Modeling and analysis of a marine bacteriophage infection with latency period

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1 Nonlinear Analysis: Real World Applications ) Modeling and analysis of a marine bacteriophage infection with latency period Edoardo Beretta a;1, Yang uang b; ;2 a Istituto di Biomatematica, Universita di Urbino, I Urbino, Italy b Department of Mathematics, Arizona State University, Tempe, AZ , USA Received 18 September 1998 eywords: Marine bacteriophage infection; Time delay; Liapunov functional; Global stability; Persistence 1. Introduction In a previous paper [4] the authors proposed a simple model to describe the epidemics induced by bacteriophages in marine bacteria populations like cyanobacteria and heterotrophic bacteria where the environment is the thermoclinic layer of the sea within which bacteriophages and bacteria are assumed to be homogeneously distributed. The main model simplication was in modeling the latent period of infected bacteria in order to describe the model with three nonlinear ordinary dierential equations. However, modeling of the latent period by suitable delay terms looks to be biologically reasonable and mathematically challenging, the ndings of which can be interesting to compare with the outcomes of our previous model [4]. Hence, in the following we rst recall the biological justication for the model and then introduce the model itself comparing it with other models on the same topic. The experimental evidence of the bacteriophage infection of marine bacteria can be found, for example, in the papers by Sieburth [14], Moebus [11], Bergh et al. [3]; Proctor and Fuhrman [12]. It is Corresponding author. Tel.: ; fax: address: kuang@asu.edu Y. uang). 1 Research supported by Gruppo Nazionale per la Fisica Matematica, C.N.R., Italy. This author has presented this paper in the frame of the research Project Con 99 Analysis of complex systems in population biology. 2 Research partially supported by NSF Grant DMS /01/$ - see front matter? 2001 Elsevier Science Ltd. All rights reserved. PII: S X99)

2 36 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) reported see [12]) that from 5 6% up to 70% of bacteria population is infected by bacteriophages. Thus, bacteriophage infection is proposed to be the main cause of bacteria mortality against the assumption that the main mortality cause is the protozoan grazing, thus implying the existence of a signicant new pathway of carbon and nitrogen cycling in marine food webs. The mechanism of bacteriophage infection is assumed to be the following. We have two populations: the bacteria whose total population density is denoted by N [N] = number of bacteria=liter); the viruses or bacteriophages whose population density is denoted by P [P] = number of viruses=liter). A.1. We assume that in the absence of viruses the bacteria population density grows according to a logistic equation with carrying capacity C [C] = number of bacteria=liter) and intrinsic growth rate constant []=d 1 ): dn t) = N t) 1 Nt) ) : 1.1) C Here, combines the growth rate constant by cellular division and bacteria mortality rate like protozoan grazing) excluding that resulting from viral infection. In the presence of viruses, we divide the total bacteria population into two subclasses: the susceptible bacteria St) and the virus infected bacteria It) [S] =[I] =[N]), i.e., N t)=st)+it): 1.2) A susceptible bacterium S becomes infected I under the attack of many virus particles on the cellular membrane see [3]) in a number ranging from one up to 5 phage=cell see, e.g., [13]), but only one virus enters its head through the bacterial membrane and then starts its replication inside the bacterium now infected) and inhibiting the further attack of other viruses on the bacterial membrane. The viruses already on the membrane return to the solution. Thus, the infecting process in the homogeneous solution of sea thermoclinic layer seems to be one infecting phage P infects one susceptible bacterium S and according to the law of mass action we assume: A.2. the rate of infection is Pt)St) 1.3) which is the number of new infected bacteria I per unit time and is the eective per bacteria phage absorption constant rate []-ml/d). The viral nucleic acid inside the infected bacterium takes control of the bacterial metabolism, inhibiting its replication by division, but directing the bacterium in the synthesis of more viral nucleic acid and other materials needed for making copies of complete virus. Hence the present model cannot account for lysogenic bacteria carrying non-replicating phages, but assumes that all phages inside bacteria are virulent. A.3. Accordingly, we assume that only susceptible bacteria S are capable of reproducing by cellular division according to the logistic growth 1.1), whereas the infected bacteria, under the genetic control of virulent phages, replicate phages inside themselves up to the death by lysis after a latency time T. However the infected bacteria I still compete with susceptible bacteria S for common resources. According to these

3 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) remarks, dst) = St) 1 Nt) ) St)Pt) 1.4) C is the balance equation for susceptible bacteria. The time elapsing from the instant of infection, i.e., when the virus injects the contents of the virus head inside the bacterium, to the instant of the bacterium cell wall-lysis, at which b copies of assembled phages are released in solution, is called latent period or incubation time of phages inside bacteria and is denoted by T [T] = days). The lysis of one infected bacterium, on the average, produces b copies of the virus particles. We denote by b the virus replication factor. A.4. For a given population of bacteria we assume that latency period T, T R +0 and virus replication factor b, b 1; + ) are constant and the same for the whole population. A.5. The infected bacteria may have mortality terms dierent from that by viral lysis like protozoan grazing. We account for these terms by the death constant rate i [ i ]=d 1 ). Let us construct the balance equation for infected bacteria It). At any time t the density of infected bacteria It) is obtained by the summation on all the rates of infection at previous times St )Pt ), 0, multiplied by the probability that infected bacteria have to survive from time t up to time t with the given mortality i, i.e., e i. The summation in the past cannot extend beyond T since the bacteria infected at t t T at time t have already left the I class by lysis. Accordingly, at any time t we have T It)= St )Pt )e i d: 1.5) 0 By the variable change = t we obtain t It)= e it ) S)P)d: 1.6) t T In the following we will use its dierential form: dit) = i It)+St)Pt) St T )Pt T )e it : 1.7) Now we consider the balance equation for viruses. A.6. We account for all kinds of possible mortality of viruses enzymatic attack, ph-dependence, UV radiation, photo-oxidation, etc.) by the death constant rate p [ p ]=d 1 ). The lysis rate of infected bacteria at time t is St T )Pt T )e it. Since each bacterium delivers b copies of the phage in solution, the input rate for phages at time t is be it St T)Pt T). The phages leave the class of free viruses

