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Mathematical Biosciences 165 (2000) 1±25 www.elsevier.com/locate/mbs Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity David Greenhalgh a,*, Odo Diekmann b, Mart C.M. de Jong c a c Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow, G1 1XH, UK b Vakgroep Wiskunde, Postbus 80.010, 3508 TA, Utrecht, The Netherlands Department of Immunology, Pathobiology and Epidemiology, Institute of Animal Science and Health, P.O. Box 65, 8200 AB, Lelystad, The Netherlands Received 7 October 1998; received in revised form 2 March 2000; accepted 3 March 2000 Abstract Many classical mathematical models for animal infections assume that all infected animals transmit the infection at the same rate, all are equally susceptible, and the course of the infection is the same in all animals. However for some infections there is evidence that seropositives may still transmit the infection, albeit at a lower rate. Animals can also experience more than one episode of the infection although those who have already experienced it have a partial immune resistance. Animals who experience a second or subsequent period of infection may not necessarily exhibit clinical symptoms. The main example discussed is bovine respiratory syncytial virus (BRSV) amongst cattle. We consider simple models with vaccination and homogeneous and proportional mixing between seropositives and seronegatives. We derive an expression for the basic reproduction number, Ro , and perform an equilibrium and stability analysis. We ®nd that it may be possible for there to be two endemic equilibria (one stable and one unstable) for Ro < 1 and in this case at Ro 1 there is a backwards bifurcation of an unstable endemic equilibrium from the infection-free equilibrium. Then the implications for control strategies are considered. Finally applications to Aujesky's disease (pseudorabies virus) in pigs are discussed. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: SISI epidemic model; Backwards bifurcation; Subcritical endemic steady states; Bovine respiratory syncytial virus; Aujesky's disease * Corresponding author. Tel.: +44-141 552 4400, ext. 3653; fax: +44-141 552 2079. E-mail address: [email protected] (D. Greenhalgh). 0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 1 2 - 2 2 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 1. Introduction Classical epidemic models usually assume that either immunity does not exist (the SIS model) or that experiencing the infection provides permanent or temporary protection against it (the SIR and SIRS models). In the SIS model a typical individual starts o susceptible, at some stage catches the infection and after an infectious period becomes completely susceptible again. SIS models are appropriate for sexually transmitted infections such as gonorrhea [1]. However, there is increasing evidence that some animal infections may provide only partial immunity and can spread amongst seropositive animals, albeit at a reduced rate. Thus seropositive animals can transmit the infection during the second and subsequent infectious periods but do not exhibit clinical symptoms of the disease during these periods. A situation where an SISI (or SIS1 I1 S1 ) model may be appropriate is the spread of bovine respiratory syncytial virus (BRSV) amongst cattle. BRSV causes respiratory tract infection, especially in young calves. Outbreaks occur each autumn and most dairy farms are aected. It is therefore often concluded that the virus is continually present on farms. One hypothesis regarding persistence of BRSV on farms is that the virus circulates amongst seropositive cattle without causing clinical signs of infection. The presumption is that seropositive cattle shed virus after infection, however no-one has yet succeeded in isolating the virus in re-infected cattle. De Jong et al. [2] examine whether the transmission of the virus amongst seropositive cattle is a plausible mechanism for the permanent persistence of BRSV in dairy herds and how likely it is with that scenario for persistence that there will be only one clinical outbreak of BRSV per year. They build a stochastic model and estimate parameters from serological data on antibodies against BRSV in sera from cattle in six dairy herds. They ®nd that, given estimated parameter values, persistence of BRSV by transmission amongst seropositive cattle would be accompanied by frequent extinctions and long infectious periods in seropositive cattle. Moreover in the model a single clinical outbreak among seronegative cattle occurred only with seasonal forcing. De Jong et al. showed that transmission of the virus amongst seropositive cattle cannot on its own account for the observed seasonal outbreaks of BRSV and some other mechanism, such as climatically determined periodicity in transmission parameters, demographic periodicity or periodicity in contacts is necessary to explain the observed data. However this does not in itself falsify the hypothesis that the infection will spread (probably at a reduced rate) amongst seropositive cattle and they conclude that persistence of the infection amongst seropositive cattle is still plausible. 2. The model Let S1 ; S2 denote respectively the numbers of ®rst time susceptible cattle and susceptible cattle who have been previously infected. Let I1 ; I2 denote respectively the numbers of ®rst time infected cattle and infected cattle who have experienced previous infection. So N S1 S2 I1 I2 is the total number of cattle. Suppose that b is the per capita birth rate per unit time and that the infectious period is exponentially distributed with parameter b1 (mean bÿ1 1 ) for ®rst time infected D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 3 cattle, and parameter b2 (mean bÿ1 2 ) for cattle infected for the second and subsequent times. During each infectious period the infectivity has a constant level and the ratio of this infectivity during the ®rst infectious period to the infectivity during subsequent infectious periods is a1 : a2 . More precisely we assume that ®rst time susceptible cattle come into contact with and are infected by ®rst time infected cattle at per capita rate a1 I1 =N and by other infectious cattle at per capita rate a2 I2 =N . For subsequent time susceptible (seropositive susceptible) cattle transmission is less ecient so these rates are reduced by a factor c 0 6 c 6 1. We use the `true mass action' transmission term aSI=N rather than the classical mass action transmission term aSI as it has been argued that this is more plausible [3]. If the population size is constant, then by rede®ning a this will make no dierence to the dynamics of the model, but will aect the formulation of results on threshold population sizes. The issue of which transmission term is best is most prominent when we are comparing the dynamics of two or more populations with dierent sizes. We assume homogeneous mixing between seropositive and seronegative cattle. We also assume that the population under consideration is of constant size N, so births balance deaths and there are no deaths from the infection. Thus the per capita death rate for all four types of cattle is b. The assumption that there are no deaths from the infection is true for BRSV [2]. A