Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol165.Issue2.Jun2000:

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

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

c_{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 ex-pression 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) forRo<1 and in this case at Ro ˆ1 there is a backwards bifurcation of an unstable endemic equilibrium from the in-fection-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:david@strath.stams.ac.uk (D. Greenhalgh).

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

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 SIS1I1S1)

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 a€ected. It is therefore often concluded that the virus is con-tinually 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 pre-sumption 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 pe-riodicity 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 se-ropositive cattle and they conclude that persistence of the infection amongst sese-ropositive cattle is still plausible.

2. The model

LetS1;S2 denote respectively the numbers of ®rst time susceptible cattle and susceptible cattle who have been previously infected. LetI1;I2denote respectively the numbers of ®rst time infected cattle and infected cattle who have experienced previous infection. So

N _ˆS1_‡S2_‡I1_‡I2

is the total number of cattle. Suppose thatbis the per capita birth rate per unit time and that the infectious period is exponentially distributed with parameterb1 (meanbÿ

1

cattle, and parameter b₂ (mean bÿ₂1) 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 isa1:a2.

More precisely we assume that ®rst time susceptible cattle come into contact with and are in-fected by ®rst time inin-fected cattle at per capita ratea1I1=N_†and by other infectious cattle at per capita rate a2I2=N_†. For subsequent time susceptible (seropositive susceptible) cattle transmis-sion is less ecient so these rates are reduced by a factor c 06c61_†. We use the `true mass action' transmission termaSI=N rather than the classical mass action transmission termaSI as it has been argued that this is more plausible [3]. If the population size is constant, then by re-de®ning a this will make no di€erence to the dynamics of the model, but will a€ect the for-mulation 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 di€erent sizes.

We assume homogeneous mixing between seropositive and seronegative cattle. We also as-sume that the population under consideration is of constant sizeN, so births balance deaths and there are no deaths from the infection. Thus the per capita death rate for all four types of cattle isb. The assumption that there are no deaths from the infection is true for BRSV [2]. A fraction /06/61_† of individuals are vaccinated at birth, these individuals immediately enter the seropositive susceptible classS2. 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 e€ect 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 tempo-rarily provide a stronger protection than the ultimate e€ect of having previously experienced the infection.

Then it is straightforward to show that the di€erential equations which describe the spread of the infection are

dS1

dt ˆb…1ÿ/†Nÿ S1

N…a1I1‡a2I2† ÿbS1;

dI1

dt ˆ S1

N…a1I1‡a2I2† ÿb1I1ÿbI1; …2:1†

dS2

dt ˆb/Nÿ

cS2

N …a1I1‡a2I2† ‡b1I1‡b2I2ÿbS2;

dI2

dt ˆ

cS2

N …a1I1‡a2I2† ÿb2I2ÿbI2

and S1‡S2‡I1‡I2 ˆN, with suitable initial conditions S1…0_†;S2…0_†;I1…0_†;I2…0_† P0 and S10_{† ‡}S20_{† ‡}I10_{† ‡}I20_{† ˆ}N.

ds1

dt ˆb…1ÿ/† ÿs1…a1i1‡a2i2† ÿbs1; …2:2a†

di1

dt ˆs1…a1i1‡a2i2† ÿ …b‡b1†i1; …2:2b†

ds2

dt ˆb/ÿcs2…a1i1‡a2i2† ‡b1i1‡b2i2ÿbs2; …2:2c†

di2

dt ˆcs2…a1i1‡a2i2† ÿ …b‡b2†i2; …2:2d† s1_‡s2_‡i1_‡i2 _ˆ1: 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 programRois 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 intondisjoint groups thenRois generally given as the largest eigenvalue of annnmatrix of secondary cases [4].

