Journal of Theoretical Biology

Journl of Theoreticl Biology 277 (2) 55 66 Contents lists vilble t ScienceDirect Journl of Theoreticl Biology journl homepge: www.elsevier.com/locte/yjtbi Prdoxicl suppression of poly-specific brodly neutrlizing ntibodies in the presence of strin-specific neutrlizing ntibodies following HIV infection Stnc M. Ciupe,, Ptrick De Leenheer b, Thoms B. Kepler c Deprtment of Mthemtics, University of Louisin t Lfyette, 433 Mxim Doucet, Lfyette, LA 754, United Sttes b Deprtment of Mthemtics, University of Florid, 4 Little Hll, P.O. Box 85, Ginesville, FL 326-85, United Sttes c Center for Computtionl Immunology, Deprtment of Biosttistics nd Bioinformtics, Duke University Medicl Center, 2424 Erwin Rod, Suite 2, Durhm, NC 277, United Sttes rticle info Article history: Received 7 August 2 Received in revised form 7 Jnury 2 Accepted 3 Jnury 2 Avilble online 6 Februry 2 Keywords: HIV infection Virl dynmics Antibody response modeling bstrct One of the first immunologic responses ginst HIV infection is the presence of neutrlizing ntibodies tht seem ble to inctivte severl HIV strins. Moreover, in vitro studies hve shown the existence of monoclonl ntibodies tht exhibit brod crossclde neutrlizing potentil. Yet their number is low nd slow to develop in vivo. In this pper, we investigte the potentil benefits of inducing poly-specific neutrlizing ntibodies in vivo throughout immuniztion. We develop mthemticl model tht considers the ctivtion of fmilies of B lymphocytes producing poly-specific nd strin-specific ntibodies nd use it to demonstrte tht, even if such fmilies re successful in producing neutrlizing ntibodies, the competition between them my limit the poly-specific response llowing the virus to escpe. We modify this model to ccount for virl evolution under the pressure of ntibody responses in nturl HIV infection. The model cn reproduce virl escpe under certin conditions of B lymphocyte competition. Using these models we provide explntions for the observed ntibody filure in controlling nturl infection nd predict quntittive mesures tht need to be stisfied for long-term control of HIV infection. & 2 Elsevier Ltd. All rights reserved.. Introduction The bility of humn immunodeficiency virus (HIV) to persist in n infected individul nd eventully cuse AIDS is dependent on its bility to void immune responses. Mny fctors fcilitte virus persistence, from high genetic diversity nd evolution (Wlker nd Korber, 2) to the bility to sty ltent in the body (Blnkson et l., 22), to the infection of immune cells, whose ctivtion by vccine cndidtes leds to n increse in the trget cell popultion (Stebbing et l., 24). The lrge-scle vccine clinicl trils (AIDSVx (Gilbert et l., 25), STEP (Priddy et l., 28) nd RV44 (Rerks-Ngrm et l., 29) tht were imed t stimulting both rms of the dptive immune system: the ntibody-medited, the cell-medited nd combined ntibody nd cell-medited immunity showed limited clinicl efficcy (Fuci et l., 28). We study the roles of ntibodies in limiting virus repliction during HIV infection. Antibodies directed ginst HIV structurl proteins re detected in the body within few weeks following Corresponding uthor. Tel.: + 337 482 5287. E-mil ddress: msc653@louisin.edu (S.M. Ciupe). nturl infection (As-Chpmn et l., 24; Richmn et l., 23). Only smll frction of them, however, neutrlize the virus, which escpes recognition by ensuing reduced ccessibility to ntibody-binding sites, hevy glycosyltion of the envelope proteins nd rpid muttion (Douek et l., 26; Prren et l., 999; Richmn et l., 23; Wytt nd Sodroski, 998). Despite the hurdles the immune system hs to overcome, neutrlizing ntibodies do develop during nturl infection (Burton et l., 25; Hynes nd Montefiori, 26; Pntophlet nd Burton, 26). Most of them re strin-specific nd preferentilly recognize nd inhibit preceding but not current virl strins (Burton et l., 24; Richmn et l., 23; Wei et l., 23). To completely control infection, the immune system hs to find wys to elicit potent, high ffinity ntibody responses cpble of brod neutrliztion, virl inctivtion nd protection ginst current infection nd/or disese (Hone et l., 22). A limited number of known brodly neutrlizing humn monoclonl ntibodies (2F5, 4E, b2, 2G2, PG9, PG6 nd VRC) hve been identified (Burton et l., 24; Zhou et l., 2). They neutrlize primry isoltes of HIV from different genetic subtypes in vitro (Buchcher et l., 994; Burton et l., 24; Li et l., 27), but re very rrely produced in vivo (Dhillon et l., 27), nd re, therefore, difficult 22-593/$ - see front mtter & 2 Elsevier Ltd. All rights reserved. doi:.6/j.jtbi.2..5

56 S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 to induce through vccintion. The filure my be due to host regultory constrints (Hynes et l., 25), incorrect epitope conformtion (Moore et l., 26), HIV induction of polyclonl B cell ctivtion nd terminl differentition (Levesque et l., 29), nd/or B cell competition (Deem nd Lee, 23; Heymn, 23). While mny different B cells clones cn recognize given HIV virus strin, only those of high ffinity (strin-specific) respond in lrge numbers to produce neutrlizing ntibodies. For series of discrete rndom infections over time (continuous immuniztion), competition mong B cell clones my led to the phenomenon of originl ntigenic sin, where B cells produced in response to first virl infection cn suppress the cretion of new immune cells in response to second infection with relted strin (Deem nd Lee, 23). For chronic infection with mutting virus, the originl ntigenic sin my be limited, since there is enough time for the immune system to crete B cells ginst the new strin. However, there is time dely in the production of ech strin-specific neutrlizing ntibody tht my cuse tht virus strin to expnd t high levels before the ntibody cn control it (Burton et l., 24; Richmn et l., 23). Most importntly, the continuous presence of strin-specific ntibodies my led to suppression of the less fit poly-specific B cell clones cpble of producing brod neutrlizing ntibodies. The limittion in number of brodly neutrlizing ntibodies my represent the gretest wekness of the immune system. Antibody-medited immune suppression hs been observed during pssive dministrtion of ntibodies s well. In this sitution, B cells re prevented from stimultion