A class of diffusive delayed viral infection models with general incidence function and cellular proliferation

We propose and analyze a new class of three dimensional space models that describes infectious diseases caused by viruses such as hepatitis B virus (HBV) and hepatitis C virus (HCV). This work constructs a Reaction–Diffusion-Ordinary Differential Equation model of virus dynamics, including absorption effect, cell proliferation, time delay, and a generalized incidence rate function. By constructing suitable Lyapunov functionals, we show that the model has threshold dynamics: if the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{0}(\tau ) \le 1 $$\end{document}R0(τ)≤1, then the uninfected equilibrium is globally asymptotically stable, whereas if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{0}(\tau ) > 1$$\end{document}R0(τ)>1, and under certain conditions, the infected equilibrium is globally asymptotically stable. This precedes a careful study of local asymptotic stability. We pay particular attention to prove boundedness, positivity, existence and uniqueness of the solution to the obtained initial and boundary value problem. Finally, we perform some numerical simulations to illustrate the theoretical results obtained in one-dimensional space. Our results improve and generalize some known results in the framework of virus dynamics.

The basic reaction-diffusion-ODE viral infection dynamics model consists of the following threedimensional system (see [4,32] and references therein): where the density of uninfected cells is represented by H (x, t) at position x at time t, the density of infected cells is represented by I (x, t) at position x at time t and the density of free virus particles by V (x, t) at position x at time t. The uninfected target cells are produced at a constant rate λ and are infected by free virus particles at a rate β H (x, t)V (x, t) which follows mass action principle. The parameters μ, α and γ represent the death rates of uninfected cells, infected host cells and free virus particles, respectively. Free virions are produced by infected cells at the rate ηI (x, t). Δ is the Laplacian operator and D is the diffusion coefficient. At this point it should be mentioned that in model (1) it is assumed that the infection rate is bilinear, that is of the form β H (x, t)V (x, t). However, this hypothesis does not always have a biological meaning. Recently, many researchers have performed virus dynamics models using various type of infection rate (or incidence function) which each time generalizes the bilinear infection rate. For example, in [35], authors studied a delayed model, in the case of HBV with diffusion and Holling-II infection rate, a virus infection model with the Crowley-Martin infection function has been studied in [16,34,37]. The Beddington-DeAngelis infection rate has been used in [10,26,36] to study a delayed in-host model with diffusion. Also in [27], authors studied a PDE-model with standard infection rate. Therefore, it is necessary to study the virus infection models with a more generalized infection rate, which can be represented by a function which has some properties and generalizes the later infection rates mentioned above.
In this work, motivated by the work done in [4], we further neglect the mobility of susceptible cells, infected cells, and we consider a delayed virus infection model with a generalized infection rate given as follows: for t > 0 and x ∈ Ω which is a bounded domain of R n representing the liver with smooth boundary ∂Ω. The original part of the model lies in the fact that the proliferation of cells due to mitotic division, and mitotic transmission obey a logistic growth. Thus the novelty of this model is that it includes both the intercellular delay in virus production, the proliferation of cells and the general incidence rate which generalizes most famous forms presented for instance in [14,16,26,27,34,36,37].
In model (2), we assume that the proliferation of cells due to mitotic division obeys a logistic growth law. The mitotic proliferation of uninfected cells is described by r 1 H (x, t) 1 − H (x,t)+I (x,t) k , and mitotic transmission occurs at a rate r 2 I (x, t) 1 − H (x,t)+I (x,t) k , which represents the mitotic division of infected cells. Some models supposed that infected hepatocytes do not proliferate; however, the effect of viral infection on hepatocytes is controversial, with conflicting data showing both proliferation induction and inhibition. With system of differential equations coupled to a reaction-diffusion equations models, we explore the impact of proliferation among infected cells in the liver. Uninfected cells and infected cells grow at the constant rate r 1 and r 2 respectively, and k is the maximal number of total cell population proliferation. The parameter a ∈ {0, 1} indicates if there is an absorption effect or not. The infection process in its general form is characterized by the term f (H, I, V )V . In this case, the incidence function f = f (H, I, V ) is assumed to be continuously differentiable in the interior of R 3 + and probes the three assumptions given by Hattaf et al. [9,15] and used in [11,13,31], that are: In the mathematical model (2) the immune response to infection is represented by an elevated death rate in infected cells, μ ≤ α, and by the destruction of free virions at rate γ . Due to the burden of supporting virus replication, infected cells may proliferate more slowly than uninfected cells, this implies that describes the newly activated infected cells at time t which are infected τ times ago. The recruitment of virus producing cells at time t is given by the number of cells that were newly infected at time t − τ and are still alive at time t. Here, m is assumed to be a constant death rate for infected but not yet virus-producing cells. Thus, the probability of surviving the time period from t − τ to t is e −τ m . We need a biologically reasonable history of the host for the system model. It is why the model (2) is supplemented with the following non-negative initial conditions: and homogeneous Neumann boundary condition It should be noted that the boundary condition in (4) imply that the free HCV virions do not move across the boundary ∂Ω.
In this paper we investigate the dynamical properties of the new model giving by (2), specifically the stability of the homogeneous equilibria. Our work is structured as follows. In the following section we discuss the existence and uniqueness, positivity and the boundedness of the solution to the IBVP with respect to the model. In Sect. 3, we start with the determination of the uninfected equilibrium, then followed by the determination of basic reproduction number R 0 (τ ) and end with the study of the local and global asymptotic stability of this equilibrium. We first determine the infected equilibrium point and then we study the local and global asymptotic stability of this point in Sect. 4. The two previous sections are followed each of them by numerical simulations where we illustrate dynamic behaviour in more detail which reinforce the theoretical results. In the last section we give brief conclusion and perspectives.

