GLOBAL STABILITY OF A 9-DIMENSIONAL HSV-2 EPIDEMIC MODEL

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 9 Number 4 Winter 0 GLOBAL STABILITY OF A 9-DIMENSIONAL HSV- EPIDEMIC MODEL Dedicated to Professor Freedman on the Occasion of his 70th Birthday ZHILAN FENG ZHIPENG QIU AND ZI SANG ABSTRACT. This paper focuses on the global stability of a 9-dimensional epidemiological model for the transmission dynamics of HSV-. The model incorporates heterosexual interactions in which a single male population and two groups of female populations with different activity levels are considered. The method of global Lyapunov functions as well as the LaSalle Invariance Principle are used to show that the basic reproduction number provides a sharp threshold which completely determines the global dynamics of the model. That is in the case when the production number is less than or equal to one the disease-free equilibrium is globally asymptotically stable; whereas in the case when the reproduction number is greater than one a unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present. Introduction HSV- is a double-stranded DNA virus that almost exclusively infects the genital region and has been recognized as the most common cause of genital ulcer disease [. An estimated 6.% oraboutoneoutofsixamericans4to49yearsofagehavegenitalhsv- infection [5. In developing countries the prevalence of infection is as high as 40 60% [3. Due to the facts that the virus has a very high transmission rates and that a lifelong infection of its host is very common the prevalence of HSV- infections has had a great impact on human health globally [5. More significantly studies have suggested that the genital HSV- infection may facilitate HIV transmission [ This has motivated an extensive use of mathematical models for understanding the transmission dynamics of HSV- [ and references This work was supported by NSFC grants Nos and ). Keywords: HSV- global stability Lyapunov functions LaSalle Invariance Principle. Copyright c Applied Mathematics Institute University of Alberta. 39

2 30 Z. FENG Z. QIU AND Z. SANG therein). However to the best of our knowledge the literature on the mathematical modeling for the study of the epidemiological synergy between HSV- and HIV is rather scant. In order to better understand this synergy we have began to consider models that incorporate the interactions between HSV- and HIV [7. The full models are very complex and their analysis require better understanding of the reduced systems in simpler cases. For example in the case when HIV is absent the full model in [7 reduces to the following 9-dimensional system of differential equation: ds i = Λ i λ A i t)s i µ i S i da.) i = λ A i t)s i +γi LL i ωi A +θi A +µ i )A i dl i = ωi A +θi A )A i γi L +µ i )L i where i = mf f and λ t) = b m cβf A A f m + c)βf A N m f A f N f ).) λ A f t) = b m β f c N f λ A f t) = b m β f c) N f. More detailed explanations about the model formulation and assumptions are provided below. In the system.).) the female population is assumed to be divided into two groups one being the high-risk group based on their sexual behaviors e.g. female sex workers) and the other being he lowrisk group e.g. the general population of females). The sexual risk in the male population is assumed to be the same. The subscripts mf and f are used to denote the groups of males the low-risk females and the high-risk females respectively. We further assume that an HSV- infection can be described by two stages: acute infection denoted by A i i = mf f ) and latent infection L i ) and that the population is homogeneous within each group in the sense that individuals in each group have the same infectious period period of immunity contact rate

