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Mathematical Biosciences 165 (2000) 1±25
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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 ˆ 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: david@strath.stams.ac.uk (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

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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 a€ected. 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 ˆ 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 ecient 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 di€erence to the dynamics of the model, but will a€ect 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 di€erent
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 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 temporarily 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
dI1
dt
dS2
dt
dI2
dt

ˆ 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
ˆ b/N ÿ
…a1 I1 ‡ a2 I2 † ‡ b1 I1 ‡ b2 I2 ÿ bS2 ;
N
cS2
ˆ
…a1 I1 ‡ a2 I2 † ÿ b2 I2 ÿ bI2
N
ˆ

…2:1†

and S1 ‡ S2 ‡ I1 ‡ I2 ˆ N , with suitable initial conditions S1 …0†; S2 …0†; I1 …0†; I2 …0† P 0 and

S1 …0† ‡ S2 …0† ‡ I1 …0† ‡ I2 …0† ˆ 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 ˆ S1 =N ; s2 ˆ S2 =N ; i1 ˆ I1 =N and i2 ˆ I2 =N , respectively the fraction of class one
susceptible, class two susceptible, class one infected and class two infected individuals. Then the
di€erential equations (2.1) can be written

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D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

ds1
ˆ b…1 ÿ /† ÿ s1 …a1 i1 ‡ a2 i2 † ÿ bs1 ;
dt
di1
ˆ s1 …a1 i1 ‡ a2 i2 † ÿ …b ‡ b1 †i1 ;
dt
ds2
ˆ b/ ÿ cs2 …a1 i1 ‡ a2 i2 † ‡ b1 i1 ‡ b2 i2 ÿ bs2 ;
dt

di2
ˆ cs2 …a1 i1 ‡ a2 i2 † ÿ …b ‡ b2 †i2 ;
dt
s1 ‡ s2 ‡ i1 ‡ i2 ˆ 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 ˆ a1 =…b1 ‡ b† and Ro2 ˆ a2 c=…b2 ‡ b†. Ro1 is the basic reproduction number for
an SIS epidemic model with no vaccination where the per capita e€ective 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 ˆ 1; a2 ˆ a1 and b2 ˆ b1 . Similarly Ro2 ˆ a2 c=…b2 ‡ b† is the basic
reproduction number for an SIS model, where the per capita e€ective 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 / ˆ 1 and S1 …0† ˆ I1 …0† ˆ 0, i.e. all individuals enter the class S2
and S1 …t† ˆ I1 …t† ˆ 0 for all t.
De®ne
Ro ˆ …1 ÿ /†Ro1 ‡ /Ro2 ;
a1
a2 c
‡/
:
ˆ …1 ÿ /†
b1 ‡ b
b2 ‡ b
To show that Ro is the basic reproduction number in our model, consider a population at the
infection-free equilibrium …S1 ; S2 † ˆ ……1 ÿ /†N; /N †. For i; j ˆ 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†
ˆ 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 ˆ

5

a1
N
a1
…1 ÿ /† ˆ
…1 ÿ /†
N b1 ‡ b
b1 ‡ b

secondary cases amongst ®rst time and
a1 c
/
m12 ˆ
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 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 ˆ 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 e€ectively
infected at birth. In this case the e€ective entry rate of susceptibles into the second class is
r ˆ b/ ‡ b1 h. For notational convenience we de®ne
s ˆ …b2 ‡ …1 ÿ c†b†=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 ˆ …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† ˆ p0 ‡ p1 a ‡ p2 a2 ;

…4:4†

where
p0 ˆ a21 b2 h2 ‡ b2 s2 ‡ 2a1 b2 hs;

p1 ˆ 2a1 bhr ÿ 2brs ÿ 4b2 b1 h and p2 ˆ r2 :

…4:5†

Also de®ne
ac1 ˆ

b ‰…b2 ‡ b†b1 h ÿ b2 c/Š
:
h …b2 ‡ b†r 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 † ˆ …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