fraction / 0 6 / 6 1 of individuals are vaccinated at birth, these individuals immediately enter the seropositive susceptible class S2 . In practice little bene®t is obtained from vaccinating individuals at birth as these individuals are protected by maternal antibodies, but individuals are vaccinated a short time after birth when the eect of maternal antibodies has waned. In the ®eld cattle are vaccinated at age 4±8 weeks, and this vaccination is repeated yearly. The aim is to prevent clinical symptoms. Note in addition that we do not incorporate that vaccination may temporarily provide a stronger protection than the ultimate eect of having previously experienced the infection. Then it is straightforward to show that the dierential equations which describe the spread of the infection are dS1 dt dI1 dt dS2 dt dI2 dt b 1 ÿ /N ÿ S1 a1 I1 a2 I2 ÿ bS1 ; N S1 a1 I1 a2 I2 ÿ b1 I1 ÿ bI1 ; N cS2 b/N ÿ a1 I1 a2 I2 b1 I1 b2 I2 ÿ bS2 ; N cS2 a1 I1 a2 I2 ÿ b2 I2 ÿ bI2 N 2:1 and S1 S2 I1 I2 N , with suitable initial conditions S1 0; S2 0; I1 0; I2 0 P 0 and S1 0 S2 0 I1 0 I2 0 N. We shall examine the behaviour of this model by means of an equilibrium and stability analysis. It is more convenient to rewrite these equations in terms of the fractions of individuals in each class. De®ne s1 S1 =N ; s2 S2 =N ; i1 I1 =N and i2 I2 =N , respectively the fraction of class one susceptible, class two susceptible, class one infected and class two infected individuals. Then the dierential equations (2.1) can be written 4 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 ds1 b 1 ÿ / ÿ s1 a1 i1 a2 i2 ÿ bs1 ; dt di1 s1 a1 i1 a2 i2 ÿ b b1 i1 ; dt ds2 b/ ÿ cs2 a1 i1 a2 i2 b1 i1 b2 i2 ÿ bs2 ; dt di2 cs2 a1 i1 a2 i2 ÿ b b2 i2 ; dt s1 s2 i1 i2 1: 2:2a 2:2b 2:2c 2:2d 2:2e 3. The basic reproduction number A key parameter in determining the behaviour of the model is the basic reproduction number, Ro . For epidemic models with a steady vaccination program Ro is de®ned as the expected number of secondary cases produced by a single typical infected case entering an infection-free population at equilibrium [4,5]. If the population is divided into n disjoint groups then Ro is generally given as the largest eigenvalue of an n n matrix of secondary cases [4]. We de®ne Ro1 a1 = b1 b and Ro2 a2 c= b2 b. Ro1 is the basic reproduction number for an SIS epidemic model with no vaccination where the per capita eective contact rate is a1 and the average infectious period, conditional on survival to the end of it, is bÿ1 1 . This model can be obtained from ours by setting c 1; a2 a1 and b2 b1 . Similarly Ro2 a2 c= b2 b is the basic reproduction number for an SIS model, where the per capita eective contact rate is a2 c and the average infectious period, conditional on survival to the end of it, is bÿ1 2 . This model can be obtained from ours by setting / 1 and S1 0 I1 0 0, i.e. all individuals enter the class S2 and S1 t I1 t 0 for all t. De®ne Ro 1 ÿ /Ro1 /Ro2 ; a1 a2 c 1 ÿ / / : b2 b b1 b To show that Ro is the basic reproduction number in our model, consider a population at the infection-free equilibrium S1 ; S2 1 ÿ /N; /N . For i; j 1; 2 de®ne mij to be the expected number of secondary cases in class j produced by a single infected class i case entering the population. The next generation matrix is ! a1 1ÿ/ a1 c/ m11 m12 b1 b b1 b 3:1 a2 1ÿ/ a2 c/ : m21 m22 b b b b 2 2 ÿ1 For example, a single infected class 1 case is infectious for time b1 b and during the infectious period transmits the disease to ®rst time and subsequent time susceptible cattle at rate a1 =N and a1 c=N, respectively. At the disease-free equilibrium there are 1 ÿ /N ®rst time and /N subsequent time susceptible cattle. Hence our case produces D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 m11 5 a1 N a1 1 ÿ / 1 ÿ / N b1 b b1 b secondary cases amongst ®rst time and a1 c / m12 b1 b cases amongst subsequent time susceptible cattle. m21 and m22 are explained similarly. Ro is the dominant eigenvalue of this matrix (3.1) which is its trace 1 ÿ /Ro1 /Ro2 , since one eigenvalue equals zero. Note that if Ro2 > 1 > Ro1 then Ro is increasing in / and so increasing the vaccination proportion / has the eect of helping the infection to spread. We shall return to this point in Section 8. 4. Equilibrium and stability results Suppose ®rst that / > 0, so some individuals are actually vaccinated. De®ne h b 1 ÿ /= b b1 ; 4:1 h is an important parameter in the model and corresponds to the fraction of individuals who die before reaching the second class if a1 is very large so that class one individuals are eectively infected at birth. In this case the eective entry rate of susceptibles into the second class is r b/ b1 h. For notational convenience we de®ne s b2 1 ÿ cb=c: 4:2 We shall later express our equilibrium and stability results in terms of the bifurcation parameter a2 . Note that Ro < 1 if and only if b > a1 h and a2 < aR2 o , where aR2 o b ÿ a1 h b2 b=b/c: 4:3 We choose the superscript Ro to indicate that, for this value of a2 ; Ro has the value one. De®ne G a p0 p1 a p2 a2 ; 4:4 where p0 a21 b2 h2 b2 s2 2a1 b2 hs; p1 2a1 bhr ÿ 2brs ÿ 4b2 b1 h and p2 r2 : 4:5 Also de®ne ac1 b b2 bb1 h ÿ b2 c/ : h b2 br ÿ b2 c/ 4:6 Note that ac1 < b=h. Our equilibrium and stability results can be summarised by the following theorem: Theorem 1. There is always an infection-free equilibrium (IFE) s1 ; i1 ; s2 ; i2 1 ÿ /; 0; /; 0. This equilibrium is locally asymptotically stable (LAS) when Ro < 1 and unstable when Ro > 1. If Ro 1 ÿ /Ro2 < 1 then this equilibrium is globally asymptotically stable (GAS). For Ro > 1 there 6 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 is a unique endemic equilibrium which is LAS. For Ro 6 1 there are no endemic equilibria when b=h > a1 P ac1 . On the other hand if Ro < 1 and ac1 > a1 , then there is an open interval of a2 -values aU2 < a2 < aR2 o for which there are precisely two endemic equilibria. Of these two equilibria the one with the highest value of i1 is LAS whilst the other is unstable. The threshold value aU2 is the largest of the two positive roots of the equation G a2 0 (which are both less than aR2 o when ac1 > a1 holds). At a2 aU2 there is a unique endemic equilibrium (which is unstable) and for smaller values of a2 no endemic equilibrium exists. At a2 aR2 o we have Ro 1, the endemic equilibrium is unique and it is LAS. Proof. Results about equilibria First of all we show the equilibrium results. Starting from Eqs. (2.2a)±(2.2e) let s1 ; s2 ; i1 and i2 denote the equilibrium values of s1 ; s2 ; i1 and i2 , respectively. At equilibrium from (2.2a)±(2.2d): b 1 ÿ / b b1 i1 ; a1 i1 a2 i2 b a1 i1 a2 i2 b/ b1 i1 b2 i2 b b2 i2 s2 c a1 i1 a2 i2 b c a1 i1 a2 i2 s1 and i2 i1 a2 4:7 4:8 b ÿ a 1 : h ÿ i1 4:9 Eq. (4.9) is obtained by solving the two expressions for s1 in (4.7) for i2 as a function of i1 . Substituting these expressions into b/ ÿ cs2 a1 i1 a2 i2 b1 i1 b2 i2 ÿ bs2 0 we ®nd after some manipulations the equation F x 0, where x h ÿ i1 and F x Ax2 Bx C: Here A ÿb1 a1 s; a2 B a1 bh b ÿ s r; a2 a2 Cÿ b2 h : a2 4:10 A solution for x corresponds to an endemic equilibrium solution if and only if 0 < x < min b=a1 ; h. Note that r s b2 h a1 x : 4:11 F x b1 x 1ÿ ÿ x ÿ x b ÿ a1 x ÿ b1 a2 a2 b D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 