We de®neRo1_ˆa1=…b1‡b†and Ro2 ˆa2c=…b2‡b†. Ro1 is the basic reproduction number for

an SIS epidemic model with no vaccination where the per capita e€ective contact rate isa1 and the

average infectious period, conditional on survival to the end of it, is bÿ₁1. This model can be obtained from ours by settingc_ˆ1; a2 ˆa1andb2 ˆb1. SimilarlyRo2ˆa2c=…b2‡b†is the basic

reproduction number for an SIS model, where the per capita e€ective contact rate isa2cand the

average infectious period, conditional on survival to the end of it, is bÿ₂1. This model can be obtained from ours by setting /_ˆ1 and S10_{† ˆ}I10_{† ˆ}0, i.e. all individuals enter the classS2

and S1t_{† ˆ}I1t_{† ˆ}0 for allt. De®ne

Ro_ˆ1_ÿ/_†Ro1_‡/Ro2;

ˆ …1_ÿ/_† a1

b_1‡b‡/

a2c

b_2‡b:

To show that Ro is the basic reproduction number in our model, consider a population at the infection-free equilibriumS1;S2_{† ˆ}1_ÿ/_†N; /N_†. For i;j_ˆ1;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

m11 m12 m21 m22

ˆ

a1…1ÿ/†

b1‡b

a1c/

b1‡b

a2…1ÿ/†

b2‡b

a2c/

b2‡b

!

: 3:1_†

For example, a single infected class 1 case is infectious for time b_1‡b_†ÿ1 and during the in-fectious period transmits the disease to ®rst time and subsequent time susceptible cattle at rate a1=Nanda1c=N, respectively. At the disease-free equilibrium there are…1ÿ/†N ®rst time and/N

m11_ˆ a1

b_1‡b…1ÿ/†

N N ˆ

a1

b_1‡b…1ÿ/†

secondary cases amongst ®rst time and

m12ˆ_ba1c 1‡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 trace1_ÿ/_†Ro1_‡/Ro2, since one eigenvalue equals zero. Note that ifRo2 >1>Ro1thenRois increasing in/and so increasing the vaccination proportion / has the e€ect 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_ˆb…1_ÿ/_†=b‡b_1†; 4:1_†

his 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 e€ectively

infected at birth. In this case the e€ective entry rate of susceptibles into the second class is r_ˆb/_‡b₁h. For notational convenience we de®ne

s_ˆb2‡ …1ÿc†b†=c: …4:2†

We shall later express our equilibrium and stability results in terms of the bifurcation parameter a2. Note thatRo <1 if and only ifb>a1hand a2<aR2o, where

aRo

2 ˆ …bÿa1h†…b2‡b†=b/c: …4:3†

We choose the superscriptRo to indicate that, for this value ofa2; Ro has the value one. De®ne

Ga_{† ˆ}p0‡p1a‡p2a2; …4:4†

where

p0ˆa21b 2_h2

‡b2s2

‡2a1b2hs; p1ˆ2a1bhrÿ2brsÿ4b2b1h and p2 ˆr2: …4:5†

Also de®ne

ac_1ˆb h

‰…b2‡b†b1hÿb2c/Š

…b_2‡b†r_ÿb2_c/ : …4:6†

Note that ac

1<…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_{† ˆ}1_ÿ/;0;/;0_†.This

equilibrium is locally asymptotically stable (LAS) when Ro<1 and unstable when Ro>1. If

is a unique endemic equilibrium which is LAS. For Ro61 there are no endemic equilibria when

…b=h_†>a1Pac1.On the other hand ifRo <1andac1>a1,then there is an open interval ofa2-values

aU

2 <a2 <a

Ro

2 for which there are precisely two endemic equilibria. Of these two equilibria the one

with the highest value ofi₁is LAS whilst the other is unstable. The threshold valueaU

2 is the largest of

the two positive roots of the equationGa2† ˆ0 (which are both less thana2Rowhenac1>a1holds).At

a2 ˆaU2 there is a unique endemic equilibrium (which is unstable) and for smaller values of a2 no

endemic equilibrium exists. At a2ˆaR2o we haveRoˆ1,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) lets₁;s₂;i₁ and i₂

denote the equilibrium values ofs1;s2;i1 andi2, respectively. At equilibrium from (2.2a)±(2.2d):

s_{1 ˆ} b…1ÿ/†

Eq. (4.9) is obtained by solving the two expressions for s

1 in (4.7) for i2 as a function of i1.

Substituting these expressions into

b/_ÿcs₂a1i_1‡a2i_{2† ‡}b₁i_1‡b₂i_2ÿbs_{2 ˆ}0

we ®nd after some manipulations the equation Fx_{† ˆ}0, where

x_ˆh_ÿi₁

A solution for x corresponds to an endemic equilibrium solution if and only if 0<x<

Fx_{† ˆ}0 has two positive real roots forxif (4.13) holds, equivalentlya2 >max…aA2;aB2†anda2 <aL2

a2 2 ‰0;1†. Hence asa2decreases these roots move towards each other and eventually co-alesce at

a2 ˆaU2. Fora2 ˆaU2 ÿd, wheredis small and positive,F…x†<0 forx2 ‰0;b=a1Š. HenceF…x†<0

It is straightforward to translate these equilibrium results into those in the statement of Theorem 1.