3 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 3 etc. Let S i denote the number of susceptible individuals in group i i = mf f ). Atransitiondiagrambetweentheepidemiologicalclasses is depicted in Figure. µisi µiai µili Λ i Si λ A i t)s i Ai ωi A +θi A)A i γi LL i L i FIGURE : Transfer diagram for the HSV- epidemiological model with treatment. WepointoutthatinFigure theinfectionrateλ t)forsusceptibles in the male population includes infections by contacts with females in both activity groups see.)). Upon becoming infected with HSV- the susceptibles in group i enter the infected class A i with acute HSV-. After the acute infection stage individuals enter the latent noninfectious) class L i. Periodically following an appropriate stimulus the latent individuals may become reactivated and enter the A i class again [3. The reactivation rate is γi L. It is assumed that there is a constant recruitment rate Λ i into the susceptible class and a per-capita natural deathrateµ i ingroupi. FinallytheantiviraltherapyagainstHSV-will be given to individuals with acute HSV- and the effective treatment rates for individuals in the A i class is denoted by θi A. In equations.) βim A βi ) i = f f are the HSV- transmission probability per partner between a female infected with acute HSV- in group i the male infected with acute HSV-) and a susceptible male the susceptible female in group i); b i i = mf f are the rates at which individuals in group i acquire a new sexual partner and c is the probability that a male individual would choose a female partner in group f. It then follows that the probability that a male individual would choose a female partner in group f is c. In the exhaustive survey the total number of the female partners in group i i = f f that the male individuals acquire should be equal to the total number of the male partners that the female individuals in group i i = f f acquire. These can be translated into the following balance conditions..3) b m cn m = b f N f b m c)n m = b f N f.

4 3 Z. FENG Z. QIU AND Z. SANG In order to ensure that constrains.3) are satisfied we assume that b m and c are constants and then b f and b f vary with N m N f and N f. The main objective of this paper is to establish rigorously a result for the global dynamics of the system.). Global stability is one of the important issues in the study of nonlinear dynamical systems. Due to the lack of generically applicable tools it is very difficult to prove the global stability especially for higher dimensional nonlinear systems. Although the application of theories in monotone dynamics system provides a helpful tool it is restricted to only cooperative or competitive systems [5. For more general nonlinear autonomous systems of ordinary differential equations a geometric approach has been developed by Li and Muldowney [9 which has been used successfully to prove the global stability of many epidemiological models [6 0 and references therein). The method of Lyapunov functions is most commonly used to prove the global stability of nonlinear dynamical systems. In this paper by using a class of global Lyaponuv functions we prove that the dynamics of system.) can be completely determined by the basic reproduction number R 0. That is if R 0 the disease-free equilibrium is globally asymptotically stable in the feasible region and the disease always dies out; whereas if R 0 > a unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present. Lyapunov functions of this type have been used in several early studies e.g. [0 5 7). More recent studies using Lyapunov functions include [3 4 in which a graph-theoretic approach to the method of global Lyapunov functions is used to analyze several classes of epidemic models. The present paper is arranged as follows. In the next section we provide a derivation of the basic reproduction number as well as the statement of the main result of this paper. The proof of the main result is given in Section 3. Section 4 includes a brief discussion of the current and some future work on modeling of joint dynamics of HSV- and HIV. Main results Let Γ = { SAL) : 0 S i A i L i S i +A i +L i Λ } i i = mf f µ i

5 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 33 where S = S f S f ) A i = A f A f ) and L i = L m L f L f ). Notice that the population sizes N i satisfy the equations dn i = Λ i µ i N i i = mf f. It follows that N i t) Λ i /µ i as t increases; and thus we know that the biologically feasible region Γ is positively invariant for the system.). Therefore in what follows we consider only solutions with initial conditions inside the region Γ in which the usual existence uniqueness of solutions and continuation results hold. One of the most important concepts in epidemiological models is the reproduction number for a pathogen. It is generally defined as the average number of secondary infections produced by a typical infected individual during the entire period of infection when introduced into a completely susceptible population [6. We now derive the basic reproduction number for HSV-. System.) always has the disease-free equilibrium Λm E 0 00 Λ f 00 Λ ) f 00. µ m µ f µ f Noting that the system.) has six variables representing individuals infected with HSV-: A i L i i = mf f it follows that using the notation of Driessche and Watmough [8) the matrices F A and V A corresponding to the new infection terms and the remaining transfer terms respectively) are given by where 0 Fmf A Fmf A F A = Ff 0 0 Ff 0 0 VA = Vm A Vf A V A f Fmf A = b mcβf A µ f Λ m m 0 Λ f µ m Fmf A = b m c)β A µ f Λ m f m 0 Λ f µ m ) ) bm cβ A Ff f 0 bm c)β m = Ff H mf A 0 m =