through reduction of vilble ntigenic determinnts (Heymn, 23). Finlly, studies of Heptitis C chronic infections hve shown tht strin-specific ntibodies my inhibit the development of polyspecific ntibodies by preventing them from recognizing ntigen (Zhng et l., 24). To investigte the competition mong strin-specific nd polyspecific ntibodies, we developed mthemticl models of virus ntibody interctions during both immuniztion nd nturl infection with HIV. We strt with the ssumption tht the immune system produces both strin-specific nd poly-specific, cross rective, neutrlizing ntibodies. The strin-specific nd poly-specific neutrlizing ntibodies trget vrible (unique to ech vrint) or conserved (shred mong vrints) epitopes, respectively, on the virus envelope. The governing hypothesis is tht while B cells producing both (strin-specific nd polyspecific) neutrlizing ntibodies re ctivted during the infection, those producing poly-specific brodly neutrlizing ntibodies re mde inefficient nd consequently kept t undetectble levels. This process is medited by their competition with the B cells tht produce more fit strin-specific ntibodies with which they shre ntigenic stimultion, kinetic prolongtion, spce in the lymph nodes nd T cell conjugtes. We use informtion from previous modeling studies of HIV virl infection (Ho et l., 995; Nowk nd My, 2; Perelson et l., 996, 997), cellulr immune responses (Ciupe et l., 26; Stfford et l., 2), ntibody formtion (Opre nd Perelson, 996; Tomrs et l., 28) nd competition (Anti et l., 998; Boer et l., 2; Borghns et l., 999; Leenheer nd Pilyugin, 28) to derive nd nlyze models of the interction between virus nd neutrlizing ntibodies. Our im is to determine the prmeter regimes tht led to ntibody filure nd virl persistence, nd to predict wys to reverse these phenomen. The pper is structured s follows. In Section 2 we develop nd nlyze the mthemticl model describing the interction between fmilies of B lymphocytes producing poly-specific nd strin-specific neutrlizing ntibodies following continuous immuniztion with severl HIV vrints. In Section 3 we expnd the model to ccount for nturl infection nd virl evolution; their nlysis is presented in two ppendices. In Section 4 we present numericl results of the two models. We conclude with discussion. 2. Model of ntibody responses following continuous immuniztion Let V ¼ð,,...,V n Þ T be viruses of specificity rirn, A ¼ðA,A 2,...,A n Þ T be strin-specific neutrlizing ntibodies of specificity rirn, nd be the poly-specific brodly neutrlizing ntibody. Viruses re introduced into the body t times t i, V i ðt i Þ¼V i,, nd do not mutte. We corse-grin the virl life-cycle, ggregting the processes of infection, integrtion nd host cell virl production into simple repliction model in which viruses replicte with different virl fitness per-cpit rtes r i. We tret the dynmics of ntibody production similrly, ssuming tht ntibody concentrtion is in qusi-equilibrium with the B cell popultion tht produces them, nd without representing the component subprocesses such s ctivtion, differentition nd ntibody secretion. The concentrtion of ntibody specific to virl strin i is denoted A i, nd tht of poly-specific ntibody.weonlyconsider the frction of the produced ntibodies tht hs neutrlizing function. In the presence of neutrlizing ntibodies viruses re removed t rtes K nd K by the strin-specific nd poly-specific neutrlizing ntibodies, respectively. We ssume tht the removl rtes re independent of strin nd tht K 4K. Strin-specific neutrlizing ntibodies re elicited t rte l by the virl strin to which they re specific. Poly-specific neutrlizing ntibodies re elicited t rte l by ll virl strins. We denote by the differences between B cells prolifertion nd deth rtes, effectively treting the ntibody t qusi-equilibrium with these cells s surviving t tht rte. Finlly, ll B cells compete with ech other (within nd between clones) for ntigen, spce in the lymph nodes, nd conjugte T-cell help. The strength of this competition is governed by prmeter b. The dynmics of the system is described by the following equtions: dv i ¼ðr i KA i K ÞV i, da i ¼ lv i þa i ð ba T Þ, d ¼ l T þ ð ba T Þ, ðþ with V i ðt i Þ¼V i,, A i ðt i Þ¼, ðt Þ¼, T ¼ T V nd A T ¼ þ T A. In Section 2. we investigte the system dynmics for the cse where strin-specific B cells re bsent. In Section 2.2 we explore the dynmics when both poly-specific nd strin-specific ntibodies re produced in response to infection. 2.. Virus dynmics during poly-specific ntibody responses Let us consider the cse where viruses V ¼ð,,...,V n Þ T re introduced into the body t times t ¼ðt,t 2,...,t n Þ T, independent of ech other. The immune system rects by producing polyspecific ntibodies,, t rte l, which neutrlize ll virus strins t rte K. For simplicity, we ssume tht ll virl strins re eqully dpted to the host nd they replicte t the sme rte r i ¼r independent of the strin i. System () becomes dv ¼ðr K ÞV, d ¼ l T þ ð b Þ, with V i ðt i Þ¼V i, nd ðt Þ¼. ð2þ

S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 57 2... Stedy-stte nlysis We investigte the long-run behvior of system (2) when ll viruses re present in the popultion. To eliminte the discontinuities in the model, we consider tht the system strts t time t¼t n leding to initil conditions V i ðt n Þ¼V i,n 4 nd ðt n Þ4. The stedy-sttes of system (2) cn be divided into three clsses:. The no-infection stedy-stte S ¼ð,,...,Þ: 2. The virl clernce stedy-stte S 2 ¼ð,,...,,=bÞ. 3. The chronic infection hyperplne S 3 ¼ð,,...,V n,r=k Þ, which exists for r 4K =b nd T ¼ Xn V i ¼ r b r : ð3þ K l K i ¼ Let us study the symptotic behvior of the stedy-sttes. The Jcobin mtrix corresponding to our system is r K... K J ¼ B @ r K... K............... r K K V n l l... l 2b : ð4þ C A Proposition. () The infection free stedy-stte S is lwys unstble. (b) The virl clernce stedy-stte S 2 is symptoticlly stble if r ok ð5þ b nd unstble if the inequlity is reversed. (c) If r 4K ð6þ b the chronic infection hyperplne S 3 exists nd ech of its stedysttes hs zero eigenvlues. Proof. () The chrcteristic eqution for the stedy-stte S, ðr LÞ n ð LÞ¼, hs positive eigenvlues L,2,...,n ¼ r nd L n þ ¼. Therefore, the infection free stedy-stte is lwys unstble. (b) The chrcteristic eqution for the stedy-stte S 2, n r K