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Studies in Mathematical Sciences
Vol. 4, No. 1, 2012, pp. 6-17
DOI: 10.3968/j.sms.1923845220120401.1369
ISSN 1923-8444 [Print]
ISSN 1923-8452 [Online]
www.cscanada.net
www.cscanada.org
On Behaviour of a Host-vector Epidemic Model with Non-linear
Incidence
Bismark Akoto1; Emmanuel Kwame Essel2,∗; Gunnar S¨oderbacka3
1Department of Mathematics and Statistics University of Cape Coast, CAPE COAST, GHANA
2Department of Mathematics and Statistics University of Cape Coast, CAPE COAST, GHANA
3Avdeling for Nærings-og sosialfag, Høgskolen i Finnmark, Follums vei 31, N-9509 ALTA, NORWAY
∗Corresponding author.
Address: Department of Mathematics and Statistics University of Cape Coast, CAPE COAST, GHANA
Received 5 October, 2011; accepted 18 January, 2012
Abstract
In this paper we find the possible phase portraits and bifurcations for a general class of host-vector epidemic
models with non-linear incidence function generalizing the Ross model.
Key words
Epidemics; Non-linear incidence; Global analysis; Bifurcations
MSC2010: 34C05, 34C23, 34D23, 92D30
Bismark Akoto, Emmanuel Kwame Essel, Gunnar S¨oderbacka (2012). On Behaviour of A Host-vector Epidemic Model with Non-
linear Incidence. Studies in Mathematical Sciences, 4(1), 6-17. Available from: URL: http://www.cscanada.net/index.php/sms/arti-
cle/view/j.sms.1923845220120401.1369 DOI: http://dx.doi.org/10.3968/j.sms.1923845220120401.1369
INTRODUCTION
In this paper we consider a modified Ross model of vector-borne diseases with non-linear incidence func-
tion. The model is given as a a two dimensional system of ODE:
x′=g1(y)h1(x)−c1µ1(x)
y′=g2(x)h2(y)−c2µ2(y)(1)
Here xand yrepresent the infective host and vector populations. The terms g1(y)h1(x) and g2(x)h2(y)
correspond to the incidence functions. In the original Ross model obtained from ideas given in [8], the
functions giand hiare linear. The effect of different non-linear incidence functions for usual (without
vector) epidemic models have been studied by many authors. Models with incidence function of type kIpSq
have been studied in [5, 6], and models with incidence function, where gi(I) are of type kIp
1+αIq, have been
studied in [3, 7, 9]. Results for some general type of non-linear incidence functions are obtained in [1, 2, 4].
1. MAIN THEOREM
Equations in (1) are considered for 0 ≤x,y≤1 because xand yare supposed to correspond to the relative
infectious population of host and vector. We assume that the functions gi,hiand µisatisfy the following
conditions.
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Condition 1.gi,hi, µiare continuously differentiable in (0,1] and continuous in [0,1]. Moreover h′
i(z)<0
and g′
i(z), µ′
i(z)>0 for i=1,2. The functions satisfy the boundary conditions gi(0) =0, µi(0) =0 and
hi(1) =0 and parameters ciare positive for i=1,2.
Let φi(z)=−z f ′
i(z)
fi(z)and θi(z)=zg′
i(z)
gi(z)where fi(z)=hi(z)
µi(z)and 0<z<1.
Condition 2.φiis increasing and θiis non-increasing and limz→0+φi(z)=biand limz→0+θi(z)=ai, where
aiand biare positive.
Conditions 1 and 2 are satisfied for many known epidemical models and incidence functions, for exam-
ple, if gi(z)=kzp
1+αzqand hi(z)=(1−z)band µiare linear.
We now formulate our main theorem.
Theorem. Suppose system (1) satisfies Conditions 1 and 2, then we have the following structurally stable
phase portraits.
Case 1. If a1a2>b1b2then there is a number cm(c1) depending on c1such that for c2>cm(c1) the origin
is a global attractor and for c2<cm(c1) there are two equilibria, a saddle Psand a stable one Peexcept the
origin. The stable set of Psdivides the phase space into two parts, one in the basin of attraction of the origin
and the other in the basin of attraction of Pe.
Case 2. If a1a2=b1b2then there are two possibilities. The first one is that there is a cdepending on c1such
that when c2<cthere is an equilibrium Pe(except the origin) attracting all trajectories except the origin
and when c2≥cthe origin is a global attractor. The second possibility is that for any c2there is exactly one
equilibrium Pe(except the origin) attracting all trajectories except the origin.