Lemma 1. Ifbb1 >a1r then12…aL2 ‡aU2†>max…aA2;aB2†:

Proof.See Appendix A.

Local Stability of IFE

Linearising about the IFEs1;i1;s2;i2† ˆ …1_ÿ/;0;/;0_†we see that the characteristic equation has roots x_{ˆ ÿ}b(twice) and the roots of the quadratic

x2_{‡ ‰}b‡b_{1† ‡}b‡b_{2† ÿ}a1…1ÿ/† ÿa2c/Šx‡ …b‡b1†…b‡b2† ÿ …b‡b1†a2c/

ÿ …b_‡b_2†a1…1ÿ/† ˆ0: …4:14†

Hence by the Routh±Hurwitz conditions necessary and sucient conditions for local stability are: (i)

b_‡b1‡b‡b2 >a1…1ÿ/† ‡a2c/; …4:15†

and (ii)

…b‡b_1†b‡b_2†>b‡b_1†a2c/‡ …b‡b2†a1…1ÿ/†: …4:16†

(4.16) can be rewritten as 1>Ro. If 1>Ro then

…b‡b_{1† ‡}b‡b_2†>_‰b‡b_{1† ‡}b‡b_2†Š a1…1ÿ/†

b_‡b₁

‡ a2c/ b_‡b₂

>a1…1ÿ/† ‡a2c/:

Hence the IFE is LAS ifRo<1 and unstable forRo>1 as required.

Global Stability of IFE when Ro_‡1_ÿ/_†Ro2<1. From Eq. (2.2a) we have thatds1=dt_†6b1_ÿ/_{† ÿ}bs1.

De®nes11 ˆlimt!1 supTPts1…T†. The solutionx1…t† of

dx1

dt ˆb…1ÿ/† ÿbx1; x1…0† ˆs1…0†

is a super-solution fors1t_†, (x1t_†Ps1t_†for allt). Sincex1t_{† !}1_ÿ/ast_{! 1}, given >0 there is t0>0 such that s1t_†61_ÿ/_‡ for t>t0. Hence s1₁ 61_ÿ/_‡. But >0 is arbitrary. So letting _!0 we deduce thats1₁ 61_ÿ/.

As 1>Ro_‡1_ÿ/_†Ro2 _ˆ1_ÿ/_†Ro1_‡Ro2, we can choose >0 small enough so that 1>1_ÿ/_‡†Ro1_‡Ro2. Then there existst0such thats161_ÿ/_‡fortPt0. From Eqs. (2.2b) and (2.2d) we have fortPt0, writing i_ˆi1;i2_†T;

di

dt6Qi;

where

Q_ˆ …1ÿ/‡†a1ÿ …b‡b1† a2…1ÿ/‡†

ca1 ca2ÿ …b‡b2†

:

x1‡M andx2‡M. Without loss of generality x1‡M is the spectral radius, so bothx1 and x2

are real,x1 >x2 and eQˆx1e. The characteristic equation ofQis

x2_‡a1x_‡a2 _ˆ0;

where

a1 _ˆb_‡b_1†1_ÿ1_ÿ/_‡†Ro1_{† ‡}b_‡b_2†1_ÿRo2_†;

and

a2 _ˆb_‡b_1†b_‡b_2†1_ÿ1_ÿ/_‡†Ro1_ÿRo2_†:

As 1>1_ÿ/_‡†Ro1‡Ro2; a1 and a2 are both strictly positive so by the Routh±Hurwitz con-ditionsx1 and x2 are both strictly negative. Moreover fortPt0

d

dt…ei†6eQiˆx1ei:

Integrating

06eit_†6eit1_†ex1…tÿt1†_:

So eit_{† !}0 ast_{! 1} which implies that both i1 and i2 tend to zero as t_{! 1} ase is strictly positive.

