GLOBAL ANALYSIS OF WITHIN HOST VIRUS MODELS WITH CELL-TO-CELL VIRAL TRANSMISSION. Hossein Pourbashash and Sergei S. Pilyugin and Patrick De Leenheer

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1 Manuscript submitted to AIMS Journals olume X, Number0X, XX200X doi:0.3934/xx.xx.xx.xx pp. X XX GLOBAL ANALYSIS OF WIHIN HOS IRUS MODELS WIH CELL-O-CELL IRAL RANSMISSION Abstract. Recent experimental studies have shown that HI can be transmitted directly from cell to cell when structures called virological synapses form during interactions between cells. In this article, we describe a new withinhost model of HI infection that incorporates two mechanisms: infection by free virions and the direct cell-to-cell transmission. We conduct the local and global stability analysis of the model. We show that if the basic reproduction number R 0, the virus is cleared and the disease dies out; if R 0 >, the virus persists in the host. We also prove that the unique positive equilibrium attracts all positive solutions under additional assumptions on the parameters. Finally, a multi strain model incorporating cell-to-cell viral transmission is proposed and shown to exhibit a competitive exclusion principle. Hossein Pourbashash and Sergei S. Pilyugin and Patrick De Leenheer Department of Mathematics, University of Florida 400 Stadium Road Gainesville, FL 326, USA Connell McCluskey Department of Mathematics, Wilfrid Laurier University 75 University Avenue West Waterloo, ON, Canada, N2L 3C5 his paper is dedicated to Chris Cosner, on the occasion of his 60th birthday. (Communicated by the associate editor name). Introduction. Mathematical modeling of within host virus models has flourished over the past few decades. hese models have been used to describe the dynamics inside the host of various infectious diseases such as HI, HC, HL, as well as the flu or even the malaria parasite. esting specific hypotheses based on clinical data is often difficult since samples cannot always be taken too frequently from patients, or because detection techniques of the virus may not be accurate. his justifies the central role played by mathematical models in this area of research. One of the early models, known as the standard model and reviewed in the next section, was used by Nowak and May [6], and by Perelson and Nelson [7] to model HI. It was successful in numerically reproducing the dynamics of the early stages of HI and its target, the CD4 + cells, following an infection event. he global behavior of the standard model was first investigated analytically in [2]. Here, the global stability of the disease steady state was proved using the powerful second compound matrix methods developed by Muldowney [5] and by Li and Muldowney [0]. Among the first successes of this geometrical method was the long 200 Mathematics Subject Classification. Primary: 37N25, 97M60; Secondary: 34A34, 34D23. Key words and phrases. within-host virus model; cell-to-cell transmission; global stability; Lyapunov function, second compound matrices.

2 2 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER sought-after proof of global stability in the classical epidemiological SEIR model []. Since epidemic models and within host virus models are structurally similar, it should not come as a surprise that the geometrical methods of [0, 5] are at the basis of the global stability proof of the standard model in [2]. It is well-known from the converse Lyapunov theorem [7], that whenever a nonlinear system of ODEs has a globally stable steady state, that then there exists a global Lyapunov function. Korobeinikov in [8] was the first to discover a Lyapunov function in the context of SEIR and SEIS models. Korobeinikov s Lyapunov function was exploited further in [3] to establish global stability for the standard model under relaxed conditions for the model parameters, as well as for within host virus models with multiple, possibly mutating, strains. Although our discussion so far was focused on the global stability of certain steady states, it has been demonstrated that more complex dynamical behavior, in particular taking the form of sustained oscillations, may arise in the standard model for specific proliferation rates of healthy cells [2, 24], or under the assumption that infected cells proliferate as well [23], a feature which is not encoded in the standard model. Based on more recent findings about additional infection processes, it is the main goal of this paper to extend the standard model in two ways:. In the standard model it is assumed that healthy cells can only be infected by viruses. However, recent work shows that cell-to-cell transmission of viruses also occurs when a healthy cell comes into contact with an infected cell [8]. In fact, in vitro experiments reported in [2] have shown that in shaken cultures, viral transfer via cell-to-cell contact is much more rapid and efficient than infection by free virus because it avoids several biophysical and kinetic barriers. In vivo, direct cell-to-cell transmission is also more potent and more efficient according to [4]. he details of the infection mechanism via cellto-cell contact are described in [3, 4, 9] and have been attributed to the formation of virological synapses (Ss), filopodia and nanotubes. he model proposed here deviates from, and in fact, expands the standard model, by including the additional cell-to-cell infection mechanism. 2. According to the standard model, viruses do not enter previously infected cells, although most HI infected splenocytes for example, are found to be multiply infected [4]. Moreover, the loss of virus upon entry of a healthy cell, has often been neglected in the literature. he model we propose here, accounts for both processes. 2. he mathematical model. he standard model of a within-host virus infection [6, 7] is = f( ) k, = k β, = Nβ ( k ), where the term in the parentheses may or may not be present. Our mathematical model has the following form: = f( ) k k 2, = k + k 2 β, = Nβ k 3 k 4. ()