Case 3. If a1a2<b1b2then there is always exactly one equilibrium Pe(except the origin) attracting all
trajectories except the origin.
In order to prove our main theorem we need the following lemmas. To formulate the lemmas we
introduce notations convenient for using in our proof.
Notation. We consider positive-valued continuously differentiable functions gdefined in an interval (0,B],
where B>0.
We denote by Q(a,∞), the set of functions gsuch that when z→0+then g(z)
za→A, where either A∈R+
or equal ∞and g(z)
zα→0 forα < a, where ais a real number.
Similarly we denote by Q(a,0), the set of functions gsuch that when z→0+then g(z)
za→A, where
either A∈R+or equals 0 and g(z)
zα→ ∞ forα > a, where ais a real number.
For these Q-classes the following is known to hold:
Lemma 1. If gis differentiable invertible with inverse g−1and a>0 then
g∈Q(a,∞)⇔g−1∈Q(a−1,0).
Lemma 2. If g∈Q(a,∞), h∈Q(b,0) and a<0<bthen g◦h∈Q(ab,∞).
If g∈Q(a,0), h∈Q(b,0) and a,b>0 then g◦h∈Q(ab,0).
Lemma 3. If g∈Q(a,0) and h∈Q(b,∞) then g
h∈Q(a−b,0).
Lemma 4. If g∈Q(a,∞) and h∈Q(b,∞) then gh ∈Q(a+b,∞).
The proofs of these lemmas are obtained by straightforward calculations. More details and also details
of other parts of this preprint are available from authors.
We now state another lemma connecting the Q-classes and the θ-function defined by θ(z)=zg′(z)
g(z). We
introduce a known lemma and givea short proof of it.
Lemma 5. If θis non-increasing and θ(z)→aas z→0+then g∈Q(a,∞).
Proof of Lemma 5. We denote by uthe function defined by u(z)=g(z)
zαand by ηthe function defined by
η(z)=zu′(z)
u(z). Then η(z)=θ(z)−α.
If α < athen η(z)>bfor some positive bin a neighbourhood of zero. Integrating the inequality u′
u>b
z
fromuto u0=u(z0) to left and from zto z0to right and using monotonicityof logarithm we obtain
u0
u>z0
zb
,
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implying u(z)→0 for z→0+.
We now assume α=a. Since ηis non-increasing and η→0 for z→0+, the derivative u′is non-positive
and the limit set of uwhen z→0+cannot contain more than one point and this cannot be zero.
We are now ready to prove our main theorem. The proof of the main theorem consists of three parts.
First part examines the number of equilibria from intersections of zero-isoclines. This part needs the lem-
mas. The second part examines the type of the equilibria found. The final part makes the global analysis
using sign analysis of the right hand sides of system 1.
Proof of main theorem. We start by finding the number of equilibria and their position in relation to each
other.
First we see that c2can be considered as a function of the x-coordinate at equilibrium and analyze the
behaviour at endpoints 0 and x1of the interval of definition. Secondly we differentiate that function to find
out the behaviour inside the interval (0,x1).
Let c1be fixed and suppose (x,y) is a point on the isocline x′=0. Then from Condition 1 it follows that
y=p1(x)=g−1
11
f1(x)is an increasing function of xand p1(0) =0 and p1(x)→ ∞ for x→1−. Thus, there
is an x1between 0 and 1 such that p1(x1)=1 and the isocline x′=0 is given by the function p1defined in
[0,x1].
In this part of the proof calculating the limit behaviour when z→0+we consider functions gias defined
only for z>0.
From Condition 2 and Lemma 5 it follows that gi∈Q(ai,∞) and fi∈Q(−bi,∞). From Lemma 3
it follows that 1
f1∈Q(b1,0) and from Lemma 1 it follows that g−1
1∈Q(1
a1,0). However from Lemma 2
(second part) it follows that p1∈Q(b1
a1,0).
We now suppose (x,y) is also on the isocline y′=0, that is (x,y) is an equilibrium point. We then
calculate c2as a function of xi.e. c2=g2(x)f2(p1(x)). From Lemma 2 it follows that the composition
of f2and p1belongs to Q(−b2b1
a1,∞) and finally Lemma 4 implies, that c2as a function of x, belongs to
Q(a2−b2b1
a1,∞).