Hence given1 >0 there exists t1Pt0 such that fortPt1, a1i1‡a2i26b1. So for tPt1,

ds1

dt Pb…1ÿ/ÿs1…1‡1††:

A similar argument to the one showing that s1₁ 61_ÿ/ now shows that

s1;1ˆlimt!1 infTPts1…T†P…1ÿ/†=…1‡1†. But1 is arbitrary so letting1 !0; s1;1P1ÿ/. Thus

1_ÿ/Ps1₁ Ps1;1P1ÿ/:

Sos1

1 ˆs1;1ˆ1ÿ/ands1 !1ÿ/ast! 1. Thuss2ˆ1ÿs1ÿi1ÿi2 !/ast! 1and the IFE is GAS.

Stability of Endemic Equilibria

Suppose thats₁;i₁;s₂;i_2†is an endemic equilibrium. Substitutings2 _ˆ1_ÿs1_ÿi1_ÿi2into Eqs. (2.2a), (2.2b) and (2.2d) where appropriate to eliminate s2 and linearising about s1;i1;i2† the

Jacobian is

J ˆ

ÿa1i_1ÿa2i_2ÿb _ÿa1s_{1 ÿ}a2s₁

a1i_1‡a2i₂ a1s_1ÿb_‡b_1† a2s₁

ÿca1i

1‡a2i2† ÿc…a1i1‡a2i2† ‡a1cs2 ÿc…a1i1‡a2i2† ‡a2cs2 ÿ …b_‡b_2†

2

6 6 4

3

7 7 5

:

Using the equilibrium versions of (2.2a) and (2.2d), expanding the characteristic equation and simplifying using the equation

which is immediate from the equilibrium equations (2.2b) and (2.2d) we deduce that the

char-from the equilibrium equations) we deduce that A1 >0; A2 >0 and A1A2 >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)

Substituting (4.23) into the equilibrium equation (2.2d) and simplifying we deduce that

i₂

Substituting (4.24) and (4.25) into (4.22) we deduce after some algebra that

A3_ˆb_‡b_1†…hÿx†

x2 ‰cb 2_h

AsAx2_‡Bx_‡C_ˆ0 we can add b_‡b_1†h_ÿx_†=x2_†a2c‰Ax2‡Bx‡CŠ toA3 without changing it

so

A3 ˆ …b‡b_1†…hÿx†

x a2c‰2Ax‡BŠ: …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 >a2Ro. Then F…0†<0 and our equilibrium results have

shown that Fx_{† ˆ}0 has exactly one root in 0<x<minb=a1;h†. This root corresponds to the

endemic equilibrium and is always stable.

(i) ifAa2†<0 then the second root ofF…x† ˆ0 lies in‰h;1†soxˆ …ÿB‡



B2_ÿ4AC p

†=2Aand 2Ax‡B>0.

(ii) ifAa2† ˆ0 thenxˆ ÿC…a2†=B…a2†>0 which implies thatB…a2†>0 so again 2Ax‡B>0.

(iii) ifA…a2†>0 then the second root ofF…x† ˆ0 lies in…ÿ1;0Šsoxˆ …ÿB‡



B2_ÿ4AC p

†=2A

and 2Ax_‡B>0.

In the case where there are two endemic equilibria for Ro<1 we have

bb₁ >a1r; b2‡b>

b2/cbÿa1h† bb₁h_ÿa1hr

andaU

2 <a2 <aR2o. ThenF…x† ˆ0 has two roots in‰0;h†. AsA…a2†<0 the smaller of these roots is

stable and the larger unstable. If a2 ˆaR2o then the larger root is xˆh but the same argument

shows that the smaller root, which is now the unique endemic equilibrium, is stable. This com-pletes the proof of Theorem 1.

It is interesting to consider the special case/_ˆ0 separately as in this case the results simplify considerably. This case means that there is no vaccination of susceptible individuals. Here

Ro_ˆRo1_ˆa1=…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 a

Ro

2 ! 1 as

/_!0. We have the following corollary:

Corollary 1. Suppose that /_ˆ0. Note that the IFE is always possible.

(I)IfRo<1 then the IFE is LAS. IfRo_‡Ro2 <1then the IFE is GAS. The equationGa2† ˆ0

has two positive real roots fora2.LetaU₂ denote the largest of these.If06a2 <aU₂ then the IFE is

the only equilibrium but foraU

2 <a2 there are two additional endemic equilibria.The one with the

highest value ofi

1 is LAS,the other is unstable. At a2 ˆaU2 there is one endemic equilibrium.

