DYNAMICS OF EVOLUTIONARY COMPETITION BETWEEN BUDDING AND LYTIC VIRAL RELEASE STRATEGIES

In this paper, we consider the evolutionary competition between budding and lytic viral release strategies, using a delay differential equation model with distributed delay. When antibody is not established, the dynamics of competition depends on the respective basic reproductive ratios of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproductive ratios of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have an evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, the lytic virus can outcompete the budding virus provided that its reproductive ratio is very high. An explicit threshold is derived.

1. Introduction.In the real world, there are mainly two types of viral release strategies: lytic and budding.Viruses can be released from the host cell by lysis, a process that kills the cell by bursting its membrane, after a period of accumulation of new virions inside the host cell.This is a feature of many bacterial and animal naked viruses, such as many types of phages, rhinoviruses and picornaviruses [2].Many viruses do not lyse their host cells; instead, progeny virions are released from the cells over a period of time by gradually budding.Enveloped viruses, such as HIV and influenza, are typically released from host cells by this strategy (see [5,9]).During this process a virus acquires its envelope from cell surface membrane.
A typical viral production process consists of viral attachment (to the host cells), penetration, uncoating, replication and release.However, lytic and budding viral strains have different life cycles.A lytic virus has a lytic cycle during which the new virions are produced and accumulated inside the host cell, and released by a burst (lysis) when the number of viruses inside becomes too large for the cell to 1092 XIULAN LAI AND XINGFU ZOU hold.A budding virus reproduces inside and escapes the host cell by constantly budding throughout the lifespan of the infected cell.
There is some research on the kinetics of viral production.Coombs [4] examined the optimal virus production schedules by considering the trade-off between viral replication and cell death rate.Burst size, defined as the expected number of virions produced over the lifetime of an infected cell, was considered as viral reproductive fitness.It was found that if viral production rate and cell mortality rate are linked, replicating at the maximal rate so that the burst size is maximized, may not be the optimal strategy for virus, even if natural selection favors viral strains whose virion production rate maximizes viral burst size.The optimal viral production rate may be lower than the maximum viral production rate, or may not be a constant, meaning that it may vary with the time or the age of the infected cell, depending on the trade-off between cell mortality and viral production.In a subsequent work, Gilchrist et al. [6] used an age-structured model of virus dynamics to study the optimal viral fitness.It was shown that trade-offs between virion production and immune system clearance of infected cells could lead natural selection to favor production rates lower than the one that maximizes burst size.Nelson et al. [10] also used an age structured model to study the influence of different profiles of nonconstant viral production rate and nonconstant infected cell death rate on HIV infection dynamics.As for lytic virus, Wang et al. (see [13,14]) studied the optimal lysis time and phage fitness.It was found that a delay in lysis time can lead to production of more progeny per infected host.Therefore, there is a trade-off between a present immediate linear gain by extending the vegetative cycle of phage and a future uncertain exponential gain derived from lysing the current host and releasing the progeny virion.
Komarova [8] studied the evolutionary competition between budding and lytic strategies.It was concluded that if all the parameters, such as the rate of viral production, cell lifespan and neutralizing capacity of antibodies, were the same for the lytic and budding viruses, the budding life-strategy would have a large evolutionary advantage because it is advantageous for an organism to reproduce earlier in life rather than later, given that the offspring is the same in both cases.However when the antibody effect is considered, the difference in removal capacity of the antibodies against budding and lytic virions could make lytic virus evolutionarily more competitive.Newly produced virions of a budding virus exit the host cell gradually and are immediately attacked by antibodies, while that of a lytic strain exit all at once, in a burst, and if there are sufficiently many of them, they can "flood" the immune system making it less effective.
Komarova [8] used the Euler-Lotka equation for the host cell population and reaction diffusion equations for antibody flooding effect.The disadvantage of the Euler-Lotka model is that it only models a steady state of viral spread, when the uninfected host cells are freely available.In this paper, we aim at providing an alternative perspective by focusing on the infection age and release strategy.More specifically, we propose a mathematical model described by ordinary differential equations with distributed delay accounting for infection age.By analyzing this structured model system, we study the evolutionary competition between these two viral productive strategies.
The rest of this paper is organized as follows.In Section 2, we present an age structured model and its simplified form with distributed delay.In Section 3, we prove that all the solutions of our model are positive and bounded.In Section 4, the equilibria of the model and their stability are discussed.In Section 5, we give two explicit forms of the viral production function and investigate the effect of antibody on evolutionary competition of budding and lytic strategies.The paper is ended by Section 6, where in addition to conclusion, some discussion is also presented.