We conclude that for x→0+we get c2(x)→0 in case 1 where a1a2>b1b2and c2(x)→ ∞ in case 3
where a1a2<b1b2and either c2(x)→ ∞ or c2(x)→c∈R+in case 2 where a1a2=b1b2.
Because h2(1) =0, we conclude that in all cases c2(x)→0 for x→x1.
We have now finished examining the behaviour of c2at the endpoints.
To find out when c2(x) is growing or decreasing we calculate the derivative.
Differentiating c1=g1(y)f1(x) with respect to xand solving for y′we get
y′=−g1(y)f′
1(x)
g′
1(y)f1(x).
Differentiating c2=g2(x)f2(y) with respect to xand substituting our expression for y′we get
dc2
dx =g′
2(x)f2(y) 1−φ1(x)φ2(y)
θ2(x)θ1(y)!.(2)
From the boundary behaviour of c2and the derivative, we make conclusions about the behaviour of c2
between 0 and x1and from there we find the number of equilibria depending on c2. From condition 1 and
2 it follows that dc2
dx is always decreasing.
We consider case 1 when a1a2>b1b2. From conditions 1 and 2 it follows that dc2
dx is positive near 0
and becomes negative near x1. Then there is an xm(c1) such that dc2
dx =0 forx=xm(c1) and dc2
dx >0 for
x<xm(c1) and dc2
dx <0 forx>xm(c1).
We denote by cm(c1), the maximum value of c2for fixed c1.
In the case when c2>cm(c1), we cannot solve for xand thus have no equilibrium apart from the origin.
If c2=cm(c1), we have only one solution for xi.e. x=xm(c1) and thus one equilibrium (in addition to
the origin).
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As c2(x)→0 for x→x1and x→0+for the case c2<cm(c1) we obtain two solutions for x, one with
x<xm(c1) and the other with x>xm(c1) and thus have two equilibria (in addition to the origin).
Let us denote the equilibrium at the origin by P0. If c2<cm(c1), we denote the equilibrium when
x>xm(c1) by Peand the equilibrium when x<xm(c1) by Ps.
The equilibrium Pstends to P0(disease-free) and Pegets it’s maximal size as c2tends to zero. At
c2=cm(c1) there is a saddle-node bifurcation with both equilibria coinciding and disappearing after that.
Next we consider case 3, where a1a2<b1b2. There dc2
dx <0 for all xand c2is decreasing from infinity to
zero when xis increasing from zero to x1. Thus, there is always exactly one non-trivial equilibrium denoted
by Pe. The equilibrium Petends to zero when c2grows to infinity.
Finally we consider case 2, where a1a2=b1b2. In this case dc2
dx ≤0 and if c2(x)→ ∞ for x→0+, we
have a situation analogous to the one in case 3. If c2(x)→cthen c2is decreasing from cto zero when x
is increasing from zero to x1. Thus for c2<cthere is always exactly one non-trivial equilibrium denoted
by Pe. The equilibrium Petends to zero when c2→c−. In this case there is no non-trivial (endemic)
equilibrium for c2>c. At c2=cthere is a transcritical bifurcation.
The first part of the proof is now complete and we begin with the second part to find the type of the
equilibria.
The Jacobian matrix for system (1) is given by
J="g1(y)h′
1(x)−c1µ′
1(x)g′
1(y)h1(x)
g′
2(x)h2(y)g2(x)h′
2(y)−c2µ′
2(y)#.
Using that hi(z)=fi(z)µi(z) and at equilibrium c1=g1(y)f1(x) and c2=g2(x)f2(y) after some calcula-
tions the Jacobian matrix becomes
J="g1(y)f′
1(x)µ1(x)g′
1(y)f1(x)µ1(x)
g′
2(x)f2(y)µ2(y)g2(x)f′
2(y)µ2(y)#
for x,y,0. We now calculate the trace and determinant of the Jacobian matrix.
From Condition 1, it follows that f′
i(z)<0 and
Trace(J)=g1(y)f′
1(x)µ1(x)+g2(x)f′
2(y)µ2(y)<0.
Calculations show that the determinant D of the Jacobian matrix is equal to
D=−µ1(x)µ2(y)g′
1(y)f1(x)dc2
dx .
We consider case 1.
We have two equilibria in the case c2<cm(c1). When x>xm(c1) then dc2
dx <0. This implies D>0.We
note that g′
i(z)>0 from Condition 1. Thus the determinant is positive at Pe. When x<xm(c1) then dc2
dx >0.