3 WIHIN HOS IRUS MODELS 3 where, and denote the concentrations of uninfected host cells, infected host cells and free virus particles, respectively. Parameters k,k 2,β,N and are all positive constants. k is the contact rate between uninfected cells and viruses. k 2 is the contact rate between uninfected cells and infected cells. he parameters β and represent the death rate of infected cells and virus particles, respectively. In the case of budding viruses, N is the average number of virus particles produced by an infected cell during its lifetime; in case of lytic viruses, N represents the average burst size of an infected cell. he parameter k 3 models the rate of absorption of free virions into healthy cells during the infection process. In some instances, this term has been neglected [6, 7]. he parameter k 4 models the absorption of free virions into already infected cells, a process which may be biologically relevant for some viruses, yet in each case it will be determined by the specific mechanism of the viral cell entry [4]. he growth rate of the uninfected cell population is modeled by the smooth function f : R + R, which is assumed to satisfy the following: 0 > 0:f( )( 0 ) < 0, = 0, and f ( ) < 0 [0, 0 ]. (2) he continuity of f implies that f( 0 ) = 0, and hence E 0 =( 0, 0, 0) is an equilibrium point of system (). Biologically, E 0 represents the disease-free equilibrium. he Jacobian matrix of () at E 0 is J 0 = f ( 0 ) k 2 0 k 0 0 k 2 0 β k 0. 0 Nβ k 3 0 he submatrix of J 0 corresponding to the infectious compartments can be split as follows: k2 J 0 = 0 β k 0 k2 = 0 k 0 β 0 =: F. Nβ k Nβ + k 3 0 A quick calculation shows that the next generation matrix is given by F = k2 0 β + Nk0 k 0 +k 3 0 +k he basic reproductive number, R 0, is defined as the spectral radius of F [22], hence R 0 := ρ(f )= k 2 0 β + Nk 0. (3) + k 3 0 One eigenvalue of J 0 is given by f ( 0 ) < 0, and the remaining two are also eigenvalues of J 0. he determinant of J 0 is given by det J 0 = (β k 2 0 )( + k 3 0 ) Nβk 0 = β( + k 3 0 ) k 2 0 β Nk 0 + k 3 0 = β( + k 3 0 )( R 0 ). Consequently, if R 0 >, then det J 0 < 0, and J 0 has a positive eigenvalue. If R 0 <, then det J 0 > 0, and k 2 0 <βwhich implies that trj 0 < 0, so that all eigenvalues of J 0 have negative real parts. We obtain our first result. heorem 2.. If R 0 <, then E 0 is locally asymptotically stable; if R 0 >, E 0 is unstable.

4 4 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER A positive equilibrium E =(,, ) must satisfy the following relations: β = f( ), = Nf( ) + k 3 + k4, =k 2 β + Nk + k 3 + k 4f( ) β := G( ). he function G is monotonically increasing, since the first term is linear in, and the reciprocal of the second term + k 3 + k 4f( ) Nk Nk Nk β is strictly decreasing in when is in (0, 0 ), by (2). A positive equilibrium corresponds to a solution of G( )=with0< < 0. Since G(0) = 0 and G( 0 )=R 0, we conclude that a unique positive equilibrium exists if and only if R 0 >. Lemma 2.2. If R 0 the equilibrium E 0 is the only equilibrium of (), and if R 0 > then E 0 and E =(,, ) are the only two equilibria of (). 3. Preliminary results for the general model. Lemma 3.. Every forward solution of () starting in R 3 +, is defined in R 3 + for all t 0, and bounded. Furthermore, system () is uniformly dissipative in R 3 +. Proof. Positive invariance of R 3 + follows from the standard argument that the vector field of system () is inward pointing on the boundary of R 3 +. o show boundedness, fix a forward solution in R 3 +. Observe that the -equation in () implies that (t) M := max( 0,(0)) for all t 0 as long as the solution is defined. Since f( ) is bounded on [0,M], there exist A, B > 0 such that f( ) A B for all [0,M]. Choose a constant 0 <α < /N and consider an auxiliary function U = + + α.dueto(), we have U f( ) β( α N) α A B β( α N) α A α 2 U, where α 2 =min(b,β( α N),) > 0. Hence, U(t) max U(0), Aα2 for all t 0. his implies that the corresponding forward solution of () is bounded in R 3 + and exists for all t 0. From the above argument, it is clear that any solution of () inr 3 + eventually satisfies (t) M 0 := 0 +, and therefore one can choose uniform constants A 0,B 0 > 0 so that the function U(t) eventually satisfies the inequality U A 0 α 0 U where α 0 =min(b 0,β( α N),) > 0. Hence, any solution of () inr 3 + eventually enters a forward invariant simplex D = (,,) R 3 + : + + α A 0 +. (4) α 0 Hence, system () is uniformly dissipative in R 3 +. At this point, we make the following (possibly trivial) remark: Consider any solution of () inr 3 +. If there exists a t 0 such that (t 0 ) < 0 then (t) < 0 for all t t 0. Our next result provides the necessary and sufficient condition for the global stability of E 0. heorem 3.2. If R 0 then the infection free equilibrium E 0 attracts all solutions in R 3 +.