4 38 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) by infecting the bacteria at the rate 1.3) or by death. Hence, dpt) = p Pt) St)Pt)+be it St T )Pt T ); 1.8) is the balance equation for viruses or phages. If the virus infection to the other bacteria population is successful, an endemic equilibrium exists, say Px;t)=P x) for all t; 1.9) where x =x; y; z) are the coordinates in the thermoclinic layer of the sea, and P x) is the equilibrium distribution of bacteriophages. This could provide a constant inow of phages from the surrounding regions accounted by the parameter x)= D p 2 P x) 1.10) if diusivity D p of phages is assumed to be independent of coordinates. A.7. For simplicity, we assume a constant rate of inow, say 0; [] = ml 1 d 1 ), of phages from the surrounding regions. is assumed to be independent of coordinates. With this last assumption the model equations are: dst) dit) dpt) = St) 1 St)+It) ) St)Pt); C = i It)+St)Pt) e it St T )Pt T ); = p Pt) St)Pt)+be it St T )Pt T ); 1.11) where b 1; + ); T R + =0; + ); i R +0 =[0; + ). In this paper we will study the mathematical properties of the solutions of 1.11) under two limiting cases: 0, i.e., the phages have a constant input from the surrounding environment; = 0, i.e., the phages are produced by the epidemic itself. In both cases we consider the solutions of 1.1) depending upon the parameters b; T ) 1; + ) R +. Estimates for from the doubling time of bacteria, for the latency time T and for the virus replication factor b can be found in the paper by Proctor et al. [13]. Estimates for can be obtained, for example, in the paper by Bergh et al. [3] or by kinetic data from the previous referred paper by Proctor et al. [13]. However, for the computer simulations in this paper we use the parameter estimates unless dierently specied) suggested by Prof. A. Okubo see also Beretta and uang [4]), i.e., C = ml 1 ;=1:34 d 1 ;T=7h =6: ml=d; p =2d ) with i =0:1 p. b is used as a varying parameter. In Appendix A the above parameters are reported in dimensionless form.

5 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) It may be interesting to compare the model equations 1.11) with models on the same topic. We start with a model by Campbell [6] with the following equations: dst) = St) 1 St) ) St)Pt); C 1.13) dpt) = bst T)Pt T) p Pt) St)Pt); where t It)= S)P)d: 1.14) t T We remark that in 1.13) the competition for common resources and additional mortality rate endured by infected bacteria is neglected. Eqs. 1.13) and 1.14) can be obtained from 1.11) when =0; i =0. Another modeling of the host-phage system is by Bremermann [5] who proposed the following simple system: dst) dit) dpt) = St) 1 St) ) St)Pt); C = St)Pt) It); = bit) Pt): 1.15) In both models the competition of infected bacteria is neglected in the logistic equation. If we do the same in 1.11), in the case with = 0, the number of model equations reduces to two: dst) dpt) = St) 1 St) ) St)Pt); c = p Pt) St)Pt)+be it St T )Pt T ) 1.16) since t It)= e it ) S)P)d: 1.17) t T In the following we will refer to 1.16), 1.17) as a Campbell-like model. Another model for host-phage system in a chemostat is introduced by Lenski and Levin [10]: drt) = DR 0 Rt)) R) St); dst) = R)St) DSt) St)Pt); 1.18) dit) = St)Pt) DIt) e DT St T )Pt T ); dpt) = DPt) St)Pt)+be DT St T )Pt T );

6 40 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) where R stands for resource concentration and S; I; P have the usual meaning of previous models. In 1.18) R) is the uninfected multiplication rate via binary ssion of susceptible bacteria and is the amount of resources for a new bacterium. Finally, D is the wash-out rate constant of the chemostat. Of course there is clear dierence with 1.11), due to the chemostat structure. However, the rate of infection is the same; and assuming that p = i =D the last two equations are the same as in 1.11) =0). To the best of our knowledge, all the models mentioned above i.e., 1.15), 1.16), 1.18)) have NOT been systematically studied from a mathematical point of view. Hence the study of the mathematical properties of the solutions of 1.11) may provide novel mathematical theory for these models. The structure of the paper is as follows: Section 2 is devoted to the main mathematical properties of the solutions of the model equations 1.11) after having supplemented it by the appropriate initial conditions. Section 3 is devoted to local stability analysis by characteristic equations of the equilibria. An analysis of the endemic equilibrium of the Campbell-like model is also performed. Section 4 deals with the global stability properties of the equilibria. Then in Section 5 we consider properties of permanence of the solutions. Section 6 ends the paper with a discussion about the qualitative features of the model and a comparison in case of = 0) with a previous model by the authors [4]. 2. Basic properties of the model In this section we will present some important properties of the solutions of 1.11), i.e., the solutions of dst) dit) = St) 1 St)+It) ) St)Pt); C = i It)+St)Pt) e it St T )Pt T ); dpt) = p Pt) St)Pt)+be it St T )Pt T ); 2.1) where the parameters ; C; ; i ; p R + ; R +0 ;b 1; + ) and T R +. The initial conditions for 2.1) at t = 0 are: S)= 1 ); P)= 3 ); [ T; 0]; I0) = 0 T e i 1 ) 3 )d; i ) C[ T; 0]): i ) 0; i 0) 0; i=1; 3: 2.2) In the following we dene R 3 +0 = {S; I; P) R3 S 0; I 0; P 0} and R 3 + = {S; I; P) R 3 S 0; I 0;P 0}.

7 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Positive invariance Note that the plane S =0 of R 3 is invariant for 2.1). Now we consider the variable P in [0;T] with initial conditions 2.2). Then, for t [0;T] the third of equations 2.1) gives dpt) = p + St))Pt)+be it 1 T t)) 3 T t)) p + St))Pt) t [0;T]: 2.3) By direct integration of 2.3) we obtain { t } Pt) P0) exp p + S)) d 0 2.4) 0 as t [0;T] and as long as t S)d +. By repeating this argument, we see that 0 the nonnegativity of S and P in [0;T] can be used to infer nonnegativity of Pt). From 2.2), It) can be written as It)= t t T e it ) S)P)d; the nonnegativity of St); Pt) on[ T; + ) implies that of It) ift 0. This shows that for initial conditions 2.2) the corresponding solution of 2.1) is such that min{st); It);Pt)} 0 in its time interval of existence Boundedness of solutions We will need the following lemmas due to Barbalat see [2]). Lemma 2.1. Let g be a real-valued dierentiable function dened on some half line [0; + ); a ; + ). If i) lim t + gt)=; + ; ii) ġt) is uniformly continuous for t a; then lim t + ġt)=0. And its integral version: Lemma 2.2. Let f be a nonnegative function dened on [0; + ) such that f is integrable on [0; + ) and uniformly continuous on [0; ). Then lim t + ft)=0. Lemma 2.3. Assume 0 and the initial conditions 2:2) satisfying S0)+I0) C. Then St)+It) C for all t 0. On the contrary; if S0) + I0) C a time t 1 0 exists such that St)+It) C for all t t 1. Proof. From the rst two equations in 2.1), we obtain d S + I)=S 1 S + I ) i I e it St T )Pt T ) C