b L ðlþþ¼ ð8þ hs eigenvlues L,2,...,n ¼ r K ð9þ b nd L n þ ¼ : ðþ When r ok =b, the eigenvlues re negtive nd the virl clernce stedy-stte is symptoticlly stble. If the inequlity is reversed, the eigenvlues re positive nd the virl clernce stedy-stte is unstble. In other words, when the ntibody response t stedy-sttes exceeds the virl production, the virus will be clered otherwise the virus persists. (c) The chrcteristic eqution for the stedy-stte S 3, L n L 2 2b r Lþr b r ¼ ðþ K K ð7þ hs eigenvlues L,...,n ¼, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b r 7 2b r 2 4r b r K K K L n,n þ ¼ : ð2þ 2 Although the ltter pir of eigenvlues re negtive for r 4K =b, there re lwys zero eigenvlues, nd hence we cnnot decide stbility of the chronic hyperplne bsed on liner nlysis. & 2..2. Globl stbility Proposition 2. When r ok =b, S 2 is globlly symptoticlly stble. Proof. Consider the function Wð,,...,V n, Þ¼ Xn i ¼ Z Vi þ K A 2 l t Z A : ð3þ Note tht for positive ð,...,v n, Þ4, W is positive semidefinite nd zero only if ð,,...,v n, Þ¼ð,,...,,=bÞ. Moreover,! _W ¼ Xn i ¼ ¼ Xn i ¼ ¼ Xn V i r K þk K A 2 V i r K K 2 þ A2 K bða l Þ 2!! K l bð Þ 2 V i ðr K Þ X K V i ð A 2 A Þ K bða i ¼ i l Þ 2 : ð4þ When r ok =b is stisfied W _ is negtive semi-definite, nd the lrgest invrint set where W _ ¼ isfð,,...,,=bþg.since W _ r, nd W is proper function (i.e. for ech c, theset fxzjwðxþrcg is compct) we hve tht ll solutions of (2) re bounded. Therefore, from Lslle s invrince principle, S 2 is globlly symptoticlly stble. & Proposition 3. When r 4K =b, S 3 is globlly symptoticlly ttrctive. Proof. T ¼ P n i ¼ V i stisfies system dt ¼ Tðr K Þ, d ¼ l T þ ð b Þ: Consider the function WðT, Þ¼l T Z T T T t ð5þ Z A þ ðk t rþ, ð6þ where ¼ r=k nd T ¼ðr=l K Þðbr=K Þ. Note tht W is positive semi-definite for ðt, Þ4, nd zero t ðt, Þ. Moreover, _W ¼ l ðr K ÞðT T Þþðl T þ ð b ÞÞðK rþ ¼ðK rþðl T þ ð b ÞÞ ¼ðK rþð ð b Þþ ð b ÞÞ, ð7þ where we used tht l T þ ð b Þ¼. Notice tht _W is product of two fctors, ech of which hs single root t ¼. For, these fctors hve opposite signs, nd thus _W r. Moreover, the lrgest invrint set where _W ¼ is fðt, Þg. Since W _ r, nd W is proper function (i.e. for ech c, the set fxzjwðxþrcg is compct) we hve tht ll solutions of (5) re bounded. From Lslle s invrince principle, the hyperplne S 3 is globlly ttrctive. &

58 S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 2.2. Virus dynmics in the presence of competition between strinspecific nd poly-specific ntibody responses Let us consider the generl cse given by the system () where immuniztion with viruses V ¼ð,,...,V n Þ T t times t ¼ðt,t 2,...,t n Þ T leds to production of both poly-specific nd strin-specific neutrlizing ntibodies, nd A ¼ðA,...,A n Þ T respectively. As before, we ssume tht ll virl strins replicte t the sme rte r i ¼r independent of the strin i. We study the viruses long term behvior s strin-specific nd poly-specific neutrlizing ntibodies compete with ech other for resources. 2.2.. Stedy-stte nlysis As before, we eliminte the discontinuities in the model, by considering system s initil condition for time t¼t n, where ll viruses re lredy present in the popultion. The stedy-sttes cn be divided into three clsses:. The infection free stedy-stte S 2 ¼ð,,...,,,,...,,Þ. 2. The clernce hyperplne S 2 2 ¼ð,,...,,A,A 2,...,A n, Þ, where A i Z for i ¼,...,n, nd Z re ny vector, respectively number, such tht A T ¼ =b. This set of stedysttes is the intersection of hyperplne of dimension ð2nþþ n ¼ n in R 2n þ, nd the non-negtive orthnt R 2n þ þ. 3. Let I f,2,...,ng, nd ssume tht #ðiþ¼mz. Then there re chronic stedy-sttes S 2 3 ¼ð,,...,V n,a,a 2,..., A n, Þ with ðv i,a i Þ4 for ll iai nd V j ¼ A j ¼ for ll j=2i if nd only if r 4 mk l þkl mðlþl Þ b : ð8þ The nonzero components of such stedy-sttes re given by lr A ¼ A i ¼ mk l þkl, V r i ¼ mk l þkl b mrðlþl Þ mk l þkl for ll iai, ¼ ml r mk l þkl : ð9þ Proposition 4. () The infection free stedy-stte S 2 is lwys unstble. (b) The virl clernce hyperplne S 2 2 hs zero eigenvlues. (c) For every m ¼,2,...,n for which (8) holds, there re ð n m Þ corresponding stedy-sttes S 3 2 of form (9) hving exctly m virus strins nd corresponding ntibodies present, nd they re ll unstble. Moreover, there is unique stedy-stte of form (9) hving exctly m¼n virus strins nd corresponding ntibodies present, nd it is symptoticlly stble. Proof. The Jcobin mtrix corresponding to the linerized system is X... K... K X 2... K... K.............................. X n... KV n K V n J ¼ l... Y ba... ba ba, l... ba 2 Y 2... ba 2 ba 2........................... B @... l ba n ba n... Y n ba n C A l l... l b b... b Z ð2þ where for i¼,2,y,n, X i ¼ r ðk þka i Þ, Y i ¼ bða T þa i Þ, Z ¼ bða T þ Þ: () The chrcteristic eqution of the stedy-stte S 2, ðr LÞ n ð LÞ n þ ¼ ð2þ ð22þ hs positive eigenvlues L,2,...,n ¼ r nd L n þ,...,2n þ ¼. Therefore the infection free stedy-stte is lwys unstble. (b) The chrcteristic eqution of the hyperplne S 2 2, L n ðþlþ Yn i ¼ fl ðr K KA i Þg ¼ ð23þ hs eigenvlues L,2,...,n ¼, L n þ,...,2n ¼ r K KA i, L 2n þ ¼. Therefore, S 2 2 is unstble when r 4K A i þka i for t lest one i. The 2 stbility of ny stedy-stte tht belongs to the hyperplne S 2 cnnot be determined from liner nlysis. (c) If (8) holds for m¼, then it lso holds for ll m¼2,3,y,n. For ech m, there will be stedy-sttes with exctly ð n m Þ virus strins nd corresponding ntibodies present tht stisfy equlities (9). We investigte the locl stbility of the chronic stedy-sttes for which m strins persist nd n m strins re clered. The stedy-sttes hve the chrcteristic eqution x y... y z rkl n m y x... y z detðj LI 2n þ Þ¼ L det............... mk l þkl B C @ y x... x z A w w... w v rkl n m ¼ L ðy xþ m fðxþðm ÞyÞvþmzwg, ð24þ mk l þkl where x ¼ L 2 þðba þba T ÞLþKAðbA T Þ, y ¼ bal, z ¼ AfbLþK ðba T Þg, w ¼ ðl þba T Þ, v ¼ naðl þba T Þ, ð25þ which simplifies to rkl n m detðj LI 2n þ Þ¼ L ðlþba mk l þkl T Þ ðl 2 þðba T ÞLþKAðbA T ÞÞ m ðal 2 þðba T þmba þb ÞLþrðbA T ÞÞ: ð26þ The first n m eigenvlues L,...,n m ¼ rkl=ðmk l þklþ re positive. Therefore the chronic stedy-sttes S 2 3 for which m strins persist nd n m strins re clered re lwys unstble. Finlly, we will show tht the unique stedy-stte with m¼n virus strins nd corresponding ntibodies present is symptoticlly stble.