7 Hence F 0 C < 0; F h b/h ÿ and F b a1 b1 b a1 b ÿ a1 h b2 bh a2 c r b ÿ : b1 a1 Therefore (a) for h P b=a1 F b=a1 > 0, (b) a1 h 1 F h bh / ÿ 1 ÿ ; b Ro2 bh Ro ÿ 1; Ro2 P 0; for Ro P 1 with equality if and only if Ro 1: Hence for Ro > 1; F min b=a1 ; h > 0 and so F x 0 has a unique root in 0 < x < min b=a1 ; h. If Ro 1 then F x 0 has a root at x h min b=a1 ; h and at most one more root with 0 < x 6 h. If Ro < 1; F x 0 has either zero or two roots in 0 < x 6 h. Note also that b ÿ a1 h b2 b : 4:12 b/c Consider the conditions for the equation F x 0 to have two positive real roots. This will be the case if and only if Ro 6 1 if and only if a2 6 aR2 o A < 0; B > 0; B2 ÿ 4AC > 0: Now A < 0 if and only if a2 > 2 aA2 4:13 a1 =b1 s; B > 0 if and only if a2 > B2 ÿ 4AC p0 1=a2 p1 1=a2 p2 aB2 1=r bs ÿ a1 bh and G a2 : a22 G a2 is a quadratic in a2 with p0 > 0 and p2 > 0. It is straightforward from Eq. (4.5) that p21 ÿ 4p0 p2 16b2 h rs bhb1 bb1 ÿ a1 r: Case 1. bb1 6 a1 r and Ro < 1. Then h < b=a1 as Ro < 1. Moreover F b=a1 P 0 so F x 0 has a root in h; b=a1 , hence cannot have two roots in 0; h. So F x 0 has no roots in 0; h. Case 2. bb1 > a1 r. (Note that we do not require that Ro < 1.) Then p1 < 0 and p21 ÿ 4po p2 > 0. So the equation G a2 0 has two positive distinct real roots for a2 , 0 < aL2 < aU2 , say, and G a2 > 0 if a2 > aU2 or a2 < aL2 ; G a2 < 0 if aL2 < a2 < aU2 . Hence 8 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 F x 0 has two positive real roots for x if (4.13) holds, equivalently a2 >max aA2 ; aB2 and a2 < aL2 or aU2 < a2 . The inequality bb1 > a1 r implies that F b=a1 < 0 so F x 0 has zero or two roots in 0; b=a1 for all a2 . For a2 very large (4.11) implies that there are two roots in 0; b=a1 , one near x 0 and one near r=b1 > h. For ®xed x 2 0; b=a1 ; F x is a monotone increasing function of a2 2 0; 1. Hence as a2 decreases these roots move towards each other and eventually co-alesce at a2 aU2 . For a2 aU2 ÿ d, where d is small and positive, F x < 0 for x 2 0; b=a1 . Hence F x < 0 for x 2 0; b=a1 for a2 6 aU2 ÿ d and F x 0 has no roots in 0; b=a1 , hence none in 0; h. For a2 6 aL2 two positive real roots appear outside 0; b=a1 . Hence for a2 P aU2 ; F x 0 has two real roots in 0; b=a1 , whilst for a2 < aU2 ; F x 0 has no real roots in 0; b=a1 . At a2 aR2 o ; F x 0 has a root at x h 2 0; b=a1 , hence two real roots in 0; b=a1 so aR2 o P aU2 . In a2 > aU2 A < 0 by Lemma 1 below. So the roots are continuous functions of a2 for a2 > aU2 . For a2 > aR2 o one root lies in 0; h and the other in h; r=b1 . At a2 aR2 o one root crosses the line x h, whilst the other is ÿ B=A ÿ h. Hence if ÿ B=A jaRo ÿh < h then for aR2 o > a2 P 2 aU2 F x 0 has two roots in 0; h whilst if ÿ B=A jaRo ÿh > h then for aR2 o > a2 P aU2 F x 0 2 has no roots in 0; h. If ÿ B=A jaRo ÿh h, then when a2 aR2 o F x has a double root at x h, 2 so aR2 o aU2 . Lemma 2 gives an alternative interpretation of the condition ÿ B=A jaRo ÿh > h. 2 To summarise the equilibrium results: (a) If a2 > aR2 o then F x 0 has exactly one root in 0 < x <min b=a1 ; h. (b) If (i) bb1 6 a1 r or (ii) bb1 > a1 r and b2 b < cb2 / b ÿ a1 h= bb1 h ÿ a1 hr, then for a2 < aR2 o ; h < b=a1 and F x 0 has no roots in 0; h. When a2 aR2 o then h 6 b=a1 , there is one root at h, and (1) if A aR2 o < 0 aR2 o > aA2 there is one root in h; 1, (2) if A aR2 o 0 aR2 o aA2 there are no other roots, (3) if A aR2 o > 0 aR2 o < aA2 there is one root in ÿ1; 0. (c) If bb1 > a1 r and b2 b b2 /c b ÿ a1 h= bb1 h ÿ a1 hr then h < b=a1 and aU2 aR2 o . If a2 6 aU2 then F x 0 has no roots in 0; h. If a2 aU2 then h is a double root of F x 0. (d) If bb1 > a1 r and b2 b > b2 /c b ÿ a1 h= bb1 h ÿ a1 hr then h < b=a1 and aU2 < aR2 o . If a2 < aU2 ; F x 0 has no roots in 0; h. In this case if a2 aU2 then there is one double root in 0; h. For aU2 < a2 < aR2 o ; F x 0 has two roots in 0; h. If a2 aR2 o there is one root in 0; h and h is a root. It is straightforward to translate these equilibrium results into those in the statement of Theorem 1. Lemma 1. If bb1 > a1 r then 12 aL2 aU2 >max aA2 ; aB2 : Proof. See Appendix A. Lemma 2. If bb1 > a1 r then (i) ÿ B=A jaRo ÿh < h if and only if b2 b > cb2 / b ÿ a1 h= h bb1 ÿ a1 r and 2 (ii) ÿ B=A jaRo ÿh P h if and only if b2 b 6 cb2 / b ÿ a1 h= h bb1 ÿ a1 r (with equality in 2 the first expression if and only if there is equality in the second). D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 9 Proof. See Appendix A. Local Stability of IFE Linearising about the IFE s1 ; i1 ; s2 ; i2 1 ÿ /; 0; /; 0 we see that the characteristic equation has roots x ÿb (twice) and the roots of the quadratic x2 b b1 b b2 ÿ a1 1 ÿ / ÿ a2 c/x b b1 b b2 ÿ b b1 a2 c/ ÿ b b2 a1 1 ÿ / 0: 4:14 Hence by the Routh±Hurwitz conditions necessary and sucient conditions for local stability are: (i) b b1 b b2 > a1 1 ÿ / a2 c/; 4:15 and (ii) b b1 b b2 > b b1 a2 c/ b b2 a1 1 ÿ /: (4.16) can be rewritten as 1 > Ro . If 1 > Ro then a1 1 ÿ / a2 c/ b b1 b b2 > b b1 b b2 b b1 b b2 4:16 > a1 1 ÿ / a2 c/: Hence the IFE is LAS if Ro < 1 and unstable for Ro > 1 as required. Global Stability of IFE when Ro 1 ÿ /Ro2 < 1. From Eq. (2.2a) we have that ds1 =dt 6 b 1 ÿ / ÿ bs1 . De®ne s1 1 limt!1 supT P t s1 T . The solution x1 t of dx1 b 1 ÿ / ÿ bx1 ; x1 0 s1 0 dt is a super-solution for s1 t, (x1 t P s1 t for all t). Since x1 t ! 1 ÿ / as t ! 1, given > 0 there is t0 > 0 such that s1 t 6 1 ÿ / for t > t0 . Hence s1 1 6 1 ÿ / . But > 0 is arbitrary. So 1 letting ! 0 we deduce that s1 6 1 ÿ /. As 1 > Ro 1 ÿ /Ro2 1 ÿ /Ro1 Ro2 , we can choose > 0 small enough so that 1 > 1 ÿ / Ro1 Ro2 . Then there exists t0 such that s1 6 1 ÿ / for t P t0 . From Eqs. (2.2b) and (2.2d) we have for t P t0 , writing i i1 ; i2 T ; di 6 Qi; dt where 1 ÿ / a1 ÿ b b1 a2 1 ÿ / : Q ca1 ca2 ÿ b b2 If M is large enough Q MI is a matrix with strictly positive entries, and so by the Perron± Frobenius theorem [6] has a simple eigenvalue equal to its spectral radius and the corresponding left eigenvector e is strictly positive. If the eigenvalues of Q are x1 and x2 , those of Q MI are 10 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 x1 M and x2 M. Without loss of generality x1 M is the spectral radius, so both x1 and x2 are real, x1 > x2 and eQ x1 e. The characteristic equation of Q is x2 a1 x a2 0; where a1 b b1 1 ÿ 1 ÿ / Ro1 b b2 1 ÿ Ro2 ; and a2 b b1 b b2 1 ÿ 1 ÿ / Ro1 ÿ Ro2 : As 1 > 1 ÿ / Ro1 Ro2 ; a1 and a2 are both strictly positive so by the Routh±Hurwitz conditions x1 and x2 are both strictly negative. Moreover for t P t0 d e i 6 eQ i x1 e i: dt Integrating 0 6 e i t 6 e i t1 ex1 tÿt1 : So e i t ! 0 as t ! 1 which implies that both i1 and i2 tend to zero as t ! 1 as e is strictly positive. Hence given 1 > 0 there exists t1 P t0 such that for t P t1 , a1 i1 a2 i2 6 b1 . So for t P t1 , ds1 P b 1 ÿ / ÿ s1 1 1 : dt now shows that A similar argument to the one showing that s1 1 61 ÿ / s1;1 limt!1 inf T P t s1 T P 1 ÿ /= 1 1 . But 1 is arbitrary so letting 1 ! 0; s1;1 P 1 ÿ /. Thus 1 ÿ / P s1 1 P s1;1 P 1 ÿ /: So s1 1 s1;1 1 ÿ / and s1 ! 1 ÿ / as t ! 1. Thus s2 1 ÿ s1 ÿ i1 ÿ i2 ! / as t ! 