6 34 Z. FENG Z. QIU AND Z. SANG V A i = ω A i +θ A i +µ i γ L i and i = mf f. Note that ω A i +θ A i ) γl i +µ i V A i ) = ) γi L +µ i ζ i ωi A +θi A ζ i γi L ζ i ω A i +θ A i +µ i ζ i where ζ i = µ i γ L i +ωa i +θ A i +µ i ). Then the next generation matrix for HSV- denoted by K A can be expressed by where K A = F A V A ) 0 Fmf A Vf A ) Fmf A V A = Ff A V m) 0 0 Ff A V m) 0 0 γf L Fmf A Vf A ) = b m cβf A µ f Λ +µ f m m ζ f Λ f µ m f ) γ f ζ f 0 0 γf L Fmf A Vf A ) = b m c)βf A µ f Λ +µ f γ f m m ζ f ζ f Λ f µ m 0 0 Ff Vm) A = b m cβmf A γ L m +µ m ζ m Ff Vm) A = b m c)βmf A γ m ζ m 0 0 γm L +µ m ζ m γ m ζ m 0 0. Noticing that RankK A ) = and that the sum of the diagonal elements in matrix K A is zero it then follows from Vieta s formulas that the basic reproduction number for HSV- infection is given by.) R 0 := ρk A ) = E K A ) = R mf m +R mf m

7 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 35 where R mfm = R mfm = b m ) c βf βmf A Λ m µ f γm L +µ m µ m Λ f µ m γm L +ωm A +θm A +µ m ) γf L +µ f µ f γf L +ωf A +θf A +µ f ) b m ) c) βf βmf A Λ m µ f µ m Λ f γf L +µ f µ f γf L +ωf A +θf A +µ f ) γ L m +µ m µ m γ L m +ω +θ +µ m ) ) and E K A ) is the sum of all principal minors of K A of order. We will explain the biological means of R mfm R mfm and R 0 in the last section. Now we are able to state our main results of the paper. Theorem.. ) If R 0 then the DFE E 0 is the unique equilibrium of system.) and it is globally stable in Γ. ) If R 0 > then the DFE E 0 is unstable and system.) has a unique endemic equilibrium E A ml ms f A f L f S f A f L f ). Moreover E is globally stable in IntΓ. 3 Proof of the main result In the section the proof of Theorem. as well as several other results required for the proof of this theorem are presented. In order to prove Theorem. we first consider the following system: ds i = Λ i λ A i t)s i µ i S i da i = λ A i t)s i +γ L i L i ω A i +θ A i +µ i )A i 3.) dl i = ω A i +θ A i )A i γ L i +µ i )L i i = mf f λ t) = b m cβf A µ f m µ f A f + c)βf A Λ m A f f Λ f ) ; λ A f t) = b m βmf A c µ f ; Λ f λ A f t) = b m βmf A c) µ f. Λ f

8 36 Z. FENG Z. QIU AND Z. SANG For ease of notation let µ f β fm = b m cβf A µ f m β fm = b m c)βf A Λ m f Λ f β mf = b m cβ f µ f Λ f β mf = b m c)β f µ f Λ f ω i = ω A i +θ i i = mf f. Then the system 3.) can be rewritten as d d dl m ds f = Λ m β fma f β fma f µ m = β fma f +β fma f +γ m L m ω m +µ m ) = ω m γ m +µ m )L m = Λ f β mf S f µ f S f 3.) da f dl f = β mf S f +γ f L f ω f +µ f )A f = ω f A f γ f +µ f )L f ds f = Λ f β mf S f µ f S f da f = β mf S f +γ f L f ω f +µ f )A f dl f = ω f A f γ f +µ f )L f. We can easily see that systems 3.) and.) have the same basic reproduction number DFE and the positive equilibrium. For the systems.) and 3.) we have the following lemmas. Lemma 3.. If R 0 then system 3.) or system.)) has no positive equilibrium; if R 0 > then system 3.) or system.)) has a