S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 59 S 2 3, stisfying (9) for m¼n, hs the chrcteristic eqution x y... y z y x... y z detðj LI 2n þ Þ¼det............... B C @ y x... x z A w w... w v ¼ðy xþ n fðxþðn ÞyÞvþnzwg, where x ¼ L 2 þðba þba T ÞLþKAðbA T Þ, y ¼ bal, z ¼ AfbLþK ðba T Þg, w ¼ ðl þba T Þ, v ¼ naðl þba T Þ, which simplifies to detðj LI 2n þ Þ¼ðLþbA T ÞðL 2 þðba T ÞLþKAðbA T ÞÞ n ðal 2 þðba T þnba þb ÞLþrðbA T ÞÞ: ð27þ ð28þ ð29þ One cn show tht ll eigenvlues hve negtive rel prts provided tht ba T 4. Therefore, if the chronic stedy-stte exists then it is stble. In other words, when the virl production exceeds the combined removl by ntibodies the viruses will persist. & 3. Model of ntibody responses following nturl infection Model () ssumes tht the viruses re introduced into the host t rndom times through, for exmple, continuous immuniztion. We re interested in how these results chnge in n individul chronic HIV infection, where the virus muttes over time. Assuming tht primry infection with leds to production of strin-specific nd poly-specific neutrlizing ntibodies A nd (for exmple, the polyspecific production my be cused by previous vccintion), nd tht muttes over time into strins V i (i¼2,y, n) t rtes m i 4, which stimulte strin-specific immune cells to produce ntibodies, A i, model () becomes dv i ¼ r i X n j ¼ m ij V j V i ðka i þk Þ, da i ¼ lv i þa i ð ba T Þ, rte m 23 4 nd so forth. The muttion is irreversible, nd the muttion mtrix describing this sitution hs the form: m 2... m 2 m 23... Q ¼ B ^ & &... ^ C @ A,... m n n with m ii þ o. (B) Viruses mutte forwrd nd bckwrd rndomly with the dditionl ssumption tht the muttion mtrix Q ¼fm ij g r i,j r n is irreducible. Note tht Q hs simple dominnt eigenvlue with corresponding non-negtive eigenvector zz such tht Qz¼z. In cse () z is n entry-wise positive vector (by the Perron Frobenius theorem) nd in cse (2) z¼(,,y,,) T. As before we ssume tht ll viruses replicte t the sme rte r i ¼r. We perform stbility nlysis of model (3) for the muttion mtrix Q stisfying condition (A). When the ntibody response is poly-specific ll viruses re clered when r ok =b nd persist otherwise. This reltion is independent on the muttion mtrix Q or the number of viruses present n (see Appendix A). When the ntibody responses re poly-specific nd strinspecific we study viruses long term behvior for the cse n¼2 nd muttion mtrix stisfying condition (A), i.e. virus muttes to produce virus t rte m nd the muttion is irreversible. System (3) becomes d d da da 2 ¼ rð mþ ðka þk Þ, ¼ rm þr ðka 2 þk Þ, ¼ l þa ð ba T Þ, ¼ l þa 2 ð ba T Þ, d ¼ l ð þ Þþ ð ba T Þ, ð3þ with ()¼,, ()¼A ()¼A 2 () ¼ ()¼. We show tht for r ominfðk l þklþ=ðlþl Þ,K Og=b both viruses re clered. Conversely, when m4kl=ðklþk l Þ nd r 4ððK l þklþ=ðl þlþþ=b t lest one virus strin persists. Similrly for mokl=ðklþk l Þ nd r 4OK =b, both viruses persist, where K K þ l 2 K l O ¼ K þ l l K K þ l 4 K l K þ3 l 2m 2 K l K þ l sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 4 K l K þ3 l 2m 2 K l K þ l 2 8ð mþ 2 2 K l K þ l K l K þ l, ð32þ @ A l d ¼ l T þ ð ba T Þ, ð3þ with ()¼,, V i ()¼A i ()¼ ()¼ fori42. Q ¼fm ij g r i,j r n is muttion mtrix with non-negtive off-digonl entries nd columns entries tht dd up to one. We consider two situtions describing HIV evolution over time: (A) The initil virus strin muttes to produce virus strin t rte m 2 4, which muttes to produce virus V 3 t s shown in Appendix B. Note tht O depends on the reltive rtio of the poly-specific nd strin-specific ntibody production nd virl removl rtes. 4. Numericl results Previous studies (Ciupe et l., 26; Stfford et l., 2) hve considered n initil HIV lod of 9 virions per ml, corresponding to the presence of smll number of virions in the inoculum. We increse this estimte to virions per ml to ccount for

6 S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 stronger continuous immuniztion. The virl dynmics is not sensitive to this vlue, s we cn show numericlly. When we vry the initil inoculum five orders of mgnitude, the pek of the virus shifts to the right by only one dy. Viruses replicte t n effective rte of r¼25 virions per dy (Ciupe et l., 26; Stfford et l., 2). In response, B cells become ctivted nd differentite into ntibody producing plsm cells. Typicl ntibody ffinities for the elicited ntigen re 5 M (Hollinger nd Ling, 2). Since ech HIV virion hs mny potentil binding sites nd ffinity mturtion my occur, we ssume the vidity of specific nd poly-specific ntibodies to be s high s 6 9 M per dy for strin-specific ntibody nd 3 9 M per dy for the poly-specific ntibody. Using the Avogdro s number we cn covert the vidity rtes mesured in inverse molrs into removl rtes mesured in ml per molecule s follows: M ¼ Liter mole ¼ 6 ml mole ¼ 6 ml 6 23 molecules ¼ ml 6 7 molecules : ð33þ Using this conversion we obtin removl rtes of K ¼ 8 ml per ntibody molecule per dy nd 5 9 ml per ntibody molecule per dy by the strin-specific nd poly-specific ntibody respectively. Initilly, t the time of immuniztion, there re neither strinspecific nor poly-specific neutrlizing ntibodies present, i.e. A i ()¼ ()¼ molecules per ml. One B cell secretes between nd 4 ntibody molecules per second, corresponding to 8 5 nd 8 8 ntibody molecules per dy (Bchmnn et l., 994). Assuming tht one B cell is ctivted by one virl epitope, we consider n verge ntibody production rte of l ¼ l ¼ 7 molecules per dy per virion (regrdless of ntibody type). Finlly, we ssume tht the difference between ntibody (B cell) prolifertion rte nd deth rte is ¼.4 per dy (Hodgkin et l., 996; Jnewy et l., 2) nd tht the source B cells compete with ech other t rte b vrying between 2.5 nd 2:5 9 ml per ntibody molecule per dy. For the continuous immuniztion model (), numericl results for the interction between four virus strins, introduced in the body t times t ¼, t 2 ¼5, t 3 ¼ nd t 4 ¼5, in the sole presence of poly-specific ntibody re presented in Fig.. Poly-specific neutrlizing ntibody is produced immeditely fter infection with virus, expnds t fst rte, nd reches its pek t the sme time s the virus. A slight decrese to its stedy-stte vlue occurs three dys lter. When r ok =b, nd ll subsequent virl infections decy exponentilly (left pnel). When r 4K =b viruses persists nd rech different stedy-stte vlues. While we know the stedy-stte position of the totl virus lod V T, we cnnot determine the position of individul virl stedy-sttes which is highly dependent on initil conditions nd inocultion times. If two or ll viruses re introduced t time zero nd replicte t the sme rte, r then they hve identicl dynmics (not shown). When, however, viruses, V 3 nd V 4 re introduced lter thn virus nd ntibody they rech stedy-sttes vlues tht my be different thn the stedy stte vlue of virus. When both