1 and the IFE is GAS. Stability of Endemic Equilibria Suppose that s1 ; i1 ; s2 ; i2 is an endemic equilibrium. Substituting s2 1 ÿ s1 ÿ i1 ÿ i2 into Eqs. (2.2a), (2.2b) and (2.2d) where appropriate to eliminate s2 and linearising about s1 ; i1 ; i2 the Jacobian is 2 3 ÿ a1 s1 ÿ a2 s1 ÿa1 i1 ÿ a2 i2 ÿ b 6 a1 i a2 i 7 a1 s1 ÿ b b1 a2 s1 1 2 7 J 6 4 ÿc a1 i a2 i ÿ c a1 i a2 i a1 cs ÿ c a1 i a2 i a2 cs 5: 1 2 1 2 2 1 2 2 ÿ b b2 Using the equilibrium versions of (2.2a) and (2.2d), expanding the characteristic equation and simplifying using the equation b b1 ÿ a1 s1 b b2 ÿ a2 cs2 a2 s1 ca1 s2 4:17 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 11 which is immediate from the equilibrium equations (2.2b) and (2.2d) we deduce that the characteristic equation is x3 A1 x2 A2 x A3 0: 4:18 Here b 1 ÿ / i2 b b1 b b2 1 ÿ a2 cs2 ÿ a1 s1 ; A1 s1 s2 1ÿ/ ÿ 1 b b1 b b b1 ÿ a1 s1 A2 b s1 b 1 ÿ / i2 i2 b b 1 cs ÿ a 2 2 b b1 ÿ a1 s1 b b2 2 s1 s2 s2 and 1ÿ/ i2 A3 b ÿ 1 b1 b b2 b 1 ÿ ca2 s2 b b1 b ÿ a1 s1 s1 s2 i i b2 b 1 2 ÿ ca2 s2 ÿ ba2 s1 a1 cs2 ÿ a2 s1 b1 b b2 2 : s2 s2 4:19 4:20 4:21 By using the facts that b b1 > a1 s1 , b b2 > a2 cs2 and 1 ÿ / > s1 (which follow immediately from the equilibrium equations) we deduce that A1 > 0; A2 > 0 and A1 A2 > A3 . Hence by the Routh±Hurwitz conditions our endemic equilibrium is locally stable if A3 > 0 and unstable if A3 < 0. By using (4.17) 1ÿ/ i2 i2 b b s ÿ 1 b b 1 ÿ ca A3 b 2 2 b b1 b ÿ a1 s1 b2 b 1 2 s1 s2 s2 i ÿ a2 s1 b1 b b2 2 : 4:22 s2 Now we express s1 ; s2 ; i1 and i2 in terms of x h ÿ i1 as in the equilibrium results. From (4.9) a1 i1 a2 i2 bi1 : h ÿ i1 4:23 Substituting (4.23) into the equilibrium equation (2.2d) and simplifying we deduce that i2 cb h ÿ x : s2 b b2 x 4:24 Using (4.7), (4.8) and (4.23) s1 b b1 x b and s2 b b2 b ÿ a1 x: cba2 Substituting (4.24) and (4.25) into (4.22) we deduce after some algebra that A3 b b1 h ÿ x 2 cb h a1 b b2 ÿ a2 b1 c ÿ a1 bcx2 : x2 4:25 12 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 As Ax2 Bx C 0 we can add b b1 h ÿ x=x2 a2 cAx2 Bx C to A3 without changing it so A3 b b1 h ÿ x a2 c2Ax B: x 4:26 The stability of our endemic equilibrium depends on the sign of A3 . There are several cases: Suppose ®rst that Ro > 1 so that a2 > aR2 o . Then F 0 < 0 and our equilibrium results have shown that F x 0 has exactly one root in 0 < x <min b=a1 ; h. This root corresponds to the endemic equilibrium and is always stable. p (i) if A a2 < 0 then the second root of F x 0 lies in h; 1 so x ÿB B2 ÿ 4AC =2A and 2Ax B > 0. 2Ax B > 0. (ii) if A a2 0 then x ÿC a2 =B a2 > 0 which implies that B a2 > 0 so again p (iii) if A a2 > 0 then the second root of F x 0 lies in ÿ1; 0 so x ÿB B2 ÿ 4AC =2A and 2Ax B > 0. In the case where there are two endemic equilibria for Ro < 1 we have bb1 > a1 r; b2 b > b2 /c b ÿ a1 h bb1 h ÿ a1 hr and aU2 < a2 < aR2 o . Then F x 0 has two roots in 0; h. As A a2 < 0 the smaller of these roots is stable and the larger unstable. If a2 aR2 o then the larger root is x h but the same argument shows that the smaller root, which is now the unique endemic equilibrium, is stable. This completes the proof of Theorem 1. It is interesting to consider the special case / 0 separately as in this case the results simplify considerably. This case means that there is no vaccination of susceptible individuals. Here Ro Ro1 a1 = b b1 , the same basic reproduction number as in a population with only the ®rst type of individual. This is independent of the parameters c; a2 and b2 . Note that aR2 o ! 1 as / ! 0. We have the following corollary: Corollary 1. Suppose that / 0. Note that the IFE is always possible. (I) If Ro < 1 then the IFE is LAS. If Ro Ro2 < 1 then the IFE is GAS. The equation G a2 0 has two positive real roots for a2 . Let aU2 denote the largest of these. If 0 6 a2 < aU2 then the IFE is the only equilibrium but for aU2 < a2 there are two additional endemic equilibria. The one with the highest value of i1 is LAS, the other is unstable. At a2 aU2 there is one endemic equilibrium. (II) If Ro 1 there is, apart from the IFE, a unique endemic equilibrium which is LAS. (III) If Ro > 1 then the IFE is unstable and there is a unique LAS endemic equilibrium. Proof. This is a straightforward modi®cation of the proof of Theorem 1. 5. Discussion The general results can be illustrated by the following bifurcation diagrams of i1 , the equilibrium value of i1 , against a2 . All bifurcation diagrams take b 0:000648/day, b1 0:1/day and D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 13 b2 0:01/day; these parameter values for BRSV are taken from de Jong et al. [2]. There are no data available on the value of c so we arbitrarily take c 0:5, so seronegative cattle are twice as susceptible as seropositive ones. a1 and / vary as indicated in the bifurcation diagrams. From the proof of Theorem 1 (or Corollary 1 if / 0) we see that i1 6 h and the endemic equilibrium value of i1 tends to h as a2 becomes large, as can indeed be observed in Figs. 1 and 2. Fig. 1 shows two bifurcation diagrams for / > 0. If b=h > ac1 > a1 then the bifurcation diagram looks like Fig. 1(a), with two endemic equilibria (the higher locally stable and the lower unstable) co-existing with the stable infection-free equilibrium for aU2 < a2 < aR2 o . At a2 aU2 these equilibria coalesce, whereas at a2 aR2 o the unstable endemic equilibrium coalesces with the infection-free one `causing' the latter to become unstable. On the other hand if a1 > ac1 the bifurcation diagram looks like Fig. 1(b). For a2 < aR2 o there is only the stable infection-free equilibrium and at a2 aR2 o a unique stable endemic equilibrium bifurcates away from the infection-free one which then loses its stability. Fig. 1(b) is the typical bifurcation diagram for classical epidemic models. Fig. 2 shows the two corresponding bifurcation diagrams when / 0. If a1 < b1 b, the bifurcation diagram looks like Fig. 2(a). This is similar to Fig. 1(a) except that for a2 > aU2 there are always two endemic equilibria, the higher locally stable and the lower unstable. If a1 > b1 b then the bifurcation diagram looks like Fig. 2(b). There is always a unique stable endemic equilibrium and the infection-free equilibrium is always unstable. Fig. 1. Bifurcation diagrams for / > 0. Parameter values: (a) / 0:1; a1 0:05/day, and (b) / 0:5; a1 0:15/day. See text for other parameter values. 