The effects of females' susceptibility on the coexistence ... - Springer Link

J. Math. Biol. (1997) 35: 503—522

The effects of females’ susceptibility on the coexistence of multiple pathogen strains of sexually transmitted diseases Carlos Castillo-Chavez1, Wenzhang Huang2,*, Jia Li2,* 1 Biometrics Unit, Center for Applied Mathematics, Mathematical Science Institute, Cornell University, Ithaca, NY 14853, USA 2 Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA Received 25 July 1995; received in revised form 6 May 1996

Abstract. We study the dynamics of sexually transmitted pathogens in a heterosexually active population, where females are divided into two different groups based on their susceptibility to two distinct pathogenic strains. It is assumed that a host cannot be invaded simultaneously by both disease agents and that when symptoms appear — a function of the pathogen, strain, virulence, and an individual’s degree of susceptibility — then individuals are treated and/or recover. Heterogeneity in susceptibility to the acquisition of infection and/or in variability in the length of the infection period of the female subpopulations is incorporated. Pathogens’ coexistence is highly unlikely on homogeneously mixing female and male populations with no heterogeneity among individuals of either gender. Variability in susceptibility in the female subpopulation makes coexistence possible albeit under a complex set of circumstances that must include differences in contact/mixing rates between the groups of females and the male population as well as differences in the lengths of their average periods of infectiousness for the three groups. Key words: Sexually transmitted disease — Pathogen strains — Coexistence

1 Introduction A central question in evolutionary biology, still demanding a satisfactory explanation, focuses on the evolutionary need/advantage of sexual versus asexual reproduction. A quick glance at the literature on the evolution of sexual reproduction reveals (not surprisingly) a tremendous amount of ————— * The work was completed while these authors were visiting the Biometrics Unit and the Center for Applied Mathematics, Cornell University.

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C. Castillo-Chavez et al.

interest and research activity on this question among theoretical and field biologists (see Maynard Smith, 1978). Most theoretical studies on the importance of sexual reproduction have been carried out within the field of theoretical population genetics and with the help of mathematical models that must necessarily incorporate mating systems at the level of the gene. This article begins with the obvious — but often ignored — assumption that the evolution of sexually transmitted diseases (STDs) among biological organisms must be closely linked to the evolution of sexual reproduction and, therefore, diseases that are transmitted through some type of sexual activity cannot be systematically studied without frameworks and models that include males and females. In previous papers (Castillo-Chavez et al., 1993, 1995), we have analyzed S-I-S STD models with multiple competing strains in an exclusively heterosexually active population and concluded that in a behaviorally and genetically homogeneous population coexistence is not possible except under very special and nongeneric circumstances. Our analysis under these assumptions is complete; that is, we have provided the global stability analysis of the stationary states for models where a host faces any number competing strains. We have also partially analyzed a model extension to deal with STD dynamics in a two-sex population that is also stratified by the individuals’ infection stages. This level of stratification does not increase the level of heterogeneity needed to rule out competitive exclusion. Biologists have been concerned with evolutionary interactions that result from changing host and pathogen populations. Advances in evolutionary biology, behavior, and social dynamics have brought to the forefront of research the importance of a multitude of factors that not only influence disease dynamics but also play a role on the evolution of virulence (Anderson and May, 1991; Blythe et al., 1993; Brauer et al., 1992; Castillo-Chavez and Hethcote et al., 1988, 1989; Ewald, 1993, 1994; Hadeler and Castillo-Chavez, 1994; Hsu-Schmitz, 1993). Host-vector interactions such as those observed in the myxoma-rabbit system (Levin and Pimentel, 1981; Levin, 1983a, 1983b; May and Anderson, 1983, 1990) argue against pathogen evolution towards reduce virulence while providing rich systems for the study of coexistence and coevolution. A useful view within the context of host-parasite systems is to think of susceptible hosts as patches available for colonization by infectious pathogens. Hence it is possible to pose general questions such as: What are the possible outcomes of coevolutionary races when different strains of the same pathogen compete for the same patches? When is competitive exclusion the rule? What happens if the quality or desirability of a patch changes over time? Mathematical models and field studies have begun to yield useful results and have helped us formulate new paradigms on which we can study the outcomes of coevolution (see Anderson and May, 1982; 1991; Beck, 1984; Bremermann and Pickering, 1983; Bremermann and Thieme, 1989; Castillo-Chavez and Hethcote et al., 1988, 1989; Dietz, 1979; Dwyer et al., 1990; Fenner and Myers, 1978; Fenner and Ratcliffe, 1965; Levin, 1983a, 1983b; Levin and Pimentel, 1981; May and Anderson, 1983).