2. Model formulation.Age structured models have been used to study the within-host dynamics for HIV (see [6,10,11]).We use an age structured model for the infected cell population.The infection age, a, is the time lapsed since a cell was infected by a virus.Suppose that T (t) is the concentration of uninfected target cells at time t; V B (t) is the concentration of virus produced by the budding strategy at time t (we call it budding virus); V L (t) is the concentration of virus produced by the lytic strategy at time t (we call it lytic virus); T * B (t, a) is the concentration of infected cells at infection age a at time t, which are infected by budding virus; T * L (t, a) is the concentration of infected cells at infection age a and at time t, which are infected by lytic virus; A(t) is the concentration of antibody at time t.By infection, budding virus and lytic virus compete for uninfected target cells.Assuming mass action infection mechanism, we propose the following system of differential equations to describe the competition dynamics of budding and lytic viruses: Here, β B and β L represent the infection rates of budding virus and lytic virus respectively.γ B (a) and γ L (a) are the virion production rates from infected cells with an infection age a by budding strategy and by lytic strategy respectively.τ B and τ L denote the ages when the infected cells begin to release new virions by budding and lysis respectively.p is the activation rate of antibodies.η B and η L are the neutralization rates of antibodies for budding virus and lytic virus respectively.Although the model (1) looks symmetric in its form for the two viruses, it is the particular forms of the two functions γ B (a) and γ L (a) that will actually characterize the nature of the production/release strategies of the two viruses, and thereby, make the model (1) asymmetric.This will be discussed in Section 5, particularly demonstrated in Fig. 4 and Fig. 5.We assume that viruses are introduced at time t = 0, meaning that there is neither virus nor infection for t ∈ [−τ * , 0), and infection occurs at t = 0 immediately after introduction of viruses.Accordingly T * B (0, a) = 0, T * L (0, a) = 0, for all a > 0. The infected cells of age zero come from the new infections, that is Now, for the budding virus, the dynamics are determined by the following initialboundary value problem: Solving this problem by the method of characteristics, we obtain Similarly, we have Substituting the above formulas for T * B (t, a) and T * L (t, a) into V B and V L equations in (1), and by our assumptions that V B (θ) = 0, V L (θ) = 0, for all θ ∈ [−τ * , 0), we obtain the following model system: In the rest of the paper, we shall investigate the dynamics of this system.3. Positivity and boundedness of solutions.Let X = C([−τ * , 0], R 4 ) be the Banach space of continuous functions with supremum norm.By the fundamental theory of FDEs [7], we know that there is a unique solution (T (t), V B (t), V L (t), A(t)) to the system with given initial conditions (T (θ), V B (θ), V L (θ), A(θ)) ∈ X. Due to the biological meanings of the unknown functions, we need to further assume that the initial functions T (θ), V B (θ), V L (θ), and A(θ) satisfy The following theorem addresses the well-posedness of the model (3).
Proof.From the first and last equations of the system (3), we have Next, we show that V B (t) > 0 for all t ∈ (0, ∞).Otherwise, there exists a first time t 1 > 0 such that V B (t 1 ) = 0 and V B (t) > 0 for t ∈ [0, t 1 ).This would lead to This implies V B (t) is negative in a small left neighborhood of t 1 , a contradiction.Therefore, V B (t) > 0 for all t > 0. Similarly, we can prove V L (t) > 0 for all t > 0.
To prove the boundedness, let γB = max By the nonnegativity of solutions, it follows that where 4. Equilibria and their stability.Let .
Here, e ξdξ denotes the age-specific survival probability of an infected cell infected by budding virus, i.e., the probability of an infected cell remaining alive at infection age a.Thus, K B is the total number of new virions produced by one infected cell, infected by budding virus, over its whole lifespan.We call K B the burst size of budding virus.Similarly, K L is the burst size of lytic virus.Notice that 1/d V is the life span of budding virus in the absence of antibody, H/d T is the cell concentration without infection, and β B is the infection rate.Hence, one budding virus, once inoculated into an environment containing H/d T uninfected cells, can lead to β B H/(d T d V ) infected cells.These infected cells then produce the amount B gives the reproductive ratio of the budding virus in the absence of antibody (also referred to as the basic reproductive number).In parallel, R L is the reproductive ratio of the lytic virus in the absence of antibody.Note that σ B accounts for the clearance rate of antibody for budding virus.Thus, R B is the reproductive ratio of budding virus when the antibody for budding virus is established.Similarly, R L is the reproductive ratio of lytic virus when the antibody for lytic virus is established.