14 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 Fig. 2. Bifurcation diagrams for / 0: (a) a1 0:05/day, and (b) a1 0:15/day. See text for other parameter values. Fig. 3 shows the regions of existence and stability of endemic equilibria given by Theorem 1 in terms of the parameters a1 and a2 . The other parameters are b 0:000648/day, b1 0:1/day, b2 0:01/day, c 0.5 and / 0.5. In this ®gure the line PRQ is the line a2 aR2 o a1 (or Ro 1) whilst the curve RS is the curve a2 aU2 a1 . P is the point 0; b2 b=c/, Q is the point b=h; 0 and R is the intersection of the line PQ with the line a1 ac1 . Theorem 1 tells us that in the area 2 bounded by the lines PR; RS and PS (including the line between P and S, but not P ; R; S or the other two lines) two endemic equilibria are possible, one of which is stable. On the line RS Fig. 3. Existence and stability of endemic equilibria in terms of a1 and a2 . See text for parameter values. D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 15 (including S but not R) there is one repeated subcritical endemic equilibrium. In the area 1 above the line PQ (including P and the line between P and R, but not R) there is one unique endemic equilibrium which is stable. In the remaining area Z (for zero) bounded by the lines OQ; QR; RS and OS (including O; Q; R, the lines OQ; QR and OS but not S) there are no endemic equilibria. Note that we have shown that the infection-free equilibrium is globally stable when Ro 1 ÿ /Ro2 1 ÿ /Ro1 Ro2 < 1. This condition implies that both Ro and Ro2 are less than 1. Hence Ro 1 ÿ /Ro2 < 1 is a sucient condition for there to be no subcritical endemic equilibria. However it is not necessary. For example in Fig. 3 the line Ro 1 ÿ /Ro2 1 is a line through Q which lies strictly beneath PQ. In the region between this line and PQ and beneath the curve RS, Ro 1 ÿ /Ro2 > 1 and yet there are still no subcritical endemic equilibria. It is tempting to conjecture that whenever there are no subcritical endemic equilibria the infection-free equilibrium is globally asymptotically stable. It is also tempting to conjecture that if Ro > 1 and infection is initially present (so i1 0 i2 0 > 0) then the system approaches the unique endemic equilibrium as time becomes large. However we have not yet been able to prove either of these conjectures. For most epidemic models Ro is a sharp threshold parameter. For Ro < 1 there is only the infection-free equilibrium whereas for Ro > 1 there is additionally a unique endemic equilibrium. Our model diers from that in that for Ro < 1 there may be two endemic equilibria, one stable and one unstable in addition to the usual stable infection-free equilibrium. This occurs if a2 lies in the range aU2 a1 ; aR2 o a1 and a1 lies in the range 0; ac1 . At Ro 1 there is a `backwards bifurcation' of an unstable endemic equilibrium from the infection-free equilibrium. It is possible to use a1 instead of a2 as a bifurcation parameter and we expect that the bifurcation diagrams look similar (see also Fig. 3). Although it is unusual this phenomenon of backwards bifurcation has been observed before by Doyle [7] and Hadeler and Castillo-Chavez [8] in multigroup models for AIDS. All three models involve segregating the population into two groups. Hadeler and Van den Driessche [9] explore this phenomenon of backwards bifurcation further in a more general context. They consider an SIRS model with two social groups corresponding to `normal' and `educated' individuals. They also show that it is possible to derive a two group SIS model from their model using a singular perturbation approach. If in our model a1 a2 and b1 b2 , so that all infectious individuals have the same average infectious period and the same infectivity then our homogeneously mixing model corresponds to a special case of this limiting SIS model. 6. Special cases It is possible to recover some special cases from our more general results: (i) First consider the case where / 1 so all cattle are vaccinated at birth. As / ! 1, b=h ! 1 and ac1 ! ÿ1 so case I b is not relevant. Thus two endemic equilibria are never possible and the usual bifurcation behaviour is observed. Ro ca2 = b2 b is a sharp threshold value and for Ro < 1 there is a unique infection-free equilibrium which is GAS whereas for Ro > 1 this equilibrium is unstable and there is an additional stable endemic equilibrium. (ii) Second if we set c 1; a1 a2 a and b1 b2 b and combine the two classes then we obtain the usual SIS epidemic model. In the proof of Theorem 1 we see A 0 so there can never 16 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 be two endemic equilibria. Thus similar bifurcation behaviour to case (i) is observed with Ro a= b b. 7. Proportional mixing The simple model which we have studied assumes homogeneous mixing between seropositive and seronegative animals. This may not always be appropriate. A simple but more realistic alternative may be proportional mixing. One way of thinking about proportional mixing is that seronegative and seropositive animals spend diering fractions of time making potentially infectious contacts. If these fractions are, respectively, n1 and n2 , then it is straightforwards to show the following corollary to Theorem 1: Corollary 2. The results of Theorem 1 and Corollary 1 hold for proportional mixing between seronegative and seropositive animals with a1 ; a2 and c replaced by a01 a1 n21 ; a02 a2 n1 n2 and c0 cn2 =n1 , respectively provided that b2 1 ÿ c0 b > 0. For realistic parameter values it is often true that the per capita birth rate b is very small compared with b2 , so this condition is likely to be satisfied. For example de Jong et al. [2] take b 0:000684/ day and b2 0:01/day in modelling BRSV amongst cattle. 8. Implications for disease control It may happen that vaccination achieves that individuals are protected from illness when subsequently infected and so, in particular, do not show clinical symptoms. Yet this does not guarantee that transmission via vaccinated animals is excluded. In fact, as noted earlier, the combination of reduced infectivity and a prolonged infectious period may lead to a value of Ro2 that exceeds Ro1 . In such a situation vaccination is helping the infectious agent to spread and we shall see that there does not exist a critical vaccination eort for eradication. Of course vaccination may still be economically bene®cial by reducing losses due to illness. When Ro2 < 1 < Ro1 it is possible to reduce Ro to below one by increasing the vaccination fraction /, and also by doing so to eliminate the infectious agent if it was originally present. On the other hand when Ro2 > max f1; Ro1 g, increasing / acts in the opposite direction, in the sense that (i) Ro increases with /, (ii) if the stable endemic equilibrium does not exist it may be created by increasing /, and (iii) we strongly expect that the stable endemic equilibrium infection level increases with /. In the rest of this section we explore these eects in more detail. We need the following Lemmas. Lemma 3. (a) For Ro1 < 1 let / denote the unique root in 0; 1 of the quadratic equation Q / 0 where b1 bc/ 1 ÿ / ÿ /Ro1 1 ÿ Ro1 ÿ /: Q / / b b2 b b1 Then ac1 > a1 if and only if 0 6 / < / , and ac1 a1 if and only if / / . D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 17 (b) If Ro1 > 1 then a1 > ac1 . (c) If Ro1 1 then / 0 is the unique root of Q / 0 in 0; 1 and a1 P ac1 with equality if and only if / 0. Proof. See Appendix A. Lemma 4. For a1 < ac1 ; aU2 aU2 / is a strictly monotone increasing function of / for 0 6 / 6 / . Note that by Lemma 3(a), aU2 / is de®ned for 0 6 / 6 / . Proof. See Appendix A. Lemma 5. We define aU2 / as the limit of aU2 / as / tends to / from below. If aU2 0 P a2 > ÿ1 aU2 / then let /~ aU2 a2 < / . (So /~ is the unique root in 0; / of the equation aU2 / a2 .) Then there are two distinct endemic equilibria if and only if Ro < 1; Ro1 < 1; Ro2 > 1 and either (i) a2 > aU2 0 and 0 6 / < / 1 ÿ Ro1 = Ro2 ÿ Ro1 , or (ii) aU2 0 P a2 > aU2 / aR2 o / and /~ < / < / . Note that by putting / 0 and h b= b b1 into Eq. (9.1) we deduce that s!2 p b b b2 b 1 aU2 0 : b 1 ÿ Ro1 c b1 Proof. See Appendix A. Corollary 3. Necessary conditions for two endemic equilibria are Ro < 1 and Ro1 < 1 < Ro2 . The condition Ro1 < 1 < Ro2 means that in the SIS model with a a1 and b b1 (so all animals in