5 WIHIN HOS IRUS MODELS 5 Proof. If (0) = 0, then (0) > 0, so (t) > 0 for all sufficiently small t>0. If (t) = (t) = 0, then it is clear that (t) 0 as t. If exactly one of (0) or (0) is zero, then its derivative at t = 0 is positive, and hence this variable is positive for all small t>0. Furthermore, none of (t), (t) or (t) can decrease to zero in finite time, hence it suffices to consider only solutions with both (t) and (t) positive for all t 0. We previously remarked that either (t) 0 for all t 0, or there exists a t 0 0 such that (t) < 0 for all t>t 0. Consider the former case, that is, (t) 0 for all t 0. hen (t) 0 for all t 0, and (t) 0 as t. Barbalat s Lemma implies that lim t (t) = 0. Noticing that by (2), f( ) f( 0 ) = 0, we use the -equation in (), to conclude that = 0, and lim t (t) =lim t (t) = 0. Now, suppose that there exists a t 0 0 such that (t) < 0 for all t>t 0. Without loss of generality, we may assume that (t) < 0 for all t 0. We introduce an auxiliary function: and evaluate Ẇ along a solution of (): Ẇ = ( + k 3 0 ) + k 0 W =( + k 3 0 ) + k 0, (5) = ( + k 3 0 )(k + k 2 β )+k 0 (Nβ k 3 k 4 ) = (k 0 Nβ +( + k 3 0 )(k 2 β)) + (k ( + k 3 0 ) k 0 ( + k 3 ) k 0 k 4 ) β( + k 3 0 ) (R 0 ) + k ( 0 ) k k 4 0 0, where all three terms are non-positive. herefore, there exists lim t W (t) = W 0. Barbalat s Lemma then implies that lim t Ẇ (t) = 0. If W = 0, it is clear that lim t (t) =lim t (t) = 0 and lim t (t) = 0. If W > 0, then there exists > 0 such that k + k 2 > 0 for all sufficiently large t. he -equation in () implies that f( ), and thus there exists 2 > 0 such that (t) 0 2 for all sufficiently large t. Since Ẇ k ( 0 ) k 2,lim t Ẇ (t) = 0 implies that lim t (t) = 0, and by Barbalat s Lemma, we have that lim t (t) = 0, and the -equation in () implieslim t (t) = 0, hence W = 0, a contradiction. herefore, W = 0, and the proof is complete. he next heorem shows that the positive equilibrium E is locally stable whenever it exists. heorem 3.3. If R 0 >, then E is locally asymptotically stable. Proof. If R 0 >, a unique positive equilibrium E exists by Lemma 2.2. For notational convenience, we will drop all bars so that all variables are at their equilbrium values. he following relations hold at E: and β = f( ), k + k 2 = f( )/, + k 3 + k 4 = Nβ β k 2 = k. = Nf( ),