This implies D<0.Thus the determinant is negative at Ps.
Since the trace is negative and the determinant is positive at Pe, it is a sink. Also, since the determinant
is negative at Ps, it is a saddle.
In cases 2 and 3 we conclude in the same way that Peis always stable when it exists.
The type of equilibrium P0cannot always be found from Jacobian matrix, as the derivatives of the
functions might not exist at 0. Anyhow a lot is known about origin from global analysis below.
The second part of proof is now complete and we begin with the last part and examine the global
behaviour. We do this by using sign analysis of x′and y′.
We start with case 1, which has the most complicated behaviour.
We notice that in the xy-space above the isocline x′=0, the sign of x′is positive and below negative.To
the left of the isocline y′=0, the sign of y′is negative and to the right it is positive.
We consider the situation when c2>cm(c1). Here the isoclines do not intersect in the xy-plane. The
isocline x′=0 is above and to the left from the isocline y′=0. The isoclines divide the phase space into
three parts as seen in Figure 1. These parts are defined as follows:
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Figure 1
Sign Analysis for System x′=y2(1 −x)−0.25x, y′=x2(1 −y)−0.3y
Figure 2
Phase Portrait and Zero-Isoclines for System x′=y2(1 −x)−0.25x, y′=x2(1 −y)−0.3y
1. The region where y′<0<x′.
2. The region where x′,y′<0.
3. The region where x′<0<y′.
In region 1 the x-coordinate of the trajectory is increasing and the y-coordinate is decreasing. Thus the
trajectory cannot remain in the region, but has to hit the isocline x′=0 entering region 2 after some time . In
region 3 the x-coordinate of the trajectory is decreasing and the y-coordinate is increasing. In that way the
trajectory cannot remain in the region, but has to hit the isocline y′=0 entering region 2 after sometime. In
region 2, the x- and y-coordinates are decreasing and the trajectory cannot escape from region 2 against the
direction field on the boundaries x′=0 and y′=0 and also on x=1 and y=1. In region 2, trajectories can
be attracted only to the origin. We conclude that in the first and the third region the trajectories hit either
the isocline x′=0 or y′=0 and afterwards they remain in region 2 where they are all attracted to P0. Thus
the disease-free origin is a global attractor. One example of such a phase portrait is given in Figure 2.
We now consider the situation c2<cm(c1). Here the isoclines intersect and divide the phase space into
five parts as shown in Figure 3. These regions are defined as follows:
1. The region where y′<0<x′.
2. The region where x′,y′<0 and xis less than the x-coordinate of Ps.
3. The region where x′,y′<0 and xis greater than the x-coordinate of Pe.
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Figure 3
Sign Analysis for System x′=y2(1 −x)−0.25x, y′=x2(1 −y)−0.15y
Figure 4
Phase Portrait and Zero-Isoclines for System x′=y2(1 −x)−0.1x, y′=x2(1 −y)−0.4y
4. The region where x′,y′>0.
5. The region where x′<0<y′.
In region 1, the x-coordinate of the trajectory is increasing and the y-coordinate is decreasing. This
means the trajectory cannot remain in the region, but has to hit either the isocline x′=0 or y′=0 entering
one of regions 2, 3 or 4 after some time, or the trajectory is attracted to Psor Pe. In region 5, the x-coordinate
of the trajectory is decreasing and the y-coordinate is increasing. This means the trajectory cannot remain
in the region, but has to hit either the isocline x′=0 or y′=0 entering one of regions 2, 3 or 4 after
some time, or the trajectory is attracted to Psor Pe. In region 2, the x- and y-coordinates are decreasing
and the trajectory cannot escape from region 2 against the direction field on the boundaries x′=0 and
y′=0. In region 2, trajectories can be attracted only to the origin. In region 4, the x- and y-coordinates
are increasing and the trajectory cannot escape from region 4 against the direction field on the boundaries
x′=0 and y′=0. In region 4, trajectories can be attracted only to Pe. In region 3, the x- and y-coordinates
are decreasing and the trajectory cannot escape from region 3 against the direction field on the boundaries
x′=0 and y′=0 and also on x=1 and y=1. In region 3, trajectories can be attracted only to Pe.