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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 † ˆ 0 (which are both less than aR2 o when ac1 > a1 holds). At
a2 ˆ aU2 there is a unique endemic equilibrium (which is unstable) and for smaller values of a2 no
endemic equilibrium exists. At a2 ˆ aR2 o we have Ro ˆ 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
;
ˆ
a1 i1 ‡ a2 i2 ‡ b a1 i1 ‡ a2 i2
b/ ‡ b1 i1 ‡ b2 i2
…b ‡ b2 †i2
s2 ˆ
ˆ
c…a1 i1 ‡ a2 i2 † ‡ b c…a1 i1 ‡ a2 i2 †
s1 ˆ

…4:7†
…4:8†

and
i2 ˆ

i1
a2




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 ˆ 0
we ®nd after some manipulations the equation F …x† ˆ 0, where
x ˆ h ÿ i1
and
F …x† ˆ Ax2 ‡ Bx ‡ C:
Here
A ˆ ÿb1 ‡

a1
s;
a2

Bˆ

a1 bh b
ÿ s ‡ r;
a2
a2

Cˆÿ

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†
1ÿ
F …x† ˆ b1 x
ÿ x ÿ x…b ÿ a1 x† ÿ
b1
a2
a2
b

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D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

Hence
F …0† ˆ C < 0;
F …h† ˆ b/h ÿ

…b ÿ a1 h†…b2 ‡ b†h
a2 c

and
F



b
a1



ˆ

b1 b
a1




r
b
:
ÿ
b1 a1

Therefore
(a) for h P …b=a1 † F …b=a1 † > 0,
(b)




a1 h 1
F …h† ˆ bh / ÿ 1 ÿ
;
b Ro2
bh
…Ro ÿ 1†;
ˆ
Ro2
P 0; for Ro P 1 with equality if and only if Ro ˆ 1:
Hence for Ro > 1; F …min…b=a1 ; h†† > 0 and so F …x† ˆ 0 has a unique root in 0 < x < min…b=a1 ; h†.
If Ro ˆ 1 then F …x† ˆ 0 has a root at x ˆ h ˆ min…b=a1 ; h† and at most one more root with
0 < x 6 h. If Ro < 1; F …x† ˆ 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† ˆ 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 ˆ

A < 0;

B > 0;

B2 ÿ 4AC > 0:

Now A < 0 if and only if a2 >
2

aA2

ˆ …a1 =b1 †s; B > 0 if and only if a2 >

B2 ÿ 4AC ˆ p0 …1=a2 † ‡ p1 …1=a2 † ‡ p2 ˆ

G…a2 †
:
a22

aB2

…4:13†

ˆ …1=r†…bs ÿ a1 bh† and

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 ˆ 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† ˆ 0 has a root in …h; b=a1 Š, hence
cannot have two roots in …0; h†. So F …x† ˆ 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 † ˆ 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

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D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