6 6 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER he Jacobian at E has the form f ( ) k k 2 k 2 k J(E) = k + k 2 k 2 β k. k 3 Nβ k 4 k 3 k 4 Using the equilibrium relations, J(E) simplifiesto f f/ k 2 k J(E) = f/ k k, k 3 ( + k 3 ) Nf where we have suppressed the arguments of f and f. he characteristic polynomial p(λ) =λ 3 + a λ 2 + a 2 λ + a 3 is given by λ f + f/ k 2 k p(λ) =det f/ λ + k k. k 3 ( + k 3 ) λ + Nf A direct computation using the equilibrium relations above to simplify a 2 and a 3, we obtain a = f/ f + k + Nf > (f/ f )+ Nf > 0, a 2 = k k 4 +( f k + f/ ) + Nf + k 2 f k k 3, a 3 = ( f + f/ )k k 4 + f/ Since f > 0, we have that Since kf a 2 >k k 4 + k f = k β, Nf k 2 Nf + k ( + k 3 ) + Nf2 + k 2f k k 3. >k 3, and f/ > k, we have that a 2 >k k 4 + k β + k 2 f>0. Simplifying a 3 in a similar fashion, we find that k k 3 β. a 3 =( f + f/ )k k 4 + k 2Nf 2 + k β > 0. Finally, since Nf >, we have that a a 2 >a 3, and J(E) satisfies the Routh- Hurwitz stability conditions. Lemma 3.4. System () is uniformly persistent in D, the interior of D, ifand only if R 0 >. Here, the set D is defined in (4). Proof. If R 0, then by heorem 3.2, E 0 attracts all positive solutions. Hence R 0 > is necessary for persistence. he sufficiency of the condition R 0 > follows from a uniform persistence result, heorem 4.3 in [6]. o demonstrate that system () satisfies all the conditions of heorem 4.3 in [6] whenr 0 >, choose X = R 3 + and E = D. he maximal invariant set N on the boundary D is the singleton {E 0 } and it is isolated. hus, the hypothesis (H) of [6] holds for (). We conclude the proof by observing that, in the setting of (), the necessary and sufficient condition for uniform persistence in heorem 4.3 of [6] is equivalent to E 0 being unstable. In the remainder of this paper, we prove that the positive equilibrium E is globally asymptotically stable in several special cases of system ().

7 WIHIN HOS IRUS MODELS 7 4. he case k 4 =0. We begin this section by introducing the method of demonstrating global stability through the use of compound matrices; see [0, 5]. Let Ω be an open subset of R n, and let f : Ω R n be a vector field that generates the flow φ(t, x). hat is, for a given x Ω, z(t) =φ(t, x) istheunique solution of the system ż = f(z) (6) with z(0) = x. A compact set K Ω is called absorbing if for each compact set F Ωthereexistsa>0suchthat φ(t, F ) K for all t. It is clear that an absorbing set must contain all equilibria of f in Ω. We say that an open set Ω R n is simply connected if each closed curve in Ω can be contracted to a singleton within Ω. Let denote the exterior product on R n R n, that is, a surjective antisymmetric bilinear transformation : R n R n R d,whered = n 2. Consider the basis of R d consisting of pairwise products e i e j, i<j n ordered lexicographically, where the vectors e i, i n are the elements of the canonical basis of R n. For a real n n matrix M, its second additive compound matrix, denoted by M [2],is defined by its action on the chosen basis as follows: M [2] (e i e j ):=(Me i ) e j + e i (Me j ), i<j n. According to this definition, M [2] is a d d matrix. For instance, if M =(m ij )is a3 3 matrix, then its second additive compound is given by M [2] = m + m 22 m 23 m 3 m 32 m + m 33 m 2. (7) m 3 m 2 m 22 + m 33 For more details on compound matrices, we refer the reader to [5, 5]. Let denote a vector norm on R d and let be the induced matrix norm on R d d, the space of all real d d matrices. For a given matrix A R d d,wedefine its Lozinskii measure with respect to the norm as µ(a) = I + ha lim. h 0 + h More details on the properties of Lozinskii measures can be found in []. For future reference we recall without proof the Lozinskii measure, associated to the l vector norm: Lemma 4.. he Lozinskii measure of a real n n matrix M with respect to the matrix norm induced by the l vector norm, is given by µ (M) = max i m ii + j=i m ji Let the map A :Ω R d d be continuously differentiable and nonsingular for all x Ω, and let µ be a Lozinskii measure on R d d,whered = n 2. Let K be a compact absorbing set for the flow φ(t, x) in Ω. Define a quantity q 2 as where q 2 =limsupsup x K t t t 0 µ(m(φ(s, x)))ds, (8) M(x) =A f (x)a (x)+a(x)j [2] (x)a (x), (9)

