteristic equation (13) is changed into Obviously, equation (14) has a negative real root We now discuss the transcendental equation as follows: Set Case 1. R 0 < 1. By direct computation, we have and for μ ≥ 0, So, there is only a negative real root for equation (14).
Assume that there be a complex root ρ + iϖ with ρ ≥ 0, then we can conclude that μ δ is contradicts to equation (15). en, all roots of equation (14) have no positive real parts yet. us, the infection-free equilibrium E 0 is locally asymptotically stable if R 0 < 1.
Case 2. R 0 > 1. Obviously, we need to only consider the space X 0 corresponding to ] 0 � 0 when q � 0. By direct computation, we get is implies that the equation H(μ, 0) � 0 must hold for some μ 0 > 0. us, there exists at least one positive root of equation (14). So, the infection-free equilibrium E 0 is not stable for R 0 > 1. en, we have the following results.
When the equilibrium of model (1) is E * � (u * , w * , v * ), the characteristic equation (13) is changed into where It is easy to see that A > 0. From the condition (H 1 ), together with g(u * , v * ) � δw * and pw * � cv * , we get Complexity 5 which yields B > 0 and C > 0. By direct computation, we have that So, we can see that all eigenvalues of (22) have no positive real parts by the Routh-Hurwitz criterion. erefore, E * is locally asymptotically stable for R 0 > 1. us, we obtain the following results.

Theorem 3.
e infection equilibrium E * of model (1) is locally asymptotically stable if R 0 > 1. (1). In this section, by constructing appropriate Lyapunov functionals, we continue to investigate the global stability of model (1). To this end, we need the following lemma.

Lemma 2.
Under homogeneous Neumann boundary condition, for a density-dependent diffusion problem, we can obtain the following result: where Ω is a bounded domain in R n and its boundary zΩ is smooth.
Proof. According to the homogeneous Neumann boundary condition, we get is completes the proof of Lemma 2.
Firstly, we discuss the global asymptotic stability of the infection-free equilibrium E 0 of model (3).
Proof. Define the following Lyapunov functional: 6 Complexity rough calculating the derivative of V(t) about time, we can get Note that (zu/zt) ≤ λ − du, then we have lim t⟶∞ u (x, t) ≤ (λ/d), which means that all omega limit points satisfy u ≤ (λ/d). So, we need to only consider solutions for which u ≤ (λ/d). Since (zg/zu) ≥ 0, we can conclude the Meanwhile, from Remark 1, we get So, we have that (dV(t)/dt) ≤ 0 when R 0 ≤ 1. Further, it is not hard to believe that (dV(t)/dt) ≤ 0 leads to either v � 0 or R 0 � 1 and u � (λ/d). It is easy to confirm that the singleton E 0 is just the largest compact invariant set in When R 0 ≤ 1, we can know that the infection-free equilibrium E 0 of model (3) is globally asymptotically stable by the LaSalle's invariance principle. e proof of eorem 4 is complete. Next, we investigate the global asymptotic stability problem of the infection equilibrium E * of model (3). To prove the following theorem, we need the following hypothesis: (3) is globally asymptotically stable.
Hence, E * is stable and (dV(t)/dt) � 0 iff u � u * , w � w * , and v � v * . It is easy to confirm that the singleton According to the LaSalle's invariance principle, we can know that the infection equilibrium E * of model (3) is globally asymptotically stable when R 0 > 1. is completes the proof of eorem 5. □

Well Posedness.
In this section, we give out some definitions and results first, which can help us to prove that the solutions to problems (7) and (42) exist and are unique and bounded. Similar to the definition method in [6,7], we can definite that two pairs of functions are called coupled upper and lower solutions to system (7) and to system (42), respectively. en, we can easily conclude that each pair of coupled upper and lower solutions U s and U s of problems (7) is also ones of (42). In what follows, we shall search a pair of constant solutions of (42), which is called upper and lower solutions of (42).

Complexity
Proof. Obviously, the constant pair L � (L 1 , L 2,i , L 3,i ) and l � (l 1 , l 2,i , l 3,i ) satisfy the homogeneous Neumann boundary conditions (40). Further, by means of the condition (H 1 ), we can confirm that the differential inequalities in (42) can also satisfied by a constant pair L and l if (43) en, the above inequalities can be fulfilled for constants l 1 and L 1 , sufficiently large L 2,i and L 3,i , and small l 2,i and To obtain the upper solution of (42), we only need to take L 1 � (λ/d) and sufficiently large L 2 and L 3 satisfying (λη i L 3,i /dδ i ) ≤ L 2,i ≤ (cL 3,i /p). Meanwhile, choosing sufficiently small l 3,i , l 2,i � (c i l 3,i /p i ), and l 1 satisfies (c i δ i /p i ) ≤ g iv (l 1 , l 3,i ) and l 1 ≤ (λ/(d + n i�1 η i )), and we can also obtain the lower solution of (42). is means that the coupled upper and lower solutions of (42) are a constant pair L and l. is completes the proof of Lemma 3.
According to the results of Lemma 3 and method in [6,7], we can conclude that problems (7) and (42) is wellposed similarly. at is, the solutions to problems (7)