the SIS model behave as ®rst time susceptible animals in our model discussed above) infection cannot persist, but that in the SIS model with a a2 and b b2 infection will persist. For a real disease it is more likely that Ro1 > Ro2 as ®rst time infected animals spread infection at a higher rate than subsequent time infected animals, so a1 > a2 , and also once infected and recovered hosts will probably defend themselves better against an infectious agent, so c < 1. However it is still ÿ1 possible for Ro2 > 1 > Ro1 if bÿ1 2 > b1 , so the average length of subsequent infectious periods exceeds that of the ®rst, as appears to be the case for BRSV in cattle [2]. We can use Lemma 5 and Theorem 1 to express the possibilities for existence, uniqueness and stability of endemic equilibria in terms of the vaccination proportion /. These are given in Theorem 2. Recall that in general if Ro < 1 then infection cannot invade an infection-free population in which a steady-state proportion / of new-born individuals are vaccinated. If Ro > 1 then infection will always invade an infection-free population and a unique stable endemic equilibrium is always possible. If two distinct endemic equilibria are possible then the one with the higher value of i1 is locally stable and the other is unstable. 18 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 Theorem 2. (I) If Ro1 > 1 and Ro2 > 1 then Ro > 1 whatever the value of /. (II) If Ro1 > 1 P Ro2 then: (a) For / < / Ro1 ÿ 1= Ro1 ÿ Ro2 we have Ro > 1. (b) For / P / we have Ro 6 1 (with equality if and only if / / ). No endemic equilibria are possible. (III) If 1 P Ro1 and 1 P Ro2 then Ro 6 1 with equality if and only if (a) Ro1 1 and / 0; b Ro2 1 and / 1 or c Ro1 Ro2 1. No endemic equilibria are possible. (IV) If Ro2 > 1 P Ro1 then for / < / 1 ÿ Ro1 = Ro2 ÿ Ro1 ; Ro < 1. For / > / we have Ro > 1. (a) If a2 > aU2 0 then / > / . For 0 6 / < / , two endemic equilibria are possible. If Ro1 < 1 and / / then there is a unique endemic equilibrium which is stable. If Ro1 1 and / / then there are no endemic equilibria. (b) If aU2 0 P a2 > aU2 / aR2 o / then / > / . No endemic equilibria are possible for ~ / , two for /~ < / < / . For / /~ < / there is a unique (repeated) subcrit0 6 / <min /; ical endemic equilibrium. If Ro1 < 1 and / / then there is a unique endemic equilibrium which is stable. If Ro1 1 and / / then there are no endemic equilibria. (c) If aU2 / P a2 then / 6 / , with equality if and only if aU2 / a2 . No endemic equilibria are possible if / 6 / . Proof. Cases I and II(a) are straightforward. In case II(b) it is straightforward that / P / implies Ro 6 1 with equality if and only if / / . By Lemma 3, a1 > ac1 so by Theorem 1 no endemic equilibria are possible. In case III it is straightforward that Ro 6 1 whatever the value of /, with equality if and only if the stated conditions are true. For / P / ; a1 P ac1 by Lemma 3 and again no endemic equilibria are possible. For / < / ; a1 < ac1 . However, the proof of Lemma 5 shows that if a2 P aU2 / then Ro2 > 1. This is a contradiction as Ro2 6 1. So a2 < aU2 / < aU2 / (using Lemma 4). Hence by Theorem 1 there are no endemic equilibria. For case IV it is straightforward to show that Ro < 1 for / < / and Ro > 1 for / > / . As in the proof of Lemma 5, aU2 / aR2 o / and for a2 > aR2 o / ; / < / . Similarly if a2 6 aR2 o / then / P / , with equality if and only if a2 aR2 o / . In case IV(a) by Lemma 5 two endemic equilibria are possible for 0 6 / < / . For / / Ro 1. Using Lemma 3 if Ro1 < 1 then as / < / ; ac1 > a1 . Theorem 1 says that there is a unique endemic equilibrium which is stable. If Ro1 1 then / / 0 and a1 ac1 . Hence when / / 0 Theorem 1 now says that there are no endemic equilibria.In case IV(b) / > / : ~ then Ro < 1 and / < /~ so aU / > a2 and there are no en(i) If Ro1 < 1 and 0 6 / <min / ; / 2 demic equilibria; (ii) If Ro1 < 1 and / /~ < / then aU2 / a2 . As /~ < / ; ac1 > a1 and so by Theorem 1 there is a unique (repeated) subcritical endemic equilibrium; (iii) If Ro1 < 1 and /~ < / < / then Lemma 5 says that there are two endemic equilibria; (iv) If Ro1 < 1 and / / then / < / and so by Lemma 3, ac1 > a1 . Hence by Theorem 1 there is a unique endemic equilibrium which is locally stable; (v) If Ro1 1 then / / 0 and so if / 0 by Lemma 3, a1 ac1 . Theorem 1 now says there are no endemic equilibria. D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 19 In case IV(c) we have / P / . For 0 6 / < / then by Lemma 3, ac1 > a1 and aU2 / > aU2 / P a2 so by Theorem 1 there are no endemic equilibria. For / P / P / by Lemma 3, a1 P ac1 so again Theorem 1 implies that there are no endemic equilibria. The situation described in Theorem 2(IV) can perhaps be clari®ed a little with the aid of Fig. 4, which illustrates in a qualitative manner the existence and stability of possible endemic equilibria for dierent values of / and a2 . In Region 2, Ro < 1 and two endemic equilibria exist, one of which is stable. In Region 1, Ro > 1 and there is a unique endemic equilibrium which is stable. In Region Z, there are no endemic equilibria. The behaviour at the boundaries of these regions can also be deduced using Theorem 2. We see that in each situation where the infection persists without vaccination, either the criterion Ro / 1 gives the correct critical vaccination proportion for infection elimination, as well as for the prevention of infection invasion into an infection-free population (case II) or infection cannot be eliminated (cases I, IV(a)). In case IV(a) the criterion Ro / > 1 (or / > / ) does not give the correct condition for the elimination of infection. This is because infection will always persist even if / < / (but in the latter case it will not invade into an initially infection-free population). In case IV(b) infection does not persist with no vaccination, but can persist at intermediate vaccination levels (with Ro < 1). Note that in Theorem 1 we showed that the infectionfree equilibrium will be globally stable provided that 1 > Ro 1 ÿ /Ro2 1 ÿ /Ro1 Ro2 . This inequality implies that Ro < 1 and Ro2 < 1 so corresponds to cases II(b) and III when no subcritical endemic equilibria are possible for Ro < 1. This is consistent with the global stability result. We can illustrate these results numerically by using parameter values for BRSV amongst cattle. We take values for b; b1 ; b2 and c as in Section 5. We consider values for a1 and a2 as in Figs. 1 and 2: (i) a1 0:05/day Ro1 0:4968; a2 0:05; 0:1 and 0:25/day Ro2 4:6957; 9:3914 and 23:4786 respectively; (ii) a1 0:15/day Ro1 1:4903; a2 0:01; 0:02 and 0:04/day Ro2 0:9391; 1:8783 and 3:7566. These values for a1 and a2 are roughly consistent with the infection mixing matrix for BRSV estimated by de Jong et al. [2]. Fig. 4. Existence of endemic equilibria when Ro2 P 1 > Ro1 for dierent values of /. See text for explanation. 20 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 For (i) we have Ro2 > 1 > Ro1 ; / 0:9438; aU2 / aR2 o / 0:02193/day and aU2 0 0:02707/day. Hence by Theorem 2 case IV(a) we have for / < / two endemic equilibria exist one of which is stable, and for / P / we have a unique endemic equilibrium which is stable. For a2 0:05/day, / 0:1198; a2 0:1/day, / 0:05658 and a2 0:25/day, / 0:02190. For (ii) if a2 0:02 or 0:04/day then whatever the value of / infection will always invade and there is always a unique endemic equilibrium which is stable (Theorem 2, case I). If a2 0:01/day, then for / < / 0:8896 infection always invades and there is a unique endemic equilibrium which is stable, whereas for / P / infection never invades and there are no endemic equilibria (Theorem 2, case II). 