UserMicrosoft Word - JAEBS-1358-5, khanJ. Appl. Environ. Biol. Sci., 6(9)38-45, 2016 |
© 2016, TextRoad Publication |
ISSN: 2090-4274 |
Journal of Applied Environmental and Biological Sciences www.textroad.com |
Stability Analysis of a General SEIRS Epidemic Model |
Roman Ullah1, Farhad Ali1, Muhammad Adil2, Arshad Ali Shah2, Zahid Hussain3, Gul Zaman4, Muhammad Altaf Khan4, Saeed Islam4 |
1 Department of Mathematics, Bacha Khan University Charsadda,Khyber Pakhtunkhwa,Pakistan |
2 Department of Management Sciences, Bacha Khan University Charsadda,Khyber Pakhtunkhwa, Pakistan |
3 Department of Mathematics, University of Malakand, ChakdaraDir, Pakistan |
4 Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan |
Received: May 5, 2016 |
Accepted: July 28, 2016 |
ABSTRACT |
In this paper, we developed a general SEIR Sepidemic model that provides knowledge about the occurrence of |
epidemic. The model can integrate the birth, death and examine the outcome mathematically. Along the way, we show how this simple SEIR Sepidemic model assists to lay a theoretical foundation for public health interventions. KEYWORDS: mathematical models, SEIR Sepidemic, dynamics |
INTRODUCTION |
It is proved that mathematical modeling plays an important role in the disease spread and control. A better |
qualitative assessment can be obtained by an appropriate mathematical model for the problem. Normally mathematical models for epidemics are comprised of system of differential equations that show the rate of change of each interacting component. Numerous advantages of avoiding invasion of infection to population can be obtained by developing a good epidemic model; therefore epidemiological models attracted the attention of many researchers [1-4].Several epidemic models are there in the literature that focuses on the dynamical properties [5-16].In this work, we consider a single host population, the mode of transmission is direct contact, stay in latency period before becoming infectious. The infectious host can be recovered if the required immunity is provided during infectious stage. |
In research literature a lot of mathematical models have been presented to study the dynamics of infectious diseases [20, 21, 23]. Khan et al. [21] presented an SEIR epidemic model with preventive vaccination. They divided the host population into four subclasses that is S-susceptible, E-exposed, I-infected and R-recovered. Kaddar et al. [20] proposed a generalized SEIRS epidemic model. The proved the global stability for a generalized SEIRS model by using the geometric approach. |
In this work, we present a general SEIRS epidemic model. In our model we assume that the infections stay in the exposed classes before becoming infectious. The term (1- o) is used which represents the number of individuals that gain natural immunity during the incubation period. Further, we two different transmission a1anda2, which respectively represent the contact rate between susceptible and exposed, and susceptible and infected individuals. We denote the total host population by N(t), subdividing into four subclasses, that’s the susceptible S, latent (exposed) E, Infectious I and recovered R. Thus the total host population can be written as N(t) = S(t) + E(t) + |