Sexually transmitted diseases

505

This paper focuses on the dynamics of sexually transmitted pathogens in a heterosexually active population, where females are divided into two groups based on their susceptibility to infection (colonization) by two distinct pathogenic strains of an STD. It is assumed that a host cannot be invaded simultaneously by both disease agents (that is, there is no superinfection) and that when symptoms appear — a function of the pathogen, strain, virulence, and an individual’s degree of susceptibility — then individuals are treated and/or recover. Heterogeneity in susceptibility to the acquisition of infection and/or in variability in the length of the infection period of female populations is incorporated in an expanded two-sex model. The presence of only a homogeneous female group and a homogeneous male group make disease coexistence impossible (Castillo-Chavez et al., 1993, 1995). Variability in susceptibility in female subpopulations (two groups) makes coexistence possible albeit under a complex sets of circumstances that must take into account differences in contact/mixing rates between females and males as well as differences in the lengths of their average periods of infectiousness. This manuscript is organized as follows: Sect. 2 introduces our model and simplifies it using some recent results on asymptotically autonomous epidemic models (Castillo-Chavez and Thieme, 1994; Thieme, 1992, 1994). The necessary thresholds are computed and the stability of the infection-free state is studied in Sect. 3. A principle of competitive exclusion for SIS models with homogeneous mixing is established in Sect. 4. Section 5 provides our coexistence results. In Sect. 6, we discuss the consequences of our results and outline some future work.

2 Model description and the generated monotone flow The use of differential equations for STD models began with Ross in 1911. Ross introduced a differential equation model for the transmission dynamics of vector-transmitted diseases which, as he recognized, was formally equivalent to a model for the transmission dynamics of STDs. Ross’ theoretical work was driven by his efforts to develop management strategies for the control of malaria. The formulation of the first explicit STD model, a gonorrhea model, is due to Cooke and Yorke (1973). An important observation made by Ross is that the average total rate of contacts between host and vectors must be conserved (Ross, 1911, p. 667). This simple conservation law of contact rates has become the basis for modeling heterogeneous contact structures (Busenberg and Castillo-Chavez, 1989, 1991; Castillo-Chavez, 1989). We use it in this section. A common but limiting assumption is that the sizes of the interacting sexually active populations are constant (Lajmanovich and Yorke, 1976; Hethcote and Yorke, 1984; and references therein). Variable population size may seriously impact the qualitative dynamics of epidemic models (CastilloChavez and Cooke et al. 1989; Huang et al., 1992). Here it is assumed that the population under consideration does not experience disease induced

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mortality. We also assume that the recruitment of new hosts (all susceptible) occurs at a constant rate and, consequently, the total population sizes of males and females (both groups) become asymptotically constant. It is therefore possible to replace the ‘‘real’’ model with an asymptotically autonomous limiting system (see Castillo-Chavez and Thieme, 1994; Thieme, 1992, 1994). Consequently, we implicitly assume (as Lajmanovich and Yorke, 1976) that social dynamics do not play any role on the qualitative dynamics of the model. This is a strong assumption/limitation which is justified, in part, because our efforts are directed to the study of the dynamics of two competing strains in a minimally heterogeneous population. To be explicit, we consider an S-I-S STD model for a heterosexually active population. The population consists of susceptibles and infecteds. Among the population, there are two different groups of female individuals in the transmission of diseases, denoted by superscripts f and c, based on their susceptibility, which are determined by their sexual behavior, genetics, or other factors. We assume that the infecteds are divided into 2 classes based on the pathogen strains in their body, and that susceptibles infected by infecteds with a certain pathogen strain have the same pathogen strain. We use Sk, k"m, f, c, to denote the susceptible males, susceptible females in two different groups, and use I k, k"m, f, c, i"1, 2, to denote the infecteds with strain i. Then the i dynamics of the spread of the disease are governed by

where

G

2 SQ k"Kk!Bk!kkSk# + c kI k , i i k"m, f, c, i"1, 2 , i/1 IQ k"B k!( kk#c k )I k , i i i i   Bmi"rm(¹ m, ¹ f, ¹ c)Sm    Bli"rl(¹ m, ¹ f, ¹ c)Slbmi

Ic If bf i #bc i , i ¹c i ¹f Im i , l"f, c , ¹m

A

with the constraint

(2.1)

B

2 Bk" + Bk , k"m, f, c , i i/1

rm(¹ m, ¹ f, ¹ c)¹ m"r f(¹ m, ¹ f, ¹ c)¹ f#rc(¹ m, ¹ f, ¹ c) ¹ c .

(2.2)

Here Kk denote the input flow (recruitment) entering the sexually active subpopulations; 1/kk are the average sexual life span for people in group k; ck are the rates of recovery; ¹ k"Sk#+2 I k are the total number of males, i i/1 i females in group f and group c, respectively; rk, as functions of ¹ m, ¹ f, and ¹ c, are the numbers of partners per individual per unit of time; and b k are the i rates of infection. The constraint indicates that the total number of female sexual partners of males per unit of time and the total number of male partners of females per unit of time given the current availability of partners must be balanced.