F …x† ˆ 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† ˆ 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 ˆ 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 ˆ aU2 . For a2 ˆ aU2 ÿ d, where d is small and positive, F …x† < 0 for x 2 ‰0; b=a1 Š. Hence F …x† < 0
for x 2 ‰0; b=a1 Š for a2 6 aU2 ÿ d and F …x† ˆ 0 has no roots in ‰0; b=a1 Š, hence none in ‰0; hŠ. For
a2 6 aL2 two positive real roots appear outside ‰0; b=a1 Š. Hence for a2 P aU2 ; F …x† ˆ 0 has two real
roots in ‰0; b=a1 Š, whilst for a2 < aU2 ; F …x† ˆ 0 has no real roots in ‰0; b=a1 Š. At a2 ˆ aR2 o ; F …x† ˆ 0
has a root at x ˆ h 2 ‰0; b=a1 †, hence two real roots in ‰0; b=a1 Š 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 ˆ aR2 o one root crosses the line
x ˆ 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† ˆ 0 has two roots in ‰0; h† whilst if ÿ…B=A† jaRo ÿh > h then for aR2 o > a2 P aU2 F …x† ˆ 0
2
has no roots in ‰0; h†. If ÿ…B=A† jaRo ÿh ˆ h, then when a2 ˆ aR2 o F …x† has a double root at x ˆ h,
2
so aR2 o ˆ 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† ˆ 0 has exactly one root in 0 < x 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† ˆ 0 has no roots in ‰0; h†. When a2 ˆ 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 † ˆ 0 …aR2 o ˆ 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 ˆ …b2 /c…b ÿ a1 h†=…bb1 h ÿ a1 hr†† then h < …b=a1 † and aU2 ˆ aR2 o . If
a2 6 aU2 then F …x† ˆ 0 has no roots in ‰0; h†. If a2 ˆ aU2 then h is a double root of F …x† ˆ 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† ˆ 0 has no roots in ‰0; h†. In this case if a2 ˆ aU2 then there is one double root
in ‰0; h†. For aU2 < a2 < aR2 o ; F …x† ˆ 0 has two roots in ‰0; h†. If a2 ˆ 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 † ˆ …1 ÿ /; 0; /; 0† we see that the characteristic equation
has roots x ˆ ÿb (twice) and the roots of the quadratic
x2 ‡ ‰…b ‡ b1 † ‡ …b ‡ b2 † ÿ a1 …1 ÿ /† ÿ a2 c/Šx ‡ …b ‡ b1 †…b ‡ b2 † ÿ …b ‡ b1 †a2 c/
ÿ …b ‡ b2 †a1 …1 ÿ /† ˆ 0:

…4:14†

Hence by the Routh±Hurwitz conditions necessary and sucient 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 †Š
‡
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 ˆ limt!1 supT P t s1 …T †. The solution x1 …t† of
dx1
ˆ b…1 ÿ /† ÿ bx1 ; x1 …0† ˆ 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 ˆ …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 ˆ …i1 ; i2 †T ;
di
6 Qi;
dt
where


…1 ÿ / ‡ †a1 ÿ …b ‡ b1 † a2 …1 ÿ / ‡ †
:
Qˆ
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

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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 ˆ x1 e. The characteristic equation of Q is
x2 ‡ a1 x ‡ a2 ˆ 0;

where
and

a1 ˆ …b ‡ b1 †…1 ÿ …1 ÿ / ‡ †Ro1 † ‡ …b ‡ b2 †…1 ÿ Ro2 †;
a2 ˆ …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 ˆ 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
A similar argument to the one showing that s1
now shows that
1 61 ÿ /
s1;1 ˆ 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 ˆ s1;1 ˆ 1 ÿ / and s1 ! 1 ÿ / as t ! 1. Thus s2 ˆ 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 ˆ 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 a2 s1
ÿ a1 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 ˆ6
4 ÿc…a1 i ‡ a2 i † ÿ c…a1 i ‡ a2 i † ‡ a1 cs ÿ c…a1 i ‡ a2 i † ‡ a2 cs 5:
2
2
1
2
2
1
2
1
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 † ˆ a2 s1 ca1 s2

…4:17†

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D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

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 ˆ 0:

…4:18†

Here



b…1 ÿ /†
i2
A1 ˆ
‡ b ‡ b1 ‡ …b ‡ b2 † 1 ‡  ÿ a2 cs2 ÿ a1 s1 ;
s1
s2


1ÿ/
ÿ 1 …b ‡ b1 † ‡ b…b ‡ b1 ÿ a1 s1 †
A2 ˆ b
s1




b…1 ÿ /†
i2
i2


‡
…b
‡
b
†
1
‡
ÿ
a
cs
2 2 ‡ …b ‡ b1 ÿ a1 s1 †…b ‡ b2 † 
2


s1
s2
s2

…4:19†

…4:20†

and






1ÿ/
i2

A3 ˆ 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

…4:21†

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
‡
b†
1
‡
ÿ
1
…b
ÿ
ca
A3 ˆ b
2 2 ‡ b…b1 ‡ b ÿ a1 s1 †…b2 ‡ b† 
1
2


s1
s2
s2

i
…4:22†
ÿ a2 s1 b1 …b ‡ b2 † 2 :
s2
Now we express s1 ; s2 ; i1 and i2 in terms of x ˆ h ÿ i1 as in the equilibrium results. From (4.9)
a1 i1 ‡ a2 i2 ˆ