I(t) + R(t). |
Model Formulation |
This section shows the mathematical formulation of the general infectious SEIR epidemic disease model. The population is categorized is four different subclasses, namely, the susceptible individualsS (t), the individuals latent (Exposed) which are not yet infectious byE(t), infected by I(t) and the individuals whose recover from infection or removed byR(t). Thus, we write,N (t) = S (t) + E(t) + I(t) + R(t) the total size of host population at any timet. The model that describes the assumptions above can be written through the following systems of differentials equations: |
dS(t) |
dt = Λ − a1S(t)E(t) − a2S(t)I(t) − μS(t) + yR(t), S (0) = So ≥ 0 |
*Corresponding Author: Roman Ullah, Department of Mathematics, Bacha Khan University Charsadda, Khyber Pakhtunkhwa, |
Pakistan. |
dE(t) |
dt |
= a1S(t)E(t) + a2S(t)I(t) − (1 − o)E(t) − μE(t), E(0) = Eo ≥ 0 |
dl(t) = (1 − o)E(t) − ( + ℰ + μ)I(t), I (0) = I |
≥ 0 (1) |
dt dR(t) |
= I(t) − yR(t) − μR(t), R (0) = R |
o |
≥ 0. |
dt o |
The host population is increased by the recruitment rate Λ, α₁and α₂respectively show the contact rate between |
susceptible-exposed and susceptible-infected individuals. The induced death rate is given by ε, natural death rate μ, recovery rate is ω (the recovery may be assumed here, natural or due to treatment). The individuals in the latent class gain immunity naturally at a rate δ while loss at a rate γ. The model (1) has the DFE, denoted by, EO = |
Λ |
(So, 0, 0, 0) and is given byE = ( |
µ |
, 0, 0, 0). |
The total dynamics is obtained by summing the equations in (1), |
dN |
= Λ − µN − EI ≤ Λ − µN. |
dt |
The feasible region for the model is the closed setΓ, which is positive invariant and bounded, given by |
Γ = {(S, E, I, R): 0 ≤ S, E, I, R, S + E + I + R ≤ Λ}. |
µ |
Basic Reproduction Number RO |
This section describes the computation of the basic reproduction number, which is defined as the number of |
secondary infections generated by single infections when an infection is introduced into a purely susceptible population. The finding of the reproduction number involves the matrices, F and V, see [17]. It follows from [17] that the matrix F and V can be obtained as: |
0 |
ℱ = a1SE + a2SI , Ѵ = |
0 |
0 |
a1SE + a2SI − Λ + μS − yR |
(1 − o)E + μE |
−(1 − o)E + ( + ℰ + μ)I . |
− I + yR + μR |
It follows from the disease free equilibrium EO |
a1Λ a2Λ |
(1 − o) + µ 0 r |
F = µ |
µ V = [−(1 − o) ( + ℰ + μ) |
0 0 |
1 |
0 |
V-1 = |
(1 − o) + µ |
(1 − o) 1 |
l ((1 − o) + µ)( + ℰ + μ) ( + ℰ + μ) J |
r a1Λ + |
a2Λ(1 − o) |
a2Λ l |
FV-1 = 1µ((1 − o) + µ) |
1 |
µ((1 − o) + µ)( + ℰ + μ) µ( + ℰ + μ)I |
I |
l 0 0 J |
Thus, the required basic reproduction number for model (1) is given by |
RO = |
a1Λ |
µ((1 − o) + µ) |
a2Λ(1 − o) |
+ |
µ((1 − o) + µ)( + ℰ + μ) |
a1Λ( + ℰ + μ) + a2Λ(1 − o) |
= . |
µ((1 − o) + µ)( + ℰ + μ) |
The next section describes the local stability of the system (1) at the DFE,EO. |
Local stability: |
The present section describes the local stability of the model (1) at the disease free and endemic equilibrium. |
Theorem 1: The model (1) is stable locally asymptotically, at the DFEE₀ whenever R₀< 1. |