8 8 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER with A f (x) := d A(φ(t, x)), dt We have the following result [0, 24]. J(x) := f x (x). heorem 4.2. Consider system (6), and suppose that Ω is simply connected and contains a compact absorbing set K. Let (6) have a unique equilibrium point x K. If A :Ω R d d is C and everywhere nonsingular, and if µ is a Lozinskii measure such that q 2 < 0, then x is globally asymptotically stable in Ω. For the proof of the global stability of E, we will need the following Lemma. Lemma 4.3. Every forward solution of () in R 3 + satisfies lim sup (t) Nβ f(0) t β + 0 =: max. Proof. Consider a nonnegative forward solution of () and let >0. he -equation in () implies that (t) 0 + for all sufficiently large t. Hence, since f ( ) < 0 on [0, 0 ]by(2), the total number of cells, tot := +, for all large t satisfies: tot = f( ) β f(0) + β β tot f(0) + β( 0 + ) β tot. It follows that (t) tot (t) f(0) β for all sufficiently large t. he -equation in () now implies that f(0) Nβ β for all sufficiently large t, thus (t) Nβ f(0) β for all sufficiently large t. Since >0 is arbitrary, the claim follows. heorem 4.4. Suppose that k 4 =0and k 3 0. Let D be the set defined in (4), and let Ω= D. SupposethatR 0 >, 2k 2 0 <β,andk 3 (f(0) + β 0 ) <β, then E is globally asymptotically stable for system () in Ω. Proof. Consider a positive solution of (). As in the proof of heorem 3.2, any solution with (t) 0 for all t 0 converges to E 0. Since R 0 >, Lemma 3.4 implies persistence, hence any positive solution must satisfy (t) < 0 for all sufficiently large t. In addition, Lemma 4.3 implies that lim sup t (t) max. he condition k 3 (f(0) + β 0 ) <βimpliesthat k 3 max <Nβ,hencek 3 (t) <Nβ for all sufficiently large t. By shifting time (which does not affect the value of the quantity q 2 ), if necessary, we will assume that (t) < 0 and k 3 (t) <Nβfor all t 0. Now, consider the Jacobian matrix of system (): f ( ) k k 2 k 2 k J = k + k 2 k 2 β k. (0) k 3 Nβ k 3 he second compound matrix corresponding to J is given by J [2] = a β k k Nβ b k 2, () k 3 k + k 2 c β

9 WIHIN HOS IRUS MODELS 9 where a = f ( ) k k 2 + k 2, b = f ( ) k k 2 k 3, c = k 2 k 3. (2) Define an auxiliary matrix function A on Ω as A := diag,,. Since, > 0everywhereinΩ,A is smooth and nonsingular. Furthermore, we find that A f = diag ( ) 2, 2, 2, A f A = diag,,, hence a β AJ [2] A = Nβ M = A f A + AJ [2] A = where M = a β M 22 = k k b k 2 k 3 k + k 2 c β = a β M 2 Nβ k, (3) k b k 2 k 3 k + k 2 c β M M 2, (4) M 22, M 2 = k b k + k 2 k, M2 = k 2 c β. Nβ k 3, Let (u, v, w) denote a vector in R 3, and define a vector norm on R 3 as (u, v, w) = max( u, v + w ) and let µ be the corresponding Lozinskii measure. hen from [, 2], we have that where µ(m) max(g,g 2 ), (5) g = µ (M )+M 2, g 2 = µ (M 22 )+M 2. M 2, M 2 are the operator norms associated to the linear mappings M 2 : R 2 R, and M 2 : R R 2 respectively, where R 2 is endowed with the l vector norm in both cases.

10 0 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER We have the following µ (M ) = a β, M 2 = k, M 2 = max Nβ,k 3, µ (M 22 ) = + max (k + k 2 + b, k 2 + c β), where Lemma 4. has been used to calculate µ (M 22 ). Since k 3 < Nβ, it follows that Nβ k 3, and M 2 = Nβ. Using (), we find that = k = Nβ Recalling the expressions for a, b and c in (2), we have that + k 2 β, (6) k 3. (7) µ (M 22 )= Nβ + max(f ( ), 2k 2 β), and therefore that g = a β + k = f ( ) k k 2, g 2 = max(f ( ), 2k 2 β). Since f ( ) k k 2 max(f ( ), 2k 2 β), we find that g g 2, and then (5) implies that µ(m) g 2. Let p = max [0, 0] f ( ), and p 2 =2k 2 0 β. he assumptions in 2 imply that p < 0, and the assumptions of this heorem imply that p 2 < 0. Hence, µ(m) p, p := max(p,p 2 ) < 0. Integrating the above inequality, we find that herefore, t lim sup t t t 0 t 0 µ(m) ds p. µ(m) ds p<0. his clearly holds for all solutions in Ω, thus q 2 p<0. he uniform persistence established in Lemma 3.4 implies the existence of a compact absorbing set K Ω. Observing that Ω is simply connected, we apply heorem 4.2 to complete the proof. 5. he case k 3 =0. Although this case is biologically relevant only if k 4 = 0, the following global stability result holds for all k 4 0. heorem 5.. Suppose that k 3 =0and k 4 0. Let D and Ω be as defined in heorem 4.4. IfR 0 >, then the equilibrium E is globally asymptotically stable for system () in Ω.