8 42 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) we see that d [St)+It) C] St)[St)+It) C] C which implies { St)+It) C +[S0) + I0) C] exp t } S)d : 2.5) C 0 Hence if S0)+I0)=C, then St)+It) C for all t 0, and if S0)+I0) C, then St)+It) C for all t 0. The case of S0)+I0)=C can give rise to the following cases: i) either a positive time t exists such that St) +It) C for all t t,or ii) St)+It)=C for all t 0. The second case implies d=)st)+it)) = 0 for all t t, which gives rise to a contradiction since if St)+It)=C, then d St)+It)) = ii e it St T )Pt T ) 0 2.6) for any t 0. Assume now that S0) + I0) C. We need only to exclude the possibility that if S0) + I0) C then St)+It) C for all t 0. If so, then on [0; + ), d St)+It) C) St)St)+It) C) 0: C Hence on [0; + ) we can dene the nonnegative function ft):= d St)+It) C); 2.7) for which S0)+I0) t fu)du=s0)+i0) St)+It)) 0 for all t [0; + ). 0 Then Lemma 2.2 implies that lim t + ft) = 0, i.e., d lim t + St)+It))=0: 2.8) However, from the rst two equations 2.1) we get d lim t + St)+It)) { S = lim t + 1 S + I ) } i I e it St T )Pt T ) : 2.9) C Then these two cases are possible i) It) +St) C; St) 0 as t ; ii) St) +It) C; It) 0ast +. The case i) is trivial, since it gives lim t + d=)st)+it)) = i C 0. The case ii) gives d lim t + St)+It)) = e it C lim Pt T ) 2.10) t +

9 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) which requires a further analysis on Pt) behavior. From the rst equation in 2.1) we see that lim sup St) C: 2.11) t + Of course there will be suciently large times T 0 such that for t T 0 ;St) 2C. Then, from the third equation in 2.1) we have dpt) p Pt) St)Pt) p +2C)Pt) 2.12) which shows that, for large t, say t T 1 T 0, then Pt) 0: 2.13) 2 p +2C) Therefore, 2.10) and 2.11) imply d lim t + St)+It)) Ce it 0; 2.14) a contradiction to 2.8). In conclusion, let t 1 := max{t ;T 1 } 0. Hence, whenever S0)+I0) C then St)+ It) C for all t t 1. This proves the lemma. The case = 0 gives rise to a dierent result with respect to Lemma 2.3, but the procedure to prove it is however similar. Furthermore, the same kind of results are presented in [1]. In the following by E f =C; 0; 0) we denote the free disease equilibrium of 2.1) which is feasible only if =0. We omit the proof of the following lemma to avoid repetition. Lemma 2.4. Assume =0 in 2:1). If we assume that at t =0 the initial conditions of 2:1) satisfy S0) + I0) C; then either i) St) +It) C for all t 0 and therefore St);It);Pt)) E f =C; 0; 0) as t + ; or ii) there exists a time; say t 1 0; such that St)+It) C for all t t 1. Finally; iii) if S0) + I0) C then St)+It) C for all t 0. We can further prove the following results regarding boundedness of I; P variables. Lemma 2.5. Assume 0. Dene the function: W t)=bit)+pt); t [0; + ): 2.15) Assume further that b b, 1+ p C 2.16) and L is any positive constant such that L m b 1)C, L 1; 2.17) where m = min{ i +b 1)C; p }. Then there is a t 1 = t 1 L) 0; such that; for all t t 1 ;Wt) L.

10 44 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Proof. From the second and third parts of Eq. 2.1), it follows that dw = i bi p P +b 1)SP: 2.18) From Lemma 2.3 we know that a time, say t 0 ;t 0 0, exists such that St) C for all t t 0. Hence, from 2.18) we obtain that, for all t t 0, dw i bi p P +b 1)CW bi) = [ i +b 1)C]bI p P +b 1)CW: 2.19) Thus we obtain dw [ m b 1)C]W: 2.20) Observe that if m = p, then 2.16) indicates that m = p b 1)C. If m = i + b 1)C, then clearly b 1)C m. Hence 2.20) implies that lim sup t + W t) This proves the lemma. L: 2.21) m b 1)C The following is an important result in the study of global stability: Theorem 2.1. Assume that =0. Then for all b 1;b =1+ p =C)) the free disease equilibrium E f =S = C; I =0;P =0) is globally asymptotically stable in R 3 +. Proof. We assume = 0. According to Lemma 2.4 we have the following. Assume that S0) + I0) C. Then either i) St);It);Pt)) E f =C; 0; 0) as t + or ii) there exists a time, say t 1 0, such that St)+It) C for all t t 1. Finally iii) if S0)+I0) C, then St)+It) C for all t 0. Thus, in case i) the theorem is proven, whereas the cases ii), iii) we can assume that a time, say t 1 0, exists such that St) C for all t t 1. Dene W t)=bit)+pt) for all t [0; + ). Then from the second and third equations of 2.1) we obtain note =0) dw t) = i bi p P +b 1)SP: 2.22) This implies see 2.19)): dw t) [ m b 1)C]W t) 2.23) for all t t 1 and where m =min{ i +b 1)C; p }. Furthermore if b 1;b );b = 1+ p =C), then m b 1)C 0. Hence 2.23) implies that W t)=bit) + Pt) 0 as t +, i.e., It) 0 and Pt) 0 as t +. Since St) is bounded, then for any 1 0 there is a T t 1 such that for t T, [ St) 1 St) ] dst) [ St) 1+ St) C C ] : 2.24)

11 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Hence 1 )C lim inf t + St) lim sup t + St) 1 + )C. Hence lim t + St) C s C and the conclusion follows by letting 0. Note that for =0, a necessary condition for the existence of the positive endemic) equilibrium E + is b b. Thus, Theorem 2.1 holds true when E + is not feasible. Our next theorem shows that there is an L 2 0 such that, regardless of the value of b and, and independent of initial conditions, lim sup t + Pt) L 2. Observe rst that, for large t, say t t 1 T; St) C+ 1. Hence for t t 1, P t) [ p + C + 1)]Pt) 2.25) which implies that Pt) Pt T )e [p+c+1)]t 2.26) thus C 1 =e [p+c+1) T ) Pt T ) Pt)e [p+c+1)]t, Pt)C 1 : 2.27) Hence, for t t 1, P t) + be it C 1 C +1)Pt): 2.28) Let C 2, be it C 1 C + 1), then for t t 0 t 1, Pt) C 1 2 +Pt 0 )+C 1 2 )e C2t t0) : 2.29) It is also easy to observe that if, for all large t, Pt) +1 then lim t + St) = 0 and lim t + Pt)=C p, in which case L 2 = C p. Let l = max{ 1 ;C 1 2 } +1; 2.30) T 1 = T + 1 ) 2bC +1) l ln p e it, T + ; 2.31) L 2 = C 1 2 +l + C 1 2 )e C2T1 +2 p : 2.32) Clearly L 2 l and L 2 2= p =2C p. We have, regardless of the value of 0) and b: Theorem 2.2. With L 2 dened above; we have lim sup t + Pt) L 2 : 2.33) Proof. Recall that it is impossible for Pt) l for all large t. Assume the theorem is not true; then there is a t t 1 + T 1, such that Pt )=L 2 ; P t ) 0; 2.34)