9. Application to Aujesky's disease Another example where it is known that the infection can spread amongst seropositive animals is pseudorabies virus (Aujeszky's disease virus) in pigs. Pseudorabies virus is a highly neurotropic alphaherpesvirus for which swine are the natural host, the sole reservoir, and the sole source of virus transmission. It is well established that virus transmission (at a reduced rate) can take place in seropositive animals who are either protected by maternal antibodies or who have been immunised [10]. Sab o and Blaskovic [11] document that pigs which have experienced an episode of infection can subsequently be re-infected. However, pseudorabies virus persisted in tonsils for a shorter period in the second infection, suggesting that the infectious period is shorter for the second and subsequent infections than for the ®rst. De Jong and Kimman [10] estimate Ro1 10:0 and Ro2 0:5 for pseudo-rabies virus. Smith and Grenfell [12] give an impressive mathematical analysis of dierent control strategies for pseudorabies virus. Their model is similar to ours in that it assumes that seropositive susceptible animals have reduced susceptibility. However, their model does not distinguish between ®rst and subsequent time infectives. (They do state that the infectious period for primary and secondary infections is probably not the same and brie¯y qualitatively discuss the eect of this.) They also allow for environmental transmission of pseudorabies virus and recrudesence. We feel that the transmission term aSI=N used in our model is more appropriate than the classical mass action transmission term aSI used by Smith and Grenfell [12] as our variables are numbers and not densities and also recent data on pseudorabies virus in experimental pig populations show that the term aSI=N is more realistic [13]. The assumption made in our model that there are no deaths from the disease is true for pseudorabies virus [12]. For pseudorabies de Jong et al. [14] eectively set b 0 as b is very small compared with the other parameters in the model. If in our general model where b is not necessarily very small we set a1 a2 b~1 N ; b1 b2 c~; c b~2 =b~1 (where b~2 < b~1 ) and / 0 then we obtain a special case of the model of Smith and Grenfell [12] for pseudorabies virus with no environmental eects and no recrudesence of disease (so in their model we must set a d 0). In this case Smith and c b which agrees with ours. We agree with Grenfell obtain a threshold parameter Ro b~1 N= ~ their conclusion that for Ro > 1 there is a unique stable endemic equilibrium. However, we have also shown that endemic equilibria are possible when Ro < 1. They miss this point as they assume that Ro must exceed one for an endemic equilibrium to exist. This may have implications for control strategies. They also obtain threshold population size results for the persistence of D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 21 pseudorabies virus in a pig population, but these do not carry over to our model because of the dierent infection transmission term assumptions. 10. Summary and conclusions Traditional SIS models for infections amongst animals assume that all infected animals transmit the infection at the same rate and the course of the infection is the same in all animals. However for some infections there is increasing evidence that seropositive animals who have experienced the infection or been vaccinated may still transmit the infection albeit at a lower rate. The examples discussed here are BRSV amongst cattle and swine pseudorabies virus. One possible defect of our model for pseudorabies virus is that it does not distinguish between pigs who are seropositive because they have experienced the infection and those who are seropositive because they have been vaccinated. In practice, although both types can be re-infected, experiencing the infection provides more eective immunity [15]. In this paper we considered a simple model where animals are divided into seronegative susceptibles, seronegative infectives, seropositive susceptibles and seropositive infectives. A fraction / of all animals are vaccinated at birth and immediately enter the seropositive class. Seropositives transmit infection at a dierent (probably lower) rate than seronegatives and the infectious periods may also dier (probably longer for seropositives). There are no deaths from the infection so the population size remains constant. We assumed homogeneous and proportional mixing between seropositives and seronegatives although evidence on BRSV indicates that this may not be realistic. We derived an expression for the basic reproduction number and performed an equilibrium and local stability analysis. Unusually, it may be possible for there to be two endemic equilibria (one stable and one unstable) for Ro < 1. The condition Ro > 1 can be thought of as a condition for whether the infection will invade an initially infection-free population. If Ro < 1 then although infection cannot invade an infection-free population it may still ultimately persist at a non-zero endemic level if it was present initially. The results were extended from homogeneous to proportional mixing. We expressed conditions for the persistence of infection and single or multiple endemic equilibria in terms of the vaccination proportion / and the infection transmission rate a2 . We found that in each situation where the infection persists without vaccination, either the criterion Ro 1 gives the correct vaccination proportion necessary for infection elimination as well as for the prevention of infection invasion, or infection cannot be eliminated. The only case where Ro 1 does not give the correct answer is where Ro1 < 1 in immunologically naive animals and Ro2 > 1 in vaccinated animals, when vaccination should not be attempted. The values of Ro1 and Ro2 can usually be determined experimentally to decide whether vaccination should be attempted. Our analysis is based on a deterministic model and hence will be valid provided that all of the population sizes under consideration are relatively large. However, as much of the spread of BRSV takes place on farms the population size may not be so large and a stochastic model may be more appropriate. In a situation where Ro1 < 1 < Ro2 and Ro 6 1 it may be possible that due to stochastic eects the infection may overcome the invasion barrier and reach the quasi-stationary state corresponding to the locally stable endemic equilibrium. So just as an infective agent may fail to create an epidemic outbreak, despite Ro being bigger than one, due to demographic stochas- 22 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 ticity, it may be able to establish itself for a very long period, despite Ro being less than one, if there is deterministic bistability. In the deterministic model the backwards bifurcation of the endemic equilibrium can occur