Preliminary results
This section is devoted to the study of existence, uniqueness, positivity and boundedness of solutions of the initial and boundary value problem (IBVP) (2)- (4). For this purpose, we first introduce the following spaces and definition: let X = C(Ω, R 3 ) be a Banach space of continuous functions from Ω to R 3 and C = C([−τ, 0], X) be the Banach space of continuous functions from [−τ, 0] to X with the usual supremum norm and let . Also, we adopt the standard notation that for τ 0 > 0, a function ϕ:

Lemma 2.1 F is Lipschitz continuous on bounded subsets of
Using the definition of F 1 , we have: From equality (5), it follows that According to the mean value theorem and assumptions made on f , there exists M > 0 and β > 0 such that employing (5) yields with P and Z given in theorem 2.4. In a similar manner, we get and Thus, F is Lipschitz continuous on bounded subsets of C + . Now, IBVP (2)-(4) can be rewritten as the following abstract functional differential equation: where k . Let us also consider the following sets We prove the following result: Moreover, we can easily obtain: We have now shown that for ρ small enough, from which we deduce that From the main results of the literature and the previous lemmas, we can state this following result.

Theorem 2.3 For any
Proof For D = (0, 0, D) , according to Theorem 1.5 of [5] the X-realisation of the operator DΔ generates an analytical semi-group T (t) on X. Applying the Corollary 4 of [18], we conclude that the IVBP (2)-(4) admits a unique mild solution In addition, Corollary 2.5 of [33] ensures that the mild solution is classic for t ≥ τ.

Boundedness of the solutions of the IBVP (2)-(4)
In this section, we establish the boundedness in time of the global solution of the IBVP (2)-(4) for x ∈ Ω and t ∈ [0, T max ) where T max > 0 is the maximal existence time for solution of the the IBVP (2)-(4).
From (7), we get: Now to have the bounds of V , we consider the following problems Using a comparison principle (see [23]), we infer that where From the above discussion, we deduce that H , I and V are bounded on Ω × [0, T max ).

Asymptotic stability analysis of the uninfected equilibrium
The aim of this section is to study the local and global stability of the uninfected equilibrium.