11 WIHIN HOS IRUS MODELS Proof. Local stability of E follows from heorem 3.3. equilibrium relations hold: k + k2 Combining (9) and (20), we find that Consider the following function on R 3 +: W = dτ + τ τ So, dw dt = d dt + he first term, A,inẆ A = = = Recall that the following = β, (8) = k, β k 2 (9) Nβ = + k 4. (20) Nβk = β k2 + k k 4. (2) d dt + k can be rewritten as (f( ) k k 2 ) (f( ) f( )) + (f( ) f( )) + dτ + k τ dτ. d dt := A + A 2 + A 3. f( ) (k + k 2 )+ (k + k 2 ) (k + k2 ) (k + k 2 ) + (k + k 2 ) = (f( ) f( )) + (k + k2 ) 2 (k + k 2 ) (k + k 2 )+ (k + k 2 ). Due to (8), the second term, A 2,inẆ takes the form A 2 = (k + k 2 β )= k + k 2 β k he third term, A 3,inẆ is A 3 = k (Nβ k 4 )= k Nβ k k k 4 Using (2), A 3 can be written as k Nβ k 2 + k + k2. + k + k k 4. A 3 = β k 2 + k k 4 k k 4 k β + k 2 k k k + k k 4.

12 2 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER Adding A + A 2 + A 3, and using (8), we obtain Ẇ = (f( ) f( )) + k + k k k he first term is always non-positive due to our assumptions on f. he second, third and forth terms are non-positive as well due to the arithmetic-geometric mean (AM- GM) inequality. Hence, Ẇ 0in R 3 +. o characterize the subset of R 3 + where Ẇ equals zero, we distinguish two cases: k 4 > 0 and k 4 = 0. If k 4 > 0, then Ẇ equals zero if and only if (,,)=E. Since W is a proper Lyapunov function in R 3 +, it follows that E is globally asymptotically stable in R 3 +.Ifk 4 = 0, then Ẇ =0if and only if = and =. Since all solutions of () in R3 + are bounded by Lemma 3., the LaSalle s invariance principle implies that any ω-limit set in R 3 + is a subset of the largest invariant set in M = {(,,) R 3 + =, = }. Any such invariant set in M must satisfy = 0, hence 0=f( ) k + k2 = f( ), which implies that = and =. herefore, the largest invariant set in M is the singleton {E}, and since W is a proper function in R 3 +, E attracts all solutions in R Competitive exclusion. he global stability result of heorem 5. can be readily extended to the model with several viral strains competing for the same pool of susceptible cells. he multistrain competition model takes the form = f( ) i ki i i ki 2i, i = k i i + k2 i i β ii, i =,...,n, (22) i = N i β i i i i, i =,...,n, on R 2n+ +, where we neglect the loss of virions due to cellular absorption. For each strain, we define the quantities i 0 = ki 2 0 β i + N ik i 0 i, i = We have the following extinction result. i β i k2 i i + N i k i β = 0 i i. heorem 5.2. Suppose that 0 > max{, 2 0,..., n 0 }. hen the equilibrium E = (,,, 0,...,0) attracts all solutions of (22) with (0) + (0) > 0. Here, := f( )/β and := N f( )/. Proof. We begin by observing that 0 > 0 i is equivalent to < i for i =2,...,n. It is also easy to check that E is indeed an equilibrium of (22). Define W as follows W = τ dτ + + i 2( i + B i i ), τ dτ + k 0 τ dτ