We conclude that in the first and the fifth region trajectories after some time either hit the isocline x′=0
or y′=0 or tend directly to some equilibrium without visiting other parts. If they go through one of the
isoclines they come into one of regions 2, 3 or 4 and remain in the region they enter. Trajectories in region
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2 are attracted to P0and in regions 3 and 4 to Pe. Thus the stable set of the saddle Psdivides the phase
space into two parts, one where trajectories are attracted to origin and another where they are attracted to
Pe. One example of such a phase portrait is given in Figure 4.
Figure 5
Sign Analysis for System x′=y2(1 −x)−0.25x, y′=x2(1 −y)−0.25y
In the case c2=cm(c1) there is a saddle-node bifuraction dividing the parameter space into two parts
with different qualitative behaviourdescribed in the two sitautions above. In this case, the isoclines intersect,
except at origin at a tangency point which is an equilibrium. The phase space is divided into four parts as
shown in Figure 5. The regions are defined as follows:
1. The region where y′<0<x′.
2. The region where x′,y′<0 and xis less than the x-coordinate of the equilibrium.
3. The region where x′,y′<0 and xis greater than the x-coordinate of the equilibrium.
4. The region where x′<0<y′.
In region 1, the x-coordinate of the trajectory is increasing and the y-coordinate is decreasing. Thus the
trajectory cannot remain in the region, but has to hit either the isocline x′=0 entering one of region 2 or
3 after some time or the trajectory is attracted to the equilibrium point at tangency of isoclines. In region 4
the x-coordinate of the trajectory is decreasing and the y-coordinate is increasing. In that way the trajectory
cannot remain in the region but has to hit after some time either the isocline y′=0 entering one of region 2
or 3 or the trajectory is attracted to the equilibrium point. Trajectories in region 2 cannot escape against the
direction field on the boundary and they are all attracted by the origin. For the same reason, the trajectories
in region 3 cannot escape and they must be attracted by the equilibrium at tangency.
We conclude that trajectories in regions 1 and 4 either hit one of the isoclines x′=0 or y′=0 after some
time or are attracted to the equilibrium at tangency. The equilibrium is a saddle-node and the boundary of
it’s stable set divides the phase space into two parts, to the left, the trajectories are in the basin of attraction
of the origin and to the right, we have the stable set of the equilibrium including the boundary of itself.
Figure 6 shows one example of such a phase portrait.
In case 2, where a1a2=b1b2and in the situation c2≥cthe sign analysis can be carried out in the same
way as in case 1 when c2>cm(c1) and we get origin as global attractor.
In case 2 and in the situation when c2<cwe have the typical situation in the endemic case in the Ross
model. The zero-isoclines divide the phase space into four regions as shown in Figure 7. The regions are
defined as:
1. The region where y′<0<x′.
2. The region where x′,y′>0 (here xis less than the x-coordinate of the endemic equilibrium).
3. The region where x′,y′<0 (here xis greater than the x-coordinate of the endemic equilibrium).
4. The region where x′<0<y′.
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Figure 6
Phase Portrait and Zero-Isoclines for System x′=y2(1 −x)−0.25x, y′=x2(1 −y)−0.25y
Figure 7
Sign Analysis for System x′=y(1 −x)−0.5x, y′=x(1 −y)−0.5y
As before we conclude that the trajectories in regions 1 and 4 hit one of the isoclines after some time
or are attracted directly by the endemic equilibrium. After hitting one isocline they either enter region 2 or
3 where they are attracted to the endemic equilibrium. Thus the endemic equilibrium is a global attractor,
attracting everything except the origin. One example of such a phase portrait is given in Figure 8.
In case 3, where a1a2<b1b2similar sign analysis as in the previous case shows that Peis a global
attractor attracting everything except the origin.
2. SOME EXAMPLES
We now study special cases of the functions gi,hiand µiin system (1). These are often used in models
with non-linear incidence. We assume the functions havethe form gi(z)=za,hi(z)=(1 −z)band µi(z)=z,
i=1,2, which gives system
x′=ya(1 −x)b−c1x
y′=xa(1 −y)b−c2y.(3)
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Figure 8
Phase Portrait and Zero-Isoclines for System x′=y(1 −x)−0.5x, y′=x(1 −y)−0.5y
We suppose a>1 which here will imply case 1, a saddle-node bifurcation and two possible main types
of phase portraits.
It is possible to prove that this system satisfies Conditions 1-2 by direct calculations.
Calculations give θi(z)=aand φi(z)=1+dz
1−z, where d=b−1 and a1=a2=aand b1=b2=1. Thus,
we can apply case 1 in our theorem.