9 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 37 unique positive equilibrium E A ml ms f A f L f S f A f L f ) where S i A i and L i i = mf f ) can be expressed as Λ i Si = µ i +Θ A i A i = Λ iγ L i +µ i ) µ i ω i +µ i +γ L i ) Θ A i µ i +Θ A i L i = Λ i ω i µ i ω i +µ i +γ L i ) Θ A i µ i +Θ A i Θ A f = β mf Λ m γm L +µ m ) µ m ω m +µ +γm) A µ m +Θ A m Θ Θ A f = β mf Λ m γm L +µ m ) Θ µ m ω m +µ +γm) A µ m +Θ A m and Θ is the unique root of the equation 3.3) Gχ) := R mf m + + β mf Λ m γm L +µ m ) µ m µ m ) ω m +µ +γm) A R mf m + + β mf Λ m γm L +µ m ) µ m µ m ) ω m +µ +γm) A ) χ µ f ) = 0. χ µ f Proof. The positive equilibria E Si A i L i ) of the system 3.) are solutions of the following equations: Λ i χ i S i µ i S i = 0 χ i S i +γ L i L i ω i +µ i )A i = 0 3.4) ω i A i γi L +µ i )L i = 0 χ m = β fma f +β fma f χ f = β mf χ f = β mf i = mf f.

10 38 Z. FENG Z. QIU AND Z. SANG From the first three equations in 3.4) we can easily obtain that 3.5) S i = Λ i µ i +χ i := S i χ i ) A i = Λ iγ L i +µ i ) µ i ω i +µ i +γ L i ) χ i µ i +χ i := A i χ i ) L i = Λ i ω i µ i ω i +µ i +γ L i ) χ i µ i +χ i := L i χ i ) where i = mf f. Substituting the expression for into the fifth and sixth equations in 3.4) yields that χ m χ f = β mf Λ m γm L +µ m ) µ m ω m +µ m +γm) L := χ f χ m ) µ m +χ m χ m χ f = β mf Λ m γm L +µ m ) µ m ω m +µ m +γm) L := χ f χ m ). µ m +χ m By substituting the expressions for A f and A f into the fourth equation in 3.4) we can obtain a self-consistent equation as follows χ m = β fma f χ f χ m ))+β fma f χ f χ m )). Rearranging the above equation we have Gχ m )χ m = 0 where the function Gχ m ) is defined in 3.3). We can easily verify that the function is a monotone decreasing function and G0) = R 0 ) lim Gχ m) =. χ m + Thus if R 0 the equation Gχ m ) = 0 has no solution on the interval Θ m 0 ) and it follows that the system 3.) has no positive equilibrium. This completes the proof of the first conclusion of Lemma 3.. If R 0 > it then follows from intermediate value theorem that the equation Gχ m ) = 0 has only one solution χ m = Θ m 0 ). Let Θ f = β mf Λ m γm L +µ m ) Θ m µ m ω m +µ m +γm) L µ m +Θ m

11 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 39 Θ f = β mf Λ m γm L +µ m ) Θ m µ m ω m +µ m +γm) L. µ m +Θ m Then by using equations 3.5) we obtain S i = L i = Λ i A i = Λ iγi L +µ i ) Θ i µ i +Θ i µ i ω i +µ i +γi L) µ i +Θ i Λ i ω i µ i ω i +µ i +γ L i ) Θ i µ i +Θ i i = mf f and Si A i and L i i = mf f satisfy the equation 3.4) i.e. the system 3.) has a unique equilibrium E Si A i L i ). This completes the proof of Theorem 3.. Lemma 3.. If R 0 then the DFE E 0 of system 3.) is globally stable in Γ. Proof. Noting that the next generation matrix K A is non-negative it then follows from Perron-Frobenius theory of positive matrices that there exists a positive eigenvector c c...c 6 ) where c i > 0 i =...6 such that c c...c 6 )K A = ρk A )c c...c 6 ). Since K A = F A V A ) direct computation yields c c...c 6 )V A = ρk A ) c c...c 6 )F A. Define the following Lyapunov function L = c +c L m +c 3 A f +c 4 L f +c 5 A f +c 6 L f. The derivative of L along solutions of system 3.) is dl = c c c 3 c 4 c 5 c 6 ) 3.) Λ m β fm Λ m A f +β fm A f +γ m L m ω m +µ m ) µ m µ m ω m γ m +µ m )L m Λ f β mf +γ f L f ω f +µ f )A f µ f ω f A f γ f +µ f )L f Λ f β mf +γ f L f ω f +µ f )A f µ f