13 WIHIN HOS IRUS MODELS 3 where B i = ki i for i =2,...,n.Evaluating Ẇ, we find that Ẇ = A +A A n, where A = ( )(f( ) f( )) +k2 2 + k 3 is the same as Ẇ in the proof of heorem 5., and A i = (k i i + k i 2 i ) β i i + B i (N i β i i i ) = i ( k2 i β i + B i N i β i )+ i ( k i B i i ) = i k2 i β i + ki N i β i i = ki 2 i + N i k i β i i Clearly, all A i s are non-positive, hence Ẇ = 0 is given by i ( i )= β i i 0 0 i ( i ). Ẇ 0. Furthermore, the set M where M = { =, =, 2 =... = n =0}, thus the largest invariant set of (22) inm is E. An application of the LaSalle s invariance principle concludes the proof. 6. he case 0 <k 3 << k 4. In this section, we use a global perturbation result [20] to show that the positive equilibrium E is globally asymptotically stable for system () in R 3 +, provided that k i > 0 for i =,...,4, and k 3 is sufficiently small. We begin with the following result. Lemma 6.. Suppose that k,k 2,k 4 > 0, k 3 =0,andR 0 >. Let W be the Lyapunov function defined in the proof of heorem 5.. hen there exist, 2 > 0 such that Ẇ (x) for all x R 3 + such that x E 2. In other words, Ẇ is bounded away from zero outside of any compact neighborhood of E in R 3 +. Proof. From the proof of heorem 5., we have that Ẇ k 3 + k k 4 o simplify the notation, let 2 + k 2. 2 A = k, A2 = k 2, A 3 = k k 4, u =, v =, w =, so that Ẇ A 3 u u v v A 2 (u ) 2 u A 3 v(w ) 2. For each v>0, the first term is maximized when u = v,hence 3 u u v v 3 2 v v = 2+ v v ( v ) 2 = 2+ v v( v + ) 2 (v )2.

14 4 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER herefore, 2+ v Ẇ A (v (u ) 2 v( v + ) 2 )2 A 2 A 3 v(w ) 2. u Now, let >0 be sufficiently small, and suppose that Ẇ ( 2, 0). hen we have (u ) 2 A 2 < 2 u <κ u <κ, for some κ > 0 which is independent of. Similarly, A 2+ v (v v( v + ) 2 )2 < 2 v <κ 2, for some κ 2 > 0 which is independent of. his implies that w A 3 ( κ 2 ) <κ 3 <κ 3, for an appropriate κ 3 > 0. Lastly, we have that = w w v (κ 2 + κ 3 ) <κ 4, κ 3 for some κ 4 > 0. Combining the above inequalities, we conclude that there exists κ 0 > 0 such that for all sufficiently small >0, the inequality 2 < Ẇ (x) < 0 implies that x E <κ 0, and the claim follows. heorem 6.2. Suppose that k,k 2,k 4 > 0, k 3 =0,andR 0 >. hen there exist δ>0 and a smooth mapping E :[0,δ) R 3 +, such that for every k 3 [0,δ) the point E(k 3 ) is a globally asymptotically stable equilibrium of () in R 3 +. Proof. First, we observe that R 0 is a continuous function of k 3 and for k 3 = 0, R 0 >. Hence, there exists δ > 0 such that R 0 > as long as k 3 [0,δ ). By Lemma 2.2, for each k 3 [0,δ ) there exists a unique positive equilibrium E(k 3 ) of () which is locally asymptotically stable by heorem 3.3. By the Implicit Function heorem, the mapping E :[0,δ ) R 3 + is smooth. Now, let, 2 > 0 be as defined in Lemma 6.. Let W be the Lyapunov function defined in the proof of heorem 5.. SinceW is a proper Lyapunov function, there exists a sufficiently small 0 > 0 such that W (x) 0 implies x E(0) 2.Define the compact set K = W ([0, 0 ]). Let δ 2 := 2k (0 + ), and suppose that k 3 [0,δ 2 ). hen for any solution of () in R 3 + it holds that (t) < 0 + for all sufficiently large t. We have that Ẇ = (f( ) f( )) + k 3 + k k k 4 2 k k 3 ( ). As long as x K and (t) < 0 +, Lemma 6. implies that Ẇ + k δ 2 (0 + ) 2 < 0, hence every positive solution enters the set K in finite time, so that K is an absorbing forward invariant compact set for () for all k 3 [0,δ 2 ). Now, letting

