Mathematical Biosciences 241 (2013) 137–144
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Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
Anthrax epizootic and migration: Persistence or extinction
Avner Friedman a, Abdul-Aziz Yakubu b,⇑
a
b
Mathematical Bioscience Institute and Mathematics Department, The Ohio State University, Columbus, Ohio 43210, United States
Department of Mathematics, Howard University, Washington DC 20059, United States
a r t i c l e
i n f o
Article history:
Received 9 June 2012
Received in revised form 21 October 2012
Accepted 24 October 2012
Available online 5 November 2012
Keywords:
Anthrax
Extinction
Migration
Persistence
a b s t r a c t
In this paper, we use an extension of the deterministic mathematical model of an anthrax epizootic of
Hahn and Furniss to study the effects of anthrax transmission, carcass ingestion, carcass induced environmental contamination, and migration rates on the persistence or extinction of animal populations. We
compute the basic reproduction number R0 for the anthrax epizootic model with and without taking into
account animal migration. We obtained conditions for an anthrax enzootic region. We demonstrate that
decreasing the levels of carcass ingestion by removal of carcases in game reserves, for example, may not
always lead to a reduction in the population of animals infected with anthrax. However, increasing levels
of carcass induced environmental contamination rates in an enzootic anthrax region can result in the catastrophic extinction of a persistent animal population.
Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction
Disease is becoming increasingly recognized as responsible for a
large number of species extinctions [2–7,10,11,13,15]. In this paper, we use an anthrax epizootic mathematical model to study the
impact of the disease on population persistence or extinction.
The anthrax disease, is an acute, febrile disease of warm-blooded
animals, including humans [3]. The disease is caused by the
gram-positive, sporulating bacterium, Bacillus anthracis. Anthrax
primarily affects herbivorous livestock and wildlife species, but
also poses serious public health risks in many parts of the world.
Carnivores may also become infected by ingesting contaminated
carcasses, but disease-associated illness and death are rarer than
in herbivores. Anthrax outbreaks in animals in nearly 200 countries are recorded by The World Anthrax Data Site, a World Health
Organization Collaborating Center for Remote Sensing and Geographic Information Systems for Public Health. Anthrax is a globally distributed disease, and has been reported by all continents
that are populated densely with animals and humans. The disease
is among the list of pathogens that could be used as a bioweapon,
and is categorized by the Centers for Disease Control as a category
A biological threat agent [3,21].
Anthrax is one of the most dramatic diseases affecting wild animals in Africa. For example, within Etosha National Park, anthrax
has killed a variety of herbivores, from elephants to ostriches. Carnivores in the park such as lion, leopard, hyena and jackal appear to
⇑ Corresponding author.
E-mail addresses: [email protected] (A. Friedman), ayakubu@howard.
edu (A.-A. Yakubu).
0025-5564/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.mbs.2012.10.004
be immune to the disease. The cause of the anthrax epidemic remains unknown [7,15]. In November 2004, attention was drawn
to an outbreak of anthrax in Jwana Game Reserve located in Jwanen, Botswana, when three captive cheetahs (Acinonyx jubatus)
died of Anthrax [15]. The three animals had been fed meat from
a dead red hartebeest (Alcelaphus buselaphus) from the reserve,
and anthrax was subsequently confirmed in the red hartebeest
[7,15].
Although the global and multihost nature of the pathogen presents epidemiologic challenges, heterogeneities in host range and
infection outcome provide opportunities for the mathematical
modeling of the disease dynamics in migrating endangered wild
animals. In this paper, we use an extension of an anthrax epizootic
mathematical model of Hahn and Furniss, that includes migrations,
births and death from other diseases, to compute the basic reproduction numbers R0 ðdÞ and R0 for the disease with and without
migration, respectively [8,9]. We use R0 and R0 ðdÞ to determine
factors that lead to anthrax local and global persistence or extinction in animal populations. We prove that, R0 ðdÞ < R0 , and
R0 ðdÞ < 1 implies the extinction of the infective population under
migration. Furthermore, we show that it is possible to have a catastrophic extinction of the total animal population when R0 > 1.
When R0 ðdÞ > 1, we obtain sufficient conditions for the existence
of a unique endemic equilibrium.
An SI epidemic model with migration and variable parameters
was considered in [1]. A general framework for defining the basic
reproduction number for infinite-dimensional population structure was developed by Thieme [20]. This concept was subsequently
used in [22] for an avian influenza, and in [17] for a malaria disease
model.
138
A. Friedman, A.-A. Yakubu / Mathematical Biosciences 241 (2013) 137–144
The paper is organized as follows: in Section 2, we introduce the
PDE susceptible(s)–infected(i)–environment(a)–carcass(c), compartmental epidemic model with diffusion of susceptible and infected
animal populations. In Sections 3 and 4, we compute the basic
reproduction numbers of the anthrax epidemic models with diffusion and without diffusion. Necessary and sufficient conditions for
the existence of an endemic equilibrium point for the model without diffusion and illustrative examples are given in Section 5. Sufficient conditions for the existence of an endemic equilibrium
point for the model with diffusion and concluding remarks are,
respectively, presented in Sections 6 and 7.