G

507

Since ¹ k"Sk#+ 2 Ik , (2.1) is equivalent to i/1 i ¹Q k"Kk!kk¹ k, k"m, f, c , Ic 2 If IQ m"!(km#cm )Im#rm(¹ m, ¹ f, ¹ c) ¹ m! + Im bf j #bc i , i i i j j ¹c j ¹f j/1 (¹l!+ 2 Il )Im j/1 j i l"f, c . IQ l"!(kl#cl )Il#rl(¹ m, ¹ f, ¹ c)bm i i i i ¹m (2.3)

A

BA

B

The asymptotic equilibrium values for ¹ k are ¹ k"Kk/kk. If we define bk " : rk(Km/km, Kf/kf, Kc/kc) ,

then it follows from the constraint (2.2) that bmKm bfKf bcKc " # , km kf kc as tPR. The limiting system of (2.1) or (2.3) is then given by the following set of equations

A

Km 2 IQ m"!(km#cm )Im#bm ! + Im i i i j km j/1

BA B

B

kf kc bf I f#bc Ic , i Kf i i Kc i

(2.4) kmblbm Kl 2 i ! + Il Im , l"f, c . IQ l"!(kl#cl )Il# i i j i i Km kl j/1 The dynamics of (2.1) or (2.3) can be qualitatively determined by those of (2.4). (See Castillo-Chavez and Thieme, 1994; Castillo-Chavez et al., 1993, and Thieme, 1992, 1994.) We will then investigate (2.4) hereafter instead of (2.1) or (2.3). To simplify the notation, we define the following quantities.

A

: ( kk#ck ), pk " i i

Kk pk " : , kk

bmbf i, : amf " i pf

bmbc i, : amc " i pc

bcb m bfb m i . i , and acm " : a fm " : i i pm pm System (2.4) can then be rewritten as

A

B

2 IQ m"!pm Im# pm! + Im (amf I f#amc Ic ) , j i i i i i i i j/1 2 IQ f"!pf I f#a fm pf! + I f I m , j i i i i i j/1 2 IQ c"!pc I c#acm pc! + I c I m . i j i i i i j/1

A A

B

B

(2.5)

508


If we now let R6 " : M(I m , I f, I c , I m, If, I c ); Ik70N , ` 1 1 1 2 2 2 i and define the subset X of R6 by `

G

H

2 X" : (I m , I f, I c , I m, If, I c ) 3 R6 ; + I k6pk , 1 1 1 2 2 2 ` i i/1 we then observe that the flow generated by (2.5) is positively invariant on X. Furthermore, the flow is monotone under the special order given below (see Castillo-Chavez et al., 1993). Definition 2.1. ¸et K"Mx"(x , . . . , x )3 R6; x 70, i"1, 2, 3, x 60, 1 6 i j j"4, 5, 6N. A type K order, denoted by ‘‘6 ’’, is defined in such a way that K x6 y if and only if x!y 3 K . K

(2.6)

Using this order, it is easy to see that the flow generated by (2.5) is monotone. Theorem 2.2. ¸et I"(I m , I f, I c , I m, If, I c ) and let I(t, I ) be a solution of (2.5) 0 1 1 1 2 2 2 with I(0, I )"I . ¹hen 0 0 I(t, Ia )6 I(t, Ib ), t70 , 0 0 K if Ia , Ib 3X and Ia 6 Ib . 0 K 0 0 0 Proof. Let Q"diag(q ) with q "q "q "1, q "q "q "!1. Then the i 1 2 3 4 5 6 matrix QJ (I)Q has nonnegative off-diagonal elements for every I3 X, where J(I) is the Jacobian matrix of (2.5) evaluated at I. It follows from Lemma 2.1 in Smith (1988) that the flow I(t, I ) preserves a type K order on X; that is, the 0 flow is monotone under this type K order.

3 Thresholds The linearization about the infection-free equilibrium of System (2.5) is

ABA

IQ m !pm pmamf pmamc i i i i f 0 " pfa fm !p f IQ i i i 0 !pc pcacm IQ c i i i

BA B A B Im i If i Ic i

Im i " : P I f , i"1, 2 . i i Ic i

(3.1)

System (3.1) consists of two decoupled systems of three equations. The diagonal elements of !P , i"1, 2, are positive and their off-diagonal elei ments are nonpositive. Hence, !P are M-matrices. It is easy to check that if i pmpfamfa fmpc#pmpcamcacmp f(pmp fpc , i i i i i i i i i


509

for both i"1 and 2, then the leading principal minors of both !P , i"1, 2, i are positive and hence it follows from M-matrix theory (see e.g. Berman et al., 1989) that the infection-free equilibrium is locally stable. If there exists i"1, or 2, such that the determinant of !P is negative, that is, i pmpfamfa fmpc#pmpcamcacmp f'pmp fpc , i i i i i i i i i the infection-free equilibrium is unstable. Define the reproductive number, R , in the ith subgroup by i pmpfamfa fmpc#pmpcamcacmp f i i i i i i R " : i pmp fpc i i i ( kc#cc ) bfb f#( kf#cf ) bcb c i i i i. "bmbm (3.2) i ( km#cm )( kf#c f )( kc#cc ) i i i We can now make the following observations. If R (1, (I m , I f, I c )P(0, 0, 0), i i i i i"1, 2. If R (1, for both i"1 and 2, then the infection-free equilibrium is i stable; that is, R (1, for both i"1 and 2, leads to the extinction of the i disease in the population. If there exists at least one strain such that R '1 i then (I m , I f, I c ) P(0, / 0, 0), that is, the disease will spread in the population. i i i Define bmb m i , Rm " : i km#c m i Then, consequently,

bfbf i , Rf " : i kf#c f i

bcb c i . Rc " : i kc#c c i

(3.3)