15 WIHIN HOS IRUS MODELS Log0 3 2 Uninfected cells Infected cells irus ime Figure. ime series of a solution ( (t), (t),(t)) of system () with parameter values as described in Example in the text. δ =min(δ,δ 2 ), we apply the global perturbation result (Proposition 2.3 in [20]) to conclude that E(k 3 ) is globally stable under () for all k 3 [0,δ). 7. Numerical examples. In this Section we numerically explore the behavior of our model when some of the conditions of the main results, heorem 4.4 and heorem 6.2, fail. In both cases, it appears that the disease steady state attracts the simulated solution, providing evidence for the conjecture that this steady state is globally stable when it exists. Example : In this example, we set k 4 = 0 and we have chosen the parameters such that both 2k 2 0 >βand k 3 (f(0) + β 0 ) >β, which implies that heorem 4.4 is not applicable. We used f( )=a b with a = 0 4 ml day and b =0.0day (which implies that 0 = 0 6 ml ), k = ml day,k 2 = 0 6 ml day,k 3 = 0 4 ml day,n = 3000, = 23day,β =day. Notice that with these assignments we have that R 0 =.44 >, 2k 2 0 β = > 0, and k 3 (f(0) + β 0 ) β = 78 > 0. he time series of the components of a simulated solution with initial condition ( (0), (0),(0)) = (0 6, 0, 0 6 ) are displayed in Figure. hey suggest convergence to a positive steady state. Example 2: In this example, we have chosen k 3 >> k 4 > 0 (in fact, k 3 = 0k 4 ), which implies that heorem 6.2 may not be applicable. Once again, we have set f( )=a b, and for a, b, k,k 2,k 3,N, and β we used the same values as in Example. On the other hand, here, k 4 = 0 5 ml day. Notice that these assignments yield the same value for R 0 =.44 as in Example because this value is independent of k 4 ;see(3). he time series of the components of a simulated solution with the same initial condition as in Example are displayed in Figure 2, and they suggest convergence to a positive steady state as before. 8. Conclusions. In this paper, we have analyzed a within-host virus model which incorporated the mechanisms of direct cell-to-cell viral transmission and the viral coinfection (absorption of free virions into already infected cells). We obtained a complete analytic description of equilibria and their local stability. We presented the basic reproductive number R 0 and proved that the infection persists when R 0 > and becomes extinct when R 0. We also obtained sufficient conditions for the global stability of the endemic equilibrium in several particular cases. We were unable to establish the global stability in the most general cases where all coefficients

16 6 H. POURBASHASH, S.S. PILYUGIN, C. MCCLUSKEY AND P. DE LEENHEER Log Uninfected cells Infected cells irus ime Figure 2. ime series of a solution ( (t), (t),(t)) of system () with parameter values as described in Example 2 in the text. are positive. In one instance, namely when the absorption of free virions into infected cells can be neglected (the case k 3 = 0), we used the geometric approach developed by Li and Muldowney to prove the global stability [0]. In another instance, when the absorption of free virions into uninfected cells can be neglected (admittedly, not a biologically plausible scenario), we employed a Lyapunov function approach with a specific function similar to that used by Li et al in [9]. If both k 3 = k 4 = 0, we extended the global stability result to include a competition of several viral strains, and proved that the competitive exclusion is the typical outcome. Moreover, as it is common in competition models, it turns out that the fittest competitor is determined by the highest value of the basic reproduction number, R 0. Another implication of this result is that a direct cell-to-cell viral transmission cannot induce coexistence of viral strains. Using the same Lyapunov function, we we were able to invoke a global perturbation result of Smith and Waltman [20] to prove the global stability of the endemic equilibrium when k 3 > 0 is a small parameter. Finally, we presented two numerical examples to illustrate the global stability of the endemic equilibrium. We conjecture that such equilibrium is globally stable whenever it exists, but the proof of this general result remains an open question. REFERENCES [] W.A. Coppel, Stability and asymptotic behavior of differential equations, Heath and Co., Boston, 965. [2] P. De Leenheer and H.L. Smith, irus dynamics: A global analysis, SIAM J. Appl. Math. 63 (2003), [3] P. De Leenheer and S.S. Pilyugin, Multistrain virus dynamics with mutations: a global analysis, Math.Med.Biol.25 (2008), [4] N. Dixit and A. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, J.irol.78 (2004), [5] M. Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech. Math. J. (974), [6] H.I. Freedman, M.X. ang and S.G. Ruan, Uniform persistence and flows near a closed positively invariant set, J.Dynam.DifferentialEquations6 (994), [7] H.K. Khalil, Nonlinear Systems, 3rd Edition, Prentice Hall, [8] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math.Med.Biol.2 (2005), [9] M.Y. Li, J.R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math.Biosci.60 (999), [0] M.Y. Li and J.S. Muldowney, A geometric approach to the global-stability problems, SIAMJ. Math. Anal. 27 (996),

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