Proof: The proof involves the linearization of the model (1) at DFE E₀ by setting equal to zero the left hand side of (1), which is given by the following Jacobean matrix: |
39 |
r−µ − |
1 |
a1Λ |
µ |
a2Λ |
− y l |
µ I |
J (E ) = 1 0 |
1 |
a1Λ |
− ((1 − o) + µ) |
µ |
a2Λ I |
0 . |
µ I |
1 0 1 − o −(w + E + µ) 0 I |
l 0 0 w −(y + µ)J |
We need to show that all the eigenvalues of J (EO) are negative. The first column of J (EO) contains only diagonal element which forms one negative eigenvalue −µ , the other three eigenvalues can be obtained from the matrix J1(EO) which is |
r |
J1(EO) = 1 |
a1Λ |
µ |
− ((1 − o) + µ) |
a2Λ |
0 l |
µ I |
1(1 − o ) − (w + E + µ) 0I |
l 0 w − (y + µ) J |
Now, again the third column ofJ1(EO)contains only diagonal element which forms negative eigenvalue−(y + µ), the remaining two eigenvalues can be obtained from the matrix J2(EO) which is |
J2(EO) = |
a1Λ |
µ |
− ((1 − o) + µ) |
a2Λ |
µ |
(1 − o) − (w + E + µ) |
The eigenvalues of J2(EO) are the roots of the characteristic equation |
a1Λ |
( |
µ |
− ((1 − o) + µ) − ,1) (−(w + E + µ) − ,1) − |
a1Λ |
a2(1 − o)Λ |
= 0 |
µ |
⇒ ,12 + (w + E + µ) − |
µ |
− ((1 − o) + µ) ,1 |
a2(1 − o)Λ + a1v(w + E + µ) |
+ ((1 − o) + µ)( + ℰ + μ) 1 − |
a1Λ |
µ(w + E + µ)((1 − o) + µ) |
= 0 |
⇒ ,12 + (w + E + µ) − |
µ |
⇒ A2,12 + A1,1 + AO = 0, |
− ((1 − o) + µ) ,1 + ((1 − o) + µ)( + ℰ + μ)(1 − RO) = 0 |
WhereA = 1, A = (w + E + µ) − alΛ − ((1 − o) + µ) , |
µ |
AO = ((1 − o) + µ)( + ℰ + μ)(1 − RO). |
The above quadratic equations will give two negative eigenvalues if and only if RO <1 and+E + µ + (1 − o) + µ > |
alΛ. We see that in the above polynomial A |
µ |
= 1, A1 |
will be positive only when w + E + µ + (1 − o) + µ > alΛ |
µ |
and AO will be positive if RO < 1. Thus for these two conditions all the roots of the polynomial will be negative. Hence the model (1) at the DFE EO is stable locally asymptotically wheneverRO < 1 and w + E + µ + (1 − o) + |
µ > alΛ . |
µ |
Endemic Equilibrium |
The endemic equilibria of the model (1) at endemic equilibriumE1 = (S∗, E∗, I∗, R∗)is given by |
S∗ = |
1 ò |
ò 1 1 2 |
, E∗ = |
I * ò |
1 |
, R∗ = |
ωI∗ |
, |
y + µ |
I∗ = |
1 ò 1 R0 1 |
ò 1 1 2 1 ò 1 |
A unique positive endemic equilibrium exists if and only if RO > 1. |
The following theorem analyzes the local stability of the endemic equilibrium. |
Theorem 2: The model (1) at the endemic equilibrium E1 is stable locally asymptotically if RO > 1 and the conditions of Routh-Hurwitz criteria is satisfied. |
Proof: At the endemic equilibrium E1we obtain the following jacobian matrix, |
−(a₁E∗ + a₂I∗ + µ) −a₁S∗ −a₂S∗ y |
J (E1) = |
a₁E∗ |
+ a₂I∗ |
a₁S∗ |
− (1 − o) − µ a₂S∗ 0 |
0 1 − o −(w + E + µ) 0 |
0 0 w −(y + µ) |
The Jacobian matrix J (E1)has the following characteristics equation: |
l1 l2 l3 l3 0, |
Where |
l2 ò 3 3 ò 2 ò 3 1 ò 3 |
S* ò 3 E* 1 ò 3 |
I 2 s 1 I ò 3 2 |
l ò 2 2 3 3 3 4 2 2 3 3 2 ò 1 2 (1 ) 2 1 |
(S* ò 2 2ò 3 2 |
E* ò 2 ò 2 1 ò 2 2ò 3 2 ) |
S* 1 2 I * ò 2 ò 2 1 ò 2 2ò 3 2 |
l4 E 1 ò E 1 S ò 1 |
S* 1 I * 1 ò I * 1 1 ò |
The characteristics equation above will give four eigenvalues with negative real parts if and only the conditions of |
2 2 |