R "R m (R f#R c ) . i i i i Here the superscripts characterize the reproduction numbers in the three different groups respectively. Following the approach in Castillo-Chavez et al. (1993), it can be shown that the infection-free equilibrium and the boundary equilibrium (given explicitly in the next section) of this model are globally stable under the appropriate threshold conditions. Here we only state the results and omit the details as the approach follows directly from the work in Castillo-Chavez et al. (1993). Lemma 3.1. ¸et E "(I m , I f, I c , 0, 0, 0) and E "(0, 0, 0, I m , I f, I c ) be equili1 1 1 1 2 2 2 2 bria of (2.5), where I m , I f , I c'0, if R '1, and I m"I f"I c"0, if R 61. ¸et i i i i i i i i m1"( pm, pf, pc, 0, 0, 0) and m2"(0, 0, 0, pm, pf, pc). ¹hen lim I(t, mi)"E , i"1, 2 . i t?= Theorem 3.2. ¸et the reproductive number R for each group be defined in (3.2). i If R 61 for both i"1, 2, then the epidemic goes extinct regardless of the initial i levels of infection. If, on the other hand, R '1 for i"1 or i"2 then the i epidemic spreads in the population.

510


4 Competitive exclusion There exist two types of endemic equilibria of the model, one consisting of only one nonzero triple (Ik'0, Ik"0, j9i) and the other having all positive j i components. We call the first type boundary equilibria and the second type coexistence equilibria.

4.1 Existence of boundary equilibria The boundary equilibrium always exists whenever the epidemic spreads in the population. We collect this result as follows. Theorem 4.1.1. Assume that R '1, i"1, 2. ¹hen the boundary equilibria i (Sk'0, Ik'0, Ik"0, j9i) exist. i j Proof. We need to solve pmI m"(pm!I m )(amfI f#amc I c ) , i i i i i i i p f I f"afm (pf!I f )I m , i i i i i pcI c"acm ( pc!I c )I m , i i i i i

(4.1.1a) (4.1.1b) (4.1.1c)

for I k , 0(I k(pk. i i First, we solve (4.1.1b) and (4.1.1c) to get pfafmI m i i , I f" i f p #afmI m i i i Substituting (4.1.2) into (4.1.1a) yields

pcacm I m i i . I c" i p c#p cmI m i i i

(4.1.2)

pfamf afm pcamc acm pm i i # i i i ! "0 . (4.1.3) f fm m m p #a I p c#acm I m pm!I i i i i i i i Define the left hand side of (4.1.3) as f (I m ). It is easy to check that f @(I m )'0 i i and that lim m m f (I m )"#R. Consequently, (4.1.3) has a unique solution i Ii ?p 0(I m(pm if and only if f (0)(0. Since i pfamfa fm pcamcacm pm pm i i # i i " i (1!R ) , f (0)" i ! i f c p p pm pm i i it follows that there exists a unique positive solution I m of (4.1.3) if and only if i R '1. This unique positive I m uniquely determines positive I f and I c via i i i i (4.1.2).

A

A

B

B

Remark. The components I m of the boundary equilibrium, or the solution of i (4.1.3), can be represented by the following explicit formula JK2!4K K !K 2 1 3 2, I m" i 2K 1


where

511

K "afmacm (p m#bm(b f#b c )) , 1 i i i i i K "p m (acm p f#afm p c )#pfa mf a fmp c#pcamcacm p f 2 i i i i i i i i i i i !pm(pfamfa fmacm#pca mca fmac ) i i i i i i bm p mp fp c b m i (Rc!R )# i (R f!R )#R , " i i i i i i bf i bc i pm i i K "p mp fp c (1!R ) . 3 i i i i

A

B

4.2 Stability of boundary equilibria The Jacobian at the equilibrium (Ik "0, Ik '0) has the form 1 2 A 0 , J" 11 A A 21 22 where

A

B

A

B

!p m!d (amf If#amcIc ) amf (pm!I m ) amc (pm!I m ) i 2i 2 2 2 2 i 2 i 2 a fm (pf!I f ) !p f!d a fmI m 0 A " : , i 2 i 2i 2 2 ii acm (pc!I c ) 0 !p c!d acm I m i 2 i 2i 2 2 i"1, 2 ,

A

B

!(amfI f#amcI c ) 0 0 2 2 2 2 A " 0 !a fmI m 0 , 21 2 2 0 0 !acmI m 2 2 with d being the Kronecker delta function. ij In order to show that the matrix A is always locally stable, we need to 22 establish the following lemma. Lemma 4.2.1. ¸et A be an n]n irreducible matrix with all off-diagonal elements non-negative and B"diag(b ) with b (0, j"1, . . . , n. If all nonzero j j eigenvalues of A have negative real parts, then all eigenvalues of A#B have negative real parts. Proof. First, from M-matrix theory, there exists a positive vector x such that Ax60 and therefore (A#B)x"Ax#(b x , . . . , b x )T(0, where ¹ de1 1 n n notes the transpose. Hence, it follows that all eigenvalues of A#B have negative real parts (Fiedler and Pta´k, 1962; Poole and Boullion, 1974). The proof is complete. Since