12 330 Z. FENG Z. QIU AND Z. SANG c c c 3 c 4 c 5 c 6 ) Λm β fma f µ m ) +β fma f Λm 0 β mf Λf µ f S f 0 β mf Λf µ f S f 0 ) ) µ m ) = c c c 3 c 4 c 5 c 6 )F A V A ) L m A f L f A f L f ) T )) ) Λm Λm c β fma f +β fma f µ m µ m )) Λf Λf c 3 β mf S f )) c 5 β mf S f µ f µ f ρk A ) )c c c 3 c 4 c 5 c 6 )V A L m A f L f A f L f ) T = ρk A) ) c c c 3 c 4 c 5 c 6 )F A L m A f L f A f L f ) T ρk A ) 0 since R 0 = ρk A ). Therefore all limit points are contained in the largest invariant subset K of G = { L m S f A f L f S f A f L f ) L = 0}. We can easily see that L = 0 if and only if S i = Λ i /µ i A i = 0 L i = 0 i = mf f. Therefore the only one compact invariant subset of the set where L = 0 is the singleton {E 0 }. By the LaSalle Invariance Principle E 0 is globally stable in Γ if R 0. This completes the proof of Lemma 3.. Lemma 3.3. If R 0 > then a unique endemic equilibrium E A m L ms f A f L f S f A f L f ) of system 3.) is globally stable in IntΓ.

13 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 33 Proof. If R 0 > it then follows from Lemma 3. that the system 3.) hasauniquepositiveequilibriume A ml ms f A f L f S f A f L f ) and S i A i L i i = mf f satisfy the following equations: Λ m = β fma f +β fma f +µ m 3.6) Λ f = β mf A ms f +µ f S f Λ f = β mf A ms f +µ f S f and 3.7) β fma f Sm +β fma f Sm = µ mω m +γ m +µ m ) L m ω m = µ mω m +γ m +µ m ) γ m +µ m A m β mf A ms f = µ f ω f +γ f +µ f ) ω f L f = µ f ω f +γ f +µ f ) γ f +µ f A f β mf A ms f = µ f ω f +γ f +µ f ) ω f L f = µ f ω f +γ f +µ f ) γ f +µ f A f. Now let us define the following Lyapunov function V = where i=mf f d i [ S i Si lns i ) +A i A i lna i )+ γ i L i L i lnl i ) γ i +µ i d m = β mf S f A mβ mf S f A m d f = β fma f β mf S f A m

14 33 Z. FENG Z. QIU AND Z. SANG d f = β fma f β mf S f A m. Differentiating V along solutions to 3.) using 3.6) and 3.7) and collecting terms we obtain 3.8) dv = d m [µ m Sm 3.) S ) m Sm S m +β fma f S Sm m +β fma f β fma f β fma f +β fma f +β fma f Sm µ mγ m +ω m +µ m ) γ m +µ m A m A m A m β fm A f β fma f γ m L m +ω m +µ m )A m L γ m ω m m +γ m L m γ m +µ m L m [ +d f µ f Sf S ) f Sf S f +β mf A S msf f Sf β mf A Sf m +β mf Sf S f µ f ω f +γ f +µ f ) ω f +µ f A f β mf S f A f A f A f γ f L f +ω f +µ f )A f A f L γ f ω f A f f +γ f L f γ f +µ f L f +d f [ µ f S f S f S f S f S f ) +β mf A ms f Sf β mf A Sf m +β mf Sf S +µ f Sf f µ f ω f +γ f +µ f ) ω f +µ f A f β mf S f A f A f

