Advanced Studies in Biology, Vol. 7, 2015, no. 2, 65 - 78
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/asb.2015.41059
Mathematical Analysis of Typhoid Model with
Saturated Incidence Rate
Muhammad Altaf Khan1 , Muhammad Parvez2 , Saeed Islam1 ,
Ilyas Khan3 , Sharidan Shafie4,∗ and Taza Gul1
1
2
Department of Mathematics, Abdul Wali Khan University Mardan
Khyber Pakhtunkhwa, Pakistan
Department of Mathematics, ISPAR, Khyber Pakhtunkhwa, Pakistan
3
4
College of Engineering Majmaah University
Majmaah, Kingdom of Saudi Arabia
Department of Mathematical Sciences, Faculty of Science
Universiti Teknologi Malaysia, Skudai, Johor, Malaysia
∗
Corresponding author
c 2014 Muhammad Altaf Khan et al. This is an open access article distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Abstract
Typhoid fever is considered one of the communicable disease, it’s infection causes
diarrhea and a rash. It is most commonly due to a type of bacterium called
Salmonella typhi (S. typhi). In many developing countries this disease is endemic
and remains one of the public health problem. In this paper, we consider a mathematical model of the type SEIR (Susceptible, Exposed, Infected and Recovered)
to understand the dynamics of this disease. We investigate the stability results for
both local and global. For the basic reproduction number R0 < 1, the disease free
equilibrium is stable both locally and globally at E 0 . When R0 > 1, the endemic
equilibrium is stable both locally and globally at E 1 . Finally, the numerical results
are obtained for analytical results.
Mathematics Subject Classification: 92D25, 49J15, 93D20
Keywords: SEIR Epidemic Model,Typhoid Fever, Reproduction number, Stability
results, Numerical simulation
66
1
Muhammad Altaf Khan et al.
Introduction
In epidemiology, the mathematical models plays an important role to understand the
complicated and nonlinear mechanism of the disease [17, 19, 22]. Typhoid fever is one of
the infectious disease which is endemic in most part of the world. It is systemic infection
caused by Salmonella enterica serotype typhi (S typhi). It is spread through contaminated
food, water or drink. The constipated food or water contains this bacteria, cause illness
by drinking or eating, it enter the body. They travel in the human intestines, and then
enter to the blood. Once they enter to the blood through lymph nodes, then, gallbladder,
spleen, liver etc. Abdominal pain, fever and general ill feeling are the symptoms of this
disease. High fever (103 F, or 39.5 C) or higher and severe diarrhea occur as the disease
gets worse. The incubation period is about 10-14 days, sometimes 3 days short or long
for 21 days. This disease is less severed is caused by S paratyphi A, B, and sometimes C.
It is endemic in Central America [1, 2, 3], Indian subcontinent [4, 5], Southeast Asia [6, 7]
and some parts of Africa [8, 9]. In 2000, it is estimated that the disease caused illness is
21.6 million and 216500 deaths globally[10].
Several mathematical models have been constructed on this disease [11, 12, 13, 14,
15, 23]. For example, in reference [15], the author proposed a mathematical model of the
type S,E, I. They divide the total human population into three subclass, i.e, susceptible,
exposed and infected. They, derive the basic reproduction, and study their dynamical
behavior. The stability results have been presented for the model. In his work, a saturated
incidence term have been used in the form of f (I) = βI/1 + αI. Motivated the work of