A

!p m!(amf I f#amcI c ) amf (pm!I m ) amc (pm!I m ) 2 2 2 2 2 2 2 2 2 A " a fm (pf!I f ) !p f!a fmI m 0 22 2 2 2 2 2 acm (pc!I c ) 0 !p c !acm I m 2 2 2 2 2

B

512


if we can show that all nonzero eigenvalues of the matrix

A

!p m amf (pm!I m ) amc (pm!I m ) 2 2 2 2 2 a fm (pf!I f ) !p f 0 JK " : 2 2 2 acm (pc!I c ) 0 !p c 2 2 2

B

have negative real parts, then it follows from Lemma 4.2.1 that A is stable. 22 It is easy to check that det JK "0 and tr JK "!+ p k (0. Let JK be ij k/m,f,c 2 the 2]2 principal minor of JK with row i and column j. Then p mp f amc I c JK " 2 2 2 2 '0 , 12 amfIf#amcI c 2 2 2 2 p mp c amf I f JK " 2 2 2 2 '0 , 13 amfI f#amcIc 2 2 2 2 JK "p fp c '0 . 23 2 2 Hence, JK has a zero eigenvalue and two eigenvalues with negative real parts, which implies that A is locally stable. Then the stability of the nontrivial 22 equilibrium (Ik"0, Ik'0) is determined from the stability of the matrix A . 1 2 11 We note that A is unstable if 11 det A "p fa cma mc ( pm!I m )( pc!I c ) 2 2 1 1 1 11 !p c (p m p f!amf a fm (pm!I m )( pf!I f )) 1 1 1 1 1 2 2 p fa cma mcp mp c Ic p c a mfa fmp mp fI f " 1 1 1 2 2 2 # 1 1 1 2 2 2 !p mp fp c '0 . 1 1 1 cm mf f mc c a (a I #a I ) a fm (amfI f#amc I c ) 2 2 2 2 2 2 2 2 2 2 A straightforward algebraic manipulation also yields

Denote

p m p fp c p mIm 1 1 1 2 2 det A " 11 (amfI f#amc I c )(p f#afmI m )(p c #acm I m ) 2 2 2 2 2 2 2 2 2 2 I m )(Rm R f!RmR f ) · (pfp f2 (p2c #acm 2 2 1 1 2 2 c f fm m m c #pc p (p #a I )(R R !RmR c )) . 2 2 2 2 1 1 2 2 (Rm R l !RmR l ) " : Dl, l"f, c . 1 1 2 2

Then if Df'0 and Dc'0, the boundary equilibrium (Ik "0, Ik '0) is 1 2 unstable. On the other hand, if Df(0, Dc(0, then det A (0. Again, since 11 the diagonal elements of A are negative and off-diagonal elements are 11 nonnegative, !A is an M-matrix. Then, since all leading principal manors 11 have correct sign, it follows from M-matrix theory that all eigenvalues of !A have positive real parts. Hence, A is stable. In addition, it follows 11 11


513

from Lemma 3.1 that if a boundary equilibrium is locally stable, it is globally stable. In summary, we arrive at the following result. Theorem 4.2.2. ¸et Rk'1 be defined as in (3.3). ¹hen if i m f R R '(() RmR f and RmRc '(() RmR c , 1 1 2 2 1 1 2 2 the boundary equilibrium (Ik '0, Ik "0) is globally stable (unstable) and 1 2 (Ik "0, Ik '0) is unstable (globally stable). 1 2 We may think of these equilibria as the result of competition for resources between two populations of pathogens. Hence, if Df · Dc'0, one of the boundary equilibria is globally stable and the other is unstable, which implies one strain wins and the other loses. Hence, under the hypotheses in Theorem 4.2.2, the principle of competitive exclusion holds for the two competing strains. 5 Coexistence 5.1 Existence and uniqueness of the positive endemic equilibrium In order to have coexistence, we need to solve  p mI m"( pm!(I m#I m ))(amf I f#amcI c ) , 1 2 i i i i  i i  p fI f"afm ( pf!(I f#If ))I m , i 1 2 i  ci ci cm ( pc!(I c #I c ))I m , p I "a  i i i 1 2 i

(5.1.1)

for Ik . i From the second equation in (5.1.1), (p f#a fm I m )I f#a fm I m I f"afmpfI m , 1 1 1 1 1 1 2 1 1 a fmI mI f#(p f#a fmI m )I f"a fmpfI m . 2 2 1 2 2 2 2 2 2 Solving (5.1.2) yields

G

(5.1.2)

pfa fmp f I m  f 1 2 1  I1 "p fp f#a fmp f I m#a fmp f I m , 1 2 1 2 1 2 1 2  pfa fmp f I m  2 .  I2f"p fp f#a fmp2f I m1#a fmp f I m 1 2 1 2 1 2 1 2

(5.1.3)

pcacmp c I m  c 1 2 1  I1"p c p c #acmp c I m#acmp c I m , 1 2 1 2 1 2 1 2   pcacmp c I m 2 .  I2c "p c p c #acmp2c I m1#a cmp c I m 1 2 1 2 1 2 1 2

(5.1.4)