15 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 333 A f γ f L f +ω f +µ f )A f A f L γ f ω f A f f +γ f L f γ f +µ f L f d m [ β fma f +β fma f β fma f β fma f +β fma f +β fma f µ mγ m +ω m +µ m ) A m β fm A f γ m +µ m A m A m β fma f γ m L m +ω m +µ m )A m L γ m ω m m +γ m L m γ m +µ m L m +d f [β mf A msf Sf β mf A Sf m +β mf Sf S f µ f ω f +γ f +µ f ) ω f +µ f A f β mf S f A f A f A f γ f L f +ω f +µ f )A f A f L γ f ω f A f f +γ f L f γ f +µ f L f +d f [β mf A msf Sf β mf A Sf m +β mf Sf S f µ f ω f +γ f +µ f ) ω f +µ f A f β mf S f A f A f A f γ f L f +ω f +µ f )A f A f L γ f ω f A f f +γ f L f γ f +µ f L f

16 334 Z. FENG Z. QIU AND Z. SANG since S i S i + Si S i i = mf f with equality holding if and only if 3.9) S i = S i. Rearranging the terms we have dv 3.0) d m [β fma f Sm +β fma f Sm 3.) µ [ mγ m +ω m +µ m ) +d f β mf Sf γ m +µ m µ f ω f +γ f +µ f ) A f ω f +µ f [ +d f β mf Sf µ f ω f +γ f +µ f ) A f ω f +µ f +d m [β fma f +β fma f +ω m +µ m )A m +γ m L m γ m L m A m L m γ m ω m S γ m +µ m L mβ fma Sm f m β fma f β fm A f A m β fma f A m +d f [β mf A ms f +ω f +µ f )A f +γ f L f Sf β mf A Sf A f m β mf S f A m S f A f γ f L f L γ f ω f A f f A f γ f +µ f L f A f +d f [β mf A ms f +ω f +µ f )A f +γ f L f S f β mf A m S f S f β mf S f A f A f

17 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 335 By using 3.7) we obverse that A f γ f L f L γ f ω f A f f A f γ f +µ f L f d m β fma f = d f µ f ω f +γ f +µ f ) ω f +µ f A f d m β fma f = d f µ f ω f +γ f +µ f ) ω f +µ f A f d f β mf S f +d f β mf S f = d m µ m γ m +ω m +µ m ) γ m +µ m. Consequently the sum of the first three terms in 3.0) are canceled and by rearranging terms it then follows that [ dv 3.) d m γ m L m L ma ) m ω m L m γ m +µ m L m. +β fma f +β fma f +ω m +µ m )A m γ m L m β fma f β fma f β fm A f A m β fma f A m +d f [ γ f L f L f A f L f A f ω f γ f +µ f A f L f ) +β mf A msf +ω f +µ f )A f γ f L f Sf β mf A Sf A f m β mf S f A m S f A f +d f [ γ f L f L f A f L f A f ω f γ f +µ f A f L f ) +β mf A msf +ω f +µ f )A f γ f L f Sf β mf A Sf A f m β mf S f A m. S f A f Noting that A i /L i = γ i +µ i )/µ i i = mf f we can easily see that L i A i L i A + ω i A i i = mf f i γ i +µ i L i

18 336 Z. FENG Z. QIU AND Z. SANG with equality holding if and only if 3.) L i = ω i γ i +µ i A i. By using 3.7) and collecting terms we have [ dv d m β fma f Sm +β fma f Sm Smβ fma Sm f 3.) since and β fma f β fm A f A m β fma f A m +d f [β mf A msf Sf β mf A Sf A f m β mf S f A m S f +d f [β mf A msf Sf β mf A Sf m = d m β fma f 0 +d m β fma f 4 S m A f S f β mf S f A f A f A f A m SmA S f S f A ) f f S f Sf A ma f 4 S m A f A m A f SmA S f S f A ) f m S f Sf A ma f + A f A m A f + S f S f + S f A f S f A ma f 4 + A f A m A f + S f S f + S f A f S f A ma f 4 with equality holding if and only if 3.) Therefore = A f A m A f = S f S f = S f A f S f A ma f = A f A m A f = S f S f = S f A f S f A ma f. dv 0 3.)