poly-specific nd strin-specific neutrlizing ntibodies re presented then both strin-specific nd poly-specific neutrlizing ntibodies re produced immeditely fter infection with virus. They expnd t fst rte, nd rech high stedystte vlues. When r oðð4k l þklþ=ð4ðlþl ÞÞÞ=b, nd ll subsequent virus strins decy exponentilly (Fig. 2, left upper pnel). The introduction of new virus strins leds to production of corresponding strin-specific ntibodies which expnd to lower stedy-stte vlues due to competition with existent ntibody. While we know the totl ntibody vlue, A T, we cnnot predict individul ntibody vlues t stedy-stte which re Virus RNA per μ l (solid lines) Poly specific ntibody per μ l (dotted lines) 5 5 highly dependent of initil conditions nd inocultion times. If one or ll strin-specific ntibody re introduces t the sme time s the poly-specific ntibody then their stedy-stte levels re identicl. The other strin-specific ntibodies re prevented from expnding due to their competition with high nd A ntibody lods t the time of their ppernce (Fig. 2, left, lower pnel). When r 4ðð4K l þklþ=ð4ðlþl ÞÞÞ=b, ll viruses persist nd undergo dmped oscilltions. This is due to the existence of complex eigenvlues with negtive rel prts in (29) (Fig. 2, right, upper pnel). Strin-specific ntibodies rech the sme stedystte which is lower thn tht of poly-specific ntibody (Fig. 2, left, lower pnel). The combined ntibody response is inefficient in controlling the infection. Putting together the nlyticl results regrding virus dynmics in the presence of poly-specific neutrlizing ntibodies lone nd in the presence of competing poly-specific nd strinspecific neutrlizing ntibodies, we find tht for ny mz2 for which, b K 4r 4 b V 3 V 4 V 3 5 5 2 25 3 2 3 4 5 Dys fter inocultion Fig.. Virus dynmics (solid lines) in the presence of poly-specific brodly neutrlizing ntibodies (dotted line): (left pnel) for the clernce condition r ok =b; (right pnel) for the chronic condition r 4K =b. Viruses re introduced s independent immuniztions every five dys. Prmeter used in the simultions re r¼25, ¼.4, K ¼5 9, l ¼ 7, nd b ¼ 2:5 (left pnel); b ¼ 2:82 (right pnel). ml K þlk mðl þlþ ð34þ viruses re clered in the presence of poly-specific neutrlizing ntibodies but persist in the presence of dditionl immune responses, in the form of strin-specific neutrlizing ntibodies (see Fig. 3). For the nturl infection model (3), we consider n initil virl lod of virions per ml, which muttes over time t rte m, vrying between % nd 8% to produce second virus. The virus is eliminted t rte 8 ml per ntibody molecule per dy by the strin-specific neutrlizing ntibodies nd 5 9 ml per ntibody molecule per dy by the poly-specific neutrlizing ntibodies (Hollinger nd Ling, 2). There re no strinspecific nd poly-specific neutrlizing ntibodies in the body t the time of infection, i.e. A i ()¼ ()¼ molecules per ml. Antibodies re produced t rte l ¼ l ¼ 7 molecules per dy per virion regrdless of the ntibody type (Bchmnn et l., 994), nd compete with ech other t rte b vrying between nd 5 ml per ntibody molecule per dy. Rtes r nd re s in the immuniztion model. Numericl results for the interction between two virus strins in the sole presence of poly-specific neutrlizing ntibodies nd stepwise muttion mtrix, Q, re presented in Fig. 4. Infection with nd mutnt strin leds to immedite production of V 4

S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 6 Virus RNA per μ l 2 V 3 V 4 V 3 V 4 4 5 5 2 25 2 3 4 Poly specific ntibody per μ l ( ) Strin specific ntibody per μ l (A i ) 5 =A A 2 A 3 A 4 A A 2 A 3 A 4 2 3 4 2 3 4 Dys fter inocultion Fig. 2. Virus dynmics (solid lines) in the presence of both poly-specific neutrlizing ntibodies (dotted line) nd strin-specific neutrlizing ntibodies (dshed lines): (left pnel) for the clernce condition r o=bðð4l K þlkþ=4ðl þlþþ; (right pnel) for the chronic condition r 4ð=bÞðð4l K þlkþ=4ðl þlþþ. Viruses re introduced s independent immuniztions every five dys. Prmeters, r, K, l re s in Fig. nd K¼ 8, l ¼ 7 nd b ¼ (left pnel) nd b ¼ 2:5 (right pnel). Virus RNA per μ l (solid lines) Strin specific ntibody per μ l (dshed lines) Non specific ntibody per μ l (dotted lines) 5 5 5 5 5 2 2 4 6 Dys fter inocultion Fig. 3. Prdoxicl results showing: (left pnel) virl clernce (solid lines) in the sole presence of poly-specific neutrlizing ntibody responses (dotted line), (right pnel): virl persistence (solid lines) in the presence of stronger (yet competing) poly-specific (dshed lines) nd strin-specific neutrlizing ntibody responses (dotted line) for prmeters stisfying ð=bþðð4l K þlkþ=4ðl þlþþor ok =b., r, K, K, l, l re s in Fig. 2 nd b ¼ 2:5. 5 Virus RNA per μ l (solid lines) Poly specific ntibody per μ l (dotted lines) 5 V 2 3 4 5 2 4 6 8 Dys fter infection Fig. 4. Virus dynmics (solid lines) following nturl infection in the presence of brodly neutrlizing poly-specific ntibodies (dotted line) when virus s muttion is described by condition (A): (left pnel) for the clernce condition r ok =b; (right pnel) for the chronic condition r 4K =b. Prmeter used in the simultions re r¼25, ¼.4, K ¼5 9, l ¼ 7, m ¼ :5 nd b ¼ (left pnel); b ¼ 5 (right pnel). poly-specific ntibody who clers both viruses when r ok =b (left pnel). When r 4K =b, only the dominnt virus lod persists (right pnel). The clernce conditions re independent of the muttion rte m, which only controls the rte t which the dominnt virus gets estblished. When both strin-specific nd poly-specific neutrlizing ntibodies re present nd for muttion mtrix, Q, stisfying condition (A), infection with nd mutnt strin leds to immedite production of both strin-specific nd poly-specific ntibody who cler both viruses when r ominfðk l þklþ=ðlþl Þ,K Og=b (Fig. 5, left pnel). Poly-specific nd strin specific ntibodies re produced immeditely fter infection nd our model predicts the totl ntibody vlue needed for clernce, A T, but not the individul ntibody vlues t stedy-stte. When r 4ððK l þklþ=ðlþl ÞÞ=b, nhe muttion rte stisfies m4kl=ðk l þklþ then single chronic infection estblishes where the dominnt virus persists nd the suboptiml virus is clered. The poly-specific nd dominnt virusspecific ntibodies hve the sme dynmic when l ¼ l, while the other strin-specific ntibody is degrded nd leves the body (Fig. 5, middle pnel). When r 4K O=b nd the muttion rte stisfies mokl=ðk l þklþ then both virus strins persist. The strin-specific ntibody rech stedy-stte levels smller thn the