2. Mathematical models of anthrax epizootic
Mathematical models of the anthrax disease have contributed
to our understanding of the disease dynamics. However, vast
majority of these models are based on the anthrax disease in humans. This is probably due to a prevailing interest in human epidemics, in which the timescale of the disease is short and host
demographic dynamics can thus be ignored. Many animals that
live in the wild, however, are affected by the fatal anthrax epizootic
disease. In [8,9], Furniss and Hahn examined how anthrax epizootic may be driven by
1. largely by environmental contamination; and
2. largely by contact between vulnerable animals and fresh (infective) carcases.
Hahn and Furniss, in [8,9], used the following anthrax epizootic
mathematical model to derive a threshold result and other analytic
properties for an epizootic under Hypothesis 1, the ‘‘contamination’’ hypothesis:
dv
dt
da
dt
dc
dt
¼ av ;
¼ aa þ bc;
¼ av dc;
9
>
=
ð2:1Þ
>
;
where at time t; v ðtÞ is the number of vulnerable animals, aðtÞ is the
environmental contamination, defined as the number of anthrax
spores effectively ingested by an animal in one day (ignoring spores
ingested that do not lead to infection), cðtÞ is the number of carcases
of animals that may have died of anthrax, a is the nonnegative constant contamination decay-rate representing the death of spores or
their removal from the environment (for example, by rain), b is the
nonnegative constant rate at which contamination is disseminated
from carcases, and d is the nonnegative constant decay-rate of
carcases.
The Hahn–Furniss model, Model (2.1), is a special case of the
very-well studied Perelson viral-dynamics model [18,19]. The following system of three differential equations describes the basic
Perelson viral-dynamics model.
9
dT
¼ k dT kVT; >
dt
=
dI
¼ kVT dI;
dt
>
;
dV
¼ pI cV;
dt
where the target cell, T, are infected at the constant rate k by virus,
V. Target cells are assumed to be produced from a source at rate k
and die at rate d per cell. The virus infection produces infected cells,
I. These I cells produce new virions at rate q, die at rate d per cell,
and free virions are cleared at rate c per virion. The Perelson model
and its extensions have been used to study the dynamics of human
immunodeficiency virus, hepatitis C virus, hepatitis B virus and
cytomegalovirus infections in vivo [18,19].
In Model (2.1), Hahn and Furniss assumed that infected animals
die immediately and there are no births or deaths from causes
other than the anthrax epizootic [8,9]. In what follows, inspired
by Model (2.1), we introduce an anthrax model that includes
migrations, births, and death also from other diseases.
2.1. SI anthrax epizootic model equations with migration
Let X be a bounded domain in R3 with boundary @ X. We consider a susceptible sðx; tÞ and infected ðx; tÞ (si epidemic model),
with animal migrations, and with added aðx; tÞ and cðx; tÞ compartments that, respectively, keep track of the environmental contamination and carcases at location x 2 X, and time t P 0. Let
n ¼ nðx; tÞ P 0 denote the total population of vulnerable animals
at location x 2 X and time t P 0. During the course of an anthrax
epizootic, the total vulnerable animal population,
nðx; tÞ ¼ sðx; tÞ þ iðx; tÞ;
splits into susceptible (s) and infected parts (i). To make the disease
fatal, we assume that there is no recovery from it. The anthrax disease transmission through contact between vulnerable migrating
animals and fresh (infective) carcases (respectively, infected animals) is assumed to be by the density-dependent (respectively, frequency-dependent) transmission rate
gc sðx; tÞcðx; tÞ respectively; gi
sðx; tÞiðx; tÞ
;
nðx; tÞ
where the transmission rates gc and gi are nonnegative constants.
The principal mode of transmission is ingestion of infective micro-organisms. For example, non-biting blowflies may contaminate
vegetation by depositing vomit droplets after feeding on a carcass
infected with anthrax. Migrating animals feeding on such vegetation then become infected. There is no vertical transmission, and
the natural mortality rate is the constant l > 0. The infected reproduces into the susceptible class and the per capita growth rate is the
logistic function
n
gðnÞ ¼ r 1 ;
K
where r is the intrinsic growth rate and K is the carrying capacity.
Since the disease is fatal, the infected suffers an additional disease-induced mortality, which is described by the constant virulence c. During severe outbreaks, the disease may be transmitted
from one animal to the other; but this is a very minor mode of
gi
transmission. Consequently, we assume throughout that cþ
l < 1.
The epizootic siac anthrax model with animal migration is described by the following system of four partial differential equations
in ðx; tÞ 2 X ½0; 1Þ:
@i
@t
9
¼ dO2 s þ rn 1 Kn as gc sc gi sin ls; >
>
>
>
>
>
>
si
2
=
¼ dO i þ as þ gc sc þ gi ci li;
@a
@t
¼ aa þ bc;
@c
@t
¼ ðc þ lÞi dðs þ iÞc;
@s
@t
n
>
>
>
>
>
>
>
;
ð2:2Þ
where d > 0 is the diffusion coefficient of the migrating animals,
and r > l. In (2.2), d > 0 is the constant rate at which the total animal population feed on the carcass. That is, we have replaced the
linear ‘‘scavenging’’ term in dc
of Model (2.1), dc, by the non-linear
dt
term, dðs þ iÞc, which represents the contact between live animals
(s and i) and carcases of animals. The infection term, gc sc, represents
the result of scavenging.
We will assume the following boundary conditions.