Basic reproduction number and Uninfected equilibrium
It is easy to verify that system (2) always has an uninfected equilibrium E 0 = (H 0 , 0, 0) with Following Wang and Zhao [30], we define the basic reproduction number of our model in the absence of spatial dependence as follows: One of the main tools in epidemic models is the basic reproduction number which is an important threshold parameter to discuss the dynamic behaviour of the epidemic model. It quantifies the infection risk. It measures the expected average number of new infected hepatocytes generated by a single virion in a completely healthy hepatocyte.

Local stability analysis of E 0
The objective of this section is to prove the local stability of the spatially homogeneous uninfected equilibrium E 0 for the Reaction-Diffusion-ODE system (2). We address local stability by analyzing the characteristic equation.
Letting I d be the 3 × 3 identity matrix, we have Thus, there exist non-trivial solutions z and X is an eigenvalue if the matrix −X I d − μ i D + A + e −τ X B has a determinant equal to zero. This calculation gives the characteristic equation as follows: Considering that E 0 verifies system (17), hence r 1 1 − H 0 k = μ − λ H 0 , and using the previous fact we can express the first factor of (11) as: which have a negative eigenvalue, and the other two eigenvalues satisfy the following transcendental polynomial where It is clear that a 2 > 0 due to the fact that r 2 ≤ r 1 and μ ≤ α. When τ = 0, equation (12) becomes We have If R 0 (0) < 1 then a 3 + b 3 (0) > 0. Furthermore, the fact that a 3 > 0, ensures that all the roots of (13) have negative real part according to Routh-Hurwitz criterion. Therefore, the uninfected equilibrium E 0 is locally asymptotically stable when τ = 0. Now, let us consider the distribution of the roots of (12) when τ > 0.
Therefore, if R 0 (τ ) < 1, the uninfected equilibrium E 0 of system (2) is locally asymptotically stable. Now we consider the case R 0 (τ ) > 1, recalling that i specifies the diffusion eigenvalue μ i , let Here, it is sufficient to consider i = 0 and the space X 0 corresponding to μ 0 = 0. We have Hence, there must exist X 0 > 0 such that F(X 0 , 0) = 0. This yields that (13) has at least one positive root. Thus, the uninfected equilibrium E 0 is unstable for R 0 (τ ) > 1.
We assume that f is under the following hypothesis: We have the following result: Then, clearly, L 1 (t) is nonnegative definite with respect to E 0 . Now, calculating the time derivative of L 1 along the solution of problem (2)-(4), where the dot ( · ) represents derivative with respect to time, we obtain Since Moreover, using the fact f is a decreasing function with respect to second and third variable, From the Divergence theorem and the homogeneous Neumann boundary conditions, we get and thus, the previous inequality becomes

Remark 3.3
The previous results confirm that the infection always dies out. These results strictly extend those of [4,11,32] in the case where their models ignored cell proliferation and the absorption effect in the asymptotic stability result of the homogeneous uninfected equilibrium.

Numerical results
In this section, we present the numerical simulations done by using Matlab software to confirm the theoretical results that we established in the previous section for a particular case of the incidence function f defined as follows: which is the Beddington-DeAngelis incidence function [13] where α 1 , α 2 and β are positive constants.

Asymptotic stability analysis of the infected equilibrium
The aim of this section is: first to determine the existence and the uniqueness of the infected equilibrium for the system (2), then to study the local and global stability of the the later, and finally to obtain numerical results. Due to the complexity of determining an infected equilibrium of the initial model, we will in this section restrict ourselves to the case a = 0.
Proof We suppose that there exists a homogeneous spatial equilibrium or constant solution (H 1 , I 1 , V 1 ) for system (2), then this constant solution satisfies: From (19) we get Moreover, by multiplying (17) by e −mτ and adding the latter to (18), we obtain the following quadratic equation in H 1 : Note that equation (21) has two real roots of opposite sign that depend on I 1 . The positive real root of (21) is given by Defining Substituting (19) and (23) into (18) yields It follows that F is a continuous real function defined on [0, +∞). Furthermore, and Finally, we get Therefore which is positive as R 0 (τ ) > 1. Note that 0 ≤ lim