Routh-Hurtwiz criteria: that is, the coefficients l1, l2, l3 and l4 are positive andl1l2l3 − l3 − l1 l4 > 0. Thus, Routh- Hurtwiz criteria ensures the system (1) at endemic equilibrium E1is stable locally asymptotically whenever RO > 1 and the Routh-Hurtwiz criteria is satisfied. |
Global Dynamics |
In the given section we reduce the model (1) by using S + E + I + R = N = 1, and making the assumptionsR = (1 − S − E − I), and then we obtain the following reduced model: |
dS(t) |
dt = Λ − a1S(t)E(t) − a2S(t)I(t) − μS(t) + y(1 − S(t) − E(t) − I(t)), S (0) = So ≥ 0 |
dE(t) |
dt = a1S(t)E(t) + a2S(t)I(t) − (1 − o)E(t) − μE(t), E(0) = Eo ≥ 0 |
dl(t) = (1 − o)E(t) − ( + ℰ + μ)I(t), I (0) = I |
≥ 0 (2) |
dt o |
Λ+y |
∗ ∗ ∗ |
The DFE and EE of the model (2) is now denoted by E2 = (µ+y , 0, 0) and E3 = (S , E , I ). For the global |
dynamics we will study the model (2).We follow [24] to present the global stability of system (2). We rewrite the model (2) in the following form |
dV = G(Y, V), G(Y, 0) = 0, |
dt |
dY |
= F(Y, V) |
dt |
where Y = S and Z = (V, I) respectively denotes the population of uninfected (susceptible) and infected individuals |
(exposed and infected) with X ∈ ℝ and Z ∈ ℝ2. The model (2) will be stable globally asymptotically when the conditions given in the following are hold. |
dY |
C1 For |
dt = F(Y, 0) = 0, YO is stable globally asymptotically. |
C2 G(Y, V) = LV − G-(Y, V), where G-(Y, V) ≥ 0, for(Y, V) ∈ Γ, |
where L = DLG(YO, 0), shows an M-matrix and Γis the biologically feasible region. Following the method in [24], we present the following theorem for the global stability of DFE of the model (2). |
Theorem 3: The DFE of the model (2) is stable globally asymptotically whenever R0<1. |
Proof: Choose Y = S and V = (E, I) and U= (Y , 0), where Y = S = A+y. |
µ+y |
The conditions mentioned above can be applied to model (2) as: |
dY = F(Y, 0) = Λ − μS |
+ y(1 − S ) which is stable globally asymptotically when t ⟶ ∞. |
dt O O |
G(Y, V) = LV − G-(Y, V) = [(1 − o + µ) + a1SO a2SO |
(1 − o) −( + ℰ + μ) |
E |
r [ I |
] − [ |
a1E(SO − S) + a2I(SO − S) ], 0 |
(1 − o + µ) + a1SO a2SO E |
a1E(SO − S) + a2I(SO − S) |
where = [ |
(1 − o) −( + ℰ + μ)r , V = [ I ] and G-(Y, V) = [ 0 ]. |
A+y |
In model (2) the total population is bounded bySO = µ+y, that is S, E, I ≤ SO, where SO represents the DFE of the |
model (2) and hence G-(Y, V) ≥ 0. Thus the two conditions presented above are satisfied. Thus, we can conclude that the DFE of the model (2) is stable globally asymptotically. |
Global stability of Endemic Equilibrium |
This section describes the global stability of the endemic equilibrium of the model (2). For the proof we use the geometric approach method [19]. Many authors used this method in his papers, see [20-23]. |
Theorem: The endemic equilibrium of the reduced model (2) is globally asymptotically stable if RO > 1. |
Proof: The endemic equilibrium of the model (2) is given by |
−µ − Ea1 − Ia2 −a1S −a2S |
J = |
Ea1 + Ia2 −1 + o − µ + a1S a2S , |
0 1 − o −E − µ − w |
The second additive compound matrix associated to J∗is |
f1 Sa2 Sa2 J[2] = (1 − o) f2 −Sa1 |
0 Ea1 + Ia2 f3 |
f1 = −µ − Ea1 − Ia2 + −1 + o − µ + Sa1, f2 = −µ − Ea1 − Ia2 − (E + µ + w), f3 = −1 + o − µ + Sa1 − (E + µ + w). |