(II)IfRo_ˆ1 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

b2 ˆ0:01/day; these parameter values for BRSV are taken from de Jong et al. [2]. There are no

data available on the value ofc so we arbitrarily takec_ˆ0:5, 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/_ˆ0) we see thati₁6hand the endemic equilibrium value ofi₁ tends to hasa2 becomes large, as can indeed be observed in Figs. 1 and 2.

Fig. 1 shows two bifurcation diagrams for />0. Ifb=h_†>ac

1>a1 then the bifurcation

dia-gram 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 foraU

2 <a2 <a2Ro. Ata2 ˆaU2 these

equilibria coalesce, whereas at a2 ˆaR2o the unstable endemic equilibrium coalesces with the

in-fection-free one `causing' the latter to become unstable. On the other hand if a1>ac1 the

bifur-cation diagram looks like Fig. 1(b). Fora2 <aR2othere is only the stable infection-free equilibrium

and ata2ˆaR2o 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 /_ˆ0. If a1 <b1‡b, the

bi-furcation diagram looks like Fig. 2(a). This is similar to Fig. 1(a) except that fora2 >aU2 there are

always two endemic equilibria, the higher locally stable and the lower unstable. Ifa1 >b1‡bthen

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. 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 bˆ0:000648/day, b1 ˆ0:1/day,

b_{2 ˆ}0:01/day,c _ˆ 0.5 and/ _ˆ 0.5. In this ®gure the linePRQis the linea2 ˆaR2o…a1†(orRoˆ1)

whilst the curveRSis the curvea2 ˆaU2…a1†.Pis the point…0;…b2‡b†=c/†,Qis the point…b=h;0†

andRis the intersection of the linePQwith the linea1ˆac1. 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. 2. Bifurcation diagrams for/ˆ0: (a)a1ˆ0:05/day, and (b)a1ˆ0:15/day. See text for other parameter values.

(includingS but notR) 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 notR) there is one unique endemic equilibrium which is stable. In the remaining areaZ (for zero) bounded by the linesOQ;QR;RS

and OS (including O;Q;R, the linesOQ;QRand OS but notS) there are no endemic equilibria. Note that we have shown that the infection-free equilibrium is globally stable when

Ro_‡1_ÿ/_†Ro2 _ˆ1_ÿ/_†Ro1_‡Ro2 <1. This condition implies that bothRoandRo2 are less than 1. Hence Ro_‡1_ÿ/_†Ro2<1 is a sucient condition for there to be no subcritical endemic equilibria. However it is not necessary. For example in Fig. 3 the lineRo_‡1_ÿ/_†Ro2_ˆ1 is a line throughQwhich lies strictly beneathPQ. In the region between this line andPQand 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 ifRo>1 and infection is initially present (soi10_{† ‡}i20_†>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 forRo>1 there is additionally a unique endemic equilibrium. Our model di€ers from that in that forRo<1 there may be two endemic equilibria, one stable and one unstable in addition to the usual stable infection-free equilibrium. This occurs ifa2lies in the

rangeaU

2…a1†;aR2o…a1††anda1 lies in the range‰0;ac1†. AtRoˆ1 there is a `backwards bifurcation'

of an unstable endemic equilibrium from the infection-free equilibrium. It is possible to use a1

instead ofa2as 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 con-text. They consider an SIRS model with two social groups corresponding to `normal' and `edu-cated' 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 ˆa2 and b1 ˆb2, 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 /_ˆ1 so all cattle are vaccinated at birth. As /_!1,

…b=h_{† ! 1} and ac

1 ! ÿ1 so case I…b† is not relevant. Thus two endemic equilibria are never

possible and the usual bifurcation behaviour is observed.Ro_ˆca2=…b2‡b† is a sharp threshold

value and forRo<1 there is a unique infection-free equilibrium which is GAS whereas forRo>1 this equilibrium is unstable and there is an additional stable endemic equilibrium.

(ii) Second if we set c_ˆ1; a1 ˆa2 ˆa and b1ˆb2ˆb and combine the two classes then we

be two endemic equilibria. Thus similar bifurcation behaviour to case (i) is observed with

Ro_ˆa=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 al-ternative may be proportional mixing. One way of thinking about proportional mixing is that seronegative and seropositive animals spend di€ering fractions of time making potentially in-fectious contacts. If these fractions are, respectively,n1andn2, 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 ˆa1n21; a02ˆa2n1n2 and

c0 _ˆcn_2†=n1, respectively provided thatb2‡ …1ÿc0†b>0.