19 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 337 for all L m S f A f L f S f A f L f ) IntΓ and that dv/ 3.) = 0 if and only if 3.9) 3.) and 3.) hold. From 3.9) 3.) and 3.) direct derivation yields that A = A f m A = A f f A = L m f L = L f m L = L f f L := ξ. f Therefore dv/ 3.) = 0 if and only if S i = S i A i = ξa i L i = L i i = mf f. Substituting S i = S i A i = ξa i L i = ξl i i = mf f into the first fourth and seventh equations of system 3.) we obtain 0 = Λ m ξβ fma f Sm ξβ fma f Sm µ m Sm 0 = Λ f ξβ mf A msf µ f Sf 0 = Λ f ξβ mf A msf µ f Sf. It then follows that ξ =. Therefore the only compact invariant subset of the set where dv/ 3.) = 0 is the singleton {E }. By the LaSalle Invariance Principle E is globally stable in IntΓ if R 0 >. This completes the proof of Lemma 3.3. We are now ready to provide the proof of Theorem.. Proof of Theorem.. Adding the equations of system.) gives dn i = Λ i µ i N i where i = mf f. The asymptotic equilibrium values for the N i are N i t) Λ i /µ i as t +. Thus system.) can be reduced to the limitingsystem3.) i.e. system3.). IfR 0 thedfee 0 ofsystem 3.) is globally asymptotically stable in Γ. By using Corollary 4.3 in paper [6 the DFE E 0 of system.) is also globally asymptotically stable is Γ. Similarly if R 0 > system.) has a unique positive equilibrium which is globally asymptotically stable in IntΓ since the unique positive equilibrium E of system 3.) is globally asymptotically stable in IntΓ if R 0 >. This completes the proof of Theorem..

20 338 Z. FENG Z. QIU AND Z. SANG 4 Discussion In this article we studied the asymptotical behavior of the 9-dimensional HSV- model.) which is a reduced system of a model for the joint disease dynamics of HIV and HSV-. The result presented in this paper will be critical for the analysis of the HIV/HSV- model which will be presented in another paper. By employing the method of global Lyapunov functions we provided a complete classification of the dynamics see Theorem.) for the model.). More specifically it is shown that the basic reproduction number R 0 completely determines whether the disease will die out which occurs when R 0 due to the global stability of the disease-free equilibrium E 0 ) or persist which occurs when R 0 > due to the global stability of the endemic equilibrium E ). Another main result of this study is the derivation of the basic reproduction number R 0 for the model.). Due to the complexity of the model the formula for R 0 also has a complicated expression: where R 0 = R mf m +R mf m R mf m = b m ) c β A f mβ f Λ m µ m µ f Λ f γ L m +µ m ) µ m γ L m +ω +θ +µ m ) γ L f +µ f µ f γ L f +ω A f +θ A f +µ f ) R mf m = b m ) c) β A f mβ f Λ m µ m µ f Λ f γ L m +µ m ) µ m γ L m +ω +θ +µ m ) γ L f +µ f µ f γ L f +ω A f +θ A f +µ f ). It is helpful to provide a detailed explanation for the biological meaning of the three quantities R mfm R mfm and R 0. We can easily see that the average time units in compartment I i that an infected individual in group i i = mf f spends can be expressed as τ i = ωi A +θi A + ωa i +θi A γi L +µ i ωi A +θi A +µ i γi L +µ i ω A + i +θi A γ L ) i ωi A +θi A +µ i γi L + +µ i