12 46 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) and for all t t ;Pt) Pt ). In addition, there is a t 0 ;t 1 t 0 t, such that Pt 0 )=l; P t 0 ) ) and l Pt) L 2 for t [t 0 ;t ]. By 2.29), we see that letting t = t ) t t 0 1 L2 + C 1 ) 2 ln C 2 l + c 1 T 1 : 2.36) 2 For t [t 0 ;t ], we have Hence S t) St) Pt)St) l )St): 2.37) St T ) St 0 )e l ) t T t 0 ) C + 1)e l )T1 T) : 2.38) This implies that P t ) p Pt )+be it St T )Pt T ) L 2 [ p bc + 1)e it e l ) ] [ = L 2 p ] p 2 = p 2 L 2 0; a contradiction to the second part of 2.34). This proves the theorem. In the rest of this paper, we dene L = max{l 1 ;L 2 }; 2.39) 2.3. Equilibria A) Case 0: In the following we denote by C p = = p the carrying capacity of bacteriophages. The equilibria of 2.1) are solutions of: S[C S + I)) CP]=0; i I + SPT )=0; p C p P)+SPT )=0; where, for the sake of simplicity, we set 2.40) T )=be it 1; T)=1 e it 2.41) with T ) 1;b 1]; T ) 0; 1) as T [0; + ). Furthermore, T )=0 at T = T, where T = 1 i log b: 2.42)

13 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Straightforward computation yields the following. Proposition 2.1. Assume T = T see 2:41)); i.e.; T )=0. Then 2:1) admits the following two equilibria: E f =0; 0;C p ) which is always feasible and the positive equilibrium C Cp ) E + = 1 + C p ) ; C ) pc C p ) ;C p ; 1 + C p ) provided that C p =. When C p = =; E + coincides with E P. The constant is dened in 2:43) below. Standard but tedious analysis yields the following. Proposition 2.2. Assume T ) 0; i.e.; T T. Then a unique positive equilibrium E + exists provided that C p =. When C p ==; E + coincides with E p =0; 0;C p ). Now let us consider the case ii) T) 0, i.e., T T. Let S 1 P), pc p P) T ) P ; S 2P), C P) 1 + P) ;, T ) : 2.43) i S 1 P) is positive on 0;C p ) vanishes as P =C p and has lim P 0 + S 1 P)=+. Furthermore, S 1 P) is monotone decreasing as P increases in 0;C p ) with positive concavity. Let us dene the function P)=S 1 P) S 2 P) on0;c p ). Hence C0;C p ]) is such that P) + as P 0 + and C p ) 0. Hence P) = 0 has at least one root P 0;C p ) and in any case the number of roots of P) = 0 on 0;C p ) must be odd. Since P) = 0 gives a second order of algebraic equation, the number of the roots of P) = 0 on 0;C p ) must be one. This proves the existence and uniqueness of P 0;C p ) such that S 1 P )=S 2 P ) and, correspondingly, of the positive equilibrium E + whose other components are S := S 1 P )=S 2 P );S 0;C) and I = S P. Of course, if C p = =k the positive equilibrium becomes E p =0; 0;C p ). Hence the following is proven: Proposition 2.3. Assume T ) 0; i.e.; T T and C p =. Then a unique positive equilibrium E + exists with P 0;C p ) and S 0;C). When C p = =k; E + coincides with E p =0; 0;C p ). As a summary we may say that if 0 then the positive equilibrium E + is feasible provided that C p =k. IfC p = =k, then E + becomes E p. Finally, if T ) 0, i.e., T T then P C p ; =k)), whereas if T ) 0, i.e., T T, then P 0;C p ). B) Case = 0: The equilibria are solutions of S[C S + I)) CP]=0; I = SP; P[ p + ST )]=0; 2.44)

14 48 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) where =T )= i ) and T ); T ) have been dened in 2.41). Assume rst T)=0, i.e., T =T. Then 2.44) shows that P =0;I =0 and S =C, in agreement with Proposition 2.1 in which we set C p == p )=0. Now let us consider the case T ) 0. It is easy to check that boundary equilibria E 0 =S =0;I =0;P =0); E f =S = C; I =0;P =0); 2.45) are both feasible for all parameter values. For the positive equilibria, if they exist, their components are: E + = S = p T ) ;I = S P ;P = C ) S ) C + S : 2.46) Clearly, if T ) 0, then S 0 and we cannot have positive equilibria. Hence we need only to consider the case T) 0, i.e., T T. If, in addition, S = p =kt)) C, the unique positive equilibrium is feasible and it is given by 2.46). When T ) is such that S = C then the positive equilibrium becomes E f =C; 0; 0), and E + is not feasible if S C.IfT = 0, then S = p b 1)), whereas for T increasing in 0;T );T =1= i )log b; S = p =T )) is a monotonically increasing function of T and that S + as T T from the left. Let T c be the incubation time at which T )= p =C, i.e., S = C: T c, 1 i log b b 2.47) where, T c T and b =1+ p =C. Some comments on 2.46) are in order. If the virus s replication factor b is such that b b =1+ p =C, then T c 0 and the endemic equilibrium E + cannot be feasible. Hence, the feasibility of the endemic equilibrium requires as a necessary condition that b b. This implies T c 0, and the endemic equilibrium is feasible only if the time T taken in replicating the phages is not too high, i.e., T T c. We summarize the above results as follows: Proposition 2.4. Assume =0. Then the boundary equilibria E 0 =0; 0; 0) and E f = C; 0; 0) are feasible for all parameter values. If it exists; the endemic equilibrium E + is unique and given by 2:46). A necessary and sucient condition for its existence is that T 0;T c ); where T c =1= i ) logb=b ) and b =1+ p =C. We would like to point out that the presence of 0, i.e., of a resource of bacteriophages from the surrounding environment, ensures the existence of a unique endemic equilibrium for any latency period or incubation time) T. While for = 0, it is necessary that b b and T 0;T c =1= i ) logb=b )) in order to have E + feasible. 3. Local stability of boundary equilibria For convenience, let xt)=colst) S ;It) I ;Pt) P ); x R 3 ;t 0: 3.1)