62 S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 Virl RNA per μ l (solid lines) Poly specific ntibody per μ l (dotted lines) Strin specific ntibody per μ l (dshed lines) 5 5 poly-specific ntibody nd their combined vidity is inefficient in controlling the virus (Fig. 5, right pnel). Tht prmeter region, b K 4r 4 b K O, mo Kl K l þkl =A 2 =A 2 2 3 A A A 5 Dys fter infection 5 Fig. 5. Virus dynmics (solid lines) following nturl infection in the presence of both poly-specific neutrlizing ntibodies (dotted line) nd strin-specific neutrlizing ntibodies (dshed lines) when virus s muttion is described by condition (A): (left pnel) for condition r ominfðk l þklþ=ðlþl Þ,K Og=b; (middle pnel) for the single-infection chronic conditions r 4ððK l þklþ=ðlþl ÞÞð=bÞ nd m4kl=ðk l þklþ; (right pnel) for the multiple-infection chronic conditions r 4K O=b nd mokl=ðk l þklþ. Prmeter r,, K, l re s in Fig. 4, K ¼ 8, l ¼ 7, nd b ¼, m ¼ :5 (left pnel); b ¼ 5, m ¼ :8 (middle pnel); b ¼ 5, m ¼ :5 (right pnel). Virl RNA per μ l (solid lines) Poly specific ntibody per μ l (dotted lines) Strin specific ntibody per μ l (dshed lines) 5 ð35þ describes the interesting sitution where viruses re clered in the presence of poly-specific immune responses nd persist in the presence of dditionl (yet competing) immune responses, in the form of strin-specific neutrlizing ntibodies (see Fig. 6). A A 2 2 3 4 5 5 5 2 Dys fter infection Fig. 6. Prdoxicl results showing: (left pnel) virl clernce (solid lines) in the sole presence of brodly neutrlizing poly-specific ntibody responses (dotted line), nd (right pnel) when virus s muttion is described by condition (A): virl persistence (solid lines) in the presence of stronger (yet competing) poly-specific (dshed lines) nd strin-specific neutrlizing ntibody responses (dotted line) for nturl infection nd prmeters stisfying K O=bor ok =b. We consider nturl infection with virus strin, which muttes over time t rte mokl=ðklþk l Þ. Prmeters used in simultions re r¼2, ¼.4, K ¼K¼5 9, l ¼ l ¼ 7, b ¼ 3 nd m ¼ :. A 2 5. Discussion We developed mthemticl models of HIV virl dynmics tht ccount for the counterintuitive hypothesis tht dditionl immune responses (in the form of strin-specific ntibody responses) my be detrimentl to the host nd led to virl persistence. We show tht this phenomenon my occur even when successful previous vccintion leds to production of polyspecific brodly neutrlizing ntibodies. For prmeter regimes for which the poly-specific brodly neutrlizing ntibody response (when operting lone, tht is, unccompnied by strin-specific B cells) re sufficient for clernce of the virus, the presence of dditionl strin-specific ntibodies leds to virl rebound nd in the end virl chronicity. In the first model, we described host virus interction during immuniztion with severl virus strins nd found tht for prmeters stisfying b K 4r 4 b ml K þlk mðl þlþ ð36þ viruses go extinct in the presence of only poly-specific brodly neutrlizing ntibodies, nd persist when both strin-specific nd poly-specific neutrlizing ntibodies re present (see Fig. 3). The results re bsed on the ssumption tht viruses replicte t the sme rte r. This ssumption is justified by recent findings by Arnott et l. (2) tht hve shown tht viruses obtined shortly fter initil infection from individuls who were not tking ntiretrovirl therpy (ART) hve higher fitness level thn ws previously believed nd therefore justifying our ssumption tht closely induced virl strins cn hve the sme repliction rtes. However, since it is known tht in the long-run fitness chnges nd viruses become more (less) fit in the bsence (presence) of ART we hve run simultions to see how this chnge ffects our results. We hve found the existence of prmeter regimes for which four viruses replicting t different rtes go extinct in the presence of only poly-specific brodly neutrlizing ntibodies, nd persist when both strin-specific nd poly-specific neutrlizing ntibodies re present (see Fig. 8). Anlyticl results tht sustin this observtion will be presented elsewhere. For nturl infection with virus tht muttes over time t rte mokl=ðk l þklþ, when b K 4r 4 b K O ð37þ viruses go extinct in the presence of only poly-specific brodly neutrlizing ntibodies, nd persist when both strin-specific nd poly-specific neutrlizing ntibodies re present (see Fig. 6). The results re mintined when viruses hve different fitness rtes (not shown). These results llow us to dvnce the ide tht one of the resons for the bsence (or inefficiency) of poly-specific brodly neutrlizing ntibodies, in vivo, is the competition between plsm lymphocyes tht produces them with the plsm lymphocytes producing strin-specific neutrlizing ntibodies. Prticulrly, even if such poly-specific ntibodies re being induced (sy through successful vccines), they my be kept t low enough levels by the more fit strin-specific ntibodies. Prdoxiclly, the presence of dditionl, specific, immune response is detrimentl to the host nd llows for the estblishment of chronic infections. The suppression of poly-specific immune cells by the more fit immune cells hs been documented both in HIV nd HCV infections (Heymn, 23; Zhng et l., 24), but hs never been implied s the reson for virl chronicity. Our study implies tht competition between immune cells leding to suppression of B cells cpble of inducing brodly neutrlizing responses my be sufficient for HIV to become chronic. The

S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 63 impliction is even more worrisome if we consider tht the contemporry strin-specific ntibodies usully recognize nd inhibit preceding but not current virl strins (Burton et l., 24; Richmn et l., 23; Wei et l., 23). Our model ssumes tht neutrlizing ntibodies to the current dominnt virl strin re produced immeditely fter virus infection, or muttion. A dely in their production by two weeks results in chronic infection with n incresed virl set point by one order of mgnitude (results not shown). The nlyticl results for systems () nd (3) when both ntibody types re present re bsed on locl nlysis. We hypothesized tht the conditions for long-term virl clernce nd persistence re independent of initil conditions. Our K 6.5 x 9 6 5.5 5 4.5 4 3.5 3 2.5 Region 2 Region Region 3 K = rβ/ K = rβ/(ω) Region 4 2..2.3.4.5.6.7.8.9 λ/λ Fig. 7. Miniml vlues of poly-specific ntibody vidity rte K for which viruses re clered by poly-specific (solid line) nd competing poly-specific nd strinspecific brodly neutrlizing ntibodies for muttion rtes m ¼ :3 s functions of l=l. For high poly-specific/strin-specific ntibody rtio clernce by polyspecific ntibodies lone is sufficient for clernce by competing poly-specific nd strin-specific neutrlizing ntibodies regrdless of muttion rtes. For similr vlues of poly-specific thn strin-specific ntibody, clernce by poly-specific ntibody lone does not insure clernce by competing poly-specific nd strinspecific brodly neutrlizing ntibodies for which we need n increse in the vidity rtes of poly-specific ntibody to control the infection. Prmeters, r, b re s in Fig. 6 nd K¼K. conclusions re bolstered by numericl results; future work is needed to prove