Similarly,

514


Set A " : pfamfafmp f /p m , A " : pcamcacmp c /p m , 1 1 1 2 1 2 1 1 2 1 D " : pfamfa fmp f /pm , D " : pcamca cmp c /pm , 1 2 2 1 2 2 2 2 1 2 f fm f f B " : a fm p f , B " : a p , B " : p p , 2 1 3 1 2 2 1 2 1 c c cm c cm C " : p p , C " : a p , C " : a pc . 1 1 2 2 1 2 3 2 1 Then, by substituting (5.1.3) and (5.1.4) into the first equation in (5.1.1), we have 1 A A  1 2  pm!(I m#I m )"B #B I m#B I m#C #C I m#C I m , 1 2 1 2 1 3 2 1 2 1 3 2 (5.1.5)  1 D D  1 2 " # ,  pm!(I m#I m ) B #B I m#B I m C #C I m#C I m 1 2 1 2 1 3 2 1 2 1 3 2 From (5.1.5), it follows that (C (A !D )#B (A !D ))I m" 2 1 1 2 2 2 1 !((B (A !D )#C (A !D ))#(B (A !D )#C (A !D ))I m ) , 1 2 2 1 1 1 3 2 2 3 1 1 2 or equivalently, b m ( b c R c Df#b fR fDc)I m#b m ( b c R c Df#b fR fDc)Im"!pm(Df#Dc) . 1 1 1 1 1 1 2 2 2 2 2 2 (5.1.6) If both Df'0 and Dc'0 or both Df(0 and Dc(0, (5.1.6) gives a line that does not go through the first quadrant and, consequently, there is no positive solution (I m'0, I m'0) for (5.1.5). Hence we have the following result. 1 2 Theorem 5.1.1. Coexistence is not possible if Df · Dc"(RmRf!RmRf ) · 1 1 2 2 (RmR c !R mR c )'0, that is, either R mR l 'R mR l or R mR l (R mR l , for both 1 1 2 2 1 1 2 2 1 1 2 2 l"f, c. Now we consider the case Df · Dc(0, which is equivalent to Rf Rm Rc 1' 2 ' 1 , Rf Rm Rc 2 1 2 or Rf Rm Rc 1( 2 ( 1 . Rf Rm Rc 2 1 2 Solving (5.1.6) for I m gives 1 I m"a I m#a , 1 1 2 2

(5.1.7)


515

where b m ( b c R c Df#b f R f Dc) 2 2 a " : ! 2 2 2 , 1 b m ( b c R c Df#b f R fDc) 1 1 1 1 1

pm(Df#Dc) a " : ! . 2 b m ( b c R c Df#b fRfDc) 1 1 1 1 1

(5.1.8)

Then, substituting (5.1.7) into (5.1.5) gives the following equation

A

A 1 1 ! m B #B a #(B a #B )I m pm!a !(a #1)I 1 2 2 2 1 3 2 2 1 2

B

A 2 # "0 . C #C a #(C a #C )I m 1 2 2 2 1 3 2

(5.1.9)

However, since a and a can be rewritten as 1 2 B C Df#B C Dc 3 1 , a "! 1 3 1 B C Df#B C Dc 1 2 2 1

B C (Df#Dc) a "! 1 1 , 2 B C Df#B C Dc 1 2 2 1

a simple calculation shows that B #B a C #C a 1 2 2" 1 2 2. B a #B C a #C 2 1 3 2 1 3 Using the above results, we see that (5.1.9) is reduced to A A 1 2 # B a #B 1 C a #C 3 2 1 3, " 2 1 B #B a pm!a !(a #1)I m 1 2 2#I m 2 1 2 2 B a #B 2 1 3 or equivalently to B #B a 1 2 2#I m 2 B a #B 2 1 3 m . pm!a !(a #1)I " 2 1 2 A A 2 1 # C a #C B a #B 2 1 3 2 1 3

(5.1.10)

If we now introduce the following combination of parameters B #B a 1 2 2, r" : B a #B 2 1 3

A A 1 2 E" : # , B a #B C a #C 2 1 3 2 1 3

(5.1.11)

by solving (5.1.10), it follows that r pm!a ! 2 E E( pm!a )!r 2 I m" " . 2 1 1#E(a #1) 1 #a #1 1 E

(5.1.12)

516


Then, the substitution of (5.1.12) into (5.1.7) yields a (Epm!r)#a (1#E) 2 I m" 1 . 1 1#E(a #1) 1 Therefore, we have established the following result.

(5.1.13)

Theorem 5.1.2. ¸et a , r, and E be defined as in (5.1.8) and (5.1.11) respectively. i ¹hen if (H1) (1#E(a #1))(E(pm!a )!r)'0 , and 1 2 (1#E(a #1))(a (Epm!r)#a (1#E))'0 , 1 1 2 there is a unique positive endemic equilibrium, and if (1#E(a #1))(E(pm!a )!r)60 , 1 2

or

(1#E(a #1))(a (Epm!r)#a (1#E))60 , 1 1 2 then there is no positive endemic equilibrium.