@s
þ rðs s0 Þ ¼ 0 and i ¼ 0 on @ X ½0; 1Þ;
@m
where m is the outward normal,
conditions
ð2:3Þ
r > 0 and nonnegative initial
139
A. Friedman, A.-A. Yakubu / Mathematical Biosciences 241 (2013) 137–144
0
sðx; 0Þ s0 ðxÞ;
iðx; 0Þ i ðxÞ;
2
aðx; 0Þ a0 ðxÞ; and cðx; 0Þ
c0 ðxÞ:
ð2:4Þ
That is, we assume that susceptible animals are confined to the region, and the anthrax epidemic starts in the interior of the region.
By standard parabolic PDE theory, the system (2.2)–(2.4) has a unique solution for all t P 0 [12].
In the absence of animal migration, System (2.2) reduces to the
following system of differential equations,
g
g
a
c l
g
9
gi sin ls; >
>
>
>
=
c þ lÞ i;
ð3:1Þ
>
>
>
>
;
l
cþl
sa
b
s
in
ð3:2Þ
9
di
¼ F 1 ði; a; c; sÞ dt
da
¼ F 2 ði; a; c; sÞ dt
dc
¼ F 3 ði; a; c; sÞ dt
1
0
0
7
5:
Lemma 3.1. All the eigenvalues of FV 1 have magnitude less than 1 if
and only if
The proof will use the following Jury’s conditions from [16] on the
roots of a third degree polynomial.
Lemma 3.2 [16]. Given a polynomial
b2 ¼ a1 a3 a2 ;
b1 ¼ a2 a3 a1
2
2
c 3 ¼ b3 b1 ;
c 2 ¼ b3 b2 b1 b2 :
Then the roots of PðkÞ have magnitude less than 1 if and only if
Pð1Þ > 0;
jb3 j > jb1 j;
jc3 j > jc2 j:
Proof of Lemma 3.1. To prove the lemma, we will show that all
conditions of Lemma 3.1 are satisfied whenever 0 <
ð3:3Þ
B
F ði; a; c; sÞ ¼ @
bc
ðc þ lÞi
1
0
B
C
A and mði; a; c; sÞ ¼ @
ðc þ lÞi
0<
gi
g
b
þ cþ
< 1:
c þ l d ad
aa
1
C
A:
dðs þ iÞc
Their Jacobian matrices evaluated at ðs0 ; 0; 0; 0Þ are respectively the
matrix of new infections,
gi
s0
0
0
cþl 0
gc s0
3
7
b 5
0
and the matrix of transfers in and out of the epidemiological
compartments,
cþl 0
0
a
0
Consequently,
0
0
3
7
0 5:
ds0
<1 .
g þb
0< c a <1
d 1 cþgil
PðkÞ ¼ k3 as þ gc sc þ gi sin
g
Note that
The characteristic polynomial of FV 1 is
where
0
gc þab
dð1cþilÞ
is equivalent to
m1 ði; a; c; sÞ;
>
>
>
>
=
m2 ði; a; c; sÞ;
>
m3 ði; a; c; sÞ;
>
>
>
;
ds
n
¼
rnð1
Þ
as
g
sc
g
si
l
s;
c
i
dt
K
6
V ¼4
b
ds0
3
ð1Þ3 Pð1Þ > 0;
makes g ðc þ lÞ < 0 and then there is no population explosion.
System (3.1) has the same disease-free equilibrium points,
DFEs0 , and the line of equilibrium points in the
ða cÞ plane; DFEs . Since many invertebrate and microbial lifeforms contribute to rapid decay of carcass, it is possible that the
second manifold of steady-states, DFEs , is pathological for realistic
carcass modeling. To determine the stability of DFEs0 , using the
next generation method, we compute the basic reproduction number, R0 , of the model without migration [20]. Thus, we rewrite
(3.1) in the form
2
d
and
di
@t
s
in
6
F¼4
gc
0
b3 ¼ 1 a23 ;
Þjs P 0g;
<1
2
a
let
where s0 ¼ Kð1 r Þ > 0. If g ðc þ lÞ > 0, then is strictly positive and the anthrax disease will increase without bound; but our
assumption made above that
gi
s0
PðkÞ ¼ k3 þ a1 k2 þ a2 k þ a3 ;
The disease-free equilibrium (DFE) points of (3.1) are
DFEs0 fðs0 ; 0; 0; 0Þg [ DFEs fð0; 0; s;
gi
cþl
6
¼4 0
g þb
0 < c a < 1:
d 1 cþgil
3. Stability/instability of the DFE without diffusion
@s
¼ rnð1 Kn Þ as c sc
@t
@i
¼ as þ c sc þ i ns ð
@t
@a
¼ a þ bc;
@t
@c
¼
ð þ Þi dðs þ iÞc:
@t
FV
1
gi 2 gc
b
k k :
cþl
d
ad
Hence,
a1 ¼ b1 ¼ gi
g
b
; a2 ¼ c ; a3 ¼ ;
cþl
d
ad
gc
d
bgi
;
adðc þ lÞ
b2 ¼ 2
bgc
g
i
;
ad2 c þ l
b3 ¼ 1 2
b
;
ad
2
c2 ¼ ðb3 b1 Þb2 and c3 ¼ b3 b1 :
gi
gc
b
The inequality Pð1Þ > 0 follows from 0 < cþ
l þ d þ ad < 1. Next,
ð1Þ3 Pð1Þ ¼ 1 þ
gi
g
b
cþ
> Pð1Þ > 0:
c þ l d ad
To obtain the last two conditions, we observe
gi
gc
gi
gc
b
implies that 0 < cþ
0 < cþ
l þ d þ ad < 1
l < 1; 0 < d < 1
0 < abd < 1. Hence,
that
and
140
A. Friedman, A.-A. Yakubu / Mathematical Biosciences 241 (2013) 137–144
jb3 j ¼ 1 b
ad
2
>1
b
gi
g
gi
b
g
þ c>
>
þ c
ad c þ l d
c þ l ad
d
In order to determine the stability of the DFE in the presence of
animal migrations, we consider the following linearized system of
System (2.2) for ði; a; cÞ about the DFE,
¼ jb1 j:
Next, we show that jc3 j > jc2 j. Observe that b3 > 0; b1 > 0 and
jb3 j > jb1 j imply that b3 b1 > 0. Since
c3 ¼ ðb3 b1 Þðb3 þ b1 Þ
9
þ ðgi ðc þ lÞÞi þ s0 a þ gc s0 c; >
=
@i
¼ dO2 i
@t
@a
¼ a
@t
@c
ð þ
¼
@t
a þ bc;
c lÞi ds0 c
>
;
ð4:1Þ
in X ½0; 1Þ, with
and
i ¼ 0 on @ X ½0; 1Þ:
c2 ¼ ðb3 b1 Þb2 ;
to complete the proof we only need to show that jb3 þ b1 j > jb2 j.