15 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Then 2.1) can be written as d xt)=fxt); xt T)) 3.2) where F : C[ T; 0]; R 3 ) R 3 is continuously dierentiable vector function. Hence, dene the matrices A; B R 3 3 [ ] A = ; B = ; T) x=0 Eqs. 2.1), linearized around 0, takes the form dxt) = Axt)+Bxt T) 3.4) and the corresponding characteristic equation is det[a + Be T I] = 0 3.5) where are the corresponding characteristic roots. It is easy to check that the characteristic equation takes the form of 3.6) below: 1 2S + I ) P C C S S P 1 e i+)t ) i +) S 1 e i+)t =0: ) P 1 be i+)t ) 0 p +) S 1 be i+)t ) 3.6) In the following we consider two cases: Case 1: 0. From 3.6) it is easy to check that the following proposition holds true. Proposition 3.1. Assume 0. The equilibrium E p =0; 0;Cp) is an asymptotically stable node if = C p ; i.e.; E + is not feasible; if = C p it becomes an unstable saddle point and as = = C p ;E p becomes stable if = C p ;E + is feasible). Case 2: = 0. In this case we have two boundary equilibria, i.e., E 0 =0; 0; 0) and E f =C; 0; 0), which are feasible for all parameter values. If T c 0, i.e., b b, the endemic equilibrium E + is feasible provided that T 0;T c ), and it is not feasible if T T c. E + coincides with E f if T = T c. We can prove: Proposition 3.2. Assume =0. The equilibrium E 0 =0; 0; 0) is always an unstable saddle point. As far as E f C; 0; 0) is concerned; it is locally asymptotically stable whenever T T c and unstable if T T c T c =1= i ) logb=b )). If T = T c ; then E f becomes critically stable. The proof of Proposition 3.2 is trivial where it concerns the equilibrium E 0,asit can be easily checked from 3.6). The proof of the remaining part of Proposition 3.2 is non trivial but still it is quite standard and therefore can be found in Appendix B. The

16 50 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) analysis of local stability properties of the endemic equilibrium for both cases 0 and = 0 are very dicult. However, for the case = 0, we may have an insight about the possible stability behavior of the endemic equilibrium E + considering equations very close to our model equations 1.11) but which enable a remarkable simplication of the characteristic equation: the Campbell-like model 1.16),1.17) with i 0 and the Campbell model 1.13), 1.14) where i =0. For both models we just have to consider two variables: S and P. Before considering the characteristic equation for the Campbell-like model Campbell model is just a particular case with i = 0) we recall that the equilibria have the same structure as in 1.11). The equilibria E 0 =0; 0) and E f =c; 0) are feasible for all parameter values. The endemic equilibrium E + =S ;P ) has components: S p = be it 1) ; P = ) 1 S 3.7) C and it is feasible provided that b b =1+ p =C) and T T c = 1 i log b b : 3.8) The same holds for the Campbell model, where i = 0; however, this implies that the endemic equilibrium E + = S p = b 1) ;P = )) 1 S 3.9) C is now independent from the latency time T, and is feasible provided that b b. For both cases i 0 in the following we set = S C = b 1 be it 1 ; p = P C = 1 ) 3.10) C with 0; 1). Remark that if i 0 then = b 1 be it 1 is a monotone increasing function of T as T is increasing in [0;T c ] and =1 at T =T c. In the case of i =0;is a monotone decreasing function of b in b ; + ) and =1 at b = b. In both cases i 0) the characteristic equation at the endemic equilibrium E + is +) C p C )e T ) )1 e T ) = ) where are the characteristic roots and )= p + C: We obtain 2 + a+be T + c + de T = ) where a = + ); b= ); c= d + p 1 ); d= 1 2)): Note that c + d 0, for 0; 1) and at T = 0 the characteristic roots have negative real parts.

17 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) When increasing T, a stability shift may occur only with a pair of purely imaginary characteristic roots, say = ±i!;! 0, crossing the imaginary axis from left to right. Hence, we look for characteristic roots = ±i!;! 0 of 3.12). We obtain that! 0 must be a solution of! 2 = 1 2 {b2 +2c a 2 ) ± [b 2 +2c a 2 ) 2 4c 2 d 2 )] 1=2 } 3.13) where it is easy to check that b 2 +2c a 2 0; 0; 1): 3.14) Of course, if c 2 d 2 there are no characteristic roots = ±i!;! 0: If d 2 c 2 there is one pair, say = ±i! + ;! + 0, such that dre) d 0: =±i!+ 3.15) Hence, by a simple analysis of the function ), d 2 ) c 2 )=1 )[ 2 2 p1 )+2 2 p 1 2))]; as [0; 1] it is simple to prove: Lemma 3.1. There exists a unique value; say 1 ; 1 0; 1 2 ) such that d2 ) c 2 ) for all 0; 1 )and d 2 ) c 2 ) in 1 ; 1)): Then we can conclude: Theorem 3.1. Let us denote by b c, b 1 1 ))= 1 : 3.16) i) If i 0 then for b b ;b c ) no stability shifts can occur and the positive equilibrium remains locally) asymptotically stable for all T [0;T c ]. ii) Assume i =0: If b b ;b c ] no stability shifts can occur and the positive equilibrium remains locally) asymptotically stable for all T 0. If b b c there exists a T 0 0; T 0 = 1 ;! ) where sin 1 = da! + b! + c! +) 2 b 2! + 2 ; cos + d 2 1 = ab!2 + +c! +)d 2 b 2! d 2 such that the positive equilibrium is locally) asymptotically stable if T 0;T 0 ); unstable if T T 0. The proof of Theorem 3.1 is in Appendix B. Comments on the interpretation of the results in the theorem are in the discussion section.

18 52 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Global stability analysis As in the previous sections we distinguish between the two cases A) = 0 and B) 0. Let us consider rst Case A) = 0: We have two results concerning the global asymptotic stability of E f : The rst is concerned with the case b b ; i.e., T c 0, in which the endemic equilibrium E + is not feasible. A second result regards the case b b where T c 0. The rst result slightly extends the one already proved in Theorem 2.1 for the case b = b. Theorem 4.1. Assume =0. Then if b b the free disease equilibrium E f =C; 0; 0) is globally asymptotically stable in R 3 +. Proof. Recall rst that Lemma 2.4 provides that, if S0) + I0) C then either St)+ It) C for all t 0 in which case St);It);Pt)) E f as t +, or a positive time, say t 1 0; exists such that St)+It) C for all t t 1. Furthermore, if S0) + I0) C then St)+It) C for all t 0: In conclusion, either St);It);Pt)) E f as t + or a nonnegative time exists, say t 0 0; such that St)+It) C for all t t 0. Hence we study below the global stability of E f assuming this last case. Let us consider the Liapunov function U : R 3 +0 R; where R 3 +0 := {S; I; P) R+0 3 S 0}, dened by U =S C log S)+wbI + P); w R + 4.1) which is lower bounded and dierentiable on R We obtain U 2:1) = C C St))[C St)+It))] w p C)Pt) wb i It) 1 wb 1))St)Pt): 4.2) Since St) +It) C for all t t 0, 4.2) shows that for all t t 0 U 2:1) 0 and U 2:1) = 0 if and only if S; I; P) coincides with E f provided that the arbitrary positive constant w in 4.1) can be chosen in such a way that C w 1 p b 1 : 4.3) Hence 4.3) becomes a sucient condition for the global asymptotic stability of E f in R 3 +: Since 4.3) reduces to b b = p =C) + 1, this proves the theorem if b b. If b = b then the unique choice for w is w p C = wb 1) 1=0: Hence, from 4.2) we obtain U 2:1) = C C St))[C St)+It))] wb iit): 4.4) In R 3 +0 let us consider the set E = {S; I; P): U =0}, i.e., E = {S; I; P) R 3 +0: I =0;S= C}. Say M is the largest invariant subset of E. Assume St)=C for all t. Then ds= = 0 thus implying Pt) = 0 for all t. Since in E; It) = 0 for all t, then M = {E f }. Hence, by the Liapunov LaSalle theorem e.g., [9]), the global asymptotic stability of E f in R 3 +0 follows.