(VI) The positive equilibrium , A (22) , where exists in any of the following cases: B , and B , and B , and B , and We now consider the stability of some equilibria.The following result suggests that if the basic reproductive ratios of both budding virus and lytic virus are less than one, then the population sizes of both budding virus and lytic virus will approach zero as t → ∞ and the antibody cannot be established.
Proof.First we consider local stability of the equilibrium E 0 .Linearizing the system (3) at equilibrium E 0 leads to ( The characteristic equation of this linear system is where It is obvious that λ 1 = −d T < 0 and λ 2 = −d A < 0 are two eigenvalues and the other eigenvalues are determined by and which are equivalent respectively to and We need to show that under the conditions of the theorem, all roots of ( 6) and ( 7) have negative real parts.Let λ = x + iy be a root of (6).We show that x < 0. Otherwise, x ≥ 0 implies a contradiction to (6).Therefore, x < 0 under R B < 1, implying that all roots of (6) have negative real parts if R (0) L < 1, then all roots of (7) also have negative real parts.It follows from [7], that the equilibrium E 0 is locally asymptotically stable if R (0) To show that E 0 is globally asymptotically stable, it is sufficient to show that E 0 is globally attractive.By the positivity of solutions, we have This implies Denote Let ε > 0 be sufficiently small such that R (0) For such an ε > 0, by (8), there exists a t * > 0 such that Thus, We consider the following auxiliary linear system Notice that the two equations in (9) are the same as the second and third equations in (5) except that H is replaced by H + ε.Thus, the characteristic equation of ( 9) is the product of two equations of the form ( 6) and ( 7) with H replaced by L (ε) < 1 ensure that all eigenvalues of (9) have negative real parts, and hence, the trivial solution of ( 9) is globally (since ( 9) is linear) asymptotically stable, meaning that every solution (w 2 (t), w 3 (t)) → (0, 0) as t → ∞.Notice that ( 9) is a co-operative delay system.By the comparison theorem [12], we conclude that lim Finally, the first and the last equations of (3) form a system which has the following autonomous system as the limit system.
The following result indicates that if the basic reproductive ratio for the budding virus is greater than one and exceeds the basic reproductive ratio for the lytic virus, then the budding virus can survive when the antibody effect is not established.
B > 1, then this equilibrium is unstable.Proof.Linearizing system (3) at equilibrium E 10 gives The characteristic equation of this linear system is λ + d T β B T (10)  β L T (10)   (10)  KB (λ) . One eigenvalue is B < 1, we also have The other eigenvalues are determined by (10)  KL (λ), (10) and Equation ( 10) is equivalent to By a similar argument to that in analyzing (6), we conclude that all roots of (10) have negative real parts if L .Equation ( 11) is equivalent to which can be further rewritten as Let λ = x + iy be a root of (13).If x ≥ 0, then by R B > 1, we have Therefore, This is a contradiction to equation (13).Therefore, if R B > 1, then x < 0 for equation (13), implying that all roots of (13) have negative real parts.
In summary, we have shown that under the assumption that R (0) B < 1, then all roots of the characteristic equation have negative real parts and hence equilibrium E 10 is locally asymptotically stable.
Parallel to Theorem 4.2, we have the following conclusion about the lytic virus when the antibody effect against lytic virus is not established.
B passes the value 1, E 10 loses its stability, giving rise to the equilibrium E 20 .The following theorem describes the stability of E 20 , characterizing the conditions under which the budding virus will persist in the presence of established andibody.
the equilibrium E 20 is locally asymptotically stable; if this equilibrium is unstable.
Proof.Firstly, note that p > η B and R (1) B > 1. Linearizing the system (3) at E 20 leads to The characteristic equation of this linear system is The roots of this equation are determined by where 16) is equivalent to which can be further rewritten as Let λ = x + iy be a root of (18).If x ≥ 0, then the left hand side of the equation (18) satisfies and the right hand side of the equation satisfies Therefore, if (14) holds, then the above two inequalities contradict to each other.Thus x < 0 if ( 14) holds, implying that all roots of (18) have negative real parts.Equation ( 17) is equivalent to Let λ = x + iy be a root of (19).If x ≥ 0, by R B > 1, the right hand side of the equation ( 19) satisfies KB (λ) and the left hand side of the equation satisfies leading to a contradiction to (19).Thus x < 0, implying that all roots of (19) have negative real parts.In summary, we conclude that if (14) holds, then the equilibrium E 20 is locally asymptotically stable. Let Then ψ(λ) → ∞ as λ → ∞.On the other hand provided that (15) holds.Therefore, there exists a λ * > 0, such that ψ(λ * ) = 0.This means the equation ( 18) has at least one positive eigenvalue, implying that the equilibrium E 20 is unstable.The proof of the theorem is completed.