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†
:
ˆ
s2 …b ‡ b2 †x

…4:24†

Using (4.7), (4.8) and (4.23)
s1 ˆ

…b ‡ b1 †
x
b

and s2 ˆ

…b ‡ b2 †
…b ÿ a1 x†:
cba2

Substituting (4.24) and (4.25) into (4.22) we deduce after some algebra that
A3 ˆ …b ‡ b1 †

…h ÿ x† 2
‰cb h ‡ …a1 …b ‡ b2 † ÿ a2 b1 c ÿ a1 bc†x2 Š:
x2

…4:25†

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D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

As Ax2 ‡ Bx ‡ C ˆ 0 we can add …b ‡ b1 †……h ÿ x†=x2 †a2 c‰Ax2 ‡ Bx ‡ CŠ to A3 without changing it
so
A3 ˆ …b ‡ b1 †

…h ÿ x†
a2 c‰2Ax ‡ BŠ:
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† ˆ 0 has exactly one root in 0 < x 0.
2Ax ‡ B > 0.
(ii) if A…a2 † ˆ 0 then x ˆ 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† ˆ 0 lies in …ÿ1; 0Š so x ˆ …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† ˆ 0 has two roots in ‰0; h†. As A…a2 † < 0 the smaller of these roots is
stable and the larger unstable. If a2 ˆ aR2 o then the larger root is x ˆ 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 / ˆ 0 separately as in this case the results simplify
considerably. This case means that there is no vaccination of susceptible individuals. Here
Ro ˆ Ro1 ˆ 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 / ˆ 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 † ˆ 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 ˆ aU2 there is one endemic equilibrium.
(II) If Ro ˆ 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 ˆ 0:000648/day, b1 ˆ 0:1/day and

D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

13

b2 ˆ 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 ˆ 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 / ˆ 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 ˆ aU2 these
equilibria coalesce, whereas at a2 ˆ 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 ˆ 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 / ˆ 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) / ˆ 0:1; a1 ˆ 0:05/day, and (b) / ˆ 0:5; a1 ˆ 0:15/day.
See text for other parameter values.

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D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

Fig. 2. Bifurcation diagrams for / ˆ 0: (a) a1 ˆ 0:05/day, and (b) a1 ˆ 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 ˆ 0:000648/day, b1 ˆ 0:1/day,
b2 ˆ 0:01/day, c ˆ 0.5 and / ˆ 0.5. In this ®gure the line PRQ is the line a2 ˆ aR2 o …a1 † (or Ro ˆ 1)
whilst the curve RS is the curve a2 ˆ 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 ˆ 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 ˆ …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 sucient 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 ˆ 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 di€ers 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 ˆ 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 ˆ a2 and b1 ˆ 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 / ˆ 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 ˆ 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 ˆ 1; a1 ˆ a2 ˆ a and b1 ˆ b2 ˆ b and combine the two classes then we
obtain the usual SIS epidemic model. In the proof of Theorem 1 we see A ˆ 0 so there can never

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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 ˆ 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 di€ering 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 ˆ a1 n21 ; a02 ˆ a2 n1 n2 and
c0 ˆ …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 ˆ 0:000684/
day and b2 ˆ 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 e€ort 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 e€ects 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…/† ˆ 0 where



b1
bc/
…1 ÿ /† ÿ
…/Ro1 ‡ …1 ÿ Ro1 †† ÿ /:
Q…/† ˆ
/‡
b ‡ b2
b ‡ b1

Then ac1 > a1 if and only if 0 6 / < /‡ , and ac1 ˆ a1 if and only if / ˆ /‡ .