21 A 9-DIMENSIONAL HSV- EPIDEMIC MODEL 339 = ω A + i +θi A ωi A +θi A +µ i γ L i +µ i µ i γ L i +ω A i +θ A i +µ i ). γ L i γ L i +µ i ) k + ) From system.) in the absence of infection the numbers of the susceptible males and females in group i i = f f are Λ m /µ m and Λ i /µ i respectively. Under these conditions the average number of infected females in groups f and f which are transmitted by per infected male per unit of time can be defined by n mf := b m cβmf A τ m = b m cβmf A γm L +µ m µ m γm L +ωm A +θm A +µ m ) n mf := b m c)βmf A τ f = b m c)βmf A γm L +µ m µ m γm L +ωm A +θm A +µ m ) respectively. Similarly the average number of the infected males transmitted by per infected female in groups f and f can be defined by n fm := b f β A f mτ f = b m c n fm := b f β A f mτ f = b m c Λ m µm Λ f βf µ f Λ m µm Λ f βf µ f γf L +µ f µ f γf L +ωf A +θf A +µ f ) γf L +µ f µ f γf L +ωf A +θf A +µ f ) respectively. Thus the number of secondary infected male cases generated by one infected male being transmitted through the females in group f or f can be expressed as R mfm = n mf n fm = b m ) c) βmf A βf A Λ m µ f γm L +µ m m µ m Λ f µ m γm L +ωm A +θm A +µ m ) = R mf m R mfm = n mf n fm γf L +µ f µ f γf L +ωf A +θf A +µ f )

22 340 Z. FENG Z. QIU AND Z. SANG = b m ) c) βmf A βf A Λ m µ f γm L +µ m m µ m Λ f µ m γm L +ωm A +θm A +µ m ) = R mf m. γf L +µ f µ f γf L +ωf A +θf A +µ f ) Consequently the total number of secondary infected male cases generated by one infected male being transmitted through the females in group f and f can be expressed as R 0 = R mf m +R mf m = R 0 ). Briefly R mfm) and R mfm) can be understood as the numbers of secondary infected males transmitted through the females in groups f and f respectively and R 0 ) denotes the total number of secondary infected male cases generated by one infected males when the system has no infected cases and is in balance. If R 0 > i.e. R 0 ) > the second conclusion in Theorem. implies that the HSV- will prevail since an infective males will be replaced with greater than one new infected case. If R 0 i.e. R 0 ) it follows from the first conclusion in Theorem. that the HSV- will be likely to fade out since an infective male will be replaced with less than one new infected male case. In this paper we have considered only the simpler case in which the female population is divided into two risk-dependent groups while the male population has a single activity level. It would be interesting to analyze a more general model in which both the female and the male populations have multiple groups based on their sexual behaviors. We conjecture that the results obtained in this paper will still hold for the general model. These results will be very helpful for our studies of models dealing with the synergy between HSV- and HIV to investigate the potential population-level impact of HSV- therapy on HIV dynamics and control. REFERENCES. S. M. Blower and H. Dowlatabadi Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model as an example Internat. Stat. Rev ) 9 43.

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24 34 Z. FENG Z. QIU AND Z. SANG 4. R. B. Schinzai Strategies to control the genital herpes epidemic Math. Biosci ) H. L. Smith Monotone Dynamical Systems: An Introduction to Theory of Competitive and Cooperative Systems Math. Surveys Monogr. 4 AMS Providence RI H. R. Thieme Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations J. Math. Biol ) H. R. Thieme Asymptotically autonomous differential equations in the plane Rocky Mountain J. Math ) P. van den Driessche and J. Watmough Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Math. Biosci ) A. Wald and K. Link Risk of human immunodeficiency virus infection in Herpes simplex virus type -seropositive persons: a meta-analysis J. Infect. Dis ) Department of Mathematics Nanjing University of information Science and Technology Nanjing 0044 P R China. Department of Mathematics Purdue University West Lafayette IN USA. address: zfeng@math.purdue.edu Department of Applied Mathematics Nanjing University of Science and Technology Nanjing 0094 P R China. Department of Mathematics Purdue University West Lafayette IN USA. address: nustqzp@mail.njust.edu.cn Department of Applied Mathematics Nanjing University of Science and Technology Nanjing 0094 P R China.