b
g
þ c
c þ l ad
d
b
gi
b
gc
>1
þ
þ
ad
c þ l ad
d
gi
gc
gi
b
g
þ c
>
þ þ
cþl d
c þ l ad
d
gi
gc b
gi
b
g
þ
þ c
þ
>
c þ l d ad
c þ l ad
d
gi
gc b
>
¼ jb2 j:
þ
c þ l d ad
jb3 þ b1 j ¼ 1 b
ad
2
þ
gi
By the Krein–Rutman Theorem [14], there exists a unique ‘‘principal’’ eigenpair ðk ; /ðxÞÞ of
9
dO2 / þ ½gi ðc þ lÞ þ k / ¼ 0 in X; >
=
/¼0
on @ X;
>
R
;
/ðxÞ > 0 in X; X /ðxÞdx ¼ 1
ðgi ðc þ lÞÞ þ k > 0:
Theorem 4.1. In System (2.2) and (2.3), let
ð3:4Þ
R0 ðdÞ gc þ ab
dk
ðcþlÞ
ð4:5Þ
:
If R0 ðdÞ < 1 and
iðx; 0Þ 6 C/ðxÞ in X for some positive constant C;
Clearly, the basic reproduction number for System (3.1), R0 , is an
increasing (respectively, a decreasing) function of the transmission
rates gc and gi as well as b, the rate at which contamination is disseminated from carcases (respectively, the decay and feeding rates
a and d).
then
Theorem 3.2. If R0 < 1 in System (3.1), then DFEs0 ,
then
l
Kð1 Þ; 0; 0; 0
r
iðx; tÞ 6 Ceet /ðxÞ in X ð0; 1Þ for some
ga
1 þ c ; ðc þ lÞ; a and 0:
b
e > 0;
ð4:6Þ
whereas if R0 ðdÞ > 1 and
iðx; 0Þ P c/ðxÞ in X for some positive constant c;
iðx; tÞ P c0 eet /ðxÞ in X ð0; 1Þ for some
is locally asymptotically stable, and if R0 > 1 then DFEs0 is unstable.
Theorem 3.2 follows from the next generation method developed in
[23]. Next, we determine the stability of DFEs . The eigenvalues of the
Jacobian matrix of (3.1) evaluated ð0; 0; s; sbaÞ are
e > 0;
ð4:7Þ
where c0 is a positive constant.
To prove the result, we first introduce the function
FðeÞ ¼ e k þ
s0 bðc þ lÞ
1
1
s0 gc ðc þ lÞ
þ
;
s0 d a
a e s0 d e
ðs0 d eÞ
ð4:8Þ
where e70. Then F 0 ð0Þ > 0 and
Hence, (3.1) has a line of non-hyperbolic equilibrium points in
the ða; cÞ plane. For each s P 0, the point ð0; 0; s; sbaÞ is weakly starl
rl
ble when s > ð1þ
gc a , while it is unstable when s <
g a ; in particuÞ
ð1þ c Þ
b
ð4:4Þ
For any other eigenpair ðk; wÞ; wðxÞ changes sign in X.
In this section, we will prove the following stability result.
g þb
R0 ¼ qðFV 1 Þ c a :
d 1 ðcþgilÞ
rls
ð4:3Þ
and
We denote by qðFV 1 Þ the spectral radius of FV 1 . Then, for System (3.1),
ð4:2Þ
b
lar, the origin, ð0; 0; 0; 0Þ, is unstable.
4. Stability/instability of the DFEs0 with diffusion
The disease free equilibrium for (2.2) and (2.3) is the solution
sðxÞ of
s
l s ¼ 0 in X;
dO2 s þ r 1 K
@s
þ rðs s0 Þ ¼ 0 on @ X;
@m
where r > 0. Clearly, O2 s > 0 at any point where s > s0 , so that, by
the maximum principle, s cannot take on X values larger than s0 .