only when Ro1 < 1 < Ro2 . As discussed in Section 8 it is probably more likely that Ro1 > Ro2 , (the reproduction number decreases for seropositive animals). However, it is still possible that Ro1 < 1 < Ro2 if the infectious period for second and higher time infected animals exceeds that for ®rst time infecteds. This appears to be the case for BRSV in cattle (de Jong et al. [2]). However, in the same paper de Jong et al. estimated Ro1 36:5 and Ro2 1:14. Hence it is probable that Ro1 > Ro2 for BRSV although the possibility that Ro2 > Ro1 remains for other diseases or BRSV in other situations. The phenomenon of backwards bifurcation of the endemic equilibrium away from the infection-free equilibrium has been observed before in two group models for the spread of HIV and AIDS by Hadeler and Castillo-Chavez [8] and Doyle [7], and also in a more abstract setting by Hadeler and Van den Driessche [9]. To our knowledge our paper is the ®rst time such local stability results for the endemic equilibria have been shown analytically. Appendix A Proof of Lemma 1 1 L p1 a2 aU2 ÿ ; 2p2 2 1 2b2 hb1 bs ÿ a1 bh : r r2 A:1 The ®rst term is aB2 so 12 aL2 aU2 > aB2 . Using (A.1) and the inequality bb1 > a1 r 1 L b2 b1 h a1 a2 aU2 > s > aA2 : 2 r2 b1 ~ ÿ h < h if and only if ~ A Proof of Lemma 2. (i) Write B~ B aR2 o and A~ A aR2 o . ÿ B= R U 1 L U A o ~ < 0. (Note that a P a > a a > a B~ 2Ah (using Lemma 1) implies that 2 2 2 2 2 2 Ro Ro Ro ~ A A a2 < 0:) Substituting A a2 and B a2 from (4.10) and simplifying it is routine to show ~ < 0 if and only if that B~ 2Ah ÿ b ÿ a1 h s a1 h b2 b ÿ b1 h b/ < 0: Ro a2 aR2 o c Substituting for aR2 o from (4.12) and simplifying yields the inequality b2 /c h b1 b ÿ a1 r < b2 b b ÿ a1 h as required, b > a1 h as b1 b > a1 r. The proof of (ii) follows similarly. D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 23 Proof of Lemma 3. (a) By the de®nition of ac1 ; a1 < ac1 if and only if b2 bb1 h ÿ b2 c/ b2 br ÿ b2 c/ b1 = b b1 1 ÿ / ÿ bc/= b b2 using the definition of h: / b1 = b b1 1 ÿ / ÿ bc/= b b2 1 ÿ /Ro1 < This is equivalent to Q / > 0. But Q / is a quadratic in / with Q 0 1 ÿ Ro1 b1 = b b1 > 0 and Q 1 ÿbc= b b2 < 0, hence has a unique root / in 0; 1 and a1 < ac1 if and only if 0 6 / < / . Similarly a1 ac1 if and only if / / . (b) Note that a1 > ac1 if and only if 1 ÿ /Ro1 > b2 bb1 h ÿ b2 c/ : b2 br ÿ b2 c/ (i) If 1 ÿ /Ro1 > 1 then 1 ÿ /Ro1 > 1 P b2 bb1 h ÿ b2 c/= b2 br ÿ b2 c/ so the result follows: (ii) If 1 P 1 ÿ /Ro1 then /Ro1 b1 b1 b1 b 1 ÿ /Ro1 ÿ Ro1 ÿ 1 /Ro1 > 0: b b1 b b1 b b1 b b1 Therefore 1 ÿ / /Ro1 b1 b1 1 ÿ /Ro1 ÿ b b1 b b1 bc/ 1 ÿ 1 ÿ /Ro1 > 0: b b2 There is strict inequality here as Ro > 1 implies either 1 > / or c > 0. Re-arranging 1 ÿ /Ro1 > b1 = b b1 1 ÿ / ÿ bc/= b b2 : / b1 = b b1 1 ÿ / ÿ bc/= b b2 So a1 > ac1 as required. (c) This is straightforward using the proofs of (a) and (b). Proof of Lemma 4. If a1 < ac1 then by Lemma 3, Ro1 < 1 and 0 6 / < / . Hence a1 < ac1 < bb1 =r so p1 < 0 and p21 ÿ 4p0 p2 > 0. Note that by the de®nition of h we have r b1 h b/ b 1 ÿ h. Therefore p ÿp1 p21 ÿ 4p0 p2 U a2 / ; 2p2 h b1 b1 h 1 2 s ÿ a1 2 1 ÿ h 1 ÿ h 1 ÿ h 1 ÿ h 1 ÿ h s bh b1 ÿ a1 h s 1 using 4:5: 1ÿh 1ÿh Now 0 6 h 6 b= b b1 and as ac1 > a1 , then as a1 < ac1 < bb1 =r and r b1 h b/ b 1 ÿ h we have b1 = 1 ÿ h ÿ a1 > 0. Therefore 24 D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 1 aU2 / 1 ÿ h (s r)2 b1 bh ÿ a1 s 1 h : 1ÿh 1ÿh A:2 As h; 1= 1 ÿ h and h= 1 ÿ h are all strictly monotone increasing functions of h 2 0; b= b b1 ; aU2 / is strictly monotone increasing as a function of h 2 b 1 ÿ / = b b1 ; b= b b1 so strictly monotone decreasing for / 2 0; / : Proof of Lemma 5. By Theorem 1 two endemic equilibria are possible if and only if Ro < 1; a1 < ac1 and a2 > aU2 /. By Lemma 3, a1 < ac1 is equivalent to Ro1 < 1 and 0 6 / < / . Moreover / / implies a1 ac1 and so by Lemma 2 ÿ B=A jaRo ÿh h. Therefore by the proof of Theorem 1 2 (p. 8.) aR2 o aU2 . Thus aR2 o / aU2 / . Suppose now that two endemic equilibria are possible. Then Ro < 1; Ro1 < 1; 0 6 / 6 / and a2 > aU2 /. So by Lemma 4 a2 > aU2 / > aU2 / b ÿ a1 h b2 b : c b/ A:3 Here h b= b b1 1 ÿ / . But b ÿ a1 h 1 ÿ 1 ÿ / Ro1 1 > 1; / b1 = b b1 1 ÿ / ÿ bc/ = b b1 b/ / as Q / 0. Hence inequality (A.3) implies Ro2 > 1 is a necessary condition for two endemic equilibria. So Ro / 1 ÿ /Ro1 /Ro2 is strictly increasing in / and the condition Ro < 1 is equivalent to / < / . As a2 > aR2 o / (using inequality (A.3)), Ro / > 1 and so / < / . As a2 > aU2 / then either (i) a2 > aU2 0 or (ii) aU2 0 P a2 > aR2 o / . In case (i) we have 0 6 / < / and in case ~ > aU / and / < / we have / > / > /. ~ Thus the conditions in Lemma 5 are (ii) as a2 aU2 / 2 necessary conditions for two distinct endemic equilibria to exist and a similar argument shows that they are sucient. References [1] H.W. Hethcote, J.A. Yorke, Gonorrhea dynamics and control, Lecture Notes in Biomathematics, vol. 56, Springer, Berlin, 1974. [2] M.C.M. de Jong, W.H.M. Van der Poel, J.A. Kramps, A. Brand, J.T. Van Oirschot, Persistence and recurrent outbreaks of bovine respiratory syncytial virus on dairy farms, Am. J. Vet. Res. 57 (1996) 628. [3] M.C.M. de Jong, O. Diekmann, J.A.P. Heesterbeek, How does transmission of infection depend on population size? in: D. Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data, Cambridge University, Cambridge, 1994, p. 84. [4] O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the de®nition and the computation of the basic reproduction number Ro in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990) 365. [5] D. Greenhalgh, K. Dietz, Some bounds on estimates for reproductive ratios derived from the age-speci®c force of infection, Math. Biosci. 124 (1994) 9. [6] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Clis, NJ, 1962. D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25 25 [7] M.T. Doyle, A constrained mixing two-sex model for the spread of HIV, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, vol. 1, Theory of Epidemics, Wuerz Publishing, Winnipeg, Canada, 1995, p. 209. [8] K.P. Hadeler, C. Castillo-Chavez, A core group model for disease transmission, Math. Biosci. 128 (1995) 41. [9] K.P. Hadeler, P. Van den Driessche, Backwards bifurcation in epidemic control, Math. Biosci. 146 (1997) 15. [10] M.C.M. de Jong, T.G. Kimman, Experimental quanti®cation of vaccine induced reduction in virus transmission, Vaccine 8 (1994) 761. [11] A. Sab o, D. Blaskovic, Resistance of pig tonsillary and throat mucosa to re-infection with a homologous pseudorabies virus strain, Acta Virol. 14 (1970) 17. [12] G. Smith, B.T. Grenfell, Population biology of pseudorabies in swine, Am. J. Vet. Res. 51 (1990) 148. [13] A. Bouma, M.C.M. de Jong, T.G. Kimman, Transmission of pseudorabies virus within pig populations is independent of the size of the population, Prev. Vet. Med. 23 (1995) 163. [14] M.C.M. de Jong, O. Diekmann, J.A.P. Heesterbeek, The computation of Ro for discrete-time epidemic models with dynamic heterogeneity, Math. Biosci. 119 (1994) 97. [15] C. Miry, M.B. Pensaert, Aujeszky's disease virus replication in tonsils and respiratory tract of non-immune and immune pigs, in: J.T. Van Oirschot (Ed.), Vaccination and Control of Aujeszky's Disease, Kluwer Academic, Dordrecht, 1989, p. 163.