For realistic parameter values it is often true that the per capita birth rate b is very small compared

withb2, so this condition is likely to be satisfied. For example de Jong et al.[2]takebˆ0:000684/

dayandb_{2 ˆ}0:01/dayin 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 exceedsRo1. In such a situation vaccination is helping the infectious agent to spread and we shall see that there does not exist a critical vaccination e€ort for eradication. Of course vacci-nation 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) Roincreases 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 e€ects in more detail.

We need the following Lemmas.

Lemma 3.

(a)For Ro1<1let/‡ denote the unique root in0;1_† of the quadratic equationQ/_{† ˆ}0 where

Q/_{† ˆ} /

‡_b b1 ‡b1…

1_ÿ/_†

ÿ_bbc/ ‡b2

…/Ro1_‡1_ÿRo1_{†† ÿ}/:

Thenac

(b)IfRo1>1 then a1>ac1.

Corollary 3.Necessary conditions for two endemic equilibria are Ro<1 andRo1 <1<Ro2.

The conditionRo1 <1<Ro2means that in the SIS model witha_ˆa1andbˆb1(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 witha_ˆa2 andbˆb2 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, soa1 >a2, and also once infected and recovered hosts

will probably defend themselves better against an infectious agent, so c<1. However it is still possible for Ro2 >1>Ro1 if b₂ÿ1 >bÿ₁1, so the average length of subsequent infectious periods exceeds that of the ®rst, as appears to be the case for BRSV in cattle [2].

Theorem 2.

then there are no endemic equilibria.

(b) If aU

ical endemic equilibrium. If Ro1<1 and /_ˆ/ then there is a unique endemic equilibrium

which is stable. IfRo1_ˆ1and/_ˆ/ then there are no endemic equilibria.

(c)IfaU

2…/‡†Pa2 then/‡6/,with equality if and only ifaU2…/‡† ˆa2. No endemic

equilib-ria are possible if /6/.

Proof.Cases I and II(a) are straightforward. In case II(b) it is straightforward that/P/implies

Ro61 with equality if and only if /_ˆ/. By Lemma 3, a1>ac₁ so by Theorem 1 no endemic

equilibria are possible. In case III it is straightforward that Ro61 whatever the value of/, with equality if and only if the stated conditions are true. For/P/‡; a1Pac1by Lemma 3 and again

no endemic equilibria are possible. For/</‡; a1 <ac1. However, the proof of Lemma 5 shows

that if a2PaU2…/‡† then Ro2>1. This is a contradiction as Ro261. So a2 <aU2…/‡†<aU2…/†

(using Lemma 4). Hence by Theorem 1 there are no endemic equilibria.

For case IV it is straightforward to show thatRo<1 for/</and Ro>1 for/>/. As in

1 >a1. Theorem 1 says that there is a unique endemic equilibrium which is stable. If Ro1 _ˆ1 then/ _ˆ/‡ _ˆ0 and a1 ˆac1. Hence when /ˆ/ˆ0 Theorem 1 now says that there

is a unique (repeated) subcritical endemic equilibrium;

(iii) IfRo1 <1 and/~</</ then Lemma 5 says that there are two endemic equilibria; (iv) IfRo1<1 and/_ˆ/then/</‡and so by Lemma 3,ac

1>a1. Hence by Theorem 1 there

is a unique endemic equilibrium which is locally stable;

(v) If Ro1_ˆ1 then /_ˆ/‡_ˆ0 and so if /_ˆ0 by Lemma 3, a1ˆac1. Theorem 1 now says

In case IV(c) we have /P/‡. For 06/</‡ then by Lemma 3, ac

1 >a1 and

aU

2…/†>aU2…/‡†Pa2 so by Theorem 1 there are no endemic equilibria. For /P/P/‡ by

Lemma 3,a1Pac₁ 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 di€erent 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 RegionZ, 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 cri-terionRo/_{† ˆ}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 criterionRo/_†>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 in-termediate vaccination levels (withRo <1). Note that in Theorem 1 we showed that the infection-free equilibrium will be globally stable provided that 1>Ro_‡1_ÿ/_†Ro2 _ˆ1_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 forRo<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 forb;b₁;b₂andcas in Section 5. We consider values fora1anda2as in Figs. 1 and

2: (i) a1 ˆ0:05/day …Ro1 ˆ0:4968†; a2ˆ0:05; 0:1 and 0:25/day …Ro2 ˆ4:6957; 9:3914 and

23:4786 respectively_†; (ii) a1 ˆ0:15/day …Ro1 ˆ1:4903†; a2 ˆ0:01; 0:02 and 0:04/day …Ro2 _ˆ0:9391; 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].