Similarly, s cannot take values smaller than s0 . It follows that the
DFE for System (2.2) and (2.3) is the same ðs0 ; 0; 0; 0Þ as for the model (3.1).
Fð0Þ ¼ k þ
bðc þ lÞ gc ðc þ lÞ
:
þ
d
ad
ð4:9Þ
We consider (4.1) and (4.2) with initial values
iðx; 0Þ P 0 in X; and bðx; 0Þ cðx; 0Þ 0 in X:
ð4:10Þ
Note that R0 ðdÞ < 1 is equivalent to Fð0Þ < 0 and R0 ðdÞ > 1 is
equivalent to Fð0Þ > 0. Also, note that R0 ðdÞ < R0 since
gi
k
> 1 cþ
ðcþlÞ
l. Consequently, R0 ðdÞ > 1 implies R0 > 1, while
R0 < 1 implies R0 ðdÞ < 1. Furthermore, R0 ðdÞ < 1 implies linear
asymptotic stability of the DFE of System (2.2) and (2.3), and, by
general theory [1,12], also asymptotic stability. We will use the following standard comparison lemma in the proof of Theorem 4.1.
Lemma 4.2. Let V and W be functions satisfying the following
parabolic inequalities in X ð0; 1Þ:
Z t
@V
6 dO2 V þ
Kðt; sÞVðx; sÞds þ qV;
@t
0
Z t
@W
P dO2 W þ
Kðt; sÞWðx; sÞds þ qW;
@t
0
A. Friedman, A.-A. Yakubu / Mathematical Biosciences 241 (2013) 137–144
where q is a constant, positive or negative, Kðt; sÞ P 0 and d is a nonnegative constant. If
V ¼W ¼0
FðeÞ > e0 ;
where
e0 > 0. From (4.13) and (4.14), we see that
FðeÞU AU > 0 if t P T;
on @ X ð0; 1Þ and Vðx; 0Þ 6 Wðx; 0Þ for x 2 X, then
where C 0 erT ¼ e0 .
We claim that
Vðx; tÞ 6 Wðx; tÞ in X ð0; 1Þ:
Proof of Theorem 4.1. From the last two equations in (4.1) we
get
Z
cðx; tÞ ¼ ðc þ lÞ
ds0 ðtsÞ
e
iðx; tÞ P c0 eet /ðxÞ for t 6 T;
iðx; sÞds
Vðx; tÞ ¼ c0 eet /ðxÞ and Wðx; tÞ ¼ iðx; tÞ
and
aðx; tÞ ¼ b
Z
t
¼b
Z
and conclude that (4.7) holds.
To prove (4.15), note that
eaðtsÞ cðx; sÞds
0
t
eaðtsÞ ðc þ lÞ
0
¼ bðc þ lÞ
Z
¼ bðc þ lÞ
s
@i
P dO2 i þ ðgi ðc þ lÞÞi
@t
eds0 ðssÞ iðx; sÞdsds
0
Z
t
iðx; sÞds
Z
t
0
Z
t
so that, by the comparison lemma,
eaðtsÞ eds0 ðssÞ ds
s
0
iðx; tÞ P n/ðxÞeðgi ðcþlÞÞt P n/ðxÞegi t if t 6 T
eaðtsÞ eds0 ðtsÞ
iðx; sÞds;
ds0 a
where we assume that ds0 a – 0; in case ds0 a ¼ 0 the formula
changes a little and the proof which follows will just slightly
change.
We conclude that
@i
¼ dO2 i þ
@t
Z
t
Kðt; sÞiðx; sÞds þ ðgi ðc þ lÞÞi in X
0
ð0; 1Þ;
ð4:11Þ
for some positive constant n. Hence, (4.15) holds with iðx; tÞ in
c0 ¼ neðgi þeÞT . h
5. Endemic equilibrium without diffusion
In this section, we show that when there are no animal migrations, then R0 > 1 may not guarantee an anthrax enzootic region.
First, we obtain necessary and sufficient conditions for the existence of a unique endemic equilibrium.
Set
where
A0 ¼
s0 bðc þ lÞ aðtsÞ
Kðt; sÞ ¼
e
eds0 ðtsÞ
s0 d a
þ s0 gc ðc þ lÞeds0 ðtsÞ :
ð4:12Þ
We compute, for any e70; jej small,
t
ð4:15Þ
if c0 is a sufficiently small positive number.
Assuming (4.15) is true, we can apply the comparison lemma,
Lemma 4.2, for t P T with
t
0
Z
141
Kðt; sÞees ds ¼ ees
0
s0 bðc þ lÞ
1
1
s0 gc ðc þ lÞ
þ
s0 d a
s0 d þ e
a þ e s0 d þ e
Theorem 5.1. Let R0> 1. Then System (3.1) has a unique endemic
equilibrium, s; i; a; c , if and only if
s0 >
where
r ¼ minfa; s0 dg > 0
ð4:13Þ
ð5:1Þ
Note that A10 is a ‘‘weighted’’ mean of the infection rates. Consequently, if A0 < 1, then the anthrax infection persists and this corresponds to R0 > 1.