19 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) We now present a global asymptotic stability result about E f b b. in the case in which Theorem 4.2. Assume =0 and b b. Then; the free disease equilibrium E f = C; 0; 0) is globally asymptotically stable in R+ 3 provided that T T 1, 1 ) b 1 log i b : 4.5) 1 Proof. Recall that either St);It);Pt)) E f as t + or there is a t 0 0 such that for t t 0 ;St) C: In the following we assume t t 0 + T. Then, from the last two equations of 2.1) we get d It)+Pt)) = iit) p Pt)+b 1)e it St T )Pt T ) i It) p Pt)+b 1)e it CPt T ): 4.6) Let us consider the functional: Ut)=It)+Pt)+b 1)e it C t t T P)d: 4.7) Then, from 4.6), 4.7), we get U t) i It) [ p b 1)e it C]Pt): 4.8) Let us dene, p b 1)e it C and assume it is positive, i.e., 0. Then T must satisfy that T 1 ) b 1 log i b, T 1 : 1 Hence if 4.5) holds true, then = min { i ;} 0, and we get U t) i It) Pt) It)+Pt)): 4.9) So Ut) is a Liapunov functional for global asymptotic stability of the equilibrium I =0;P = 0) of the last two equations in 2.1), i.e., It);Pt)) 0; 0) as t + : This in turn implies St) C as t + : Note that T 1 =1= i ) log[b 1)=b 1)] T c =1= i ) logb=b ) whenever T c 0. Therefore it is yet to be investigated what happens when T T c ;T 1 ]. At the moment, this problem is open. In the case of 0, the following results hold: Lemma 4.1. Assume 0 and initial conditions 2.2) such that S0) 0; P0) =. Furthermore; if C p 1+ S0) ) 4.10) p then the corresponding solutions are such that St);It);Pt)) E p =0; 0;C p ) as t + : 4.11)

20 54 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Proof. Consider the rst and third of equations 2.1): S t)= Pt) ) St) C St)St)+It)); P t)= p Pt) C p ) St)Pt)+be it St T )Pt T ): 4.12) Let us rst assume that Pt) = for all t 0: Thus, we can dene the nonnegative function ft)= S t) on[0; + ) since the rst of 4.12) shows that S t) 0 for all t 0: Furthermore, remark that t f)d exists for all t [0; + ) since negativity 0 of S t) implies S0) t 0 f)d = S0) St) 0 for all t 0: 4.13) Therefore, Barbalat s Lemma 2.2 implies lim t + ft) = 0: In other words, lim t + S t) = 0, and nonnegativity of Pt) = requires that lim t + St) =0: Since Pt) is bounded, this implies that lim t + [P t)+ p Pt) C p )]=0: 4.14) From 4.14) we can say that for all 0; T 0 exists such that P t)+ p Pt) C p ) p 4.15) for all t T, or equivalently, lim sup t + Pt) C p. Letting 0 we obtain lim Pt)=C p: t + Finally, recalling that It)= t t T 4.16) e it ) S)P)d 4.17) we get lim t It) = 0. In conclusion, every solution of 2.1), satisfying that Pt) = for all t 0, is such that St);It);Pt)) E p =0; 0;C p )ast +. Since P0) = we denote by t ;t 0, the rst time at which Pt) assumes the value =. Att = t, the second of 4.12) gives P t )= p C p ) St )+be it St T )Pt T ): Since S t) 0in0;t ];St ) S0) and therefore P t ) p C p ) S0) + be it St T )Pt T ): 4.18) Thus 4.10), 4.18) imply P t ) be it St T)Pt T ) 0; 4.19) i.e., Pt) cannot cross the =k value at any t 0. Hence Pt) =k for all t 0 and the rst part of the proof implies the theorem.

21 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Theorem 4.3. Assume 0. If p + C) 1; 4.20) then E p =0; 0;C p ) is globally asymptotically stable. If = p + C)) 1; then C p [ 1+ C )] ) p p + C) implies the global asymptotic stability of E p =0; 0;C p ). Proof. In Lemma 2.3, we stated that a time t 0 ;t 0 0, exists such that St) C for all t t 0. This remark is used in the third of equations 2:1) to imply dpt) p + C)Pt) 4.22) for all t t 0. Hence for all 0, a time exists, say t 1 ); t 1 ) t 0, such that inf Pt) : 4.23) t t 1) p + C From the rst of equations 2:1) we obtain S t)= St) Pt) [ St) ) C St)It)+St)) p + C) 1 ) ] ; 4.24) for all t t 1 ). Since 4.20) holds true, we choose 0 ) p + C) ) thus obtaining S t) 0 for all t t 1 ): 4.26) By Barbalat s Lemma 2.2, 4.26) implies lim t + St) = 0, which in turn implies lim t + Pt)=C p and lim t + It) = 0. For details, see Lemma 4.1.) Assume now = p + C)) 1. Hence 4.23) implies that for all 0, there exists t 1 ): { S t) St) 1 } St) Pt) C { St) 1 ) p + C St) } C 4.27) holds true for all t t 1 ). Hence, for all 0, [ lim sup St) C 1+ ] t + : p + C) 4.28)