globl stbility nlyticlly. In the model of nturl infection where the virus muttes, the nlyticl results re presented for the prticulr cse of two virus two strin-specific ntibody popultions. Numericlly, we cn show tht virl clernce in the presence of brodly neutrlizing poly-specific ntibodies nd virl persistence in the presence of both strin-specific nd poly-specific neutrlizing ntibodies in the sme prmeter regime occurs for ny number of new virl strins. Moreover, the nlyticl results for the muttion mtrix stisfying condition (A) cn be obtined for muttion mtrix stisfying condition (B), s long s Q, is irreducible (results not shown). The condition of virl clernce in the sole presence of the polyspecific ntibody is given by r ok =b. SinceK =boðk l þklþ= ðlþl Þ=b, virl chronicity in the presence of competing strinspecific nd poly-specific neutrlizing ntibodies hppens when the double chronic infection stedy-stte(but not the single infection one) exists (prmeters described in cse 3). If we cn induce polyspecific neutrlizing ntibody of equl vidity to tht of strin-specific ntibody, then for known virus verge production nd ntibody life spn we cn determine the correltion between the poly-specific ntibody levels nd the minimum vidity rtes needed for this condition to fil. As seen in Fig. 7, clernce in the presence of poly-specific brodly neutrlizing ntibody lone (regions nd 4) is mintined for competing poly-specific nd strin-specific ntibodies when poly-specific ntibody levels l re much higher thn strinspecific ntibody levels l (region 2). When their levels re similr then we cn insure clernce in the competition model by incresing the vidity of the poly-specific brodly neutrlizing ntibody K (region 4). 6. Conclusions In this study, we investigted the hypothesis tht poly-specific brodly neutrlizing ntibodies develop longside strin-specific neutrlizing ntibodies. Using mthemticl modeling pproch, we hve determined the prmeter regions where competition between poly-specific nd strin-specific ntibodies Virus RNA per μ l (solid lines) Poly specific ntibody per μ l (dshed lines) Strin specific ntibody per μ l (dotted lines) 5 8 V V 3 V 4 6 5 2 3 4 2 3 4 2 V 3 V 4 8 A A A A 2 3 4 6 5 5 5 2 2 3 4 Dys fter inocultion Fig. 8. Prdoxicl results showing: (left pnel) virl clernce (solid lines) in the sole presence of poly-specific neutrlizing ntibody responses (dotted line), nd (right pnel): virl persistence (solid lines) in the presence of stronger (yet competing) poly-specific (dshed lines) nd strin-specific neutrlizing ntibody responses (dotted line) for viruses with different repliction rtes. Prmeters, K, K, l, l, b re s in Fig. 3 nd r ¼27, r 2 ¼25, r 3 ¼2, r 4 ¼2.

64 S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 cn help the virus escpe nd predicted wys of preventing it, providing insight into the ultimte roles of ntibody responses in controlling HIV infection. We predict tht in preventive vccintion design, one cn prevent/dely virl chronicity by inducing polyspecific ntibodies of high vidity, incresing the overll poly-specific ntibody levels nd by speeding the rte t which viruses mutte. Acknowledgments S.M.C. cknowledges support from NSF Grnt DMS-22865. We thnk the nonymous reviewers for their comments nd suggestions. Appendix A Virus evolution during poly-specific ntibody response: We ssume tht primry inocultion leds to infection by single virus strin. This virus muttes over time under pressure from poly-specific immune response,, giving rise to n geneticlly distinct virus strins V¼(,,y,V n ) T t rtes given by the muttion mtrix Q, stisfying condition (A) or (B). As before, we consider the repliction rte r i ¼r to be independent of the virus strin i. The model describing the virus-host interction becomes dv ¼ðrQ K I n ÞV, d ¼ l T þ ð b Þ, ð38þ with ðþ4 nd V i ()¼A()¼, for i¼2,y,n. Since TðÞ4, we cn define f(t) ¼(V/T)(t), for i¼,y,n. Then the following result follows. Proposition A.. The dynmics of (38) re equivlent to the dynmics of df ¼ rðq I nþf, dt ¼ðr K ÞT, d ¼ l T þ ð b Þ: for muttion mtrix Q stisfying condition (A) or (B). Proof. Note tht ð39þ df ðtþ¼ðrq K I n Þf ðr K Þf ¼ rðq I n Þf, ð4þ with f ðþz nd T f ()¼. We see tht there is bijective correspondence between solutions of (38) with VðÞ, nd solutions of (39). & Stedy-stte nlysis: Notice tht the lst two equtions of (39) re decoupled from the first n. We study the stbility of stedy-sttes of this plnr subsystem next. System (38) hs three stedy-sttes:. The infection free stedy-stte S 3 ¼(,). 2. The virl clernce stedy-stte S 3 2 ¼ð,=bÞ. 3. The chronic stedy-stte S 3 3 ¼ððr=K l Þðbr=K Þ,r=K Þ, which exists for r 4K =b. Proposition A.2. () The infection free stedy-stte S 3 is lwys unstble. (b) If r ok =b, then S 2 3 it is GAS with respect to initil conditions TðÞ4. (c) If r 4K =b, then S 3 3 it is GAS with respect to initil conditions TðÞ4. Proof. The proof is similr to tht of Propositions 2 nd 3. Theorem A.3. If r 4K =b, then every solution (f(t),t(t), (t)) of (39) with TðÞ stisfies lim ðf ðtþ,tðtþ,ðtþþ ¼ ð= T r zþz, b r r,, ð4þ t- þ K l K K where zz is the dominnt eigenvector of the simple eigenvlue of the mtrix Q (Qz¼z). Proof. The proof follows from Proposition 6 if we show tht f ðtþ-ð= T zþz. From the eigenvector expnsion we hve tht f ðtþ¼c zþ X j e j 4 2c rjt z j, ð42þ where the rel prt of ll r j s is negtive nd c 4. Then lim f ðtþ¼c z ð43þ t- þ nd since T f(t)¼ for ll t, it follows tht c ¼ = T z. & Corollry A.4. Under the conditions of Theorem every solution (V(t), (t)) of (38) stisfies lim ðvðtþ,aðtþþ ¼ r b r T r z z, : ð44þ t- þ K l K K Proof. This is immedite from Theorem since f(t)¼v(t)/ T(t). & Proposition A.5. If rok =b, then every solution (V(t), (t)) of (38) stisfies lim VðtÞ¼: t- þ ð45þ Proof. This is immedite from Proposition 6 since rvðtþ. & Appendix B Virus evolution during both strin-specific nd poly-specific ntibody responses: Let us consider the generl cse given by system (3), where virus strin muttesovertimeunderthepressureofboth specific nd poly-specific immune responses A nd to produce n geneticlly distinct virus strins V i (i¼2,y,n) nd their corresponding strin-specific neutrlizing ntibodies, A i.moreover, the muttion mtrix, Q ¼fm ij g r i,j r n, stisfies condition (A). Similr results follow for condition (B) nd will not be shown here. Stedy-stte nlysis: We study viruses long term behvior for the cse n¼2 nd muttion mtrix stisfying condition (A), i.e. virus muttes to produce virus t rte m nd the muttion is irreversible. System (3) becomes d ¼ rð mþ ðka þk Þ, d da da 2 ¼ rm þr ðka 2 þk Þ, ¼ l þa ð ba T Þ, ¼ l þa 2 ð ba T Þ, d ¼ l ð þ Þþ ð ba T Þ, with ()¼,, ()¼A ()¼A 2 ()¼ ()¼. & ð46þ Indeed, for jz2, we hve tht ¼ T ðq I nþz j ¼ðr j Þ T z j, nd thus T z j ¼. This implies tht o T f ðþ¼c T z nd thus tht c 4.