Similarly, for lytic virus we have the following result when the antibody effect is established.
the equilibrium E 02 is locally asymptotically stable; if this equilibrium is unstable.
The conditions for the existence and stability of some equilibria are summarized in Table 2.We see that if p < η B and p < η L , there may only be three equilibria E 0 , E 01 , and E 10 (except the equilibrium line Ê), whose stability are determined by basic reproductive ratios R (0) B , E 01 is locally asymptotically stable.In this case, the antibody does not play a role in the long-term virus dynamics, this is because when p is too small (p < η B and p < η L ), activation of new antibodies cannot satisfy the demand on antibodies involved in neutralization of the virus for both strains.In the following discussion, we always assume that p > η B or p > η L .
Table 2.The conditions for the existence and stability of the equilibria Existence L.A.S. Outcompetes The bifurcation diagrams in different cases are given by figures in Fig. 1 to Fig. 3, representing the three cases For the case η B > η L , the bifurcation diagram is shown in Fig. 1, in which, the first quadrant of R (0) L plane is divided into sub-regions with appropriate shadings, representing the stability of the different equilibria.The shadings are given in such a way that regions with the same shading pattern share the same stable equilibrium.For example, in Fig. 1-(a) where σ B > σ L , the stability regions of equilibria E ij (i, j = 0, 1, 2) with respect to R (0) B and R (0) L are denoted by S i ; i = 1, ..., 13.Here the solid diagonal line is R In the regions shaded with vertical lines (S 1 , S 3 and S 5 ), E 10 is locally asymptotically; in regions shaded with southwest-northeast lines (S 6 , S 7 and S 8 ), E 20 is locally asymptotically stable.In these six regions, budding virus outcompetes.Similarly, in the regions shaded with horizontal lines (i.e., S 2 and S 4 ), E 01 is locally asymptotically stable; in the regions shaded with northwestsoutheast lines (i.e., S 9 , S 10 and S 11 ), E 02 is locally asymptotically stable.In these five regions, lytic virus outcompetes.In the regions with overlap shadings, there are two locally asymptotically stable equilibria.For instance, in S 13 both E 20 and E 02 are locally asymptotically stable; and in S 12 , both E 10 and E 01 are locally asymptotically stable.In these two regions with overlap shadings (S 12 and S 13 ), in addition to the two locally stable boundary equilibria (E 10 and E 01 , or E 20 and E 02 ) there is also a positive equilibrium E 22 whose stability is undetermined.In the region S 0 , the disease-free equilibrium E 0 is locally asymptotically stable.Table 3 summarizes the situations in these regions.
In Fig. 1-(b), σ B < σ L .As is shown in this figure, existence and stability of equilibria are the same as Fig. 1-(a) in all regions other than S 5 , S 12 and S 13 , the situation of which is shown in Table 4: in both regions S 12 and S 13 there is no stable equilibrium except the positive equilibrium E 22 whose stability is undetermined.By symmetry, we have Fig. 2 for the case η B < η L which is parallel to Fig. 1.Tables parallel to Table 3 and Table 4 can be drawn but are omitted here.
If η B = η L (see Fig. 3), the positive equilibrium E 22 does not exist.Furthermore, there are no regions S 12 and S 13 .The properties of equilibria in all other regions are same as Fig. 1-( L > max{1, R B }, then E 01 or E 02 is locally asymptotically stable.

5.
Influence of antibody effect on the evolutionary competition between budding and lytic strategies.From the above results on the dynamics of the model (3), we see that the impact of the production/release strategies for new virions is reflected by the dependence of the reproductive ratios on K B and K L , the burst sizes of budding virus and lysis virus under the respective production/release strategies represented by γ B (a) and γ L (a) and the initial releasing time τ B and τ L .In this section, we consider two particular forms for the release strategy functions γ B (a) and γ B (a), by which we hope to obtain more information on the impact of antibody and the release strategy.To this end, we assume that the total number of virions replicated from an infected cell without considering cell death is the same for all strategies, that is, The first possible candidate for the viral production kernel function is the one used in [10] which has the form Here m 2 controls how rapidly the saturation level m 1 is reached, while τ ≥ 0 is the initial releasing time.For this production kernel, the constraint equation defines a trade-off relation of τ, m 1 and m 2 , which is or equivalently, Fig. 4 demonstrates the strategy function γ(a) with N = 100 and τ = 30 fixed and m 1 determined by the trade-off equation ( 26) for some values of m 2 and τ .As is shown in Fig. 4, larger m 2 will make γ(a) to approach the saturation level m 1 faster.Smaller τ represents the budding strategy and larger τ accounts for lytic strategy.