[15], we construct a mathematical model of the type SEIR, and the saturated incidence
has been proposed of the type βI/φ(I), φ(I) = 1/1 + I 2 , where φ represent a positive
function, with φ(0) = 1 and φ′ ≥ 0. We formulate the mathematical model and define all
the parameters therein. The basic reproduction is obtained, to study the local and global
stability of the disease free and endemic equilibrium.
2
Basic Model Formulation
On the basis of the individualss epidemiological status, the human population is denoted
by N(t), which subdivided into the following classes: susceptible S, latently infected
(exposed) individuals E, infectious individuals I and those recovered from infection, called
recovered individuals R. Thus, N(t) = S(t) + E(t) + I(t) + R(t). Assuming homogeneous
mixing of the population, the model is given by
dS(t)
β(1 − κ)S(t)I(t)
= Π−
− dS(t) + γE(t) + ξI(t),
dt
φ(I)
dE(t)
β(1 − κ)S(t)I(t)
= q
− (d + γ + µ + η)E(t),
dt
φ(I)
dI(t)
β(1 − κ)S(t)I(t)
= (1 − q)
+ ηE(t) − (d + ξ + µ1 + δ)I(t),
dt
φ(I)
Mathematical analysis of typhoid model
dR(t)
= δI(t) − dR(t),
dt
67
(1)
Subject to initial conditions
S(ψ) = ̺1 (ψ), E(ψ) = ̺2 (ψ), I(ψ) = ̺3 (ψ), R(ψ) = ̺4 (ψ),
̺1 (ψ) ≥ 0, ̺2 (ψ) ≥ 0, ̺3 (ψ) ≥ 0, ̺4 (ψ) ≥ 0.
(2)
The parameters with their descriptions are presented in Table 1. The total population
size of the individuals is denoted by N(t). Thus, the total dynamics of the population is
given by
dN
= Π − dN(t) − µE − µ1 I,
dt
(3)
In the absence of disease related death at E and I, we write
n
Πo
4
Ω = (S, E, I, R) ∈ R+ : 0 ≤ S + E + I + R ≤
d
(4)
is the required feasible region of the system (1). Since the fourth equation of the system
(1) is independent of the rest, so we omit it, and thus, we obtain the following reduced
dynamical system:
dS(t)
β(1 − κ)S(t)I(t)
= Π−
− dS(t) + γE(t) + ξI(t),
dt
φ(I)
dE(t)
β(1 − κ)S(t)I(t)
= q
− (d + γ + µ + η)E(t),
dt
φ(I)
dI(t)
β(1 − κ)S(t)I(t)
= (1 − q)
+ ηE(t) − (d + ξ + µ1 + δ)I(t),
dt
φ(I)
(5)
with initial conditions
S(ψ) = ̺1 (ψ), E(ψ) = ̺2 (ψ), I(ψ) = ̺3 (ψ),
̺1 (ψ) ≥ 0, ̺2 (ψ) ≥ 0, ̺3 (ψ) ≥ 0.
2.1
(6)
Basic results
This subsection, includes the basic properties of the system (5), that will helps in the
proceeding sections.
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Muhammad Altaf Khan et al.
Table 1: Parameters values of the model with their descriptions.
2.2
Notation
Parameter Description
Value
Π
β
d
γ
ξ
µ
η
µ1
δ
q
κ
Growth rate of the population
Disease contact rate
Natural mortality rate
Rate of flow from class E to class S
Rate of flow from class I to class S
Disease related death rate at class E
The rate at which the latent individuals infected
Disease related death rate at class I
Rate of recovery from infection
Proportion of individuals joining the class E
Educational adjustment
6
10
0.02
0.01
0.02
0.01
0.2
0.02
0.1
0<q<1
0.2
Disease free equilibrium (DFE)
The system (5), has the disease free equilibrium denoted by E 0 = (S 0 , 0, 0), is
Π
0
0
, 0, 0 .
E = (S , 0, 0) =
d
(7)
Basic reproduction number R0
The basic reproductive number is defined as the number of secondary cases generated by a
primary case when the virus is introduced in a population of fully susceptible individuals
at a demographic steady state [20]. Following van den Driessche et. al [18], and using
the notation defined therein, the matrices F and V for the new infection terms and the
remaining transfer terms are, respectively, given by