22 56 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Hence, 4.28) and the third part of 2.1) imply { ) P t) p + C [ 1+ p + C) ]} Pt) for all t t 1 ). Again, for all 0 there exists t 2 );t 2 ) t 1 ) and for t t 2 ) inf Pt) t t 2 ) C p 1+ C p [1 + ) ] : 4.29) p+c) From the rst equation of 2:1), we obtain { ) } S C p t) St) 1+ C p [1 + ) ] p+c) = St) { Cp C [1 + p 1 1+ C p [1 + ) p+c) )] C p+c) ] p ) } 4.30) for all t t 2 ). Since inequality 4:21) holds true, then we can choose : { 0 C p [ 1+ C )]} p 1 p p + C) C ; 4.31) for which the choice of : 0 C p C [1 + p 1 1+ C p [1 + ) p+c) )] C p ] 4.32) p+c) implies that see 4.30)) S t) 0 for all t t 2 ). Again the Barbalat Lemma 2.2 implies lim t + St) = 0 and the global asymptotic stability of E p follows. Let us remark that = p + C) 1 is the same as C p =) p + C). Here we address our attention to some particular cases which are not included among the results of Lemma 4.1 and Theorem 4.3. Proposition 4.1. Assume initial conditions 2:2) such that 1 ) S0) 0; 3 ) ; P0) ; 0 [ T; 0]: 4.33) Denote by t 0 the rst time at which eventually) Pt )==. Then; if p C p ) St )T ) 0; 4.34) where T )=be it 1; all the solutions of 2:1) with initial conditions 4:33) satisfy that St);It);Pt)) E p =0; 0;C p ) as t +. Proof. In Lemma 4.1 we proved that if Pt) = for all t 0 then St);It);Pt)) E p as t +. Hence, it is sucient to prove P t ) 0ift is the rst time at

23 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) which Pt) assumes the = value. Assume t T. Then the second equation 4:12) gives P t )= p C p ) St )+be it St T )Pt T ): 4.35) Since S t) 0in0;t ], then S0) St T ) St ). Furthermore Pt T ) Pt ) = =. Hence, from 4:35) we have P t ) p C p ) + St )be it 1): 4.36) Assume now t 0;T]. Eq. 4.12) gives P t )= p C p ) St )+be it 1 t T ) 3 t T ): Due to the initial condition 4.33) we obtain P t ) p C p ) St )+be it S0) : Since S t) 0in0;t ], S0) St ) and again we obtain 4.36). Hence, if 4.34) holds true Pt) cannot cross the = value for all t 0 and the proposition follows. Recall that T )=be it 1 0ifT [0;T ); T ) 0ifT T ; + ) and nally T )=0 if T = T = i 1 log b. In the following we consider some cases in which Proposition 4.1 applies: i) C p = and T) 0, i.e., T [0;T ). Remark that as C p == the endemic equilibrium collapses into E p. ii) C p = and T T such that ) St )be it St )+ p C p ; from which we have T T, 1 { log i bst ) St )+ p =) C p ) } : 4.37) In 4.37), of course, T T and T 0 provided that St ) is such that bst )= St )+ p =) C p ))) 1. Recall that in this case the endemic equilibrium E + is feasible. iii) C p = and T ) 0. Then 4.34) becomes p C p ) St ) T) p C p ) S0) T ) ; since St ) S0). Hence if p C p ) S0) T ) 0; 4.38)

24 58 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Proposition 4.1 holds true. It is easy to check that 4.38) is satised if T T T 1, 1 ) bs0) log ; 4.39) i S0) + p C p =)) and S0) p C p =)). Thus, we have proved: Corollary 4.1. Assume C p = but that inequality 4:10) is reversed; i.e. S0) p C p ) : 4.40) Then if the latency time T satises 4:39); all the solutions of 2:1) with initial conditions 4:33) converge to E p =0; 0;C p ) as t +. As far as the global asymptotic stability of the endemic equilibrium is concerned, we consider the standard Liapunov functional approach on the equations centered on E +. However, to avoid many simple but) tedious algebraic computations, we decided to work on the dimensionless form of the equations as they appear in Appendix A see A.1)). We can prove the following: Theorem 4.4. Assume that the parameters of Eq. A:1) satisfy a + p 2 3 ; m i 1 2 : 4.41) Hence, for any incubation time T satisfying { p T max m 1 ) +1 bp i log ;m 1 + 2)a + p ) )+a i log m i 1=2) ma + p ; ) ba + p m 1 )} )+2a i log 3=2a + p 4.42) ) 1 the endemic equilibrium E + ; if feasible, is globally asymptotically stable in R+. 3 Proof. Before proceeding with the proof we need to center the model equations A.1) at the endemic equilibrium E + =s ;i ;p ). Denoted by u 1 = s s ;u 2 = i i and u 3 = p p, we obtain the following: u 1 = s{ a + p )u 1 au 2 su 3 }; u 2 = p u 1 m i u 2 + su 3 + p e mit u 1 t T ) e mit s)u 3 T ); u 3 = p u 1 m + s))u 3 + bp 4.43) e mit u 1 T ) + bs T )e mit u 3 T ); where E + has been transformed into 0; 0; 0). Let us consider R 3, {u R 3 u 1 s ;u 2 i ;u 3 p }; 4.44) on which we dene V : R 3 R +0 ;V C 1 R ), 3 such that V u)=u 1 s u1 + s ) log + 1 2! 2u2 2 +! 3 u3); ) s

25 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) where! i R + ;i=2; 3, are at the moment arbitrary real constants. Now it is tedious but straightforward to check that: V 4:43) [ a + p ) ! 3p ] u1) 2! 2 [ ] mi 1 2 u 2 2 )! 3 mu3) 2 s)! a=p )) ) u3) ! 2p + 1)e mit u 2 2)+ 1 2! 3bp + 1)e mit u 2 3) p e mit b! 3 +! 2)u 2 1 T ) e mit b! 3 +! 2)u 2 3 T ) 4.46) where the inequalities u i u j 1=2)u 2 i + u 2 j ) have been used and the constant! 2 has been chosen at the value! 2 =a=p ). The structure 4.46) suggests to choose! 3 such that a + p 2p! 3 2a + p ) 1 p 4.47) which requires a + p 2=3. Hence, assume 4.41) holds true and choose for the following! 3 at the value:! 3 =a + p )=2p. We are now in a position to dene the Liapunov functional: Uu )=V u)) p b! 3 +! 2)e mit b! 3 +! 2)e mit T u 2 1)d T u 2 3)d: 4.48) From 4.46) and 4.48), suitably reordering the terms, we obtain U 4:43) {[ a + p ) ! 3p ) ] 1 2 p b! 3 +! 2)e mit} u1) 2! 2 [ ) mi 1 2 p + 1)e mit] u2) 2 [ m! b! 3p +2)+! 2)e mit] u3) ) which is negative denite provided that 4.42) holds true. Hence, U satises all the assumptions of Corollary 5:2 in [9] and the theorem follows. The theorem applies without further restrictions to the case 0 or C p 0) in which, provided that = C p or C C p ), the endemic equilibrium is feasible for all the values of b and T. When =0 C p = 0) the endemic equilibrium is feasible if b b =m+1 and T T c =m 1 i logb=b ). Of course, Theorem 4.4 applies only if T c is larger than the value estimated at the right side of 4.42). Further, 4.41) excludes the case m i =0.