5.2 Stability of the positive endemic equilibrium We establish the stability of the positive endemic equilibrium in this section. Theorem 5.2.1. ¸et E* " : (I m'0, I f'0, I c '0, I m'0, I f'0, I c '0) be 1 1 1 2 2 2 the coexistence equilibrium give by (5.1.3), (5.1.4), (5.1.12), and (5.1.13). ¸et the following matrix

A

j j 12 13 j j 0 21 22 j 0 j 33 J*" 31 j 0 0 41 0 j 0 52 0 0 j 63 be evaluated at E* where

pc Ic 1 1, j " : 31 Im 1

11

j

0 0 14 0 j 0 25 0 0 j 36 j j j 44 45 46 j j 0 54 55 j 0 j 64 66

B

p mamf I m p mamc I m 1 1 1, 1 1 1 , " : j " : 12 13 G G 1 1 p fIf 1 1, j " : j " : !p f!a fmIm, j " : a fmI m , 21 22 1 1 1 25 1 1 Im 1

j " : !p m!G , 11 1 1 j " : G , 14 1

j

j

33

j

" : !p c !acmI m , 1 1 1

j "acmI m , 36 1 1

j " : G , 41 2


517

p mamfI m p mamcI m 2 2 2, 2 2 2, " : j " : 46 G G 2 2 p fI f 2 2, j " : j " : !p f!a fmIm , 54 55 2 2 2 Im 2 pc I c 2 2, j " : !p c !acmI m , j " : 66 2 2 2 64 Im 2

j " : !p m!G , 44 2 2 j " : afmI m , 52 2 2 j " : acmI m , 63 2 2

j

45

with G " : amfI f#amcI c , 1 1 1 1 1

G " : amfI f#amc I c . 2 2 2 2 2

¹hen, E* is locally asymptotically stable if the determinant of J* is positive and E* is unstable if the determinant J* is negative. Proof. The Jacobian matrix of (2.5) evaluated at E* is the same as J* except that the 3]3 off-diagonal submatrices have an opposite sign. Then, it is easy to see that they have the same eigenvalues. Hence, the stability of E* can be determined by J*. Denote the leading principal minors of J* with i rows by J* . Then, by i straightforward but tedious algebraic manipulations, p mp famc Ic J *" 1 1 1 1#p ma fmI m#(p f#a fm I m )G '0 , 2 1 1 1 1 1 1 1 G 1 pmpc amfa fmImI f J*"! 1 1 1 1 1 1 !pma fmacmImIm!(p c #a cmIm )(p f#a fmIm )G (0 , 1 1 1 1 1 1 1 1 1 1 1 1 3 G 1 p mp c amf a fmI m I f 1 1 1 1 1 1 #p ma fm acmI m I m '0 , J *"!p mJ *#G 1 1 1 1 1 4 2 3 2 G 1

A

B

and

A

G J *"!p c p mp famf (p f#a fm I m )I c 1!p fp mamca fmI c I m 1 1 2G 1 2 2 2 1 1 2 1 2 1 2 5 2 p mp mp famfafmamcI mI fI c ! 1 2 2 1 1 2 1 1 2 G G 1 2 p mp fp mp famcamcI c I c p mp famcI c !acmI m 1 1 2 2 1 2 1 2# 1 1 1 1 (p f#(1#p m )afmI m ) 1 1 2 2 2 2 G G G 1 2 1 p mp famcI c # 2 2 2 2 (p f#(1#p m )afmI m )#p fp mamcafmI c I m 1 1 1 1 1 2 1 2 1 2 G 2

B

G

H

#p mp fafma mcI mI c #p ma fma fmI mI mG (0 . 1 2 1 2 1 2 1 1 2 1 2 2

518


Note that all diagonal elements of J* are negative and all of its off-diagonal elements are nonnegative. !J* is an M-matrix. As is shown above, the leading principal minors (!1)iJ* of !J* all have positive sign for i from 1 to i 5. Then, if the determinant of J* is positive, which is the same as that of !J*, it follows from M-matrix theory that all the eigenvalues of J* have negative real parts and thus E* is stable. On the other hand, if the determinant of J* is negative then J* has at least one positive eigenvalue. The conclusions of the theorem therefore follow. We end this section with two examples. Example 1 Set the following set of parameters. kk"0.025, bc"4, b c "0.1, 2 c c "0.7, 1 Then and

k"m, f, c,

Km"Kf"1000,

bm"6,

bf"4 ,

b m"0.25, b m"0.2, b f"0.05, b f"0.1, b c "0.05 , 1 2 1 2 1 m m f f c "0.02, c "0.1, c "0.02, c "0.6, 1 2 1 2 c c "0.03 . 2 (1#E(a #1))(E ( pm!a )!r)"1982582 , 1 2

(1#E(a #1))(a (Epm!r)#a (1#E))"3877073 . 1 1 2 Hence (H1) is satisfied and there exists a positive equilibrium. In fact, this endemic equilibrium can be solved numerically as I m"24156, I f"36233, I c "2634 , 1 1 1 I m"12352, I f"1067, I c "14204 . 2 2 2 Since det J*"0.004, then Theorem 5.2.1 implies the local asymptotic stability of the coexistence equilibrium. This can be confirmed numerically by a direct computation of the eigenvalues of J* which are j "!1.46, j "!0.38, j "!1.98 , 1 2 3 j "!0.84, j "!0.37, j "!1.17 . 4 5 6 In this example, the two female groups have different recovery rates or incubation periods to the two strains. More specifically, infected females in group f with strain 1 have a longer incubation period or need longer time to recover than females in group c infected with strain 1. However the situation is reversed for strain 2, namely, females in group c with strain 2 have a longer incubation period than their counterparts from group f. We can also interpret these differences in ck not as directly linked to the incubation period distribui tions but rather to the ability of these strains to conceal themselves (asymptomatic individuals) in different populations to ‘‘retard’’ treatment. Here, females in group f with strain 1 and in group c with strain 2 can be thought as having the longer strain-specific asymptomatic periods.