q β(1−κ)S(t)I(t)
,
φ(I)


Q1 E(t),


F =
(8)
 and V =
−ηE(t)
+
Q
I(t).
 (1 − q) β(1−κ)S(t)I(t)

,
2
φ(I)
F = Jacobian of F at DF E =
and
0
qβ(1 − κ)S 0
0 (1 − q)β(1 − κ)S 0
,
69
Mathematical analysis of typhoid model
V
= Jacobian of V at DF E =
Q1 0
−η Q2
.
The next generation matrix for the system (5) is

ηqβ(1 − κ)S 0
Q1 qβ(1 − κ)S 0


 η(1 − q)β(1 − κ)S 0 Q1 (1 − q)β(1 − κ)S 0
(9)




(10)
where S 0 represents the disease free equilibrium. The basic reproduction number is given
by
ηqβ(1 − κ)Π (1 − q)β(1 − κ)Π
+
.
(11)
dQ1 Q2
dQ2
Endemic Equilibria: To obtained the positive endemic equilibrium state of the system
(5), let us consider the following system:

Q1 Q2 φ(I)
S ∗ = β(1−κ)((1−q)Q
,


1 +ηq)




qQ2 I
E ∗ = ((1−q)Q
,
(12)
1 +ηq)





 Π − β(1−κ)SI − dS + γE + ξI = 0,
φ(I)
R0 =
Making use of the expression S and E in the last equation of the system (12), the equation
for I is given by:
Q1 Q2
γqQ2
Φ(I) = Π − I
−
+ξ
−
((1 − q)Q1 + ηq)
((1 − q)Q1 + ηq)
dQ1 Q2 φ(I)
= 0.
(13)
β(1 − κ)((1 − q)Q1 + ηq)
′
Q1 Q2
γqQ2
Since ((1−q)Q1 +ηq) − ((1−q)Q1 +ηq) + ξ > 0, and φ ≥ 0, the Φ is an increasing function.
Moreover
Q1 Q2
Φ(I) < Π − I
−
((1 − q)Q1 + ηq)
then
γqQ2
+ξ
((1 − q)Q1 + ηq)
,
(14)
lim Φ(I) = −∞.
(15)
dQ1 Q2
1
Φ(0) = Π −
=Π 1−
.
β(1 − κ)((1 − q)Q1 + ηq)
R0
(16)
t−→∞
If φ(0) = 1 it follows that
Thus, a unique positive zero for Φ exists if and only if Φ(0) > 0, i,e., R0 > 1.
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3
Muhammad Altaf Khan et al.
Disease Free stability (DFE)
In this section, we investigate the stability of the system (5) at (DFE) point E 0 , both
locally and globally.
Theorem 3.1:If R0 < 1, then the (DFE) of the system (5) at E 0 is locally asymptotically
stable.
Proof: At the disease free equilibrium point E 0 of system (5) the corresponding Jacobian
matrix is given by


−d
γ
β(1 − κ)S 0 + ξ




0
 0 −Q1

qβ(1 − κ)S


J0 = 
(17)



 0
η
(1 − q)β(1 − κ)S 0 − Q2 
In echelon form the Jacobian matrix J0 is

−d
γ
β(1 − κ)S 0 + ξ


 0 −Q1
qβ(1 − κ)S 0
J0 = 


 0
0
Q1 (1 − q)β(1 − κ)S 0 − Q1 Q2 + ηqβ(1 − κ)S 0








(18)
The eigenvalues associated to the Jacobian matrix J0 at (18) are: λ1 = −d < 0,
λ2 = −Q1 < 0 and λ3 = Q1 Q2 (R0 − 1) < 0 if and only if R0 < 1. All the eigenvalues of the Jacobian matrix (18) has negative real parts. So, the disease free equilibrium of
the system (5) at E 0 is locally asymptotically stable. This completes the proof.
.
Next we prove the global stability of the disease free equilibrium. Before we show the
global stability of the disease free equilibrium of the system (5), we first give a brief detail
of the method presented by Castillo-Chavez et al.[21]. The system (5) can be expressed
in the following way:
dX
= F1 (X, Z),
dt
dZ
= F2 (X, Z),
dt
F2 (X, 0) = 0.
(19)
where X ∈ R1 shows the number of uninfected individuals while Z ∈ R2 is the number
of infected individuals. The disease free equilibrium is now shown by P 0 = (X 0 , 0). The
basic two conditions that guarantee the global stability of disease free equilibrium are:
H1 :
F or
dX
= F1 (X, 0), X 0 is globally asymptotically stable,
dt
71
Mathematical analysis of typhoid model
H2 : F2 (X, Z) = AZ − F 2 (X, Z),
(20)
where F2 (X, Z) = 0, for (X, Z) ∈ Ω. The matrix A = Dz F2 (X 0 , 0) denotes an M-matrix.1 .
The symbol Ω represent the region where the model makes the biological sense. Now
H1 :
dX
dt
= F1 (X, 0) = [Π − dS],
H2 : F 2 (X, Z) = AZ − F2 (X, Z),