Aðx; tÞ;
0 < A < C 0 ert ;
1
:
þ cþgil
gc
b
ad þ d
Kc
ð1 A0 Þ
r
ð5:2Þ
and then
Uðx; tÞ ¼ eet /ðxÞ
Kc
ð1 A0 Þ; s ¼ A0 n;
r
bðc þ lÞi
ðc þ lÞi
a¼
; and c ¼
:
dn
adn
for e70. Since, by (4.3),
Proof: Any endemic equilibrium point ðs; i; a; cÞ must satisfy
n ¼ s þ i ¼ s0 and C 0 is a positive constant.
Consider the function
n
si
as gc sc gi ls ¼ 0;
rn 1 K
n
si
as þ gc sc þ gi ðc þ lÞi ¼ 0
n
U t þ dO2 U ¼ ðe ðgi ðc þ lÞÞ k ÞU;
we get, recalling (4.8),
U t þ dO2 U þ
Z
0
t
Kðt; sÞUðx; sÞds þ ðgi ðc þ lÞÞU
¼ FðeÞU AU:
Vðx; tÞ ¼ iðx; tÞ and Wðx; tÞ ¼ Ce
et
e > 0 and we can apply
c¼
ðc þ lÞi
;
dn
a¼
bc
a
¼
bðc þ lÞi
:
adn
Adding (5.4) to (5.5), we get
n
rnð1 Þ ln ci ¼ 0
K
/ðxÞ
to conclude that (4.6) holds.
Consider next the case Fð0Þ > 0. Then for any small
ð5:4Þ
ð5:5Þ
and
ð4:14Þ
Hence, if Fð0Þ < 0 then FðeÞ < 0 for some
Lemma 4.2 with
ð5:3Þ
e>0
and since i ¼ n s, we obtain
ð5:6Þ
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A. Friedman, A.-A. Yakubu / Mathematical Biosciences 241 (2013) 137–144
s
rn c þ l r
:
¼
þ
n cK
c
ð5:7Þ
On the other hand, from (5.5) and (5.6) we get
ðc þ lÞs
bðc þ lÞs
s
þ gi ¼ c þ l
þ gc
dn
n
adn
so that
s
¼ A0 :
n
Substituting this into (5.7) we find that
l K c
n¼K 1
ð 1 A0 Þ
r
r
and, by (5.2), n > 0, and (5.3) follows h
When R0 > 1 and the disease-free susceptible equilibrium
population is sufficiently high, then System (3.1) exhibits a unique
endemic equilibrium point. Next, we use a specific example to
illustrate that in System (3.1) it is possible to have the catastrophic
extinction of both susceptible and anthrax infected animals when
R0 > 1.
c ¼ 3; d ¼ 0:4;
Then,
s0 ¼ 1;
DFEs0 ¼ ð1; 0; 0; 0Þ and R0 ¼ 1:7496 > 1:
However,
s0 ¼ 1 <
5.1. Stability of endemic equilibrium without diffusion
By Theorem 5.1, System (3.1) has the unique endemic
equilibrium
bðc þ lÞð1 A0 Þ ðc þ lÞð1 A0 Þ
EE ¼ A0 n; ð1 A0 Þn;
;
d
ad
whenever R0 > 1 and s0 > Krc ð1 A0 Þ. The Jacobian matrix at EE is
Example 5.1. In System (3.1), let
r ¼ K ¼ 2; a ¼ 2; b ¼ 1:1962;
l ¼ 1; gc ¼ 0:1 and gi ¼ 0:01:
Fig. 2. In Example 1, the solution with initial condition ð1; 0:0001; 0:0001; 0:0001Þ
limits onð1:8147; 3:0341Þ in the ða; cÞ plane projection.
Kc
ð1 A0 Þ ¼ 1:2835
r
and the system has no endemic equilibrium (Theorem 5.1). Figs. 1
and 2 show that the solution of the system with initial condition
ð1; 0:0001; 0:0001; 0:0001Þ limits on the non-hyperbolic diseasefree equilibrium point
DFEs¼1:8147 ¼ ð0; 0; 1:8147; 3:0341Þ:
That is, the solution limits on ð0; 0Þ in the ðs; iÞ plane (see
Fig. 1), while it limits on ð1:8147; 3:0341Þ in the ða; cÞ plane
(see Fig. 2).
In Example 5.1, the disease-free susceptible equilibrium population is small, and the presence of a small number of animals infected with anthrax led to the extinction of the entire population.
Thus, in order to protect biodiversity we must be very careful to
prevent anthrax infection in endangered animal populations.
Fig. 1. In Example 1, the solution with initial condition ð1; 0:0001; 0:0001; 0:0001Þ
limits on ð0; 0Þ in the ðs; iÞ plane projection, while the solution with initial
condition ð0:001; 0; 0; 0Þ on the s axis limits on s0 ¼ 1.
2
6
6
J¼6
6
4
AB
A þ l gi A20
B
gi A20 ðc þ lÞ
A0 n
0
0
a
ðc þ lÞð1 A0 Þ
ðc þ lÞA0
0
A0 n gc A0 n
3
7
gc A0 n 7
7;
7
b
5
dn
where
A¼rl
2rs0
þ 2cð1 A0 Þ
K
and
B ¼ ð 1 A0 Þ
cþl
A0
gi A0 :
EE is locally asymptotically stable if and only if all the eigenvalues J have negative real parts.