26 60 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) Permanence results We say system 2.1) with initial condition 2.2) is permanent or uniformly persistent) if there exist positive constants, independent of initial conditions, m;m;m M, such that for solutions of 2.1) 2.2), we have { } min lim inf St); lim inf It); lim inf Pt) m 5.1) t + t + t + and { max lim sup t + St); lim sup t + } It); lim sup Pt) t + M: 5.2) In view of the fact that St) and Pt) are eventually bounded, so must be It) since 1.5)). Also, we see that if there is an m 1 0 such that { } min lim inf St); lim inf Pt) m 1 ; 5.3) t + t + then, by 1.5), we have lim inf It) t + m2 11 e it )= i, m 2 : 5.4) If i = 0, then m 2 = Tm 2 1. Hence 5.1) holds for m = min{m 1;m 2 }. Therefore, to establish permanence for 2.1) 2.2), we need only to nd m 1 0, such that 5.3) holds. We consider rst the case when 0. From 4.22), we see that lim inf Pt) = p + C): 5.5) t + Thus we need only to show that there is an m 0, such that lim inf St) m: 5.6) t + We shall prove the following: Theorem 5.1. If 0 and C p =; then 2:1) with 2:2) is permanent. To prove the above, we need a few preparations. Lemmas 2.3, 2.5 and Theorem 2.2 imply that for any L L 1, there is a t 1 0 such that, for t t 1, St) C and Pt) L: 5.7) Let l 0;= p + C)). Due to 5.5), and for convenience, we assume below that for t t 2 t 1 + T, Pt) l: 5.8) Lemma 5.1. Assume t 4 T t 3 t 2 and for t [t 3 ;t 4 ];St) 1. Then for t [t 3 + T; t 4 ]; Pt) L + ) be it L 1 e pt t3 T ) + + be it L 1, f 1 t; 1 ) 5.9) p p

27 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) and Pt) l p + 1 ) e p+1)t t3 T ) +, f 2 t; 1 ): 5.10) p + 1 Proof. Eq. 5.9) follows directly from the fact that for t [t 3 + T; t 4 + T ]; Pt) L, and P t) + be it L 1 p P and 5.10) follows directly from the fact that for t [t 3 ;t 4 ];Pt) l and P t) p P 1 P: It is clear that lim f 1 t; 1 ) = lim f 2 t; 1 )= = C p ; p t + t + uniformly. Assume C p = and let = 1 ) 4 C p : 5.11) We dene { T i =0; g i )= i 1 1 e it ) i 0: Let 1 = 1 ) 0 such that 1 1 and 1 [1 + Lg i )] C ) { max C p + } be it L 1 p ; C p p ) and T 1 = T 1 ) 0, such that { max L + } be it L 1 p ; e pt1 l p + 1 : 5.14) e p+1)t1 Hence, if t 4 t 3 + T + T 1, then for t [t 3 + T + T 1 ;t 4 ], max{ f 1 t; 1 ) C p ; f 2 t; 1 ) C p } 2: 5.15) This together with 5.12), implies that for t [t 3 + T + T 1 ;t 4 ], max{ f 1 t; 1 ) ; f 2 t; 1 ) } 2: 5.16) Now we are ready to present the proof of Theorem 5.1. Proof of Theorem 5.1. Let be dened by 5.12) and 2 = 1 e L2T +T1). We claim that lim inf ST ) 2: 5.17) t +

28 62 E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) If not, there exists a t T + T 1 )+t 2, such that St ) 2 and S t ) 0: 5.18) Note that for t t 2, S t) SP LS which implies that for t t 0 t 2 St) St 0 )e Lt t0) and hence St 0 ) St)e Lt t0) : 5.19) Therefore for t [t T T 1 ;t ], St) 2 e LT +T1) = 1 e LT 1 : 5.20) Applying Lemma 5.1 with t 4 = t ;t 3 = t T T 1, we have Pt ) f 1 t ; 1 ) C p +2: 5.21) However, S t ) 0 implies that [ ) 0 St ) Pt ) [St ] )+It )) C St )[2 C It ))]: 5.22) Note that It ) t t T e it ) 1 L d: For i =0, we have It ) TL 1 and for i 0, we have It ) i 1 L1 e it ) 1. Hence we have It ) Lg i ) 1 : Therefore 5.22) implies that 0 St )[2 C Lg i )) 1 ] St )[2 ]=St ); a contradiction. Let m = 2. Then 5.6) holds and hence the theorem. The argument above in fact shows that we can with quite some computation) nd an explicit expression in terms of the parameters of 2.1)) for m in 5.1). This can be very useful in practice. To save space, we choose not to do it here. When = 0 and i 0, the persistence or permanence question becomes very challenging. The diculty is that even though we know that T T c = i 1 lnb=b )) implies the instability of E f, we cannot locate all the eigenvectors of the innitely many eigenvalues with negative real part the root of g) = 0 of 3.12)). Other methods e.g., those used in [7,8,15]) are equally dicult to implement.

29 6. Discussion E. Beretta, Y. uang / Nonlinear Analysis: Real World Applications ) In this paper we proposed a delay dierential equation model which describes the bacteriophage infection of marine bacteria in the thermoclinic level of the sea during the warm season; the experimental evidence has been reported by several authors e.g., in [12]). We have to describe the time evolution of three population densities assumed spatially homogeneous), namely the susceptible bacteria St), the phage-infected bacteria It) and the infecting agent: the phages Pt). The justication for the equations is given in the introductory Section 1. Here we recall that one goal of the authors in this paper is that of providing a better description of the infected class of bacteria with respect to a previous mathematical model see [4]), in which the infection was described by a system of three nonlinear odes. Here by T we denote the average latency time. In the previous model the time evolution of the density of infected bacteria was described by the o.d.e. dit) = St)Pt) It) 6.1) where, the lysis death constant rate of infected bacteria was just assumed to be =1=T. In the present model see Section 1 for its justication) we substitute the above ode with the delay dierential equation dit) = i It)+St)Pt) e it St T )Pt T ) 6.2) or with its integral form see 1.6)) It)= t t T e it ) S)P)d; 6.3) where i is a mortality rate constant, which accounts for possible extra-mortality besides lysis of infected bacteria. However, attention must be paid to the equivalence between 6.2) and 6.3), since any function It)=C + t t T e it ) S)P)d 6.4) where C R is a constant, of course satises 6.2). However, when St) S ;Pt) P as t +, then It) does not converge to I =kt )= i )S P as t + but to the value I + C. Since at t =0 C = I0) 0 T e i 1 ) 3 )d; 6.5) we see that if we apply initial conditions 2.2), then necessarily C = 0. Of course, all the mathematical results obtained in this paper assume initial conditions 2.2) for which C = 0 in 6.4) and 6.5). We still have to remark that the mathematical analysis of this model has been performed with i R +, i.e., i 0. The case i = 0 is in fact to be studied with care since the equivalence between dierential form 6.2) for I and its integral form 6.3) is lost. Assume i = 0. Eq. 6.2) gives di = St)Pt) St T)Pt T ) 6.6)