S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 65 System (46) hs t most four types of non-negtive stedysttes:. The infection free stedy-stte S 4 ¼(,,,,). 2. The virl clernce hyperplne S 4 2 ¼ð,,A,A 2, Þ, such tht A T ¼ =b. 3. The chronic single infection stedy-stte S 4 3 ¼, r K l þkl b rðlþl Þ K l þkl rl,, K l þkl, rl, K l þkl ð47þ which exists when r 4ððK l þklþ=ðl þlþþ=b. 4. The chronic co-infection stedy-stte S 4 4 ¼ð,,A,A 2, Þ, where x ¼ K K þ l l, y ¼ 2 K K þ l, l ¼ r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2xyK 2xþy 2myþ ð2xþy 2myÞ 2 8ð mþ 2 xy, A ¼ rð mþ K A 2 ¼ K K þ l l K K, rð mþ, K ¼ l b þ l rð mþ A l K ¼ l b þ l K A l K þ l l S 4 4 K K, rð mþ K : ð48þ exists when mokl=ðklþk l Þ¼ð2x yþ=x nd r 4K O=b, where 2xy O ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðy xþðþ2x yþð2xþy 2myþ ð2xþy 2myÞ 2 8ð mþ 2 xyþ ð49þ Cse : When : m4kl=ðklþk l Þ nd r oððk l þklþ=ðl þlþþ=b or b: mokl=ðklþk l Þ nd r ominfððk l þklþ=ðl þlþþ=b,k O=bg then S 4 nd S 2 4 exist. Cse 2: When 2: m4kl=ðklþk l Þ nd r 4ððK l þklþ=ðl þlþþ=b or 2b: mokl=ðklþk l Þ nd K O=b4r4ððK l þklþ=ðl þlþþ=b then S 4, S 2 4 nd S 3 4 exist. Cse 3: When mokl=ðklþk l Þ nd K O=boroððK l þ KlÞ=ðl þlþþ=b then S 4, S 2 4 nd S 4 4 exist. Cse 4: When mokl=ðklþk l Þ nd r 4mxfððK l þklþ= ðl þlþþ=b,k O=bg then ll four stedy-sttes exist. Proposition B.. () The infection free stedy-stte S 4 is lwys unstble. (b) The virl clernce hyperplne S 2 4 hs zero eigenvlues. (c) The chronic single infection stedy-stte S 3 4 is symptoticlly stble for prmeters stisfying condition 2 nd unstble for prmeters stisfying condition 2b. (d) The chronic co-infection stedy-stte S 4 4 is symptoticlly stble whenever it exists (for prmeters stisfying conditions 3 nd 4). Proof. The Jcobin mtrix corresponding to the linerized system is Y k K rm Y 2 K K J ¼ l ba T ba ba ba, ð5þ B C @ l ba 2 ba T ba 2 ba 2 A l l b b ba T b with Y ¼ rð mþ K KA, Y 2 ¼ r K KA 2 : () The chrcteristic eqution of S 4, ðrð mþ LÞðr LÞð LÞ 3 ¼ ð5þ ð52þ hs positive eigenvlues L ¼ rð mþ, L 2 ¼ r nd L 3,4,5 ¼. Therefore, the infection free stedy-stte is lwys unstble. (b) The chrcteristic eqution of hyperplne S 4 2, L 2 ðþlþðl ðr K KA 2 ÞÞðL ðrð mþ K KA ÞÞ ¼ ð53þ hs eigenvlues L,2 ¼, L 3 ¼, L 4 ¼ r K KA 2 nd L 5 ¼ rð mþ K KA. Therefore, S 2 4 is unstble for r 4minfK þka 2,=ð mþðk þka Þg. Since some eigenvlues re zero, the stbility cnnot be determined from liner nlysis. We will show numericlly the hyperplne is symptoticlly stble for prmeters stisfying condition (A). (c) The chronic single infection stedy-stte S 3 4 hs the chrcteristic eqution rð mþ rk l K l þkl L ð bz LÞ 2 ðl 2 þlð2bz ÞþðbZ ÞKZÞ¼, where Z ¼ rðl þlþ=ðk l þklþ. The eigenvlue ð54þ L ¼ rð mþ rk l ð55þ K l þkl is negtive when m4kl=ðklþk l Þ nd positive otherwise. The rest of the eigenvlues L 2,3 ¼ b rðl þlþ K l þkl, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L 4,5 ¼ 2 2bZ7 ð 2bZÞ 2 4KZðbZ Þ ð56þ re negtive when r 4ð=bÞðK l þklþ=ðlþl ÞÞ. This hppens every time stedy-stte S 3 4 exists. (d) Let o ¼ ba T ¼ br=k O. We will show tht when the chronic stedy-stte S 4 4 exists (respectively, when o is positive) it is symptoticlly stble. The chrcteristic eqution corresponding to the chronic stedy-stte is ðlþba T ÞðL 4 þ l 3 þ 2 L 2 þ 3 Lþ 4 Þ¼, where ¼ 2oþbA T þrm, 2 ¼ðK l þklþð þ ÞþoðoþbA T Þþrm ð2oþba T Þ, ð57þ

66 S.M. Ciupe et l. / Journl of Theoreticl Biology 277 (2) 55 66 3 ¼ðK l þklþ oð þ Þþrm þðrmk l þ2ðlþl ÞkbA 2 Þ Þ,! þ ðrmoðoþba T Þ 4 ¼ Kl ð2k l þklþþrmo ðk l þklþ þk l : ð58þ The eigenvlue L ¼ o is negtive for positive o. By the Ruth Hurwitz condition, the remining eigenvlues solving Eq. (57) hve negtive rel prts if nd only if 4, 4 4, B 2 3 4, nd B2 B 3 4 4: ð59þ Vribles, 4 nd B ¼ oðoþba T Þð2oþbA T Þþoð þ ÞðK l þklþþrmkl þklba þ 2 2 þk l ba T ð þ Þ ð6þ þ re polynomils in o with positive coefficients. The proof of the lst reltion, B 2 4, is tedious nd cn be verified using mple for prmeters where S 4 4 exists. & References As-Chpmn, M., Hymn, A., Newton, P., et l., 24. Development of the ntibody response in cute HIV- infection. AIDS 8, 37 38. Anti, R., Pilyugin, S., Ahmed, R., 998. Models of immune memory: on the role of cross-rective stimultion, competition, nd homeostsis in mintining immune memory. Proc. Ntl. Acd. Sci. USA 95, 4926 493. 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