1 0 0 |
1 0 0 |
0 0 0 |
E l r -lEl+Ell l |
Choose the function H = 0 |
l , and H-1 = |
E , Hf = 1 |
l2 0 I, |
0 0 E |
l |
0 0 0 |
0 0 l E |
1 |
l0 0 |
-lEl+EllI |
l2 J |
r El ll l |
So that Hf H-1 = 10 |
1 |
E − l 0 I. Then HJ[2]H-1 = |
El llI |
l0 0 E − l |
a2 S I |
a2 S I |
−1 + o − 2µ − Ea1 + Sa1 − Ia2 |
(1-8)E |
l |
E |
−E − 2µ − w − Ea1 |
− Ia2 |
E |
−Sa1 . |
0 Ea1 + Ia2 −1 + o − E − 2µ − w + Sa1 |
/11 |
So M = Hf H-1 + HJ[2]H-1 = (1-8)E |
l |
a2 S I |
E |
/22 |
a2 S I |
E |
−Sa1 |
where |
0 Ea1 + Ia2 /33 |
El ll |
/11 = −1 + o − 2µ − Ea1 + Sa1 − Ia2, /22 = −E − 2µ − w − Ea1 − Ia2 + E − l , |
Er Ir |
/33 = −1 + o − E − 2µ − w + Sa1 + E − I . |
Let M = [M11 M12r, where M |
= −1 + o − 2µ − Ea |
+ Sa |
− Ia , M |
= max[a2 S I , a2 S I] |
M21 M22 11 |
1 1 |
El ll |
2 12 E E |
(1-8)E T |
−E − 2µ − w − Ea1 − Ia2 + E − l −Sa1 |
M21 = [ |
, 0] |
l |
, M22 = |
El ll . |
Ea1 + Ia2 −1 + o − E − 2µ − w + Sa1 + E − l |
Now consider the norm in R3 as |(m1, m2, m3)| = max{|m1|, |m2| + |m3|}, where (m1, m2, m3) represent the vector in R3. The Lozinski associated to the above norm is shown by x. Thus it follows from [18]: |
x(M) ≤ sup{b1, b2} = sup{x1(M11) + |M12|, x1(M22) + |M21|}. Therefore |
a2S I Er Er |
b1 = x1(M11) + |M12| = −(1 − o + µ) − µ − Ea1 + Sa1 − Ia2 + |
l |
≤ − µ − Ia2 − Ea1 ≤ − µ. |
E E E |
Using the fact E |
= a2S l + Sa |
− (1 − o + µ). |
E E 1 |
El ll |
El ll |
And b2 = x1 (M22) + |M21| = max { −(E + µ + w) − µ + E − l , −(1 − o + µ) − µ − (E + µ + w + E − l } + |
(1-8)E ≤ E |
l (1-8)E El |
− − µ + − E + µ + w ≤ |
− µ. |
l E l l E |
l |
Using the fact l |
l |
(1-8)E |
= − E + µ + w . |
l |
So, x(M) = sup{b1, b2} = |
E − µ. |
E |
Every solution (S(t), E(t), I(t)) of proposed system (2) with S(0), E(0), I(0) belong to some compact absorbing set |
(say Θ). It follows |
Er 1 t 1 E(t) µ |
Numerical results |
x(M) = sup{b1, b2} = E − µ = t x(M)ds ≤ t ln E(0) − µ ≤ − 2. |
We find the numerical solution of the proposed model (1) by choosing the base line for the susceptible population |
S=50, Exposed population E=10, Infected population I=10, Recovered population R=10. The parameters and their values are given asμ = 0.01, A = 0.05, a1 = 0.005, a2 = 0.0025, E = 0.078, w = 0.2, y = 0.4 and o = 0.4. Figure 1 shows the behavior of distinct classes of the model. |
Figure 1: Dynamical behavior of the proposed model. |
Conclusion |
In this work, we studied a general SEIRS epidemic model of infectious disease. The transmission rate between susceptible-exposed and susceptible-infected was assumed. We investigated that the model is stable at the infection free state when the associated basic reproduction number less than unity. A stable endemic equilibrium was obtained for the case when the basic reproduction number exceeds than unity. Further, the stability of the reduced model was investigated. The disease free stability is examined by Castillo-Chavez method. Using the geometric approach method, the endemic equilibrium of the reduced model is derived, which is found to be stable globally asymptotically when the basic reproduction number exceeds than unity. |
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