where A = 

−Q1
η
qβ(1 − κ)S 0
(21)


 and
(1 − q)β(1 − κ)S − Q2 
0

qβ(1 − κ)S 0 I
−q β(1−κ)SI
φ(I)


F 2 (X, Z) = 
 (1 − q)β(1 − κ)S 0 I (1 − q) β(1−κ)SI
φ(I)

S
qβ(1 − κ)I Π/d − φ(I)


= 
S
(1 − q)β(1 − κ)I Π/d − φ(I)




,


 ≥ 0.

(22)
S
As Π/d ≥ φ(I)
, because at E 0 and φ(I) = φ(0) = 1. The above results can be summarized
as follows:
Theorem 3.2: If R0 < 1. Then the (DFE) of the system (5) is globally asymptotically
stable. 4
Endemic Equilibrium Stability
In this section, we find out the local and global stability of the endemic equilibrium around
E 1 . First, we present the local stability of the endemic equilibrium.
Theorem 4.1: If R0 > 1, then the endemic equilibrium E1 of the system (5) is locally
asymptotically stable.
1
A matrix whose their off diagonal entries are non-negative.
72
Muhammad Altaf Khan et al.
Proof: The Jacobian matrix J ∗ of the system (5) given by:

β(1−κ)S ∗
Iφ′ (I)
β(1−κ)I ∗
γ
− φ(I)
1 − φ(I) + ξ
 − φ(I ∗ ) − d




β(1−κ)I ∗
β(1−κ)S ∗
Iφ′ (I)

q φ(I ∗ )
−Q1
q φ(I)
1 − φ(I)
J∗ = 




∗
β(1−κ)S ∗
Iφ′ (I)
 (1 − q) β(1−κ)I
η
(1 − q) φ(I)
1 − φ(I) − Q2
φ(I ∗ )















By eliminatory row operation, we get

−d
−(d + µ)
−(d + µ1 + δ)


′ (I)

∗
)
dq β(1−κ)
(1 − I φφ(I)
)S ∗ + q β(1−κ)
(ξ − Q2 )
−(dQ1 + q β(1−κ)(d+µ)I
J = 0
φ(I)
φ(I)
φ(I)


∗
0 (ηd − (µ + d)(1 − q) β(1−κ)I
)
D∗
φ(I)
where D ∗ = d((1 − q) β(1−κ)
(1 −
φ(I)
Iφ′ (I)
)S ∗
φ(I)
− Q2 ) + (1 − q) β(1−κ)I
(ξ − Q2 ).
φ(I)
(23)







β(1 − κ)I
β(1 − κ)I
I((1 − q)Q1 + ηq) − (1 − q)
×
φ(I)
φ(I)
h
Iφ′ (I) i
) < 0,
(d + µ1 + δ)I − Q2 d ((1 − q)Q1 + ηq) − (1 − q)Q1 Q2 (1 −
φ(I)
trace(J ∗ ) = −(dQ1 + q(d + µ))
and
h
β(1 − κ)
β(1 − κ)I
detJ = dQ1 + q
(d + µ)I (d + µ1 + δ)(1 − q)
I
φ(I)
φ(I)
Iφ′ (I) i
+Q2 d((1 − q)Q1 + ηq) − d(1 − q)Q1 Q2 (1 −
)
φ(I)
′ (I)
h
dQ1 Q2 (1 − Iφφ(I)
)
β(1 − κ)I ∗ i
+(d(1 − η) + µ)
I > 0.
((1 − q)Q1 + ηq) + q(d + µ1 + δ)
φ(I)
Iφ′ (I)
∗
The determinant detJ > 0, when ((1 − q)Q1 + ηq) − d(1 − q)Q1 Q2 (1 − φ(I) ) > 0, and
h
i
′ (I)
traceJ ∗ < 0 when ((1 − q)Q1 + ηq) − (1 − q)Q1 Q2 (1 − Iφφ(I)
) > 0. Thus, the system (5)
at E1 has eigenvalues, that contains negative real part. So, we conclude that the system
(5) is locally asymptotically stable. Next, we show the global stability of the endemic
equilibrium.
Theorem 4.2: If R0 > 1. Then the endemic equilibrium E 1 of the system (5) is globally
∗
73
Mathematical analysis of typhoid model
asymptotically stable.
Proof: The second additive compound matrix J [2] of the system (5) is given by:


β(1−κ)S ∗
Iφ′ (I)
β(1−κ)S ∗
Iφ′ (I)
[∗∗]
J
q φ(I)
1 − φ(I)
1
−
−
ξ
φ(I)
φ(I)








[∗∗∗]
[2]

,
η
J
γ
J =





∗
β(1−κ)I ∗
[∗∗∗∗]

 −(1 − q) β(1−κ)I
q
J
∗
φ(I )
φ(I)
(24)
where
β(1 − κ)I ∗
− d − Q1 ,
φ(I ∗ )
β(1 − κ)I ∗
β(1 − κ)S ∗
Iφ′ (I)
= −
− d + (1 − q)
1−
− Q2 ,
φ(I ∗ )
φ(I)
φ(I)
Iφ′ (I)
β(1 − κ)S ∗
1−
− Q2 .
= −Q1 + (1 − q)
φ(I)
φ(I)
J [∗∗] = −
J [∗∗∗]
J [∗∗∗∗]
(25)
Consider the function
E E
P = P (S, E, I) = diag 1, ,
.
I I
Then
P
−1
I I
= diag 1, ,
E E
Ė E I˙ Ė E I˙
Pf = diag 0, − 2 , − 2 .
I
I I
I
And
Pf P
−1
Ė I˙ Ė I˙
= diag 0, − , −
.
E I E I
And

P J [2] P −1
[∗∗]
J




η EI
=



∗
E
 −(1 − q) β(1−κ)I
φ(I ∗ ) I
I
E
β(1−κ)S ∗
q φ(I)
1−
J [∗∗∗]
q β(1−κ)S
φ(I)
Iφ′ (I)
φ(I)
I
E
β(1−κ)S ∗
φ(I)
1−
Iφ′ (I)
φ(I)
γ
∗
J [∗∗∗∗]
−ξ





.




74
Muhammad Altaf Khan et al.
Then
B = Pf P −1 + P J [2] P −1 ,

[11]
J




η EI
= 



∗
E
 −(1 − q) β(1−κ)I
φ(I ∗ ) I
I
E
β(1−κ)S ∗
q φ(I)
1−
Iφ′ (I)
φ(I)
I
E
β(1−κ)S ∗
φ(I)
1−
J [22]
Iφ′ (I)
φ(I)
−ξ