For example, in System (3.1) we let a ¼ 2:5 and keep all the
other parameters fixed at their current values in Example 5.1. Then
R0 ¼ 1:4833 > 1 and s0 ¼ 1 > Krc ð1 A0 Þ ¼ 0:929 19. As predicted
by Theorem 5.1, System (3.1) has a unique stable endemic
equilibrium,
s; i; a; c ¼ ð0:0489; 0:0219; 1:4820; 3:0973Þ:
To study the impact of increasing levels of carcass ingestion by the
total population on an anthrax enzootic region, we keep all the
parameters fixed at their current values while varying the carcass
feeding rate, d, between 0:4 and 0:8. Fig. 3 shows that, the infective
population first increases to a maximum value before it declines to
zero with increasing values of d. Thus, decreasing the levels of carcass feeding by removal of carcases in game reserves, for example,
may not always lead to a reduction in the population of animals infected with anthrax.
By differentiating the endemic equilibrium point, EE, with respect to the carcass feeding rate, d, we observe that, decreasing
the carcass feeding rate decreases the susceptible population while
increasing the environmental contamination in the enzootic anthrax region. Fig. 3 shows that this increase in environmental contamination can give rise to an increase in the number of anthrax
infected animals while the carcass feeding rate is decreasing.
Next, to study the impact of increasing levels of carcass induced
environmental contamination rates on enzootic anthrax, we let
143
A. Friedman, A.-A. Yakubu / Mathematical Biosciences 241 (2013) 137–144
n
dO2 n þ rnð1 Þ ln ci ¼ 0
K
ð6:3Þ
and from the equation for i, after substituting (5.5), (5.6) and using
¼ 1 ni ,
s
n
dO2 i þ
ðc þ lÞi
i
¼ 0:
1 A0 A0
n
ð6:4Þ
Consider the function
w ¼ dð1 A0 Þn di:
Clearly,
Fig. 3. The infective population first increases to a maximum before it declines to
zero with increasing values of d.
a ¼ 2:5 and keep all the other parameters fixed at their current values in Example 5.1 while varying the constant rate at which contamination is disseminated from carcases, b, between 1.1962 and
1.3. Fig. 4 shows that, in this case, the total (susceptible and infected) population monotonically decreases to zero with increasing
values of b. Thus, it is possible for increasing levels of carcass induced environmental contamination rates to lead to the extinction
of a persistent susceptible and anthrax infected animal
populations.
These examples show that, to maintain biodiversity, conservation efforts in game reserves, for example, must include reducing
both animal exposure to environmental contaminants and carcass
feeding.
O2 w ¼ ð1 A0 ÞdO2 n dO2 i
n o
ðc þ lÞi
i
n
ð1 A0 Þ rn 1 ln ci :
1 A0 ¼
A0
n
K
At points where w 6 0, we can write i ¼ ð1 A0 Þnn where n P 1,
and then
1
n
O2 w ¼ ðc þ lÞð1 A0 Þð1 nÞ rð1 Þ þ l þ cð1
ð1 A0 Þn
K
A0 Þn
f ðnÞ:
From (6.1), we deduce that f ð1Þ 6 0 (f ð1Þ ¼ 0 if equality holds in
(6.1)) and since f 0 ðnÞ < 0 it follows that f ðnÞ 6 0 if n P 1. Hence,
O2 w 6 0 whenever w 6 0:
We can now use the maximum principle to conclude that either
w 0 in X, or w > 0 in X, and (6.2) follows. h
6. Endemic equilibrium with diffusion
In Theorem 5.1, we considered the case R0 > 1 without animal
migration and proved that there exists a unique equilibrium ði; nÞ
satisfying:
n ¼ s0 Kc
ð1 A0 Þ;
r
i ¼ ð1 A0 Þn:
In the case of animal migration we have the following partial result:
Theorem 6.1. If R0 > 1 and ði; nÞ is an endemic equilibrium such that
nðxÞ 6 s0 Kc
ð1 A0 Þ in X;
r
ð6:1Þ
then
iðxÞ 6 ð1 A0 ÞnðxÞ in X:
ð6:2Þ
Proof. Adding the first two equations in (2.2) we get, in the steady
state,
7. Conclusion
We have used an extension of an anthrax epizootic mathematical model of Hahn and Furniss [8,9] that includes migrations,
births and death from other diseases to study the impact of anthrax transmission rates (gc and gi ), carcass feeding (d), carcass induced environmental contamination (b) and migration rate (d) on
the persistence or extinction of animal populations. When there
are no animal migrations, we show that the basic reproduction
number for the system, R0 , is an increasing (respectively, a
decreasing) function of gc ; gi and b (respectively, the decay and
feeding rates a and d). We prove that R0 < 1 implies anthrax epizootic extinction and the animal population persists without the
disease. Furthermore, we show that it is possible to have a catastrophic extinction of both healthy susceptible and anthrax infected animals with R0 > 1. When there are animal migrations,
we determine the basic reproduction number for the system,
R0 ðdÞ. When R0 ðdÞ > 1, we obtain an estimate for enzootic anthrax
region.
Our simulations demonstrate that increasing levels of carcass
feeding by the total animal population can first lead to a monotone
increase in the number of anthrax infected animals followed by a
monotone decrease of the infected population to zero as the carcass feeding rate increases. Also, we show that increasing levels
of carcass induced environmental contamination rates can lead
to the catastrophic extinction of a persistent total population of
susceptible and anthrax infected animal populations. These results
may have serious implications on animal conservation programs.