For (i) we have Ro2 >1>Ro1; /‡ _ˆ0:9438; aU

2…/‡† ˆa

Ro

2 …/‡† ˆ0:02193/day and

aU

2…0† ˆ0:02707/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 ˆ0:05/day, / ˆ0:1198; a2ˆ0:1/day, / ˆ0:05658 and a2ˆ0:25/day, /ˆ0:02190.

For (ii) ifa2 ˆ0:02 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). Ifa2 ˆ0:01/day, then

for/</_ˆ0:8896 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 im-munised [10]. Sabo and Bla skovic [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] estimateRo1_ˆ10:0 and Ro2 _ˆ0:5 for pseudo-rabies virus.

Smith and Grenfell [12] give an impressive mathematical analysis of di€erent control strategies for pseudorabies virus. Their model is similar to ours in that it assumes that seropositive sus-ceptible 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 e€ect 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 termaSI used by Smith and Grenfell [12] as our variables are numbers and not densities and also recent data on pseudorabies virus in experimental pig pop-ulations show that the termaSI=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] e€ectively setb_ˆ0 asbis 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 ˆa2 ˆb~1N; b1 ˆb2 ˆ~c; cˆb~2=b~1 (where b~2<b~1) and/ˆ0 then we obtain a special case

pseudorabies virus in a pig population, but these do not carry over to our model because of the di€erent 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 e€ective immunity [15].

In this paper we considered a simple model where animals are divided into seronegative sus-ceptibles, 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 di€erent (probably lower) rate than seronegatives and the infectious pe-riods may also di€er (probably longer for seropositives). There are no deaths from the infection so the population size remains constant. We assumed homogeneous and proportional mixing be-tween 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 equi-librium and local stability analysis.

Unusually, it may be possible for there to be two endemic equilibria (one stable and one un-stable) forRo<1. The conditionRo>1 can be thought of as a condition for whether the infection will invade an initially infection-free population. IfRo<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 equi-libria in terms of the vaccination proportion/ and the infection transmission ratea2. We found

that in each situation where the infection persists without vaccination, either the criterionRo_ˆ1 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_ˆ1 does not give the correct answer is whereRo1 <1 in immunologically naive animals andRo2>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.

stochas-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 whenRo1<1<Ro2. As discussed in Section 8 it is probably more likely thatRo1 >Ro2, (the reproduction number decreases for seropositive animals). However, it is still possible that

Ro1 <1<Ro2if 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 _ˆ36:5 and Ro2_ˆ1:14. Hence it is probable that

Ro1 >Ro2for BRSV although the possibility thatRo2>Ro1remains 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

2 † from (4.10) and simplifying it is routine to show

that B~_‡2A~h<0 if and only if

2 from (4.12) and simplifying yields the inequality

b2/c b_2‡b<

hb₁bÿa1r† b_ÿa1h

Proof of Lemma 3.(a) By the de®nition of ac

There is strict inequality here asRo >1 implies either 1>/or c>0. Re-arranging

…1_ÿ/_†Ro1 > …b1=…b‡b1††…1ÿ/† ÿ …bc/=…b‡b2††

…/_‡b₁=b‡b_1††1_ÿ/_{†† ÿ}bc/=b‡b_2††: Soa1>ac1 as required.

(c) This is straightforward using the proofs of (a) and (b).

aU₂/_{† ˆ} 1

as Q/‡_{† ˆ}0. Hence inequality (A.3) implies Ro2>1 is a necessary condition for two endemic equilibria.

necessary conditions for two distinct endemic equilibria to exist and a similar argument shows that they are sucient.

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

numberRoin 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.

[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. Sabo, 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 ofRofor discrete-time epidemic models with

dynamic heterogeneity, Math. Biosci. 119 (1994) 97.