,




γ
q β(1−κ)I
φ(I)
∗
J [33]
where
β(1 − κ)I ∗
− d − Q1 ,
φ(I ∗ )
Ė I˙ β(1 − κ)I ∗
β(1 − κ)S ∗
Iφ′ (I)
=
− −
− d + (1 − q)
1−
− Q2 ,
E I
φ(I ∗ )
φ(I)
φ(I)
β(1 − κ)S ∗
Iφ′ (I)
Ė I˙
− − Q1 + (1 − q)
1−
− Q2 .
(26)
=
E I
φ(I)
φ(I)
J [11] = −
J [22]
J [33]
Let
B=
B 11 B 12
B 21 B 22
,
(27)
where
β(1 − κ)I ∗
B 11 = −
− d − Q1 ,
φ(I ∗ )
β(1−κ)S ∗
Iφ′ (I)
β(1−κ)S ∗
I
I
B 12 =
q φ(I)
1 − φ(I)
,E
1−
E
φ(I)
B 22
h
η EI , −(1 − q) β(1−κ)I
φ(I ∗ )
"
#
J [22]
γ
∗
=
.
q β(1−κ)I
J [33]
φ(I)
B 21 =
∗
E
I
iT
Iφ′ (I)
φ(I)
−ξ
,
,
(28)
Suppose the norm in R3 as:
|(w1 , w2 , w3 )| = max{|w1 |, |w2| + |w3 |},
(29)
where (w1 , w2 , w3 ) represents the vector in R3 and denoted by ℑ the Lozinskii measure
with respect to this norm, follows [16]:
ℑ(B) ≤ sup(g1, g2 ) = sup{ℑ(B (11) + |B 12 |), ℑ(B (22) + |B 21 |)},
(30)
75
Mathematical analysis of typhoid model
where |B 21 | and |B 12 | are the matrix norms with respect to the ℑ norm.
Then,
g1 = ℑ(B 11 ) + |B 12 |,
β(1 − κ)S I
β(1 − κ)I β(1 − κ)SI 2 φ′ (I)
I
= q
− d − Q1 − max
,q
, −ξ
,
φ(I) E
φ(I)
(φ(I))2 E
E
≤
Ė
β(1 − κ)S I
Ė
− d. F rom system (5), Eq.2, = q
− Q1 .
E
E
φ(I) E
(31)
Again,
g2 = ℑ(B (22) ) + |B 21 |,
Ė I˙ (1 − q)β(1 − κ)S ηE
(1 − q)β(1 − κ)SIφ′ (I)
=
− +
+
− Q2 − d − max
E I
φ(I)
I
(φ(I))2
β(1 − κ)I
β(1 − κ)E
+
, (1 − q)
,
φ(I)
φ(I)
Ė
I˙
(1 − q)β(1 − κ)S
E
− d. F rom third Eq.system(5) =
+ η − Q2 .
≤
E
I
φ(I)
I
Therefore
Ė
ℑ(B) ≤ sup{g1 , g2} = − d.
E
We have
Z
1
1 t
E(t)
d
q=
ℑ(Bds ≤ log
− d ≤ − < 0.
t 0
t
E(0)
2
+ Q1
(32)
(33)
(34)
Thus, the endemic equilibrium E1 of the system (5) is globally asymptotically stable.
5
Discussion
In this section, we investigate the numerical solution of the proposed model (1) with the
specified initial conditions (2). We obtain the numerical solution of the system (1) by
the Runge-Kutta method. We use the time level t = 30 and the parameter values are
shown in the Table 1. Figure 1, shows the population behavior of susceptible, exposed,
infected and individuals, with different initial conditions. We have successfully presented a
mathematical model of Typhoid fever. The basic model construction and their properties
derived and discussed. The basic reproduction number R0 , for the model is obtained. It
is found that the disease free equilibrium is stable locally and globally when R0 < 1. On
the other hand, when R0 > 1, the endemic equilibrium is stable both locally and globally
and unstable endemic equilibrium exists when R0 < 1. Finally, the numerical results of
the proposed model are obtained and presented in the form of graphics.
76
Muhammad Altaf Khan et al.
Dynamical Behavior of SEIR Model
Dynamical Behavior of SEIR Model
140
90
S
E
80
120
100
Exposed individuals
Susceptible individuals
70
80
60
60
50
40
30
20
40
10
20
0
5
10
15
Time(day)
20
25
0
30
0
5
Dynamical Behavior of SEIR Model
10
15
Time(day)
20
25
30
Dynamical Behavior of SEIR Model
70
50
I
R
45
60
40
Recovered individuals
Infected individuals
50
40
30
20
35
30
25
20
15
10
10
5
0
0
5
10
15
Time(day)
20
25
30
0
0
5
10
15
Time(day)
20
25
30
Figure 1: Population behavior of susceptible, exposed, infected and recovered individuals with different
initial conditions.
Mathematical analysis of typhoid model
77
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Received: October 30, 2014; Published: December 12, 2014