Uniqueness
Proposition 4.2 If A = H 2 0 r 2 − λk ≤ 0, then the infected equilibrium point E 1 = (H 1 , I 1 , V 1 ) is unique. Proof It is sufficient to show that F(I 1 ) is a strictly decreasing function. We have Note that F (I 1 ) depends on h (I 1 ). To calculate h (I 1 ) we first rewrite the Eq. (21) as Using implicit differentiation we get The previous expression of h (I 1 ) can be rewritten as and Using Eqs. (18) and (19) we rewrite A and B respectively as Note that From (27), we deduce that From (17), we have from which we deduce that Equation (18), allows us to have: The fact that F is a strictly monotonic function allows us to conclude that the point I 1 is unique. The uniqueness of I 1 results in that of H 1 and V 1 . Thus we conclude that the equilibrium point (H 1 ,  (2) subject to homogeneous Neumann boundary condition (4) must solve the following system Investigation of the existence and stability of such spatially-inhomogeneous equilibria will be the concern of a forthcoming work via an in-depth analysis of the above system.

Local stability analysis of E 1
In this section we deal with the local stability of the infected equilibrium E 1 .
Proof Note that this proof is analogous to the proof of Proposition 3.1.
The characteristic equation of system (2) at the infected equilibrium is of the form where a 2 (τ ) = λ From (30) and (31), we get Moreover, note that the characteristic Eq. (28) is equivalent to It follows from the above that: and Hence, we conclude that ω is a positive root of the following equation: that is: Let z = ω 2 then (33) becomes the third order equation in z Suppose that (34) has at least one positive root, let z 0 be the smallest value for these roots. Then (33) has the root ω 0 = √ z 0 then, according to (30) and (31), we obtain the value of τ associated with this ω 0 such that X = ωi is an purely imaginary root of (32), known as Then we have the following result, from Lemma 2.1 from Ruan [24].: (H (1)),

Global stability analysis of E 1
Inserting the previous equation in the expression of ∂ L + (x,t) and Ω ΔV = 0.
Then combining the final expressions of ∂L(x,t) ∂t and ∂ L + (x,t) ∂t , and furthermore, using assumption − 1   [4,11,32] in that the cellular proliferation and absorption effect were ignored in establishing the result on the asymptotic stability of the homogeneous infected equilibrium.

Conclusion
In this work, we have proposed and analyzed a class of three dimensional spatio-temporal model describing infectious diseases caused by viruses such as the human immunodeficiency virus (HIV), hepatitis C virus and the hepatitis B virus (HBV). The infection transmission process is modeled by a general incidence function which includes several forms existing in the literature. In addition, the global analysis of the proposed model is rigorously investigated. Furthermore, biological findings of our analytical results are presented. Moreover, mathematical virus models and results presented in many previous studies are extended and generalized.
To study the mechanism of viral infection and replication, we performed the mathematical analysis of a dynamic model of diffusive in-host virus with a general non-linear incidence function. The well-posedness and the stability of the equilibria of this model are examined. The basic reproductive number R 0 (τ ) which is a threshold value that predicts extinction and persistence of the viral infection is given. It is shown that the global stability of the equilibria is determined by R 0 (τ ) with some other conditions: if R 0 (τ ) < 1, the uninfected equilibrium is globally asymptotically stable, which means that the virus is finally cleared and the infection dies; if R 0 (τ ) > 1, then the infected equilibrium is globally asymptotically stable. Our results also imply that diffusion coefficients have no influence on the global behaviour of such a virus dynamics model with homogeneous Neumann boundary conditions. Furthermore, the model proposed in this work is an extension of some previous work and the results obtained improve some known results. It is interesting to improve the present work by integrating several delays and searching for the spatially heterogeneous equilibria. In addition, we can undertake the study of the existence of the Hopf bifurcation, and knowing that The memory is an important characteristic of biological systems, It will be more interesting to examine the memory effect on the spatiotemporal dynamics of our model by using the new generalized fractional derivative presented in [8].