Acknowledgments
Fig. 4. The total population of anthrax infective and susceptible populations both
monotonically decline to zero with increasing values of b.
This research has been supported in part by the Mathematical
Biosciences Institute of The Ohio State University, Department of
Homeland Security, DIMACS and CCICADA of Rutgers University
144
A. Friedman, A.-A. Yakubu / Mathematical Biosciences 241 (2013) 137–144
and the National Science Foundation under grants DMS 0931642,
0832782 and 1205185.
References
[1] L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady
state for an SIS epidemic patch model, SIAM J. Appl. Math. 67 (2007) 1283.
[2] H.H. Berry, Surveillance and control of anthrax and rabies in wild herbivores
and carnivores in Namibia, Rev. Sci. Tech. Off. Int. Epiz. 12 (1) (1993) 137.
[3] T.H. Conger, Anthrax epizootic Texas, Summer of 2001, in: Proceedings of the
One Hundred and Fifth Annual Meeting, United States Animal Health
Association: Hershey, Pennsylvania. Virginia: United States Animal Health
Association, vol. 207, 2001.
[4] P.L. Daszak, A.A. Berger, A.D. Cunningham, D.E. Hyatt, G.R. Speare, Emerging
infectious diseases and amphibian population declines, Emerg. Infect. Dis. 5
(1999) 735.
[5] A. Friedman, A.A. Yakubu, Fatal disease and demographic Allee effect:
population persistence and extinction, J. Biol. Dyn. 6 (2) (2012) 495.
[6] L.R. Gerber, H. McCallum, K.D. Lafferty, J.L. Sabo, A. Dobson, Exposing
extinction risk analysis to pathogens: is disease just another form of density
dependence?, Ecol Appl. 15 (2005) 1402.
[7] K.M. Good, C. Marobela, A.M. Houser, A report of anthrax in cheetahs
(Acinonyx jubatus) in Botswana, J. S. Afr. Vet. Assoc. 76 (2005) 186.
[8] B.D. Hahn, P.R. Furniss, A deterministic model of an anthrax epizootic:
threshold results, Ecol. Model. 20 (1983) 233.
[9] B.D. Hahn, P.R. Furniss, A mathematical model of an anthrax epizootic in the
Kruger National Park, Appl. Math. Model. 5 (1981) 130.
[10] C.D. Harvell, C.E. Mitchell, J.R. Ward, S. Altizer, A.P. Dobson, R.S. Ostfeld, M.D.
Samuel, Climate warming and disease risks for terrestrial and marine biota,
Science 296 (2002) 2158.
[11] D.T. Haydon, M.K. Laurenson, C. Sillero-Zubiri, Integrating epidemiology into
population viability analysis: managing the risk posed by rabies and canine
distemper to the Ethiopian wolf, Conserv. Biol. 16 (2002) 1372.
[12] D. Henry, Geometric Theory of Semilinear Parabolic Equations, SpringerVerlag, New York, 1981.
[13] F.M. Hilker, M. Langlais, H. Malchow, The Allee effect and Infectious diseases:
extinction, multistability, and the (dis-) appearance of oscillations, Am. Nat.
173 (1) (2009) 72.
[14] M.G. Kreinm, M.A. Rutman, Linear operators leaving invariant a cone in Banach
space, Am. Math. Soc. Transl. 10 (1962) 199.
[15] T. Lembo, K. Hampson, H. Auty, C.A. Beesley, P. Bessell, C. Packer, J. Halliday, R.
Fyumagwa, R. Hoare, E. Ernest, C. Mentzel, T. Mlengeya, K. Stamey, P.P.
Wilkins, S. Cleaveland, Serologic surveillance of anthrax in the Serengeti
ecosystem, Tanzania, 1996–2009, Emerging Infectious Diseases, vol. 17, No. 3,
March 2011, <www.cdc.gov/eid>.
[16] E.R. Lewis, Network models in population biology, Biomathematics, vol. 7,
Springer-Verlag, New York, 1977.
[17] Y. Lou, X.-Q. Zhao, A reaction–diffusion model with incubation period in the
vector population, J. Math. Biol. 62 (2011) 543.
[18] A.U. Neumann, N.P. Lam, H. Dahari, D.R. Gretch, T.E. Wiley, T.J. Layden, A.S.
Perelson, Hepatitis C Viral dynamics in vivo and the antiviral efficacy of
interferon-a therapy, Science 282 (1998) 103.
[19] A.S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. 2
(2002) 28.
[20] H. Thieme, Spectral bound and reproduction number for infinite-dimensional
population structure and time heterogeneity, SIAM J. Appl. Math 70 (2009)
188.
[21] N.A. Twenhafel, Pathology of inhalational anthrax animal models, Vet. Pathol.
47 (5) (2010) 819.
[22] N.K. Vaidya, F.-B. Wang, X. Zou, Avian influenza dynamics in wild birds with
bird mobility and spatial heterogeneous environment, Discrete and
Continuous Dynamical Systems – Series B 17 (8) (2012) 2829–2848.
[23] P. van den Driessche, J. Watmough, Further notes on the basic reproduction
number, Mathematical Epidemiology (Chapter 6), in: F. Brauer, P. van
Driessche, J. Wu (Eds.), Springer Lecture Notes in Mathematics, vol. 1945,
Berlin, 2008, pp. 159–178.