Competitive Exclusion.
To describe a single-strain i-infected cell which generates the average number of newly infected cells when the disease has just entered the body, we define which is called the basic reproduction number of strain i in model (7). Obviously, E n 0 � ((λ/d), 0, 0) is always an infection-free steady state of system (7), where 0 is an n-dimensional vector of zeros. Similar to the analysis and discussion in [27], together with the basic reproduction number R i , it is not hard to see that for each strain i, model (7) has only a corresponding single-strain steady state as follows: iff R i > 1 and it satisfies Here, we have to point out that the nonzero components w * i is in position i + 1 and v * i are in position n + i + 1. Now, we begin to investigate the competitive exclusion problem of model (7). From the results in [36], to discuss the competitive exclusion problem, we only need to discuss the globally asymptotical stability of steady state E 1 of strain one. Next, we shall verify that the globally asymptotical stability of the infected steady state E 1 of strain one by constructing an appropriate Lyapunov functional. Here, we also assume that for i � 1, 2, . . . , n, g i (u, v i ) satisfies the condition (H 1 ) and η k < η 1 , k � 2, . . . , n. Moreover, we need the following condition:  (7) is globally asymptotically stable.
rough calculating the derivative of V(t) about time, we get  Note that λ � du * which is called the Lyapunov functional. e rest of the proof is similar to eorem 4. So, we omit it. is completes the proof of eorem 7. □ Remark 2. From eorem 7, when the basic reproductive number of each strain is less than 1, we can see that all viral strains die out.

Numerical Calculations and Simulations
In this section, we perform some numerical simulations that illustrate and supplement the analytic results given in the previous sections. In what follows, we perform the numerical simulations from the two aspects of global threshold dynamics and competitive exclusion, respectively.
Firstly, we carry out the numerical simulation for global threshold dynamics. Taking the incidence function as g (u(x, t), v(x, t)) � (βu(x, t)v(x, t))/ (1+ au(x, t) + bv  (x, t) + abu(x, t)v(x, t)), we can change model (1) into the following form: , with Neumann boundary conditions (2) and initial conditions (3). From (10), together with direct computation, we get the basic reproduction number R 0 � (pλβ/(cδ(d+ aλ))) for system (59). Meanwhile, the equilibria are given out by is a unique positive solution of the following quadratic equation: It is not hard to verify that if R 0 > 1, then λ − du * > 0. So, For simplicity of the numerical illustration, we consider only one spatial dimension with spatial domain x ∈ Ω � [0, 6]. e numerical simulations are observed for a time duration of 500 or 1000 days. Here, we have to point out that some parameter values are taken from the literature [24]. en, we set choose the parameter values as m 1 � m 2 � m 3 � 2, λ � 10, d � 0.09, β � 0.0005, δ � 0.4, c � 2.4, p � 300, a � 0.5, and b � 0.4. rough direct computation, we obtain R 0 � 0.3070 < 1 and E 0 � (111.1111, 0, 0). From eorem 4, we can know that infection-free equilibrium E 0 of model (59) is globally asymptotically stable, see Figure 1. Further, we choose the parameter values as rough direct computation, we obtain R 0 � 74.4048 > 1 and E * � (104.9054, 1.3963, 174.5354). From eorem 5, we can know that the infection equilibrium E * of model (59) is globally asymptotically stable, see Figure 2.

Conclusion
In this paper, we investigate the global threshold dynamics in single-strain viral infection models (1) and competitive exclusion in multistrain viral infection models (7) under the homogeneous Neumann boundary conditions. For model (1), by constructing a Lyapunov functional, we obtain that the infection-free equilibrium E 0 of model (1) is globally asymptotically stable if the basic reproduction number R 0 < 1, and the infection equilibrium E * of model (1) is globally asymptotically stable if the basic reproduction number R 0 > 1. ese facts are numerically confirmed in Figures 1 and 2, respectively. For model (7), by constructing a Lyapunov functional, we also obtain that the steady state E 1 of model (7) is globally asymptotically stable if R 1 > 1 and R 1 > R k for k � 2, . . . , n, which implies that strain one persists, while strains k � 2, . . . , n die out. ese facts are numerically confirmed in Figures 3 and 4. Meanwhile, the steady state E n 0 of model (7) is globally asymptotically stable if R i < 1 for i � 1, . . . , n, which implies that all viral strains die out. ese facts are numerically confirmed in Figure 5. Here, we have to point out that the incidence function in the present paper is general and performed the numerical simulation for global threshold dynamics and competitive exclusion. So, our results in the present paper are new and can also be applied to other reaction-diffusion systems. Of course, we hope that our work could be instructive to study the dynamics of the viral infection model.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.