D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

17

(b) If Ro1 > 1 then a1 > ac1 .
(c) If Ro1 ˆ 1 then /‡ ˆ 0 is the unique root of Q…/† ˆ 0 in ‰0; 1† and a1 P ac1 with equality if and
only if / ˆ 0.
Proof. See Appendix A.
Lemma 4. For a1 < ac1 ; aU2 ˆ 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 /~ ˆ …aU2 † …a2 † …< /‡ †. (So /~ is the unique root in ‰0; /‡ † of the equation
aU2 …/† ˆ 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 / < / ˆ …1 ÿ Ro1 †=…Ro2 ÿ Ro1 †, or (ii) aU2 …0† P a2 > aU2 …/‡ †
ˆ aR2 o …/‡ † and /~ < / < / .
Note that by putting / ˆ 0 and h ˆ b=…b ‡ b1 † into Eq. (9.1) we deduce that
s!2
p

b
‡
b
b2 ‡ b
1
aU2 …0† ˆ
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 ˆ a1 and b ˆ 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 ˆ a2 and b ˆ 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.

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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 / < / ˆ …Ro1 ÿ 1†=…Ro1 ÿ Ro2 † we have Ro > 1.
(b) For / P / we have Ro 6 1 (with equality if and only if / ˆ / ). 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 ˆ 1 and
/ ˆ 0; …b† Ro2 ˆ 1 and / ˆ 1 or …c† Ro1 ˆ Ro2 ˆ 1. No endemic equilibria are possible.
(IV) If Ro2 > 1 P Ro1 then for / < / ˆ …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 / ˆ / then there is a unique endemic equilibrium which is stable. If Ro1 ˆ 1 and / ˆ /
then there are no endemic equilibria.
(b) If aU2 …0† P a2 > aU2 …/‡ † ˆ aR2 o …/‡ † then /‡ > / . No endemic equilibria are possible for
~ / †, two for /~ < / < / . For / ˆ /~ < / there is a unique (repeated) subcrit0 6 / 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 …/‡ † ˆ aR2 o …/‡ † and for a2 > aR2 o …/‡ †; / < /‡ . Similarly if a2 6 aR2 o …/‡ †
then / P /‡ , with equality if and only if a2 ˆ aR2 o …/‡ †. In case IV(a) by Lemma 5 two endemic
equilibria are possible for 0 6 / < / . For / ˆ / Ro ˆ 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 ˆ 1 then / ˆ /‡ ˆ 0 and a1 ˆ ac1 . Hence when / ˆ / ˆ 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 / 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 / ˆ / 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 ˆ 1 then / ˆ /‡ ˆ 0 and so if / ˆ 0 by Lemma 3, a1 ˆ 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 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
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 …/† 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 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 ˆ …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 ˆ 0:05/day …Ro1 ˆ 0:4968†; a2 ˆ 0:05; 0:1 and 0:25/day …Ro2 ˆ 4:6957; 9:3914 and
23:4786 respectively†; (ii) a1 ˆ 0:15/day …Ro1 ˆ 1:4903†; a2 ˆ 0:01; 0:02 and 0:04/day
…Ro2 ˆ 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 di€erent values of /. See text for explanation.

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D. Greenhalgh et al. / Mathematical Biosciences 165 (2000) 1±25

For (i) we have Ro2 > 1 > Ro1 ; /‡ ˆ 0:9438; aU2 …/‡ † ˆ aR2 o …/‡ † ˆ 0:02193/day and
aU2 …0† ˆ 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 ˆ 0:05/day, / ˆ 0:1198; a2 ˆ 0:1/day, / ˆ 0:05658 and a2 ˆ 0:25/day, / ˆ 0:02190.
For (ii) if a2 ˆ 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 ˆ 0:01/day, then
for / < / ˆ 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 ˆ 10:0
and Ro2 ˆ 0:5 for pseudo-rabies virus.
Smith and Grenfell [12] give an impressive mathematical analysis of di€erent control strategies
for pseudorabies virus. Their mo