Another candidate for γ(a) has the following form which was used in [1]: This function is not monotone, and it has the maximum Here k 2 determines how rapidly the maximum is reached.For this function, the calculation gives  thus, the constraint (24) reads Similarly, Fig. 5 shows the behavior of γ(a) given by ( 27) for some values of the parameters.As is in Fig. 4, we also fix N = 100 and τ = 30.k 2 is fixed at k 2 = 2 in Fig. 5-(a) and at k 2 = 20 in Fig. 5-(b), the plots are for τ = 5, 10, 15, 20, 25 respectively with k 1 determined by (28).We can see from Fig. 5 that with small k 2 , γ(a) reaches the maximum rapidly.Furthermore, for small τ , there is a small surge in viral production with a subsequent long period of low level viral production; in contrast, for large τ , there is a big surge in viral production.Again, small τ accounts for the scenario of budding strategy while large τ explains lytic strategy.For the above two concrete forms of γ(a), if we further assume that the death rate of infected cells is constant: d T * (a) = d T * , we can calculate the total number of new virions produced/released by an infected cell under the strategy γ(a) as for γ(a) given by (23), and for γ(a) given by ( 27).One can explore the dependence of K(τ ) on τ , as well as on m 2 and k 2 , to obtain more information.For example, numeric plots show that these two functions are both decreasing in τ (see Fig. 6).When d T * (a) is not constant, it is generally difficult to obtain an explicit formula for K, but numerical calculation can give some information.To illustrate this, we consider the following death rate function proposed in [10]: where δ 0 is the background death rate, τ 0 is the delay between infection and the onset of cell-mediated killing or the beginning of cell death due to the viral cytopathic effects, δ 0 + δ 1 is the maximal death rate and δ 2 controls how quickly it approaches the saturation level.Fig. 7 shows some plots of this function for some parameter values.
For this death function, if τ 0 ≤ τ , the burst size reads  an ordinary differential equation model with distributed delays accounting for the release strategies.Our model can also predict a possibility that lytic virus may have an evolutionary advantage when the efficacies of antibodies are different for the two viruses.Thus this work offers an alternative view point for the scenario that a lytic virus can also outcompete budding virus under certain circumstances.

Fig. 1 -
Fig. 1-(c) covers the case σ B = σ L .Comparing Fig. 1-(c) with Fig. 1-(a) and Fig. 1-(b), there is no region S 5 , S 12 and S 13 .The existence and stability of equilibria in all other regions remain the same as Fig. 1-(a) and Fig. 1-(b).By symmetry, we have Fig.2for the case η B < η L which is parallel to Fig.1.Tables parallel to Table3and Table4can be drawn but are omitted here.If η B = η L (see Fig.3), the positive equilibrium E 22 does not exist.Furthermore, there are no regions S 12 and S 13 .The properties of equilibria in all other regions are same as Fig.1-(a) for Fig. 3-(a), Fig 1-(b) for Fig. 3-(b), Fig. 1-(c) for Fig. 3-(c).
Fig. 1-(c) covers the case σ B = σ L .Comparing Fig. 1-(c) with Fig. 1-(a) and Fig. 1-(b), there is no region S 5 , S 12 and S 13 .The existence and stability of equilibria in all other regions remain the same as Fig. 1-(a) and Fig. 1-(b).By symmetry, we have Fig.2for the case η B < η L which is parallel to Fig.1.Tables parallel to Table3and Table4can be drawn but are omitted here.If η B = η L (see Fig.3), the positive equilibrium E 22 does not exist.Furthermore, there are no regions S 12 and S 13 .The properties of equilibria in all other regions are same as Fig.1-(a) for Fig. 3-(a), Fig 1-(b) for Fig. 3-(b), Fig. 1-(c) for Fig. 3-(c).

Figure 2 .L 5 S
Figure 2. The stability regions of equilibria with respect to R (0) B

Table 1 .
(3) descriptions of the parameters in the model(3)