519

While (H1) is satisfied, it follows from Theorem 5.1.2 that there exists a positive endemic equilibrium. However, it may be unstable. Example 2 We now use the following set of parameters. km"kf"0.025, bm"4,

bf"2,

bm"0.24, 1 bc "0.15 , 2 m c "0.15, 1 c c "0.1 . 2

kc"0.015,

Km"200,

Kf"100 ,

bc"4 ,

b m"0.2, 2

bf"0.05, 1

b f"0.1, 2

bc "0.4 , 1

c m"0.08, 2

c f"0.1, 1

c f"0.15, 2

c c "0.15 , 1

Then and

(1#E(a #1))(E(pm!a )!r)"1859598 , 1 2

(1#E(a #1))(a (Epm!r)#a (1#E))"1110 . 1 1 2 Hence, the coexistence equilibrium exists and its components are Im"8.6, If"0.6, I c "6.7 , 1 1 1 m f I "14318.4, I "549.8, I c "13435.2 . 2 2 2 However, since det J*"!0.8]10~7, it follows from Theorem 5.2.1 that this coexistence equilibrium is unstable. The corresponding set of eigenvalues of J*, j "!0.95, j "!0.78, j "!0.32 , 1 2 3 j "!0.12, j "!0.5, j "!0.5]10~5 , 4 5 6 corroborates our conclusion. In one-sex models as studied in Blythe et al. (1994), whenever a coexistence equilibrium exists, it is always stable. This is not the case for our two-sex model. Hence, a heterosexual structure may have a destabilizing effect on coexistence.

6 Concluding remarks An important principle in theoretical biology is that of competitive exclusion which states that no two species can forever occupy the same ecological niche. Clarifications on the meaning of competitive exclusion and niche have been central to theoretical ecology (Butler et al., 1983; Levin, 1970; May, 1975; Maynard Smith, 1978). Sexually transmitted diseases like gonorrhea have incredibly high incidences throughout the world providing the necessary environment and opportunities for the evolution of new strains (see Hethcote

520


and Yorke, 1984, and references therein). The coexistence of gonorrhea strains has become an increasingly serious problem. Understanding the mechanisms that lead to coexistence or competitive exclusion is central to the development of disease management strategies as well as to our increased understanding of STD-dynamics. We previously formulated heterosexual models where two strains (or any number of strains) competed for ‘‘identical’’ hosts as we included only one (hence homogeneous) female group in the population. The outcome of those models was always the same: competitive exclusion (Castillo-Chavez et al., 1993, 1995). In this article, we study a heterosexual model where two ‘‘genetically’’ different female groups interact with a homogeneous (genetically uniform) male population in the presence of two competing strains of a venereal disease. We find out that, under various situations, both competitive exclusion and coexistence may occur. We see that, as expected, if one of the strains has higher transmissibility in both groups or both female groups are more susceptible to one of the strains, we have competitive exclusion. Mathematically, the result follows from the inequalities R m R f'R mR f and Rm R c'R mR c. In this case, i i j j i i j j strain i wins and strain j loses. On the other hand, if the transmissibility of one strain is higher, let’s say in group f than group c, then the transmissibility must be reversed for the other strain to make coexistence likely. A similar level of asymmetry must exist between both groups of females regarding their average periods of infection (or their average asymptomatic periods) to increase the likelihood of coexistence. The necessary conditions for coexistence require that inequalities, such as Rm R f'R mR f and Rm R c'R mR c , be satisfied. Coexistence is possible only i i j j i i j j under a set of complex relationships between these factors as illustrated in the examples. Furthermore, we observe that the existence of a coexistence equilibrium does not guarantee its stability. It is clear that the limited heterogeneity available in the system investigated in this article makes the satisfaction of the conditions for coexistence difficult. It is even more difficult to meet the conditions for stable coexistence. We have been unable to provide transparent and specific necessary and sufficient biological conditions guaranteeing stable coexistence in the system. Nevertheless, it is important to reiterate a key feature of our investigations, that coexistence is indeed theoretically possible as soon as a minimal level of diversity is introduced. Preliminary results have shown that superinfection as defined in the works of Levin (1970, 1983a, 1983b) provides a clear biological mechanism for coexistence. We plan to concentrate our efforts in elucidating the role of superinfection as a mechanism that supports pathogens’ diversity in a minimally heterogeneous host population. Acknowledgements. This research was partially supported by NSF grant DEB-925370, Presidential Faculty Fellowship Award, to Carlos Castillo-Chavez, by the Center for Applied Mathematics at Cornell University, and by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University (Contract DAAL03-91-C-0027). The authors also thank the referees for their valuable comments and suggestions.


521

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