Global properties of virus dynamics with B-cell impairment

Abstract In this paper we construct a class of virus dynamics models with impairment of B-cell functions. Two forms of the incidence rate have been considered, saturated and general. The well-posedness of the models is justified. The models admit two equilibria which are determined by the basic reproduction number R0. The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle’s invariance principle. The theoretical results are illustrated by numerical simulations.


Introduction
The study of within-host virus dynamics using mathematical modeling has been an interesting topic to research in the last decades. A proper model could provide insights of a better understanding of the virus dynamics and clinical treatments used to ght against it. In an infection process, the interaction between viruses and cells can be seen as an ecological system within the infected host. A wide of mathematical models focused on exploring the interaction between three basic compartments, uninfected cells (U), infected cells producing viruses (I) and viruses (P). A basic model of virus dynamics was originally developed by Nowak and Bangham [1] which has become highly used by experimentalists and theorists (see e.g., Nowak and May [2]). The model presented in [1] is given by: where U, I and P are the concentrations of uninfected cells, infected cells and viruses, respectively. The parameters ϱ, γ, ω, β, κ and ξ are positive. The full description of the model was given in [1]. A huge number of papers have been published as extension of the basic model (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]). The immune response plays a critical role in controlling the virus spreading. The speci city and memory in adaptive immune responses are the responsibility of lymphocytes. B cells and T cells are the two main types of lymphocytes. The function of T cells is to recognize and kill infected cells, while the function of B cells is to produce antibodies which bind to virus particles and mark it as a foreign structure for elimination by other cells of the immune system. Antibody alone can neutralize, and thus protect against, viruses [23]. The virus dynamics model with B cell immune response was presented by Murase et al. [24] aṡ where C is the concentration of B cells. Many extended models are developed with B cell immune response (see, e.g., [25][26][27][28][29][30][31][32][33][34][35][36]). In certain circumstances, some viruses can suppress immune response or even destroy it especially when the load of viruses is too high. Models with T cell immune impairment were studied several times (see, e.g., [37][38][39][40]). In addition, there are factors a ect B cell function and cause the impairment of B cell [41][42][43]. These factors include the following; malnutrition, tumors, cytotoxic drugs, irradiation, aging, trauma, some diseases (e.g., diabetes) and immunosuppression by microbes, e.g., malaria, measles virus but especially HIV [23]. In a very recent work, Miao et al. [44] have proposed a virus dynamics model which includes: humoral impairment, time delay, reaction-di usion, and logistic growth of the target cells. Due to the complexity of the model presented in [44], the global stability analysis of the model's equilibria did not studied. Studying the global stability of equilibria for virus dynamics models will give us a detailed information and enhances our understanding about the virus dynamics. Therefore, many mathematician have paid great e orts to study global stability of systems in virology (see, e.g., [7][8][9][10][11][12][13][14][15][16][17][18][19] and [45][46][47][48][49][50][51][52][53][54]) and epidemiology (see, e.g., [55][56][57]).
In [44], the incidence rate of infection is given by bilinear. In reality, the bilinear incidence may not accurate to characterize the virus dynamics during di erent stages of infection especially when the concentration of the viruses is high [8]. Therefore, in the present paper, we propose viral infection model with B-cell impairment and with two nonlinear forms of the incidence rate, saturation and general. We show that the solutions of the model are nonnegative and bounded. The global stability of the equilibria is established by constructing Lyapunov functions and applying LaSalle's invariance principle.

Model with saturation
In this section we propose a virus dynamics model including B-cell impairment and saturated incidence as: where, ϑPC is the B-cell impairment rate and α ≥ is a saturation constant.

. Basic properties
We de ne the compact set where s i > , i = , , .
Then the equilibrium EP exists when R > .

. Global properties
De ne a function G(u) = u − − ln u. Clearly G(u) ≥ , for u > and G( ) = . The global stability analysis of the two equilibria of model (8)-(11) will be established in the next theorems.
Calculating dL dt as: Since R < , then for all U, P, C > we have dL dt ≤ . Moreover, dL dt = when U(t) = U and P(t) = C(t) = .
Let D = (U, I, P, C) : dL dt = and M be the largest invariant subset of D . The trajectory of model (8)- (11) tends to M [58]. All the elements of M satisfy U(t) = U and P(t) = C(t) = . Then Eq. (10) we geṫ Hence, M = {EP }. From LaSalle's invariance principle, we derive that if R < , then EP is globally asymptotically stable.
Proof. Construct a Lyapunov function L (U, I, P, C) as Note that from the equilibrium condition Eq. (16) that Then dL dt is given by: From the equilibrium conditions, we have: Utilizing the conditions of EP , we get Simplifying the result, we obtain .
Using geometrical mean (GM) and arithmetical mean (AM) inequality we get Thus for all U, I, P, C > we have LaSalle's invariance principle we obtain that if R > , then EP is globally asymptotically stable.

Model with general incidence rate
In this section we propose a model with more general incidence rate function Θ(U, P) as: We need the following Assumptions of the function Θ(U, P): Proof. At any equilibrium EP(U, I, P, C) we have εP − µC − ϑCP = .

. Global stability of equilibria
The global stability analysis of the two equilibria of model (18)-(21) will be investigated in this section.

Theorem 3. Let R G > , then the infection-free equilibrium EP of model (18)-(21) is globally asymptotically stable.
Proof. Construct a Lyapunov function Z (U, I, P, C) as Calculating dZ dt as: From the Assumptions, we have the rst term is less than or equal to zero. In addition, for all U > . Then κΘ(U, P) βξR G P It implies that Therefore, if R G < , then dZ dt ≤ for all U, P, C > . Similar to the proof of Theorem 1, one can show that EP is globally asymptotically stable.
Then dZ dt can be calculated as: Using the equilibrium condition, εP − µC − ϑP C = , we have From the equilibrium conditions, we have Applying these conditions, we obtain From Assumptions (A2) and (A4) we have Therefore, using inequality (17) we get that for all U, I, P, C > we have dZ dt ≤ and dZ dt = if and only if U = U , I = I , P = P and C = C . Applying LaSalle's invariance principle, we obtain that if R G > , then EP is globally asymptotically stable.

Case(1): E ect of ω on the stability of equilibria.
For this case, we take α = α = and ϑ = . . We choose three di erent initial conditions as: We consider two values of the parameter ω as: (i) ω = .
, then we compute R G = .
, then we compute R G = .

Case(2): E ect of the saturation infection on the virus dynamics.
In this case, we take α = and ϑ = . . We choose ω = .
, and α varied. Moreover we consider the initial condition IC2. Figure 2 shows that as α is increased, the concentrations of the uninfected target cells is increased, while the the concentration of infected cells, virus particles and B cells are decreased. We note that the parameter α has no e ect on the stability of equilibria
, and ϑ = . then Θ(U, P) represents the Holling type-II. Let us choose the initial condition IC2. We suggest di erent values of α to see its e ect on the model as we can see in Figure 3. Moreover, we have the following cases: (i) EP is globally asymptotically stable when ≤ α < . , (ii) EP is globally asymptotically stable when α > .
. This means that α can play the role of controller which can be designed to stabilize the system around the infection-free equilibrium EP .

Case(4): E ect of the B cell impairment parameter ϑ.
In this case, we take α = . and α = .
, and ϑ varied. Moreover, we consider the following initial condition IC4: U( ) = , I( ) = , P( ) = , C( ) = . Figure 4 shows that as ϑ is increased, the concentrations of infected cells and virus particles are increased, while the concentration of uninfected cells is decreased. We note that the parameter ϑ has no e ect on the stability of equilibria.