The saddle-node bifurcation occurs when dc2
dx =0 in (2) thereby giving us
φ1(x)φ2(y)=θ2(x)θ1(y) (4)
Equation (4) must be satisfied for an equilibrium point (x,y) in order to get a saddle-node bifurcation.
Equation (4) for bifurcation in our example (3) becomes
1+dx
1−x
1+dy
1−y=a2,(5)
and solving for ywe get:
y=−(d+a2)x−1+a2
(d2−a2)x+d+a2.(6)
For any equilibrium in system (3) we must have
c1=ya(1 −x)b
x,c2=xa(1 −y)b
y.(7)
Substituting (6) into (7) we obtain a parameter representation for the saddle-node bifurcation curve in
the c1c2-space if aand bare known. Some examples of such bifurcation curves are shown in Figures 9
and 10.
In some special cases, it is possible to get algebraic formulas for calculating c1and c2or the equilibrium
(x,y) at bifurcation.
In the case where b=1, it is possible to calculate c1,c2and the equilibrium at bifurcation if the product
c|c2is given.
In this case bifurcation equation (5) becomes
(1 −x)(1 −y)=1
a2.(8)
Multiplying equalities (7) we get
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Figure 9
Bifurcation Curves for a= 1.5,3,4,6and b= 1 of System 3.
Figure 10
Bifurcation Curves for b= 0.25,0.5,1,1.5,2,3and a= 2 of System 3.
c1c2=(xy)a−1(1 −x)(1 −y) (9)
and using (8), we obtain an expression for xy:
xy =a2c1c21
a−1.(10)
Using expression (10) in expansion of (8) gives
x+y=1−1
a2+a2c1c21
a−1.(11)
Solving for yfrom (8) and substituting into (11), we obtain a second order equation for xif aand c1c2
are known. Knowing xwe can solve yfrom (8) and finally we can calculate c1and c2from equalities (7).
Finally we consider a special case where a=2 and b=1. Here it is possible to get an expression for c1
or c2at bifurcation, if we know one of them. Also the bifurcation point (x,y) can be easily calculated from
explicit algebraic expressions afterwards.
In this case equality (10) takes on a simple form
xy =4c1c2,(12)
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and (11) takes on the form
4x+4y=3+16c1c2.(13)
Subsituting a=2 and b=1 into the expression for yin (6) we get
y=3−4x
4(1 −x).(14)
From (7) using (14) we see that
c1x
y2=1−x=3−4x
4y,(15)
which simplifies to
3y−4xy =4c1x.(16)
Pluging (12) in (16) we obtain
−4c1x+3y=16c1c2.(17)
Solving for xand yfrom (13) and (17) we obtain expressions
x=9−16c1c2
16c1+12 ,y=(16c2
1+16c1)c2+3c1
4c1+3(18)
Substituting now expressions (18) for xand yinto (12) we see after simplifications that the bifurcation
curve in the c1c2-space satisfies the condition
256(c2
1c2
2+c1c2
2+c2
1c2)+288c1c2=27.(19)
From this we can easily solve for c1or c2from a second order equation, if one of them is known. And
knowing c1and c2, we can calculate the coordinates for the equilibrium at bifurcation from formulas (18).
CONCLUSION
We have examined a generalized Ross model for a large class of non-linear incidence functions and found
possible phase portraits and bifurcations. Many known incidence functions are inside this class. There are
three types of structurally stable types of phase portraits. One type has the disease-free origin as a global
attractor. A second one has the endemic equilibrium as a global attractor. In the third type both disease-free
origin and endemic equilibrium are attractors and there is a saddle equilibrium with stable set forming the
boundary between the basins of attractions of the both attractors. The possible bifurcations are the usual
saddle-node and transcritical bifurcations.
ACKNOWLEDGEMENTS
We wish to thank Professors V A Osipov and O Staffans for checking some details in the proofs for mis-
prints. We also thank I Hauge and T Utsi, the dean and the head of Department of Natural Science of
University College of Finnmark for support in cooperation. Many sincere thanks also go to the Institute of
Mathematical Sciences, Ghana, and the Department of Mathematics and Statistics of University of Cape
Coast, Ghana, for ensuming a fruitful collaboration with the University College of Finnmark.
16
Bismark Akoto; Emmanuel Kwame Essel; Gunnar S¨oderbacka/Studies in Mathematical Sciences Vol.4
No.1, 2012
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